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| {{DISPLAYTITLE:Cactus Language}} | | {{DISPLAYTITLE:Cactus Language}} |
| + | |
| + | ==Inquiry Driven Systems 1.3.10.8 – 1.3.10.13== |
| + | |
| <pre> | | <pre> |
| o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
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| | | |
| Which was to be shown. | | Which was to be shown. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | </pre> |
| + | |
| + | ==Notes Found in a Cactus Patch== |
| + | |
| + | <pre> |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | IDS, NKS -- CL |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | CL. Cactus Language |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | CL. Note 1 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Table 13 illustrates the "existential interpretation" |
| + | of cactus graphs and cactus expressions by providing |
| + | English translations for a few of the most basic and |
| + | commonly occurring forms. |
| + | |
| + | Even though I do most of my thinking in the existential interpretation, |
| + | I will continue to speak of these forms as "logical graphs", because |
| + | I think it is an important fact about them that the formal validity |
| + | of the axioms and theorems is not dependent on the choice between |
| + | the entitative and the existential interpretations. |
| + | |
| + | The first extension is the "reflective extension of logical graphs" (RefLog). |
| + | It is obtained by generalizing the negation operator "(_)" in a certain way, |
| + | calling "(_)" the "controlled", "moderated", or "reflective" negation operator |
| + | of order 1, then adding another such operator for each finite k = 2, 3, ... . |
| + | In sum, these operators are symbolized by bracketed argument lists as follows: |
| + | "(_)", "(_,_)", "(_,_,_)", ..., where the number of slots is the order of the |
| + | reflective negation operator in question. |
| + | |
| + | The cactus graph and the cactus expression |
| + | shown here are both described as a "spike". |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o | |
| + | | | | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | ( ) | |
| + | o---------------------------------------o |
| + | |
| + | The rule of reduction for a lobe is: |
| + | |
| + | x_1 x_2 ... x_k |
| + | o-----o--- ... ---o |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | @ = @ |
| + | |
| + | if and only if exactly one of the x_j is a spike. |
| + | |
| + | In Ref Log, an expression of the form "(( e_1 ),( e_2 ),( ... ),( e_k ))" |
| + | expresses the fact that "exactly one of the e_j is true, for j = 1 to k". |
| + | Expressions of this form are called "universal partition" expressions, and |
| + | they parse into a type of graph called a "painted and rooted cactus" (PARC): |
| + | |
| + | e_1 e_2 ... e_k |
| + | o o o |
| + | | | | |
| + | o-----o--- ... ---o |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | @ |
| + | |
| + | |
| + | | ( x1, x2, ..., xk ) = [blank] |
| + | | |
| + | | iff |
| + | | |
| + | | Just one of the arguments x1, x2, ..., xk = () |
| + | |
| + | The interpretation of these operators, read as assertions |
| + | about the values of their listed arguments, is as follows: |
| + | |
| + | 1. Existential Interpretation: "Just one of the k argument is false." |
| + | 2. Entitative Interpretation: "Not just one of the k arguments is true." |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | o-------------------o-------------------o-------------------o |
| + | | Graph | String | Translation | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | @ | " " | true. | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | o | | | |
| + | | | | | | |
| + | | @ | ( ) | untrue. | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | r | | | |
| + | | @ | r | r. | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | r | | | |
| + | | o | | | |
| + | | | | | | |
| + | | @ | (r) | not r. | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | r s t | | | |
| + | | @ | r s t | r and s and t. | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | r s t | | | |
| + | | o o o | | | |
| + | | \|/ | | | |
| + | | o | | | |
| + | | | | | | |
| + | | @ | ((r)(s)(t)) | r or s or t. | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | r implies s. | |
| + | | r s | | | |
| + | | o---o | | if r then s. | |
| + | | | | | | |
| + | | @ | (r (s)) | no r sans s. | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | r s | | | |
| + | | o---o | | r exclusive-or s. | |
| + | | \ / | | | |
| + | | @ | (r , s) | r not equal to s. | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | r s | | | |
| + | | o---o | | | |
| + | | \ / | | | |
| + | | o | | r if & only if s. | |
| + | | | | | | |
| + | | @ | ((r , s)) | r equates with s. | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | r s t | | | |
| + | | o--o--o | | | |
| + | | \ / | | | |
| + | | \ / | | just one false | |
| + | | @ | (r , s , t) | out of r, s, t. | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | r s t | | | |
| + | | o o o | | | |
| + | | | | | | | | |
| + | | o--o--o | | | |
| + | | \ / | | | |
| + | | \ / | | just one true | |
| + | | @ | ((r),(s),(t)) | among r, s, t. | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | genus t over | |
| + | | r s | | species r, s. | |
| + | | o o | | | |
| + | | t | | | | partition t | |
| + | | o--o--o | | among r & s. | |
| + | | \ / | | | |
| + | | \ / | | whole pie t: | |
| + | | @ | ( t ,(r),(s)) | slices r, s. | |
| + | o-------------------o-------------------o-------------------o |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Table 13. The Existential Interpretation |
| + | o-------------------o-------------------o-------------------o |
| + | | Cactus Graph | Cactus Expression | Existential | |
| + | | | | Interpretation | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | @ | " " | true. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | o | | | |
| + | | | | | | |
| + | | @ | ( ) | untrue. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a | | | |
| + | | @ | a | a. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a | | | |
| + | | o | | | |
| + | | | | | | |
| + | | @ | (a) | not a. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b c | | | |
| + | | @ | a b c | a and b and c. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b c | | | |
| + | | o o o | | | |
| + | | \|/ | | | |
| + | | o | | | |
| + | | | | | | |
| + | | @ | ((a)(b)(c)) | a or b or c. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | | | a implies b. | |
| + | | a b | | | |
| + | | o---o | | if a then b. | |
| + | | | | | | |
| + | | @ | (a (b)) | no a sans b. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b | | | |
| + | | o---o | | a exclusive-or b. | |
| + | | \ / | | | |
| + | | @ | (a , b) | a not equal to b. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b | | | |
| + | | o---o | | | |
| + | | \ / | | | |
| + | | o | | a if & only if b. | |
| + | | | | | | |
| + | | @ | ((a , b)) | a equates with b. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b c | | | |
| + | | o--o--o | | | |
| + | | \ / | | | |
| + | | \ / | | just one false | |
| + | | @ | (a , b , c) | out of a, b, c. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b c | | | |
| + | | o o o | | | |
| + | | | | | | | | |
| + | | o--o--o | | | |
| + | | \ / | | | |
| + | | \ / | | just one true | |
| + | | @ | ((a),(b),(c)) | among a, b, c. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | | | genus a over | |
| + | | b c | | species b, c. | |
| + | | o o | | | |
| + | | a | | | | partition a | |
| + | | o--o--o | | among b & c. | |
| + | | \ / | | | |
| + | | \ / | | whole pie a: | |
| + | | @ | ( a ,(b),(c)) | slices b, c. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Table 14. The Entitative Interpretation |
| + | o-------------------o-------------------o-------------------o |
| + | | Cactus Graph | Cactus Expression | Entitative | |
| + | | | | Interpretation | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | @ | " " | untrue. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | o | | | |
| + | | | | | | |
| + | | @ | ( ) | true. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a | | | |
| + | | @ | a | a. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a | | | |
| + | | o | | | |
| + | | | | | | |
| + | | @ | (a) | not a. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b c | | | |
| + | | @ | a b c | a or b or c. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b c | | | |
| + | | o o o | | | |
| + | | \|/ | | | |
| + | | o | | | |
| + | | | | | | |
| + | | @ | ((a)(b)(c)) | a and b and c. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | | | a implies b. | |
| + | | | | | |
| + | | o a | | if a then b. | |
| + | | | | | | |
| + | | @ b | (a) b | not a, or b. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b | | | |
| + | | o---o | | a if & only if b. | |
| + | | \ / | | | |
| + | | @ | (a , b) | a equates with b. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b | | | |
| + | | o---o | | | |
| + | | \ / | | | |
| + | | o | | a exclusive-or b. | |
| + | | | | | | |
| + | | @ | ((a , b)) | a not equal to b. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b c | | | |
| + | | o--o--o | | | |
| + | | \ / | | | |
| + | | \ / | | not just one true | |
| + | | @ | (a , b , c) | out of a, b, c. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a b c | | | |
| + | | o--o--o | | | |
| + | | \ / | | | |
| + | | \ / | | | |
| + | | o | | | |
| + | | | | | just one true | |
| + | | @ | ((a , b , c)) | among a, b, c. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | | | | | |
| + | | a | | | |
| + | | o | | genus a over | |
| + | | | b c | | species b, c. | |
| + | | o--o--o | | | |
| + | | \ / | | partition a | |
| + | | \ / | | among b & c. | |
| + | | o | | | |
| + | | | | | whole pie a: | |
| + | | @ | ( a ,(b),(c)) | slices b, c. | |
| + | | | | | |
| + | o-------------------o-------------------o-------------------o |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | Graph | String | Entitative | Existential | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | @ | " " | untrue. | true. | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | o | | | | |
| + | | | | | | | |
| + | | @ | ( ) | true. | untrue. | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | r | | | | |
| + | | @ | r | r. | r. | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | r | | | | |
| + | | o | | | | |
| + | | | | | | | |
| + | | @ | (r) | not r. | not r. | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | r s t | | | | |
| + | | @ | r s t | r or s or t. | r and s and t. | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | r s t | | | | |
| + | | o o o | | | | |
| + | | \|/ | | | | |
| + | | o | | | | |
| + | | | | | | | |
| + | | @ | ((r)(s)(t)) | r and s and t. | r or s or t. | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | r implies s. | |
| + | | | | | | |
| + | | o r | | | if r then s. | |
| + | | | | | | | |
| + | | @ s | (r) s | not r, or s | no r sans s. | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | r implies s. | |
| + | | r s | | | | |
| + | | o---o | | | if r then s. | |
| + | | | | | | | |
| + | | @ | (r (s)) | | no r sans s. | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | r s | | | | |
| + | | o---o | | |r exclusive-or s.| |
| + | | \ / | | | | |
| + | | @ | (r , s) | |r not equal to s.| |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | r s | | | | |
| + | | o---o | | | | |
| + | | \ / | | | | |
| + | | o | | |r if & only if s.| |
| + | | | | | | | |
| + | | @ | ((r , s)) | |r equates with s.| |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | r s t | | | | |
| + | | o--o--o | | | | |
| + | | \ / | | | | |
| + | | \ / | | | just one false | |
| + | | @ | (r , s , t) | | out of r, s, t. | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | | |
| + | | r s t | | | | |
| + | | o o o | | | | |
| + | | | | | | | | | |
| + | | o--o--o | | | | |
| + | | \ / | | | | |
| + | | \ / | | | just one true | |
| + | | @ | ((r),(s),(t)) | | among r, s, t. | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | | | | | genus t over | |
| + | | r s | | | species r, s. | |
| + | | o o | | | | |
| + | | t | | | | | partition t | |
| + | | o--o--o | | | among r & s. | |
| + | | \ / | | | | |
| + | | \ / | | | whole pie t: | |
| + | | @ | ( t ,(r),(s)) | | slices r, s. | |
| + | o-----------------o-----------------o-----------------o-----------------o |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Differential Logic |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 1 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | One of the first things that you can do, once you |
| + | have a really decent calculus for boolean functions |
| + | or propositional logic, whatever you want to call it, |
| + | is to compute the differentials of these functions or |
| + | propositions. |
| + | |
| + | Now there are many ways to dance around this idea, |
| + | and I feel like I have tried them all, before one |
| + | gets down to acting on it, and there many issues |
| + | of interpretation and justification that we will |
| + | have to clear up after the fact, that is, before |
| + | we can be sure that it all really makes any sense, |
| + | but I think this time I'll just jump in, and show |
| + | you the form in which this idea first came to me. |
| + | |
| + | Start with a proposition of the form x & y, which |
| + | I graph as two labels attached to a root node, so: |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | x y | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | x and y | |
| + | o---------------------------------------o |
| + | |
| + | Written as a string, this is just the concatenation "x y". |
| + | |
| + | The proposition xy may be taken as a boolean function f(x, y) |
| + | having the abstract type f : B x B -> B, where B = {0, 1} is |
| + | read in such a way that 0 means "false" and 1 means "true". |
| + | |
| + | In this style of graphical representation, |
| + | the value "true" looks like a blank label |
| + | and the value "false" looks like an edge. |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | true | |
| + | o---------------------------------------o |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o | |
| + | | | | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | false | |
| + | o---------------------------------------o |
| + | |
| + | Back to the proposition xy. Imagine yourself standing |
| + | in a fixed cell of the corresponding venn diagram, say, |
| + | the cell where the proposition xy is true, as pictured: |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o o | |
| + | | / \ / \ | |
| + | | / \ / \ | |
| + | | / · \ | |
| + | | / /%\ \ | |
| + | | / /%%%\ \ | |
| + | | / /%%%%%\ \ | |
| + | | / /%%%%%%%\ \ | |
| + | | / /%%%%%%%%%\ \ | |
| + | | o x o%%%%%%%%%%%o y o | |
| + | | \ \%%%%%%%%%/ / | |
| + | | \ \%%%%%%%/ / | |
| + | | \ \%%%%%/ / | |
| + | | \ \%%%/ / | |
| + | | \ \%/ / | |
| + | | \ · / | |
| + | | \ / \ / | |
| + | | \ / \ / | |
| + | | o o | |
| + | | | |
| + | o---------------------------------------o |
| + | |
| + | Now ask yourself: What is the value of the |
| + | proposition xy at a distance of dx and dy |
| + | from the cell xy where you are standing? |
| + | |
| + | Don't think about it -- just compute: |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | dx o o dy | |
| + | | / \ / \ | |
| + | | x o---@---o y | |
| + | | | |
| + | o---------------------------------------o |
| + | | (x + dx) and (y + dy) | |
| + | o---------------------------------------o |
| + | |
| + | To make future graphs easier to draw in Ascii land, |
| + | I will use devices like @=@=@ and o=o=o to identify |
| + | several nodes into one, as in this next redrawing: |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | x dx y dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ | |
| + | | @=@ | |
| + | | | |
| + | o---------------------------------------o |
| + | | (x + dx) and (y + dy) | |
| + | o---------------------------------------o |
| + | |
| + | However you draw it, these expressions follow because the |
| + | expression x + dx, where the plus sign indicates (mod 2) |
| + | addition in B, and thus corresponds to an exclusive-or |
| + | in logic, parses to a graph of the following form: |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | x dx | |
| + | | o---o | |
| + | | \ / | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | x + dx | |
| + | o---------------------------------------o |
| + | |
| + | Next question: What is the difference between |
| + | the value of the proposition xy "over there" and |
| + | the value of the proposition xy where you are, all |
| + | expressed as general formula, of course? Here 'tis: |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | x dx y dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ x y | |
| + | | o=o-----------o | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | ((x + dx) & (y + dy)) - xy | |
| + | o---------------------------------------o |
| + | |
| + | Oh, I forgot to mention: Computed over B, |
| + | plus and minus are the very same operation. |
| + | This will make the relationship between the |
| + | differential and the integral parts of the |
| + | resulting calculus slightly stranger than |
| + | usual, but never mind that now. |
| + | |
| + | Last question, for now: What is the value of this expression |
| + | from your current standpoint, that is, evaluated at the point |
| + | where xy is true? Well, substituting 1 for x and 1 for y in |
| + | the graph amounts to the same thing as erasing those labels: |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | dx dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ | |
| + | | o=o-----------o | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | ((1 + dx) & (1 + dy)) - 1·1 | |
| + | o---------------------------------------o |
| + | |
| + | And this is equivalent to the following graph: |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | dx dy | |
| + | | o o | |
| + | | \ / | |
| + | | o | |
| + | | | | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | dx or dy | |
| + | o---------------------------------------o |
| + | |
| + | Have to break here -- will explain later. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 2 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | We have just met with the fact that |
| + | the differential of the "and" is |
| + | the "or" of the differentials. |
| + | |
| + | x and y --Diff--> dx or dy. |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | dx dy | |
| + | | o o | |
| + | | \ / | |
| + | | o | |
| + | | x y | | |
| + | | @ --Diff--> @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | x y --Diff--> ((dx)(dy)) | |
| + | o---------------------------------------o |
| + | |
| + | It will be necessary to develop a more refined analysis of |
| + | this statement directly, but that is roughly the nub of it. |
| + | |
| + | If the form of the above statement reminds you of DeMorgan's rule, |
| + | it is no accident, as differentiation and negation turn out to be |
| + | closely related operations. Indeed, one can find discussions of |
| + | logical difference calculus in the Boole-DeMorgan correspondence |
| + | and Peirce also made use of differential operators in a logical |
| + | context, but the exploration of these ideas has been hampered |
| + | by a number of factors, not the least of which being a syntax |
| + | adequate to handle the complexity of expressions that evolve. |
| + | |
| + | For my part, it was definitely a case of the calculus being smarter |
| + | than the calculator thereof. The graphical pictures were catalytic |
| + | in their power over my thinking process, leading me so quickly past |
| + | so many obstructions that I did not have time to think about all of |
| + | the difficulties that would otherwise have inhibited the derivation. |
| + | It did eventually became necessary to write all this up in a linear |
| + | script, and to deal with the various problems of interpretation and |
| + | justification that I could imagine, but that took another 120 pages, |
| + | and so, if you don't like this intuitive approach, then let that be |
| + | your sufficient notice. |
| + | |
| + | Let us run through the initial example again, this time attempting |
| + | to interpret the formulas that develop at each stage along the way. |
| + | |
| + | We begin with a proposition or a boolean function f(x, y) = xy. |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o o | |
| + | | / \ / \ | |
| + | | / \ / \ | |
| + | | / · \ | |
| + | | / /`\ \ | |
| + | | / /```\ \ | |
| + | | / /`````\ \ | |
| + | | / /```````\ \ | |
| + | | / /`````````\ \ | |
| + | | o x o`````f`````o y o | |
| + | | \ \`````````/ / | |
| + | | \ \```````/ / | |
| + | | \ \`````/ / | |
| + | | \ \```/ / | |
| + | | \ \`/ / | |
| + | | \ · / | |
| + | | \ / \ / | |
| + | | \ / \ / | |
| + | | o o | |
| + | | | |
| + | o---------------------------------------o |
| + | | | |
| + | | x y | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | f = x y | |
| + | o---------------------------------------o |
| + | |
| + | A function like this has an abstract type and a concrete type. |
| + | The abstract type is what we invoke when we write things like |
| + | f : B x B -> B or f : B^2 -> B. The concrete type takes into |
| + | account the qualitative dimensions or the "units" of the case, |
| + | which can be explained as follows. |
| + | |
| + | 1. Let X be the set of values {(x), x} = {not x, x}. |
| + | |
| + | 2. Let Y be the set of values {(y), y} = {not y, y}. |
| + | |
| + | Then interpret the usual propositions about x, y |
| + | as functions of the concrete type f : X x Y -> B. |
| + | |
| + | We are going to consider various "operators" on these functions. |
| + | Here, an operator F is a function that takes one function f into |
| + | another function Ff. |
| + | |
| + | The first couple of operators that we need to consider are logical analogues |
| + | of those that occur in the classical "finite difference calculus", namely: |
| + | |
| + | 1. The "difference" operator [capital Delta], written here as D. |
| + | |
| + | 2. The "enlargement" operator [capital Epsilon], written here as E. |
| + | |
| + | These days, E is more often called the "shift" operator. |
| + | |
| + | In order to describe the universe in which these operators operate, |
| + | it will be necessary to enlarge our original universe of discourse. |
| + | We mount up from the space U = X x Y to its "differential extension", |
| + | EU = U x dU = X x Y x dX x dY, with dX = {(dx), dx} and dY = {(dy), dy}. |
| + | The interpretations of these new symbols can be diverse, but the easiest |
| + | for now is just to say that dx means "change x" and dy means "change y". |
| + | To draw the differential extension EU of our present universe U = X x Y |
| + | as a venn diagram, it would take us four logical dimensions X, Y, dX, dY, |
| + | but we can project a suggestion of what it's about on the universe X x Y |
| + | by drawing arrows that cross designated borders, labeling the arrows as |
| + | dx when crossing the border between x and (x) and as dy when crossing |
| + | the border between y and (y), in either direction, in either case. |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o o | |
| + | | / \ / \ | |
| + | | / \ / \ | |
| + | | / · \ | |
| + | | / dy /`\ dx \ | |
| + | | / ^ /```\ ^ \ | |
| + | | / \`````/ \ | |
| + | | / /`\```/`\ \ | |
| + | | / /```\`/```\ \ | |
| + | | o x o`````o`````o y o | |
| + | | \ \`````````/ / | |
| + | | \ \```````/ / | |
| + | | \ \`````/ / | |
| + | | \ \```/ / | |
| + | | \ \`/ / | |
| + | | \ · / | |
| + | | \ / \ / | |
| + | | \ / \ / | |
| + | | o o | |
| + | | | |
| + | o---------------------------------------o |
| + | |
| + | We can form propositions from these differential variables in the same way |
| + | that we would any other logical variables, for instance, interpreting the |
| + | proposition (dx (dy)) to say "dx => dy", in other words, however you wish |
| + | to take it, whether indicatively or injunctively, as saying something to |
| + | the effect that there is "no change in x without a change in y". |
| + | |
| + | Given the proposition f(x, y) in U = X x Y, |
| + | the (first order) 'enlargement' of f is the |
| + | proposition Ef in EU that is defined by the |
| + | formula Ef(x, y, dx, dy) = f(x + dx, y + dy). |
| + | |
| + | In the example f(x, y) = xy, we obtain: |
| + | |
| + | Ef(x, y, dx, dy) = (x + dx)(y + dy). |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | x dx y dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ | |
| + | | @=@ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Ef = (x, dx) (y, dy) | |
| + | o---------------------------------------o |
| + | |
| + | Given the proposition f(x, y) in U = X x Y, |
| + | the (first order) 'difference' of f is the |
| + | proposition Df in EU that is defined by the |
| + | formula Df = Ef - f, or, written out in full, |
| + | Df(x, y, dx, dy) = f(x + dx, y + dy) - f(x, y). |
| + | |
| + | In the example f(x, y) = xy, the result is: |
| + | |
| + | Df(x, y, dx, dy) = (x + dx)(y + dy) - xy. |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | x dx y dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ x y | |
| + | | o=o-----------o | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Df = ((x, dx)(y, dy), xy) | |
| + | o---------------------------------------o |
| + | |
| + | We did not yet go through the trouble to interpret this (first order) |
| + | "difference of conjunction" fully, but were happy simply to evaluate |
| + | it with respect to a single location in the universe of discourse, |
| + | namely, at the point picked out by the singular proposition xy, |
| + | in as much as if to say, at the place where x = 1 and y = 1. |
| + | This evaluation is written in the form Df|xy or Df|<1, 1>, |
| + | and we arrived at the locally applicable law that states |
| + | that f = xy = x & y => Df|xy = ((dx)(dy)) = dx or dy. |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | dx dy | |
| + | | ^ | |
| + | | o | o | |
| + | | / \ | / \ | |
| + | | / \|/ \ | |
| + | | /dy | dx\ | |
| + | | /(dx) /|\ (dy)\ | |
| + | | / ^ /`|`\ ^ \ | |
| + | | / \``|``/ \ | |
| + | | / /`\`|`/`\ \ | |
| + | | / /```\|/```\ \ | |
| + | | o x o`````o`````o y o | |
| + | | \ \`````````/ / | |
| + | | \ \```````/ / | |
| + | | \ \`````/ / | |
| + | | \ \```/ / | |
| + | | \ \`/ / | |
| + | | \ · / | |
| + | | \ / \ / | |
| + | | \ / \ / | |
| + | | o o | |
| + | | | |
| + | o---------------------------------------o |
| + | | | |
| + | | dx dy | |
| + | | o o | |
| + | | \ / | |
| + | | o | |
| + | | | | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Df|xy = ((dx)(dy)) | |
| + | o---------------------------------------o |
| + | |
| + | The picture illustrates the analysis of the inclusive disjunction ((dx)(dy)) |
| + | into the exclusive disjunction: dx(dy) + dy(dx) + dx dy, a proposition that |
| + | may be interpreted to say "change x or change y or both". And this can be |
| + | recognized as just what you need to do if you happen to find yourself in |
| + | the center cell and desire a detailed description of ways to depart it. |
| + | |
| + | Jon Awbrey -- |
| + | |
| + | Formerly Of: |
| + | Center Cell, |
| + | Chateau Dif. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 3 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Last time we computed what will variously be called |
| + | the "difference map", the "difference proposition", |
| + | or the "local proposition" Df_p for the proposition |
| + | f(x, y) = xy at the point p where x = 1 and y = 1. |
| + | |
| + | In the universe U = X x Y, the four propositions |
| + | xy, x(y), (x)y, (x)(y) that indicate the "cells", |
| + | or the smallest regions of the venn diagram, are |
| + | called "singular propositions". These serve as |
| + | an alternative notation for naming the points |
| + | <1, 1>, <1, 0>, <0, 1>, <0, 0>, respectively. |
| + | |
| + | Thus, we can write Df_p = Df|p = Df|<1, 1> = Df|xy, |
| + | so long as we know the frame of reference in force. |
| + | |
| + | Sticking with the example f(x, y) = xy, let us compute the |
| + | value of the difference proposition Df at all of the points. |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | x dx y dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ x y | |
| + | | o=o-----------o | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Df = ((x, dx)(y, dy), xy) | |
| + | o---------------------------------------o |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | dx dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ | |
| + | | o=o-----------o | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Df|xy = ((dx)(dy)) | |
| + | o---------------------------------------o |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o | |
| + | | dx | dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / o | |
| + | | \| |/ | | |
| + | | o=o-----------o | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Df|x(y) = (dx) dy | |
| + | o---------------------------------------o |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o | |
| + | | | dx dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / o | |
| + | | \| |/ | | |
| + | | o=o-----------o | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Df|(x)y = dx (dy) | |
| + | o---------------------------------------o |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o o | |
| + | | | dx | dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / o o | |
| + | | \| |/ \ / | |
| + | | o=o-----------o | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Df|(x)(y) = dx dy | |
| + | o---------------------------------------o |
| + | |
| + | The easy way to visualize the values of these graphical |
| + | expressions is just to notice the following equivalents: |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | x | |
| + | | o-o-o-...-o-o-o | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / x | |
| + | | \ / o | |
| + | | \ / | | |
| + | | @ = @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | (x, , ... , , ) = (x) | |
| + | o---------------------------------------o |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o | |
| + | | x_1 x_2 x_k | | |
| + | | o---o-...-o---o | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / | |
| + | | \ / x_1 ... x_k | |
| + | | @ = @ | |
| + | | | |
| + | o---------------------------------------o |
| + | | (x_1, ..., x_k, ()) = x_1 · ... · x_k | |
| + | o---------------------------------------o |
| + | |
| + | Laying out the arrows on the augmented venn diagram, |
| + | one gets a picture of a "differential vector field". |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | dx dy | |
| + | | ^ | |
| + | | o | o | |
| + | | / \ | / \ | |
| + | | / \|/ \ | |
| + | | /dy | dx\ | |
| + | | /(dx) /|\ (dy)\ | |
| + | | / ^ /`|`\ ^ \ | |
| + | | / \``|``/ \ | |
| + | | / /`\`|`/`\ \ | |
| + | | / /```\|/```\ \ | |
| + | | o x o`````o`````o y o | |
| + | | \ \`````````/ / | |
| + | | \ o---->```<----o / | |
| + | | \ dy \``^``/ dx / | |
| + | | \(dx) \`|`/ (dy)/ | |
| + | | \ \|/ / | |
| + | | \ | / | |
| + | | \ /|\ / | |
| + | | \ / | \ / | |
| + | | o | o | |
| + | | | | |
| + | | dx | dy | |
| + | | o | |
| + | | | |
| + | o---------------------------------------o |
| + | |
| + | This really just constitutes a depiction of |
| + | the interpretations in EU = X x Y x dX x dY |
| + | that satisfy the difference proposition Df, |
| + | namely, these: |
| + | |
| + | 1. x y dx dy |
| + | 2. x y dx (dy) |
| + | 3. x y (dx) dy |
| + | 4. x (y)(dx) dy |
| + | 5. (x) y dx (dy) |
| + | 6. (x)(y) dx dy |
| + | |
| + | By inspection, it is fairly easy to understand Df |
| + | as telling you what you have to do from each point |
| + | of U in order to change the value borne by f(x, y). |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 4 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | We have been studying the action of the difference operator D, |
| + | also known as the "localization operator", on the proposition |
| + | f : X x Y -> B that is commonly known as the conjunction x·y. |
| + | We described Df as a (first order) differential proposition, |
| + | that is, a proposition of the type Df : X x Y x dX x dY -> B. |
| + | Abstracting from the augmented venn diagram that illustrates |
| + | how the "models", or the "satisfying interpretations", of Df |
| + | distribute within the extended universe EU = X x Y x dX x dY, |
| + | we can depict Df in the form of a "digraph" or directed graph, |
| + | one whose points are labeled with the elements of U = X x Y |
| + | and whose arrows are labeled with the elements of dU = dX x dY. |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | x · y | |
| + | | | |
| + | | o | |
| + | | ^^^ | |
| + | | / | \ | |
| + | | (dx)· dy / | \ dx ·(dy) | |
| + | | / | \ | |
| + | | / | \ | |
| + | | v | v | |
| + | | x ·(y) o | o (x)· y | |
| + | | | | |
| + | | | | |
| + | | dx · dy | |
| + | | | | |
| + | | | | |
| + | | v | |
| + | | o | |
| + | | | |
| + | | (x)·(y) | |
| + | | | |
| + | o---------------------------------------o |
| + | | | |
| + | | f = x y | |
| + | | | |
| + | | Df = x y · ((dx)(dy)) | |
| + | | | |
| + | | + x (y) · (dx) dy | |
| + | | | |
| + | | + (x) y · dx (dy) | |
| + | | | |
| + | | + (x)(y) · dx dy | |
| + | | | |
| + | o---------------------------------------o |
| + | |
| + | Any proposition worth its salt, as they say, |
| + | has many equivalent ways to look at it, any |
| + | of which may reveal some unsuspected aspect |
| + | of its meaning. We will encounter more and |
| + | more of these alternative readings as we go. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 5 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | The enlargement operator E, also known as the "shift operator", |
| + | has many interesting and very useful properties in its own right, |
| + | so let us not fail to observe a few of the more salient features |
| + | that play out on the surface of our simple example, f(x, y) = xy. |
| + | |
| + | Introduce a suitably generic definition of the extended universe of discourse: |
| + | |
| + | Let U = X_1 x ... x X_k and EU = U x dU = X_1 x ... x X_k x dX_1 x ... x dX_k. |
| + | |
| + | For a proposition f : X_1 x ... x X_k -> B, |
| + | the (first order) 'enlargement' of f is the |
| + | proposition Ef : EU -> B that is defined by: |
| + | |
| + | Ef(x_1, ..., x_k, dx_1, ..., dx_k) = f(x_1 + dx_1, ..., x_k + dx_k). |
| + | |
| + | It should be noted that the so-called "differential variables" dx_j |
| + | are really just the same kind of boolean variables as the other x_j. |
| + | It is conventional to give the additional variables these brands of |
| + | inflected names, but whatever extra connotations we might choose to |
| + | attach to these syntactic conveniences are wholly external to their |
| + | purely algebraic meanings. |
| + | |
| + | For the example f(x, y) = xy, we obtain: |
| + | |
| + | Ef(x, y, dx, dy) = (x + dx)(y + dy). |
| + | |
| + | Given that this expression uses nothing more than the "boolean ring" |
| + | operations of addition (+) and multiplication (·), it is permissible |
| + | to "multiply things out" in the usual manner to arrive at the result: |
| + | |
| + | Ef(x, y, dx, dy) = x·y + x·dy + y·dx + dx·dy. |
| + | |
| + | To understand what this means in logical terms, for instance, as expressed |
| + | in a boolean expansion or a "disjunctive normal form" (DNF), it is perhaps |
| + | a little better to go back and analyze the expression the same way that we |
| + | did for Df. Thus, let us compute the value of the enlarged proposition Ef |
| + | at each of the points in the universe of discourse U = X x Y. |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | x dx y dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ | |
| + | | @=@ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Ef = (x, dx)·(y, dy) | |
| + | o---------------------------------------o |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | dx dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ | |
| + | | @=@ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Ef|xy = (dx)·(dy) | |
| + | o---------------------------------------o |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o | |
| + | | dx | dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ | |
| + | | @=@ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Ef|x(y) = (dx)· dy | |
| + | o---------------------------------------o |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o | |
| + | | | dx dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ | |
| + | | @=@ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Ef|(x)y = dx ·(dy) | |
| + | o---------------------------------------o |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | o o | |
| + | | | dx | dy | |
| + | | o---o o---o | |
| + | | \ | | / | |
| + | | \ | | / | |
| + | | \| |/ | |
| + | | @=@ | |
| + | | | |
| + | o---------------------------------------o |
| + | | Ef|(x)(y) = dx · dy | |
| + | o---------------------------------------o |
| + | |
| + | Given the sort of data that arises from this form of analysis, |
| + | we can now fold the disjoined ingredients back into a boolean |
| + | expansion or a DNF that is equivalent to the proposition Ef. |
| + | |
| + | Ef = xy · Ef_xy + x(y) · Ef_x(y) + (x)y · Ef_(x)y + (x)(y) · Ef_(x)(y). |
| + | |
| + | Here is a summary of the result, illustrated by means of a digraph picture, |
| + | where the "no change" element (dx)(dy) is drawn as a loop at the point x·y. |
| + | |
| + | o---------------------------------------o |
| + | | | |
| + | | x · y | |
| + | | (dx)·(dy) | |
| + | | -->-- | |
| + | | \ / | |
| + | | \ / | |
| + | | o | |
| + | | ^^^ | |
| + | | / | \ | |
| + | | / | \ | |
| + | | (dx)· dy / | \ dx ·(dy) | |
| + | | / | \ | |
| + | | / | \ | |
| + | | x ·(y) o | o (x)· y | |
| + | | | | |
| + | | | | |
| + | | dx · dy | |
| + | | | | |
| + | | | | |
| + | | o | |
| + | | | |
| + | | (x)·(y) | |
| + | | | |
| + | o---------------------------------------o |
| + | | | |
| + | | f = x y | |
| + | | | |
| + | | Ef = x y · (dx)(dy) | |
| + | | | |
| + | | + x (y) · (dx) dy | |
| + | | | |
| + | | + (x) y · dx (dy) | |
| + | | | |
| + | | + (x)(y) · dx dy | |
| + | | | |
| + | o---------------------------------------o |
| + | |
| + | We may understand the enlarged proposition Ef |
| + | as telling us all the different ways to reach |
| + | a model of f from any point of the universe U. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 6 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | To broaden our experience with simple examples, let us now contemplate the |
| + | sixteen functions of concrete type X x Y -> B and abstract type B x B -> B. |
| + | For future reference, I will set here a few tables that detail the actions |
| + | of E and D and on each of these functions, allowing us to view the results |
| + | in several different ways. |
| + | |
| + | By way of initial orientation, Table 0 lists equivalent expressions for the |
| + | sixteen functions in a number of different languages for zeroth order logic. |
| + | |
| + | |
| + | Table 0. Propositional Forms On Two Variables |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | |
| + | | | | | | | | |
| + | | Decimal | Binary | Vector | Cactus | English | Vulgate | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | | x = 1 1 0 0 | | | | |
| + | | | y = 1 0 1 0 | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | | | | | | | |
| + | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | |
| + | | | | | | | | |
| + | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | |
| + | | | | | | | | |
| + | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | |
| + | | | | | | | | |
| + | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | |
| + | | | | | | | | |
| + | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | |
| + | | | | | | | | |
| + | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | |
| + | | | | | | | | |
| + | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | |
| + | | | | | | | | |
| + | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | |
| + | | | | | | | | |
| + | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | |
| + | | | | | | | | |
| + | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | |
| + | | | | | | | | |
| + | | f_10 | f_1010 | 1 0 1 0 | y | y | y | |
| + | | | | | | | | |
| + | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | |
| + | | | | | | | | |
| + | | f_12 | f_1100 | 1 1 0 0 | x | x | x | |
| + | | | | | | | | |
| + | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | |
| + | | | | | | | | |
| + | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | |
| + | | | | | | | | |
| + | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | |
| + | | | | | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | |
| + | |
| + | The next four Tables expand the expressions of Ef and Df |
| + | in two different ways, for each of the sixteen functions. |
| + | Notice that the functions are given in a different order, |
| + | here being collected into a set of seven natural classes. |
| + | |
| + | |
| + | Table 1. Ef Expanded Over Ordinary Features {x, y} |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | | f | Ef | xy | Ef | x(y) | Ef | (x)y | Ef | (x)(y)| |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_0 | () | () | () | () | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | (dx)(dy) | |
| + | | | | | | | | |
| + | | f_2 | (x) y | dx (dy) | dx dy | (dx)(dy) | (dx) dy | |
| + | | | | | | | | |
| + | | f_4 | x (y) | (dx) dy | (dx)(dy) | dx dy | dx (dy) | |
| + | | | | | | | | |
| + | | f_8 | x y | (dx)(dy) | (dx) dy | dx (dy) | dx dy | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_3 | (x) | dx | dx | (dx) | (dx) | |
| + | | | | | | | | |
| + | | f_12 | x | (dx) | (dx) | dx | dx | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_6 | (x, y) | (dx, dy) | ((dx, dy)) | ((dx, dy)) | (dx, dy) | |
| + | | | | | | | | |
| + | | f_9 | ((x, y)) | ((dx, dy)) | (dx, dy) | (dx, dy) | ((dx, dy)) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_5 | (y) | dy | (dy) | dy | (dy) | |
| + | | | | | | | | |
| + | | f_10 | y | (dy) | dy | (dy) | dy | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_7 | (x y) | ((dx)(dy)) | ((dx) dy) | (dx (dy)) | (dx dy) | |
| + | | | | | | | | |
| + | | f_11 | (x (y)) | ((dx) dy) | ((dx)(dy)) | (dx dy) | (dx (dy)) | |
| + | | | | | | | | |
| + | | f_13 | ((x) y) | (dx (dy)) | (dx dy) | ((dx)(dy)) | ((dx) dy) | |
| + | | | | | | | | |
| + | | f_14 | ((x)(y)) | (dx dy) | (dx (dy)) | ((dx) dy) | ((dx)(dy)) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_15 | (()) | (()) | (()) | (()) | (()) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | |
| + | |
| + | Table 2. Df Expanded Over Ordinary Features {x, y} |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)| |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_0 | () | () | () | () | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) | |
| + | | | | | | | | |
| + | | f_2 | (x) y | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy | |
| + | | | | | | | | |
| + | | f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) | |
| + | | | | | | | | |
| + | | f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_3 | (x) | dx | dx | dx | dx | |
| + | | | | | | | | |
| + | | f_12 | x | dx | dx | dx | dx | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) | |
| + | | | | | | | | |
| + | | f_9 | ((x, y)) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_5 | (y) | dy | dy | dy | dy | |
| + | | | | | | | | |
| + | | f_10 | y | dy | dy | dy | dy | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_7 | (x y) | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy | |
| + | | | | | | | | |
| + | | f_11 | (x (y)) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) | |
| + | | | | | | | | |
| + | | f_13 | ((x) y) | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy | |
| + | | | | | | | | |
| + | | f_14 | ((x)(y)) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_15 | (()) | () | () | () | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | |
| + | |
| + | Table 3. Ef Expanded Over Differential Features {dx, dy} |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | | f | T_11 f | T_10 f | T_01 f | T_00 f | |
| + | | | | | | | | |
| + | | | | Ef| dx·dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)| |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_0 | () | () | () | () | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) | |
| + | | | | | | | | |
| + | | f_2 | (x) y | x (y) | x y | (x)(y) | (x) y | |
| + | | | | | | | | |
| + | | f_4 | x (y) | (x) y | (x)(y) | x y | x (y) | |
| + | | | | | | | | |
| + | | f_8 | x y | (x)(y) | (x) y | x (y) | x y | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_3 | (x) | x | x | (x) | (x) | |
| + | | | | | | | | |
| + | | f_12 | x | (x) | (x) | x | x | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) | |
| + | | | | | | | | |
| + | | f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_5 | (y) | y | (y) | y | (y) | |
| + | | | | | | | | |
| + | | f_10 | y | (y) | y | (y) | y | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) | |
| + | | | | | | | | |
| + | | f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) | |
| + | | | | | | | | |
| + | | f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) | |
| + | | | | | | | | |
| + | | f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_15 | (()) | (()) | (()) | (()) | (()) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | |
| + | | Fixed Point Total | 4 | 4 | 4 | 16 | |
| + | | | | | | | |
| + | o-------------------o------------o------------o------------o------------o |
| + | |
| + | |
| + | Table 4. Df Expanded Over Differential Features {dx, dy} |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | | f | Df| dx·dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)| |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_0 | () | () | () | () | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_1 | (x)(y) | ((x, y)) | (y) | (x) | () | |
| + | | | | | | | | |
| + | | f_2 | (x) y | (x, y) | y | (x) | () | |
| + | | | | | | | | |
| + | | f_4 | x (y) | (x, y) | (y) | x | () | |
| + | | | | | | | | |
| + | | f_8 | x y | ((x, y)) | y | x | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_3 | (x) | (()) | (()) | () | () | |
| + | | | | | | | | |
| + | | f_12 | x | (()) | (()) | () | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_6 | (x, y) | () | (()) | (()) | () | |
| + | | | | | | | | |
| + | | f_9 | ((x, y)) | () | (()) | (()) | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_5 | (y) | (()) | () | (()) | () | |
| + | | | | | | | | |
| + | | f_10 | y | (()) | () | (()) | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_7 | (x y) | ((x, y)) | y | x | () | |
| + | | | | | | | | |
| + | | f_11 | (x (y)) | (x, y) | (y) | x | () | |
| + | | | | | | | | |
| + | | f_13 | ((x) y) | (x, y) | y | (x) | () | |
| + | | | | | | | | |
| + | | f_14 | ((x)(y)) | ((x, y)) | (y) | (x) | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_15 | (()) | () | () | () | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | |
| + | |
| + | If the medium truly is the message, |
| + | the blank slate is the innate idea. |
| + | |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 7 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | If you think that I linger in the realm of logical difference calculus |
| + | out of sheer vacillation about getting down to the differential proper, |
| + | it is probably out of a prior expectation that you derive from the art |
| + | or the long-engrained practice of real analysis. But the fact is that |
| + | ordinary calculus only rushes on to the sundry orders of approximation |
| + | because the strain of comprehending the full import of E and D at once |
| + | whelm over its discrete and finite powers to grasp them. But here, in |
| + | the fully serene idylls of ZOL, we find ourselves fit with the compass |
| + | of a wit that is all we'd ever wish to explore their effects with care. |
| + | |
| + | So let us do just that. |
| + | |
| + | I will first rationalize the novel grouping of propositional forms |
| + | in the last set of Tables, as that will extend a gentle invitation |
| + | to the mathematical subject of "group theory", and demonstrate its |
| + | relevance to differential logic in a strikingly apt and useful way. |
| + | The data for that account is contained in Table 3. |
| + | |
| + | Table 3. Ef Expanded Over Differential Features {dx, dy} |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | | f | T_11 f | T_10 f | T_01 f | T_00 f | |
| + | | | | | | | | |
| + | | | | Ef| dx·dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)| |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_0 | () | () | () | () | () | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) | |
| + | | | | | | | | |
| + | | f_2 | (x) y | x (y) | x y | (x)(y) | (x) y | |
| + | | | | | | | | |
| + | | f_4 | x (y) | (x) y | (x)(y) | x y | x (y) | |
| + | | | | | | | | |
| + | | f_8 | x y | (x)(y) | (x) y | x (y) | x y | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_3 | (x) | x | x | (x) | (x) | |
| + | | | | | | | | |
| + | | f_12 | x | (x) | (x) | x | x | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) | |
| + | | | | | | | | |
| + | | f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_5 | (y) | y | (y) | y | (y) | |
| + | | | | | | | | |
| + | | f_10 | y | (y) | y | (y) | y | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) | |
| + | | | | | | | | |
| + | | f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) | |
| + | | | | | | | | |
| + | | f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) | |
| + | | | | | | | | |
| + | | f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_15 | (()) | (()) | (()) | (()) | (()) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | |
| + | | Fixed Point Total | 4 | 4 | 4 | 16 | |
| + | | | | | | | |
| + | o-------------------o------------o------------o------------o------------o |
| + | |
| + | The shift operator E can be understood as enacting a substitution operation |
| + | on the proposition that is given as its argument. In our immediate example, |
| + | we have the following data and definition: |
| + | |
| + | E : (U -> B) -> (EU -> B), |
| + | |
| + | E : f(x, y) -> Ef(x, y, dx, dy), |
| + | |
| + | Ef(x, y, dx, dy) = f(x + dx, y + dy). |
| + | |
| + | Therefore, if we evaluate Ef at particular values of dx and dy, |
| + | for example, dx = i and dy = j, where i, j are in B, we obtain: |
| + | |
| + | E_ij : (U -> B) -> (U -> B), |
| + | |
| + | E_ij : f -> E_ij f, |
| + | |
| + | E_ij f = Ef | <dx = i, dy = j> = f(x + i, y + j). |
| + | |
| + | The notation is a little bit awkward, but the data of the Table should |
| + | make the sense clear. The important thing to observe is that E_ij has |
| + | the effect of transforming each proposition f : U -> B into some other |
| + | proposition f' : U -> B. As it happens, the action is one-to-one and |
| + | onto for each E_ij, so the gang of four operators {E_ij : i, j in B} |
| + | is an example of what is called a "transformation group" on the set |
| + | of sixteen propositions. Bowing to a longstanding local and linear |
| + | tradition, I will therefore redub the four elements of this group |
| + | as T_00, T_01, T_10, T_11, to bear in mind their transformative |
| + | character, or nature, as the case may be. Abstractly viewed, |
| + | this group of order four has the following operation table: |
| + | |
| + | o----------o----------o----------o----------o----------o |
| + | | % | | | | |
| + | | · % T_00 | T_01 | T_10 | T_11 | |
| + | | % | | | | |
| + | o==========o==========o==========o==========o==========o |
| + | | % | | | | |
| + | | T_00 % T_00 | T_01 | T_10 | T_11 | |
| + | | % | | | | |
| + | o----------o----------o----------o----------o----------o |
| + | | % | | | | |
| + | | T_01 % T_01 | T_00 | T_11 | T_10 | |
| + | | % | | | | |
| + | o----------o----------o----------o----------o----------o |
| + | | % | | | | |
| + | | T_10 % T_10 | T_11 | T_00 | T_01 | |
| + | | % | | | | |
| + | o----------o----------o----------o----------o----------o |
| + | | % | | | | |
| + | | T_11 % T_11 | T_10 | T_01 | T_00 | |
| + | | % | | | | |
| + | o----------o----------o----------o----------o----------o |
| + | |
| + | It happens that there are just two possible groups of 4 elements. |
| + | One is the cyclic group Z_4 (German "Zyklus"), which this is not. |
| + | The other is Klein's four-group V_4 (German "Vier"), which it is. |
| + | |
| + | More concretely viewed, the group as a whole pushes the set |
| + | of sixteen propositions around in such a way that they fall |
| + | into seven natural classes, called "orbits". One says that |
| + | the orbits are preserved by the action of the group. There |
| + | is an "Orbit Lemma" of immense utility to "those who count" |
| + | which, depending on your upbringing, you may associate with |
| + | the names of Burnside, Cauchy, Frobenius, or some subset or |
| + | superset of these three, vouching that the number of orbits |
| + | is equal to the mean number of fixed points, in other words, |
| + | the total number of points (in our case, propositions) that |
| + | are left unmoved by the separate operations, divided by the |
| + | order of the group. In this instance, T_00 operates as the |
| + | group identity, fixing all 16 propositions, while the other |
| + | three group elements fix 4 propositions each, and so we get: |
| + | Number of orbits = (4 + 4 + 4 + 16) / 4 = 7. Amazing! |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 8 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | We have been contemplating functions of the type f : U -> B |
| + | studying the action of the operators E and D on this family. |
| + | These functions, that we may identify for our present aims |
| + | with propositions, inasmuch as they capture their abstract |
| + | forms, are logical analogues of "scalar potential fields". |
| + | These are the sorts of fields that are so picturesquely |
| + | presented in elementary calculus and physics textbooks |
| + | by images of snow-covered hills and parties of skiers |
| + | who trek down their slopes like least action heroes. |
| + | The analogous scene in propositional logic presents |
| + | us with forms more reminiscent of plateaunic idylls, |
| + | being all plains at one of two levels, the mesas of |
| + | verity and falsity, as it were, with nary a niche |
| + | to inhabit between them, restricting our options |
| + | for a sporting gradient of downhill dynamics to |
| + | just one of two, standing still on level ground |
| + | or falling off a bluff. |
| + | |
| + | We are still working well within the logical analogue of the |
| + | classical finite difference calculus, taking in the novelties |
| + | that the logical transmutation of familiar elements is able to |
| + | bring to light. Soon we will take up several different notions |
| + | of approximation relationships that may be seen to organize the |
| + | space of propositions, and these will allow us to define several |
| + | different forms of differential analysis applying to propositions. |
| + | In time we will find reason to consider more general types of maps, |
| + | having concrete types of the form X_1 x ... x X_k -> Y_1 x ... x Y_n |
| + | and abstract types B^k -> B^n. We will think of these mappings as |
| + | transforming universes of discourse into themselves or into others, |
| + | in short, as "transformations of discourse". |
| + | |
| + | Before we continue with this intinerary, however, I would like to highlight |
| + | another sort of "differential aspect" that concerns the "boundary operator" |
| + | or the "marked connective" that serves as one of the two basic connectives |
| + | in the cactus language for ZOL. |
| + | |
| + | For example, consider the proposition f of concrete type f : X x Y x Z -> B |
| + | and abstract type f : B^3 -> B that is written "(x, y, z)" in cactus syntax. |
| + | Taken as an assertion in what Peirce called the "existential interpretation", |
| + | (x, y, z) says that just one of x, y, z is false. It is useful to consider |
| + | this assertion in relation to the conjunction xyz of the features that are |
| + | engaged as its arguments. A venn diagram of (x, y, z) looks like this: |
| + | |
| + | o-----------------------------------------------------------o |
| + | | U | |
| + | | | |
| + | | o-------------o | |
| + | | / \ | |
| + | | / \ | |
| + | | / \ | |
| + | | / \ | |
| + | | / \ | |
| + | | o x o | |
| + | | | | | |
| + | | | | | |
| + | | | | | |
| + | | | | | |
| + | | | | | |
| + | | o--o----------o o----------o--o | |
| + | | / \%%%%%%%%%%\ /%%%%%%%%%%/ \ | |
| + | | / \%%%%%%%%%%o%%%%%%%%%%/ \ | |
| + | | / \%%%%%%%%/ \%%%%%%%%/ \ | |
| + | | / \%%%%%%/ \%%%%%%/ \ | |
| + | | / \%%%%/ \%%%%/ \ | |
| + | | o o--o-------o--o o | |
| + | | | |%%%%%%%| | | |
| + | | | |%%%%%%%| | | |
| + | | | |%%%%%%%| | | |
| + | | | |%%%%%%%| | | |
| + | | | |%%%%%%%| | | |
| + | | o y o%%%%%%%o z o | |
| + | | \ \%%%%%/ / | |
| + | | \ \%%%/ / | |
| + | | \ \%/ / | |
| + | | \ o / | |
| + | | \ / \ / | |
| + | | o-------------o o-------------o | |
| + | | | |
| + | | | |
| + | o-----------------------------------------------------------o |
| + | |
| + | In relation to the center cell indicated by the conjunction xyz, |
| + | the region indicated by (x, y, z) is comprised of the "adjacent" |
| + | or the "bordering" cells. Thus they are the cells that are just |
| + | across the boundary of the center cell, as if reached by way of |
| + | Leibniz's "minimal changes" from the point of origin, here, xyz. |
| + | |
| + | The same sort of boundary relationship holds for any cell of origin that |
| + | one might elect to indicate, say, by means of the conjunction of positive |
| + | or negative basis features u_1 · ... · u_k, with u_j = x_j or u_j = (x_j), |
| + | for j = 1 to k. The proposition (u_1, ..., u_k) indicates the disjunctive |
| + | region consisting of the cells that are just next door to u_1 · ... · u_k. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 9 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might conceivably have |
| + | | practical bearings you conceive the objects of your |
| + | | conception to have. Then, your conception of those |
| + | | effects is the whole of your conception of the object. |
| + | | |
| + | | Charles Sanders Peirce, "The Maxim of Pragmatism, CP 5.438. |
| + | |
| + | One other subject that it would be opportune to mention at this point, |
| + | while we have an object example of a mathematical group fresh in mind, |
| + | is the relationship between the pragmatic maxim and what are commonly |
| + | known in mathematics as "representation principles". As it turns out, |
| + | with regard to its formal characteristics, the pragmatic maxim unites |
| + | the aspects of a representation principle with the attributes of what |
| + | would ordinarily be known as a "closure principle". We will consider |
| + | the form of closure that is invoked by the pragmatic maxim on another |
| + | occasion, focusing here and now on the topic of group representations. |
| + | |
| + | Let us return to the example of the so-called "four-group" V_4. |
| + | We encountered this group in one of its concrete representations, |
| + | namely, as a "transformation group" that acts on a set of objects, |
| + | in this particular case a set of sixteen functions or propositions. |
| + | Forgetting about the set of objects that the group transforms among |
| + | themselves, we may take the abstract view of the group's operational |
| + | structure, say, in the form of the group operation table copied here: |
| + | |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | · % e | f | g | h | |
| + | | % | | | | |
| + | o=========o=========o=========o=========o=========o |
| + | | % | | | | |
| + | | e % e | f | g | h | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | f % f | e | h | g | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | g % g | h | e | f | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | h % h | g | f | e | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | |
| + | This table is abstractly the same as, or isomorphic to, the versions with |
| + | the E_ij operators and the T_ij transformations that we discussed earlier. |
| + | That is to say, the story is the same -- only the names have been changed. |
| + | An abstract group can have a multitude of significantly and superficially |
| + | different representations. Even after we have long forgotten the details |
| + | of the particular representation that we may have come in with, there are |
| + | species of concrete representations, called the "regular representations", |
| + | that are always readily available, as they can be generated from the mere |
| + | data of the abstract operation table itself. |
| + | |
| + | For example, select a group element from the top margin of the Table, |
| + | and "consider its effects" on each of the group elements as they are |
| + | listed along the left margin. We may record these effects as Peirce |
| + | usually did, as a logical "aggregate" of elementary dyadic relatives, |
| + | that is to say, a disjunction or a logical sum whose terms represent |
| + | the ordered pairs of <input : output> transactions that are produced |
| + | by each group element in turn. This yields what is usually known as |
| + | one of the "regular representations" of the group, specifically, the |
| + | "first", the "post-", or the "right" regular representation. It has |
| + | long been conventional to organize the terms in the form of a matrix: |
| + | |
| + | Reading "+" as a logical disjunction: |
| + | |
| + | G = e + f + g + h, |
| + | |
| + | And so, by expanding effects, we get: |
| + | |
| + | G = e:e + f:f + g:g + h:h |
| + | |
| + | + e:f + f:e + g:h + h:g |
| + | |
| + | + e:g + f:h + g:e + h:f |
| + | |
| + | + e:h + f:g + g:f + h:e |
| + | |
| + | More on the pragmatic maxim as a representation principle later. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 10 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might conceivably have |
| + | | practical bearings you conceive the objects of your |
| + | | conception to have. Then, your conception of those |
| + | | effects is the whole of your conception of the object. |
| + | | |
| + | | Charles Sanders Peirce, "The Maxim of Pragmatism, CP 5.438. |
| + | |
| + | The genealogy of this conception of pragmatic representation is very intricate. |
| + | I will delineate some details that I presently fancy I remember clearly enough, |
| + | subject to later correction. Without checking historical accounts, I will not |
| + | be able to pin down anything like a real chronology, but most of these notions |
| + | were standard furnishings of the 19th Century mathematical study, and only the |
| + | last few items date as late as the 1920's. |
| + | |
| + | The idea about the regular representations of a group is universally known |
| + | as "Cayley's Theorem", usually in the form: "Every group is isomorphic to |
| + | a subgroup of Aut(S), the group of automorphisms of an appropriate set S". |
| + | There is a considerable generalization of these regular representations to |
| + | a broad class of relational algebraic systems in Peirce's earliest papers. |
| + | The crux of the whole idea is this: |
| + | |
| + | | Consider the effects of the symbol, whose meaning you wish to investigate, |
| + | | as they play out on "all" of the different stages of context on which you |
| + | | can imagine that symbol playing a role. |
| + | |
| + | This idea of contextual definition is basically the same as Jeremy Bentham's |
| + | notion of "paraphrasis", a "method of accounting for fictions by explaining |
| + | various purported terms away" (Quine, in Van Heijenoort, page 216). Today |
| + | we'd call these constructions "term models". This, again, is the big idea |
| + | behind Schönfinkel's combinators {S, K, I}, and hence of lambda calculus, |
| + | and I reckon you know where that leads. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 11 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might 'conceivably' |
| + | | have practical bearings you 'conceive' the |
| + | | objects of your 'conception' to have. Then, |
| + | | your 'conception' of those effects is the |
| + | | whole of your 'conception' of the object. |
| + | | |
| + | | Charles Sanders Peirce, |
| + | | "Maxim of Pragmaticism", CP 5.438. |
| + | |
| + | Continuing to draw on the reduced example of group representations, |
| + | I would like to draw out a few of the finer points and problems of |
| + | regarding the maxim of pragmatism as a principle of representation. |
| + | |
| + | Let us revisit the example of an abstract group that we had befour: |
| + | |
| + | Table 1. Klein Four-Group V_4 |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | · % e | f | g | h | |
| + | | % | | | | |
| + | o=========o=========o=========o=========o=========o |
| + | | % | | | | |
| + | | e % e | f | g | h | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | f % f | e | h | g | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | g % g | h | e | f | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | h % h | g | f | e | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | |
| + | I presented the regular post-representation |
| + | of the four-group V_4 in the following form: |
| + | |
| + | Reading "+" as a logical disjunction: |
| + | |
| + | G = e + f + g + h |
| + | |
| + | And so, by expanding effects, we get: |
| + | |
| + | G = e:e + f:f + g:g + h:h |
| + | |
| + | + e:f + f:e + g:h + h:g |
| + | |
| + | + e:g + f:h + g:e + h:f |
| + | |
| + | + e:h + f:g + g:f + h:e |
| + | |
| + | This presents the group in one big bunch, |
| + | and there are occasions when one regards |
| + | it this way, but that is not the typical |
| + | form of presentation that we'd encounter. |
| + | More likely, the story would go a little |
| + | bit like this: |
| + | |
| + | I cannot remember any of my math teachers |
| + | ever invoking the pragmatic maxim by name, |
| + | but it would be a very regular occurrence |
| + | for such mentors and tutors to set up the |
| + | subject in this wise: Suppose you forget |
| + | what a given abstract group element means, |
| + | that is, in effect, 'what it is'. Then a |
| + | sure way to jog your sense of 'what it is' |
| + | is to build a regular representation from |
| + | the formal materials that are necessarily |
| + | left lying about on that abstraction site. |
| + | |
| + | Working through the construction for each |
| + | one of the four group elements, we arrive |
| + | at the following exegeses of their senses, |
| + | giving their regular post-representations: |
| + | |
| + | e = e:e + f:f + g:g + h:h |
| + | |
| + | f = e:f + f:e + g:h + h:g |
| + | |
| + | g = e:g + f:h + g:e + h:f |
| + | |
| + | h = e:h + f:g + g:f + h:e |
| + | |
| + | So if somebody asks you, say, "What is g?", |
| + | you can say, "I don't know for certain but |
| + | in practice its effects go a bit like this: |
| + | Converting e to g, f to h, g to e, h to f". |
| + | |
| + | I will have to check this out later on, but my impression is |
| + | that Peirce tended to lean toward the other brand of regular, |
| + | the "second", the "left", or the "ante-representation" of the |
| + | groups that he treated in his earliest manuscripts and papers. |
| + | I believe that this was because he thought of the actions on |
| + | the pattern of dyadic relative terms like the "aftermath of". |
| + | |
| + | Working through this alternative for each |
| + | one of the four group elements, we arrive |
| + | at the following exegeses of their senses, |
| + | giving their regular ante-representations: |
| + | |
| + | e = e:e + f:f + g:g + h:h |
| + | |
| + | f = f:e + e:f + h:g + g:h |
| + | |
| + | g = g:e + h:f + e:g + f:h |
| + | |
| + | h = h:e + g:f + f:g + e:h |
| + | |
| + | Your paraphrastic interpretation of what this all |
| + | means would come out precisely the same as before. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 12 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Erratum |
| + | |
| + | Oops! I think that I have just confounded two entirely different issues: |
| + | 1. The substantial difference between right and left regular representations. |
| + | 2. The inessential difference between two conventions of presenting matrices. |
| + | I will sort this out and correct it later, as need be. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 13 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might 'conceivably' |
| + | | have practical bearings you 'conceive' the |
| + | | objects of your 'conception' to have. Then, |
| + | | your 'conception' of those effects is the |
| + | | whole of your 'conception' of the object. |
| + | | |
| + | | Charles Sanders Peirce, |
| + | | "Maxim of Pragmaticism", CP 5.438. |
| + | |
| + | Let me return to Peirce's early papers on the algebra of relatives |
| + | to pick up the conventions that he used there, and then rewrite my |
| + | account of regular representations in a way that conforms to those. |
| + | |
| + | Peirce expresses the action of an "elementary dual relative" like so: |
| + | |
| + | | [Let] A:B be taken to denote |
| + | | the elementary relative which |
| + | | multiplied into B gives A. |
| + | | |
| + | | Peirce, 'Collected Papers', CP 3.123. |
| + | |
| + | And though he is well aware that it is not at all necessary to arrange |
| + | elementary relatives into arrays, matrices, or tables, when he does so |
| + | he tends to prefer organizing dyadic relations in the following manner: |
| + | |
| + | | A:A A:B A:C | |
| + | | | |
| + | | B:A B:B B:C | |
| + | | | |
| + | | C:A C:B C:C | |
| + | |
| + | That conforms to the way that the last school of thought |
| + | I matriculated into stipulated that we tabulate material: |
| + | |
| + | | e_11 e_12 e_13 | |
| + | | | |
| + | | e_21 e_22 e_23 | |
| + | | | |
| + | | e_31 e_32 e_33 | |
| + | |
| + | So, for example, let us suppose that we have the small universe {A, B, C}, |
| + | and the 2-adic relation m = "mover of" that is represented by this matrix: |
| + | |
| + | m = |
| + | |
| + | | m_AA (A:A) m_AB (A:B) m_AC (A:C) | |
| + | | | |
| + | | m_BA (B:A) m_BB (B:B) m_BC (B:C) | |
| + | | | |
| + | | m_CA (C:A) m_CB (C:B) m_CC (C:C) | |
| + | |
| + | Also, let m be such that |
| + | A is a mover of A and B, |
| + | B is a mover of B and C, |
| + | C is a mover of C and A. |
| + | |
| + | In sum: |
| + | |
| + | m = |
| + | |
| + | | 1 · (A:A) 1 · (A:B) 0 · (A:C) | |
| + | | | |
| + | | 0 · (B:A) 1 · (B:B) 1 · (B:C) | |
| + | | | |
| + | | 1 · (C:A) 0 · (C:B) 1 · (C:C) | |
| + | |
| + | For the sake of orientation and motivation, |
| + | compare with Peirce's notation in CP 3.329. |
| + | |
| + | I think that will serve to fix notation |
| + | and set up the remainder of the account. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 14 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might 'conceivably' |
| + | | have practical bearings you 'conceive' the |
| + | | objects of your 'conception' to have. Then, |
| + | | your 'conception' of those effects is the |
| + | | whole of your 'conception' of the object. |
| + | | |
| + | | Charles Sanders Peirce, |
| + | | "Maxim of Pragmaticism", CP 5.438. |
| + | |
| + | I am beginning to see how I got confused. |
| + | It is common in algebra to switch around |
| + | between different conventions of display, |
| + | as the momentary fancy happens to strike, |
| + | and I see that Peirce is no different in |
| + | this sort of shiftiness than anyone else. |
| + | A changeover appears to occur especially |
| + | whenever he shifts from logical contexts |
| + | to algebraic contexts of application. |
| + | |
| + | In the paper "On the Relative Forms of Quaternions" (CP 3.323), |
| + | we observe Peirce providing the following sorts of explanation: |
| + | |
| + | | If X, Y, Z denote the three rectangular components of a vector, and W denote |
| + | | numerical unity (or a fourth rectangular component, involving space of four |
| + | | dimensions), and (Y:Z) denote the operation of converting the Y component |
| + | | of a vector into its Z component, then |
| + | | |
| + | | 1 = (W:W) + (X:X) + (Y:Y) + (Z:Z) |
| + | | |
| + | | i = (X:W) - (W:X) - (Y:Z) + (Z:Y) |
| + | | |
| + | | j = (Y:W) - (W:Y) - (Z:X) + (X:Z) |
| + | | |
| + | | k = (Z:W) - (W:Z) - (X:Y) + (Y:X) |
| + | | |
| + | | In the language of logic (Y:Z) is a relative term whose relate is |
| + | | a Y component, and whose correlate is a Z component. The law of |
| + | | multiplication is plainly (Y:Z)(Z:X) = (Y:X), (Y:Z)(X:W) = 0, |
| + | | and the application of these rules to the above values of |
| + | | 1, i, j, k gives the quaternion relations |
| + | | |
| + | | i^2 = j^2 = k^2 = -1, |
| + | | |
| + | | ijk = -1, |
| + | | |
| + | | etc. |
| + | | |
| + | | The symbol a(Y:Z) denotes the changing of Y to Z and the |
| + | | multiplication of the result by 'a'. If the relatives be |
| + | | arranged in a block |
| + | | |
| + | | W:W W:X W:Y W:Z |
| + | | |
| + | | X:W X:X X:Y X:Z |
| + | | |
| + | | Y:W Y:X Y:Y Y:Z |
| + | | |
| + | | Z:W Z:X Z:Y Z:Z |
| + | | |
| + | | then the quaternion w + xi + yj + zk |
| + | | is represented by the matrix of numbers |
| + | | |
| + | | w -x -y -z |
| + | | |
| + | | x w -z y |
| + | | |
| + | | y z w -x |
| + | | |
| + | | z -y x w |
| + | | |
| + | | The multiplication of such matrices follows the same laws as the |
| + | | multiplication of quaternions. The determinant of the matrix = |
| + | | the fourth power of the tensor of the quaternion. |
| + | | |
| + | | The imaginary x + y(-1)^(1/2) may likewise be represented by the matrix |
| + | | |
| + | | x y |
| + | | |
| + | | -y x |
| + | | |
| + | | and the determinant of the matrix = the square of the modulus. |
| + | | |
| + | | Charles Sanders Peirce, 'Collected Papers', CP 3.323. |
| + | |'Johns Hopkins University Circulars', No. 13, p. 179, 1882. |
| + | |
| + | This way of talking is the mark of a person who opts |
| + | to multiply his matrices "on the rignt", as they say. |
| + | Yet Peirce still continues to call the first element |
| + | of the ordered pair (I:J) its "relate" while calling |
| + | the second element of the pair (I:J) its "correlate". |
| + | That doesn't comport very well, so far as I can tell, |
| + | with his customary reading of relative terms, suited |
| + | more to the multiplication of matrices "on the left". |
| + | |
| + | So I still have a few wrinkles to iron out before |
| + | I can give this story a smooth enough consistency. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 15 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might 'conceivably' |
| + | | have practical bearings you 'conceive' the |
| + | | objects of your 'conception' to have. Then, |
| + | | your 'conception' of those effects is the |
| + | | whole of your 'conception' of the object. |
| + | | |
| + | | Charles Sanders Peirce, |
| + | | "Maxim of Pragmaticism", CP 5.438. |
| + | |
| + | I have been planning for quite some time now to make my return to Peirce's |
| + | skyshaking "Description of a Notation for the Logic of Relatives" (1870), |
| + | and I can see that it's just about time to get down tuit, so let this |
| + | current bit of rambling inquiry function as the preamble to that. |
| + | All we need at the present, though, is a modus vivendi/operandi |
| + | for telling what is substantial from what is inessential in |
| + | the brook between symbolic conceits and dramatic actions |
| + | that we find afforded by means of the pragmatic maxim. |
| + | |
| + | Back to our "subinstance", the example in support of our first example. |
| + | I will now reconstruct it in a way that may prove to be less confusing. |
| + | |
| + | Let us make up the model universe $1$ = A + B + C and the 2-adic relation |
| + | n = "noder of", as when "X is a data record that contains a pointer to Y". |
| + | That interpretation is not important, it's just for the sake of intuition. |
| + | In general terms, the 2-adic relation n can be represented by this matrix: |
| + | |
| + | n = |
| + | |
| + | | n_AA (A:A) n_AB (A:B) n_AC (A:C) | |
| + | | | |
| + | | n_BA (B:A) n_BB (B:B) n_BC (B:C) | |
| + | | | |
| + | | n_CA (C:A) n_CB (C:B) n_CC (C:C) | |
| + | |
| + | Also, let n be such that |
| + | A is a noder of A and B, |
| + | B is a noder of B and C, |
| + | C is a noder of C and A. |
| + | |
| + | Filling in the instantial values of the "coefficients" n_ij, |
| + | as the indices i and j range over the universe of discourse: |
| + | |
| + | n = |
| + | |
| + | | 1 · (A:A) 1 · (A:B) 0 · (A:C) | |
| + | | | |
| + | | 0 · (B:A) 1 · (B:B) 1 · (B:C) | |
| + | | | |
| + | | 1 · (C:A) 0 · (C:B) 1 · (C:C) | |
| + | |
| + | In Peirce's time, and even in some circles of mathematics today, |
| + | the information indicated by the elementary relatives (I:J), as |
| + | I, J range over the universe of discourse, would be referred to |
| + | as the "umbral elements" of the algebraic operation represented |
| + | by the matrix, though I seem to recall that Peirce preferred to |
| + | call these terms the "ingredients". When this ordered basis is |
| + | understood well enough, one will tend to drop any mention of it |
| + | from the matrix itself, leaving us nothing but these bare bones: |
| + | |
| + | n = |
| + | |
| + | | 1 1 0 | |
| + | | | |
| + | | 0 1 1 | |
| + | | | |
| + | | 1 0 1 | |
| + | |
| + | However the specification may come to be written, this |
| + | is all just convenient schematics for stipulating that: |
| + | |
| + | n = A:A + B:B + C:C + A:B + B:C + C:A |
| + | |
| + | Recognizing !1! = A:A + B:B + C:C to be the identity transformation, |
| + | the 2-adic relation n = "noder of" may be represented by an element |
| + | !1! + A:B + B:C + C:A of the so-called "group ring", all of which |
| + | just makes this element a special sort of linear transformation. |
| + | |
| + | Up to this point, we are still reading the elementary relatives of |
| + | the form I:J in the way that Peirce reads them in logical contexts: |
| + | I is the relate, J is the correlate, and in our current example we |
| + | read I:J, or more exactly, n_ij = 1, to say that I is a noder of J. |
| + | This is the mode of reading that we call "multiplying on the left". |
| + | |
| + | In the algebraic, permutational, or transformational contexts of |
| + | application, however, Peirce converts to the alternative mode of |
| + | reading, although still calling I the relate and J the correlate, |
| + | the elementary relative I:J now means that I gets changed into J. |
| + | In this scheme of reading, the transformation A:B + B:C + C:A is |
| + | a permutation of the aggregate $1$ = A + B + C, or what we would |
| + | now call the set {A, B, C}, in particular, it is the permutation |
| + | that is otherwise notated as: |
| + | |
| + | ( A B C ) |
| + | < > |
| + | ( B C A ) |
| + | |
| + | This is consistent with the convention that Peirce uses in |
| + | the paper "On a Class of Multiple Algebras" (CP 3.324-327). |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 16 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might 'conceivably' |
| + | | have practical bearings you 'conceive' the |
| + | | objects of your 'conception' to have. Then, |
| + | | your 'conception' of those effects is the |
| + | | whole of your 'conception' of the object. |
| + | | |
| + | | Charles Sanders Peirce, |
| + | | "Maxim of Pragmaticism", CP 5.438. |
| + | |
| + | We have been contemplating the virtues and the utilities of |
| + | the pragmatic maxim as a hermeneutic heuristic, specifically, |
| + | as a principle of interpretation that guides us in finding a |
| + | clarifying representation for a problematic corpus of symbols |
| + | in terms of their actions on other symbols or their effects on |
| + | the syntactic contexts in which we conceive to distribute them. |
| + | I started off considering the regular representations of groups |
| + | as constituting what appears to be one of the simplest possible |
| + | applications of this overall principle of representation. |
| + | |
| + | There are a few problems of implementation that have to be worked out |
| + | in practice, most of which are cleared up by keeping in mind which of |
| + | several possible conventions we have chosen to follow at a given time. |
| + | But there does appear to remain this rather more substantial question: |
| + | |
| + | Are the effects we seek relates or correlates, or does it even matter? |
| + | |
| + | I will have to leave that question as it is for now, |
| + | in hopes that a solution will evolve itself in time. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 17 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might 'conceivably' |
| + | | have practical bearings you 'conceive' the |
| + | | objects of your 'conception' to have. Then, |
| + | | your 'conception' of those effects is the |
| + | | whole of your 'conception' of the object. |
| + | | |
| + | | Charles Sanders Peirce, |
| + | | "Maxim of Pragmaticism", CP 5.438. |
| + | |
| + | There a big reasons and little reasons for caring about this humble example. |
| + | The little reasons we find all under our feet. One big reason I can now |
| + | quite blazonly enounce in the fashion of this not so subtle subtitle: |
| + | |
| + | Obstacles to Applying the Pragmatic Maxim |
| + | |
| + | No sooner do you get a good idea and try to apply it |
| + | than you find that a motley array of obstacles arise. |
| + | |
| + | It seems as if I am constantly lamenting the fact these days that people, |
| + | and even admitted Peircean persons, do not in practice more consistently |
| + | apply the maxim of pragmatism to the purpose for which it is purportedly |
| + | intended by its author. That would be the clarification of concepts, or |
| + | intellectual symbols, to the point where their inherent senses, or their |
| + | lacks thereof, would be rendered manifest to all and sundry interpreters. |
| + | |
| + | There are big obstacles and little obstacles to applying the pragmatic maxim. |
| + | In good subgoaling fashion, I will merely mention a few of the bigger blocks, |
| + | as if in passing, and then get down to the devilish details that immediately |
| + | obstruct our way. |
| + | |
| + | Obstacle 1. People do not always read the instructions very carefully. |
| + | There is a tendency in readers of particular prior persuasions to blow |
| + | the problem all out of proportion, to think that the maxim is meant to |
| + | reveal the absolutely positive and the totally unique meaning of every |
| + | preconception to which they might deign or elect to apply it. Reading |
| + | the maxim with an even minimal attention, you can see that it promises |
| + | no such finality of unindexed sense, but ties what you conceive to you. |
| + | I have lately come to wonder at the tenacity of this misinterpretation. |
| + | Perhaps people reckon that nothing less would be worth their attention. |
| + | I am not sure. I can only say the achievement of more modest goals is |
| + | the sort of thing on which our daily life depends, and there can be no |
| + | final end to inquiry nor any ultimate community without a continuation |
| + | of life, and that means life on a day to day basis. All of which only |
| + | brings me back to the point of persisting with local meantime examples, |
| + | because if we can't apply the maxim there, we can't apply it anywhere. |
| + | |
| + | And now I need to go out of doors and weed my garden for a time ... |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 18 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might 'conceivably' |
| + | | have practical bearings you 'conceive' the |
| + | | objects of your 'conception' to have. Then, |
| + | | your 'conception' of those effects is the |
| + | | whole of your 'conception' of the object. |
| + | | |
| + | | Charles Sanders Peirce, |
| + | | "Maxim of Pragmaticism", CP 5.438. |
| + | |
| + | Obstacles to Applying the Pragmatic Maxim |
| + | |
| + | Obstacle 2. Applying the pragmatic maxim, even with a moderate aim, can be hard. |
| + | I think that my present example, deliberately impoverished as it is, affords us |
| + | with an embarassing richness of evidence of just how complex the simple can be. |
| + | |
| + | All the better reason for me to see if I can finish it up before moving on. |
| + | |
| + | Expressed most simply, the idea is to replace the question of "what it is", |
| + | which modest people know is far too difficult for them to answer right off, |
| + | with the question of "what it does", which most of us know a modicum about. |
| + | |
| + | In the case of regular representations of groups we found |
| + | a non-plussing surplus of answers to sort our way through. |
| + | So let us track back one more time to see if we can learn |
| + | any lessons that might carry over to more realistic cases. |
| + | |
| + | Here is is the operation table of V_4 once again: |
| + | |
| + | Table 1. Klein Four-Group V_4 |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | · % e | f | g | h | |
| + | | % | | | | |
| + | o=========o=========o=========o=========o=========o |
| + | | % | | | | |
| + | | e % e | f | g | h | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | f % f | e | h | g | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | g % g | h | e | f | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | | % | | | | |
| + | | h % h | g | f | e | |
| + | | % | | | | |
| + | o---------o---------o---------o---------o---------o |
| + | |
| + | A group operation table is really just a device for |
| + | recording a certain 3-adic relation, to be specific, |
| + | the set of triples of the form <x, y, z> satisfying |
| + | the equation x·y = z where · is the group operation. |
| + | |
| + | In the case of V_4 = (G, ·), where G is the "underlying set" |
| + | {e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G |
| + | whose triples are listed below: |
| + | |
| + | | <e, e, e> |
| + | | <e, f, f> |
| + | | <e, g, g> |
| + | | <e, h, h> |
| + | | |
| + | | <f, e, f> |
| + | | <f, f, e> |
| + | | <f, g, h> |
| + | | <f, h, g> |
| + | | |
| + | | <g, e, g> |
| + | | <g, f, h> |
| + | | <g, g, e> |
| + | | <g, h, f> |
| + | | |
| + | | <h, e, h> |
| + | | <h, f, g> |
| + | | <h, g, f> |
| + | | <h, h, e> |
| + | |
| + | It is part of the definition of a group that the 3-adic |
| + | relation L c G^3 is actually a function L : G x G -> G. |
| + | It is from this functional perspective that we can see |
| + | an easy way to derive the two regular representations. |
| + | Since we have a function of the type L : G x G -> G, |
| + | we can define a couple of substitution operators: |
| + | |
| + | 1. Sub(x, <_, y>) puts any specified x into |
| + | the empty slot of the rheme <_, y>, with |
| + | the effect of producing the saturated |
| + | rheme <x, y> that evaluates to x·y. |
| + | |
| + | 2. Sub(x, <y, _>) puts any specified x into |
| + | the empty slot of the rheme <y, >, with |
| + | the effect of producing the saturated |
| + | rheme <y, x> that evaluates to y·x. |
| + | |
| + | In (1), we consider the effects of each x in its |
| + | practical bearing on contexts of the form <_, y>, |
| + | as y ranges over G, and the effects are such that |
| + | x takes <_, y> into x·y, for y in G, all of which |
| + | is summarily notated as x = {(y : x·y) : y in G}. |
| + | The pairs (y : x·y) can be found by picking an x |
| + | from the left margin of the group operation table |
| + | and considering its effects on each y in turn as |
| + | these run across the top margin. This aspect of |
| + | pragmatic definition we recognize as the regular |
| + | ante-representation: |
| + | |
| + | e = e:e + f:f + g:g + h:h |
| + | |
| + | f = e:f + f:e + g:h + h:g |
| + | |
| + | g = e:g + f:h + g:e + h:f |
| + | |
| + | h = e:h + f:g + g:f + h:e |
| + | |
| + | In (2), we consider the effects of each x in its |
| + | practical bearing on contexts of the form <y, _>, |
| + | as y ranges over G, and the effects are such that |
| + | x takes <y, _> into y·x, for y in G, all of which |
| + | is summarily notated as x = {(y : y·x) : y in G}. |
| + | The pairs (y : y·x) can be found by picking an x |
| + | from the top margin of the group operation table |
| + | and considering its effects on each y in turn as |
| + | these run down the left margin. This aspect of |
| + | pragmatic definition we recognize as the regular |
| + | post-representation: |
| + | |
| + | e = e:e + f:f + g:g + h:h |
| + | |
| + | f = e:f + f:e + g:h + h:g |
| + | |
| + | g = e:g + f:h + g:e + h:f |
| + | |
| + | h = e:h + f:g + g:f + h:e |
| + | |
| + | If the ante-rep looks the same as the post-rep, |
| + | now that I'm writing them in the same dialect, |
| + | that is because V_4 is abelian (commutative), |
| + | and so the two representations have the very |
| + | same effects on each point of their bearing. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 19 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might 'conceivably' |
| + | | have practical bearings you 'conceive' the |
| + | | objects of your 'conception' to have. Then, |
| + | | your 'conception' of those effects is the |
| + | | whole of your 'conception' of the object. |
| + | | |
| + | | Charles Sanders Peirce, |
| + | | "Maxim of Pragmaticism", CP 5.438. |
| + | |
| + | So long as we're in the neighborhood, we might as well take in |
| + | some more of the sights, for instance, the smallest example of |
| + | a non-abelian (non-commutative) group. This is a group of six |
| + | elements, say, G = {e, f, g, h, i, j}, with no relation to any |
| + | other employment of these six symbols being implied, of course, |
| + | and it can be most easily represented as the permutation group |
| + | on a set of three letters, say, X = {A, B, C}, usually notated |
| + | as G = Sym(X) or more abstractly and briefly, as Sym(3) or S_3. |
| + | Here are the permutation (= substitution) operations in Sym(X): |
| + | |
| + | Table 2. Permutations or Substitutions in Sym_{A, B, C} |
| + | o---------o---------o---------o---------o---------o---------o |
| + | | | | | | | | |
| + | | e | f | g | h | i | j | |
| + | | | | | | | | |
| + | o=========o=========o=========o=========o=========o=========o |
| + | | | | | | | | |
| + | | A B C | A B C | A B C | A B C | A B C | A B C | |
| + | | | | | | | | |
| + | | | | | | | | | | | | | | | | | | | | | | | | | | |
| + | | v v v | v v v | v v v | v v v | v v v | v v v | |
| + | | | | | | | | |
| + | | A B C | C A B | B C A | A C B | C B A | B A C | |
| + | | | | | | | | |
| + | o---------o---------o---------o---------o---------o---------o |
| + | |
| + | Here is the operation table for S_3, given in abstract fashion: |
| + | |
| + | Table 3. Symmetric Group S_3 |
| + | |
| + | | _ |
| + | | e / \ e |
| + | | / \ |
| + | | / e \ |
| + | | f / \ / \ f |
| + | | / \ / \ |
| + | | / f \ f \ |
| + | | g / \ / \ / \ g |
| + | | / \ / \ / \ |
| + | | / g \ g \ g \ |
| + | | h / \ / \ / \ / \ h |
| + | | / \ / \ / \ / \ |
| + | | / h \ e \ e \ h \ |
| + | | i / \ / \ / \ / \ / \ i |
| + | | / \ / \ / \ / \ / \ |
| + | | / i \ i \ f \ j \ i \ |
| + | | j / \ / \ / \ / \ / \ / \ j |
| + | | / \ / \ / \ / \ / \ / \ |
| + | | ( j \ j \ j \ i \ h \ j ) |
| + | | \ / \ / \ / \ / \ / \ / |
| + | | \ / \ / \ / \ / \ / \ / |
| + | | \ h \ h \ e \ j \ i / |
| + | | \ / \ / \ / \ / \ / |
| + | | \ / \ / \ / \ / \ / |
| + | | \ i \ g \ f \ h / |
| + | | \ / \ / \ / \ / |
| + | | \ / \ / \ / \ / |
| + | | \ f \ e \ g / |
| + | | \ / \ / \ / |
| + | | \ / \ / \ / |
| + | | \ g \ f / |
| + | | \ / \ / |
| + | | \ / \ / |
| + | | \ e / |
| + | | \ / |
| + | | \ / |
| + | | ¯ |
| + | |
| + | By the way, we will meet with the symmetric group S_3 again |
| + | when we return to take up the study of Peirce's early paper |
| + | "On a Class of Multiple Algebras" (CP 3.324-327), and also |
| + | his late unpublished work "The Simplest Mathematics" (1902) |
| + | (CP 4.227-323), with particular reference to the section |
| + | that treats of "Trichotomic Mathematics" (CP 4.307-323). |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Work Area |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 20 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might 'conceivably' |
| + | | have practical bearings you 'conceive' the |
| + | | objects of your 'conception' to have. Then, |
| + | | your 'conception' of those effects is the |
| + | | whole of your 'conception' of the object. |
| + | | |
| + | | Charles Sanders Peirce, |
| + | | "Maxim of Pragmaticism", CP 5.438. |
| + | |
| + | By way of collecting a shot-term pay-off for all the work -- |
| + | not to mention the peirce-spiration -- that we sweated out |
| + | over the regular representations of V_4 and S_3 |
| + | |
| + | Table 2. Permutations or Substitutions in Sym_{A, B, C} |
| + | o---------o---------o---------o---------o---------o---------o |
| + | | | | | | | | |
| + | | e | f | g | h | i | j | |
| + | | | | | | | | |
| + | o=========o=========o=========o=========o=========o=========o |
| + | | | | | | | | |
| + | | A B C | A B C | A B C | A B C | A B C | A B C | |
| + | | | | | | | | |
| + | | | | | | | | | | | | | | | | | | | | | | | | | | |
| + | | v v v | v v v | v v v | v v v | v v v | v v v | |
| + | | | | | | | | |
| + | | A B C | C A B | B C A | A C B | C B A | B A C | |
| + | | | | | | | | |
| + | o---------o---------o---------o---------o---------o---------o |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Note 21 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | | Consider what effects that might 'conceivably' |
| + | | have practical bearings you 'conceive' the |
| + | | objects of your 'conception' to have. Then, |
| + | | your 'conception' of those effects is the |
| + | | whole of your 'conception' of the object. |
| + | | |
| + | | Charles Sanders Peirce, |
| + | | "Maxim of Pragmaticism", CP 5.438. |
| + | |
| + | problem about writing |
| + | |
| + | e = e:e + f:f + g:g + h:h |
| + | |
| + | no recursion intended |
| + | need for a work-around |
| + | ways way explaining it away |
| + | |
| + | action on signs not objects |
| + | |
| + | math def of rep |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Zeroth Order Logic |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Here is a scaled-down version of one of my very first applications, |
| + | having to do with the demographic variables in a survey data base. |
| + | |
| + | This Example illustrates the use of 2-variate logical forms |
| + | for expressing and reasoning about the logical constraints |
| + | that are involved in the following types of situations: |
| + | |
| + | 1. Distinction: A =/= B |
| + | Also known as: logical inequality, exclusive disjunction |
| + | Represented as: ( A , B ) |
| + | Graphed as: |
| + | | |
| + | | A B |
| + | | o---o |
| + | | \ / |
| + | | @ |
| + | |
| + | 2. Equality: A = B |
| + | Also known as: logical equivalence, if and only if, A <=> B |
| + | Represented as: (( A , B )) |
| + | Graphed as: |
| + | | |
| + | | A B |
| + | | o---o |
| + | | \ / |
| + | | o |
| + | | | |
| + | | @ |
| + | |
| + | 3. Implication: A => B |
| + | Also known as: entailment, if-then |
| + | Represented as: ( A ( B )) |
| + | Graphed as: |
| + | | |
| + | | A B |
| + | | o---o |
| + | | | |
| + | | @ |
| + | |
| + | Example of a proposition expressing a "zeroth order theory" (ZOT): |
| + | |
| + | Consider the following text, written in what I am calling "Ref Log", |
| + | also known as the "Cactus Language" synpropositional logic: |
| + | |
| + | | ( male , female ) |
| + | | (( boy , male child )) |
| + | | (( girl , female child )) |
| + | | ( child ( human )) |
| + | |
| + | Graphed as: |
| + | |
| + | | boy male girl female |
| + | | o---o child o---o child |
| + | | male female \ / \ / child human |
| + | | o---o o o o---o |
| + | | \ / | | | |
| + | | @ @ @ @| |
| + | |
| + | Nota Bene. Due to graphic constraints -- no, the other |
| + | kind of graphic constraints -- of the immediate medium, |
| + | I am forced to string out the logical conjuncts of the |
| + | actual cactus graph for this situation, one that might |
| + | sufficiently be reasoned out from the exhibit supra by |
| + | fusing together the four roots of the severed cactus. |
| + | |
| + | Either of these expressions, text or graph, is equivalent to |
| + | what would otherwise be written in a more ordinary syntax as: |
| + | |
| + | | male =/= female |
| + | | boy <=> male child |
| + | | girl <=> female child |
| + | | child => human |
| + | |
| + | This is a actually a single proposition, a conjunction of four lines: |
| + | one distinction, two equations, and one implication. Together these |
| + | amount to a set of definitions conjointly constraining the logical |
| + | compatibility of the six feature names that appear. They may be |
| + | thought of as sculpting out a space of models that is some subset |
| + | of the 2^6 = 64 possible interpretations, and thereby shaping some |
| + | universe of discourse. |
| + | |
| + | Once this backdrop is defined, it is possible to "query" this universe, |
| + | simply by conjoining additional propositions in further constraint of |
| + | the underlying set of models. This has many uses, as we shall see. |
| + | |
| + | We are considering an Example of a propositional expression |
| + | that is formed on the following "alphabet" or "lexicon" of |
| + | six "logical features" or "boolean variables": |
| + | |
| + | $A$ = {"boy", "child", "female", "girl", "human", "male"}. |
| + | |
| + | The expression is this: |
| + | |
| + | | ( male , female ) |
| + | | (( boy , male child )) |
| + | | (( girl , female child )) |
| + | | ( child ( human )) |
| + | |
| + | Putting it very roughly -- and putting off a better description |
| + | of it till later -- we may think of this expression as notation |
| + | for a boolean function f : %B%^6 -> %B%. This is what we might |
| + | call the "abstract type" of the function, but we will also find |
| + | it convenient on many occasions to represent the points of this |
| + | particular copy of the space %B%^6 in terms of the positive and |
| + | negative versions of the features from $A$ that serve to encase |
| + | them as logical "cells", as they are called in the venn diagram |
| + | picture of the corresponding universe of discourse X = [$A$]. |
| + | |
| + | Just for concreteness, this form of representation begins and ends: |
| + | |
| + | <0,0,0,0,0,0> = (boy)(child)(female)(girl)(human)(male), |
| + | <0,0,0,0,0,1> = (boy)(child)(female)(girl)(human) male , |
| + | <0,0,0,0,1,0> = (boy)(child)(female)(girl) human (male), |
| + | <0,0,0,0,1,1> = (boy)(child)(female)(girl) human male , |
| + | ... |
| + | <1,1,1,1,0,0> = boy child female girl (human)(male), |
| + | <1,1,1,1,0,1> = boy child female girl (human) male , |
| + | <1,1,1,1,1,0> = boy child female girl human (male), |
| + | <1,1,1,1,1,1> = boy child female girl human male . |
| + | |
| + | I continue with the previous Example, that I bring forward and sum up here: |
| + | |
| + | | boy male girl female |
| + | | o---o child o---o child |
| + | | male female \ / \ / child human |
| + | | o---o o o o--o |
| + | | \ / | | | |
| + | | @ @ @ @ |
| + | | |
| + | | (male , female)((boy , male child))((girl , female child))(child (human)) |
| + | |
| + | For my master's piece in Quantitative Psychology (Michigan State, 1989), |
| + | I wrote a program, "Theme One" (TO) by name, that among its other duties |
| + | operates to process the expressions of the cactus language in many of the |
| + | most pressing ways that we need in order to be able to use it effectively |
| + | as a propositional calculus. The operational component of TO where one |
| + | does the work of this logical modeling is called "Study", and the core |
| + | of the logical calculator deep in the heart of this Study section is |
| + | a suite of computational functions that evolve a particular species |
| + | of "normal form", analogous to a "disjunctive normal form" (DNF), |
| + | from whatever expression they are prebendered as their input. |
| + | |
| + | This "canonical", "normal", or "stable" form of logical expression -- |
| + | I'll refine the distinctions among these subforms all in good time -- |
| + | permits succinct depiction as an "arboreal boolean expansion" (ABE). |
| + | |
| + | Once again, the graphic limitations of this space prevail against |
| + | any disposition that I might have to lay out a really substantial |
| + | case before you, of the brand that might have a chance to impress |
| + | you with the aptitude of this ilk of ABE in rooting out the truth |
| + | of many a complexly obscurely subtly adamant whetstone of our wit. |
| + | |
| + | So let me just illustrate the way of it with one conjunct of our Example. |
| + | What follows will be a sequence of expressions, each one after the first |
| + | being logically equal to the one that precedes it: |
| + | |
| + | Step 1 |
| + | |
| + | | g fc |
| + | | o---o |
| + | | \ / |
| + | | o |
| + | | | |
| + | | @ |
| + | |
| + | Step 2 |
| + | |
| + | | o |
| + | | fc | fc |
| + | | o---o o---o |
| + | | \ / \ / |
| + | | o o |
| + | | | | |
| + | | g o-------------o--o g |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | @ |
| + | |
| + | Step 3 |
| + | |
| + | | f c |
| + | | o |
| + | | | f c |
| + | | o o |
| + | | | | |
| + | | g o-------------o--o g |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | @ |
| + | |
| + | Step 4 |
| + | |
| + | | o |
| + | | | |
| + | | c o o c o |
| + | | | | | |
| + | | o o c o o c |
| + | | | | | | |
| + | | f o---o--o f f o---o--o f |
| + | | \ / \ / |
| + | | g o-------------o--o g |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | @ |
| + | |
| + | Step 5 |
| + | |
| + | | o c o |
| + | | c | | |
| + | | f o---o--o f f o---o--o f |
| + | | \ / \ / |
| + | | g o-------------o--o g |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | @ |
| + | |
| + | Step 6 |
| + | |
| + | | o |
| + | | | |
| + | | o o o |
| + | | | | | |
| + | | c o---o--o c o c o---o--o c |
| + | | \ / | \ / |
| + | | f o-------------o--o f f o-------------o--o f |
| + | | \ / \ / |
| + | | \ / \ / |
| + | | \ / \ / |
| + | | \ / \ / |
| + | | \ / \ / |
| + | | \ / \ / |
| + | | g o---------------------------o--o g |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | @ |
| + | |
| + | Step 7 |
| + | |
| + | | o o |
| + | | | | |
| + | | c o---o--o c o c o---o--o c |
| + | | \ / | \ / |
| + | | f o-------------o--o f f o-------------o--o f |
| + | | \ / \ / |
| + | | \ / \ / |
| + | | \ / \ / |
| + | | \ / \ / |
| + | | \ / \ / |
| + | | \ / \ / |
| + | | g o---------------------------o--o g |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | \ / |
| + | | @ |
| + | |
| + | This last expression is the ABE of the input expression. |
| + | It can be transcribed into ordinary logical language as: |
| + | |
| + | | either girl and |
| + | | either female and |
| + | | either child and true |
| + | | or not child and false |
| + | | or not female and false |
| + | | or not girl and |
| + | | either female and |
| + | | either child and false |
| + | | or not child and true |
| + | | or not female and true |
| + | |
| + | The expression "((girl , female child))" is sufficiently evaluated |
| + | by considering its logical values on the coordinate tuples of %B%^3, |
| + | or its indications on the cells of the associated venn diagram that |
| + | depicts the universe of discourse, namely, on these eight arguments: |
| + | |
| + | <1, 1, 1> = girl female child , |
| + | <1, 1, 0> = girl female (child), |
| + | <1, 0, 1> = girl (female) child , |
| + | <1, 0, 0> = girl (female)(child), |
| + | <0, 1, 1> = (girl) female child , |
| + | <0, 1, 0> = (girl) female (child), |
| + | <0, 0, 1> = (girl)(female) child , |
| + | <0, 0, 0> = (girl)(female)(child). |
| + | |
| + | The ABE output expression tells us the logical values of |
| + | the input expression on each of these arguments, doing so |
| + | by attaching the values to the leaves of a tree, and acting |
| + | as an "efficient" or "lazy" evaluator in the sense that the |
| + | process that generates the tree follows each path only up to |
| + | the point in the tree where it can determine the values on the |
| + | entire subtree beyond that point. Thus, the ABE tree tells us: |
| + | |
| + | girl female child -> 1 |
| + | girl female (child) -> 0 |
| + | girl (female) -> 0 |
| + | (girl) female child -> 0 |
| + | (girl) female (child) -> 1 |
| + | (girl)(female) -> 1 |
| + | |
| + | Picking out the interpretations that yield the truth of the expression, |
| + | and expanding the corresponding partial argument tuples, we arrive at |
| + | the following interpretations that satisfy the input expression: |
| + | |
| + | girl female child -> 1 |
| + | (girl) female (child) -> 1 |
| + | (girl)(female) child -> 1 |
| + | (girl)(female)(child) -> 1 |
| + | |
| + | In sum, if it's a female and a child, then it's a girl, |
| + | and if it's either not a female or not a child or both, |
| + | then it's not a girl. |
| + | |
| + | Brief Automata |
| + | |
| + | By way of providing a simple illustration of Cook's Theorem, |
| + | that "Propositional Satisfiability is NP-Complete", here is |
| + | an exposition of one way to translate Turing Machine set-ups |
| + | into propositional expressions, employing the Ref Log Syntax |
| + | for Prop Calc that I described in a couple of earlier notes: |
| + | |
| + | Notation: |
| + | |
| + | Stilt(k) = Space and Time Limited Turing Machine, |
| + | with k units of space and k units of time. |
| + | |
| + | Stunt(k) = Space and Time Limited Turing Machine, |
| + | for computing the parity of a bit string, |
| + | with Number of Tape cells of input equal to k. |
| + | |
| + | I will follow the pattern of the discussion in the book of |
| + | Herbert Wilf, 'Algorithms & Complexity' (1986), pages 188-201, |
| + | but translate into Ref Log, which is more efficient with respect |
| + | to the number of propositional clauses that are required. |
| + | |
| + | Parity Machine |
| + | |
| + | | 1/1/+1 |
| + | | -------> |
| + | | /\ / \ /\ |
| + | | 0/0/+1 ^ 0 1 ^ 0/0/+1 |
| + | | \/|\ /|\/ |
| + | | | <------- | |
| + | | #/#/-1 | 1/1/+1 | #/#/-1 |
| + | | | | |
| + | | v v |
| + | | # * |
| + | |
| + | o-------o--------o-------------o---------o------------o |
| + | | State | Symbol | Next Symbol | Ratchet | Next State | |
| + | | Q | S | S' | dR | Q' | |
| + | o-------o--------o-------------o---------o------------o |
| + | | 0 | 0 | 0 | +1 | 0 | |
| + | | 0 | 1 | 1 | +1 | 1 | |
| + | | 0 | # | # | -1 | # | |
| + | | 1 | 0 | 0 | +1 | 1 | |
| + | | 1 | 1 | 1 | +1 | 0 | |
| + | | 1 | # | # | -1 | * | |
| + | o-------o--------o-------------o---------o------------o |
| + | |
| + | The TM has a "finite automaton" (FA) as its component. |
| + | Let us refer to this particular FA by the name of "M". |
| + | |
| + | The "tape-head" (that is, the "read-unit") will be called "H". |
| + | The "registers" are also called "tape-cells" or "tape-squares". |
| + | |
| + | In order to consider how the finitely "stilted" rendition of this TM |
| + | can be translated into the form of a purely propositional description, |
| + | one now fixes k and limits the discussion to talking about a Stilt(k), |
| + | which is really not a true TM anymore but a finite automaton in disguise. |
| + | |
| + | In this example, for the sake of a minimal illustration, we choose k = 2, |
| + | and discuss Stunt(2). Since the zeroth tape cell and the last tape cell |
| + | are occupied with bof and eof marks "#", this amounts to only one digit |
| + | of significant computation. |
| + | |
| + | To translate Stunt(2) into propositional form we use |
| + | the following collection of propositional variables: |
| + | |
| + | For the "Present State Function" QF : P -> Q, |
| + | |
| + | {p0_q#, p0_q*, p0_q0, p0_q1, |
| + | p1_q#, p1_q*, p1_q0, p1_q1, |
| + | p2_q#, p2_q*, p2_q0, p2_q1, |
| + | p3_q#, p3_q*, p3_q0, p3_q1} |
| + | |
| + | The propositional expression of the form "pi_qj" says: |
| + | |
| + | | At the point-in-time p_i, |
| + | | the finite machine M is in the state q_j. |
| + | |
| + | For the "Present Register Function" RF : P -> R, |
| + | |
| + | {p0_r0, p0_r1, p0_r2, p0_r3, |
| + | p1_r0, p1_r1, p1_r2, p1_r3, |
| + | p2_r0, p2_r1, p2_r2, p2_r3, |
| + | p3_r0, p3_r1, p3_r2, p3_r3} |
| + | |
| + | The propositional expression of the form "pi_rj" says: |
| + | |
| + | | At the point-in-time p_i, |
| + | | the tape-head H is on the tape-cell r_j. |
| + | |
| + | For the "Present Symbol Function" SF : P -> (R -> S), |
| + | |
| + | {p0_r0_s#, p0_r0_s*, p0_r0_s0, p0_r0_s1, |
| + | p0_r1_s#, p0_r1_s*, p0_r1_s0, p0_r1_s1, |
| + | p0_r2_s#, p0_r2_s*, p0_r2_s0, p0_r2_s1, |
| + | p0_r3_s#, p0_r3_s*, p0_r3_s0, p0_r3_s1, |
| + | p1_r0_s#, p1_r0_s*, p1_r0_s0, p1_r0_s1, |
| + | p1_r1_s#, p1_r1_s*, p1_r1_s0, p1_r1_s1, |
| + | p1_r2_s#, p1_r2_s*, p1_r2_s0, p1_r2_s1, |
| + | p1_r3_s#, p1_r3_s*, p1_r3_s0, p1_r3_s1, |
| + | p2_r0_s#, p2_r0_s*, p2_r0_s0, p2_r0_s1, |
| + | p2_r1_s#, p2_r1_s*, p2_r1_s0, p2_r1_s1, |
| + | p2_r2_s#, p2_r2_s*, p2_r2_s0, p2_r2_s1, |
| + | p2_r3_s#, p2_r3_s*, p2_r3_s0, p2_r3_s1, |
| + | p3_r0_s#, p3_r0_s*, p3_r0_s0, p3_r0_s1, |
| + | p3_r1_s#, p3_r1_s*, p3_r1_s0, p3_r1_s1, |
| + | p3_r2_s#, p3_r2_s*, p3_r2_s0, p3_r2_s1, |
| + | p3_r3_s#, p3_r3_s*, p3_r3_s0, p3_r3_s1} |
| + | |
| + | The propositional expression of the form "pi_rj_sk" says: |
| + | |
| + | | At the point-in-time p_i, |
| + | | the tape-cell r_j bears the mark s_k. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~INPUTS~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Here are the Initial Conditions |
| + | for the two possible inputs to the |
| + | Ref Log redaction of this Parity TM: |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~INPUT~0~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Initial Conditions: |
| + | |
| + | p0_q0 |
| + | |
| + | p0_r1 |
| + | |
| + | p0_r0_s# |
| + | p0_r1_s0 |
| + | p0_r2_s# |
| + | |
| + | The Initial Conditions are given by a logical conjunction |
| + | that is composed of 5 basic expressions, altogether stating: |
| + | |
| + | | At the point-in-time p_0, M is in the state q_0, and |
| + | | At the point-in-time p_0, H is on the cell r_1, and |
| + | | At the point-in-time p_0, cell r_0 bears the mark "#", and |
| + | | At the point-in-time p_0, cell r_1 bears the mark "0", and |
| + | | At the point-in-time p_0, cell r_2 bears the mark "#". |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~INPUT~1~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Initial Conditions: |
| + | |
| + | p0_q0 |
| + | |
| + | p0_r1 |
| + | |
| + | p0_r0_s# |
| + | p0_r1_s1 |
| + | p0_r2_s# |
| + | |
| + | The Initial Conditions are given by a logical conjunction |
| + | that is composed of 5 basic expressions, altogether stating: |
| + | |
| + | | At the point-in-time p_0, M is in the state q_0, and |
| + | | At the point-in-time p_0, H is on the cell r_1, and |
| + | | At the point-in-time p_0, cell r_0 bears the mark "#", and |
| + | | At the point-in-time p_0, cell r_1 bears the mark "1", and |
| + | | At the point-in-time p_0, cell r_2 bears the mark "#". |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~PROGRAM~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | And here, yet again, just to store it nearby, |
| + | is the logical rendition of the TM's program: |
| + | |
| + | Mediate Conditions: |
| + | |
| + | ( p0_q# ( p1_q# )) |
| + | ( p0_q* ( p1_q* )) |
| + | |
| + | ( p1_q# ( p2_q# )) |
| + | ( p1_q* ( p2_q* )) |
| + | |
| + | Terminal Conditions: |
| + | |
| + | (( p2_q# )( p2_q* )) |
| + | |
| + | State Partition: |
| + | |
| + | (( p0_q0 ),( p0_q1 ),( p0_q# ),( p0_q* )) |
| + | (( p1_q0 ),( p1_q1 ),( p1_q# ),( p1_q* )) |
| + | (( p2_q0 ),( p2_q1 ),( p2_q# ),( p2_q* )) |
| + | |
| + | Register Partition: |
| + | |
| + | (( p0_r0 ),( p0_r1 ),( p0_r2 )) |
| + | (( p1_r0 ),( p1_r1 ),( p1_r2 )) |
| + | (( p2_r0 ),( p2_r1 ),( p2_r2 )) |
| + | |
| + | Symbol Partition: |
| + | |
| + | (( p0_r0_s0 ),( p0_r0_s1 ),( p0_r0_s# )) |
| + | (( p0_r1_s0 ),( p0_r1_s1 ),( p0_r1_s# )) |
| + | (( p0_r2_s0 ),( p0_r2_s1 ),( p0_r2_s# )) |
| + | |
| + | (( p1_r0_s0 ),( p1_r0_s1 ),( p1_r0_s# )) |
| + | (( p1_r1_s0 ),( p1_r1_s1 ),( p1_r1_s# )) |
| + | (( p1_r2_s0 ),( p1_r2_s1 ),( p1_r2_s# )) |
| + | |
| + | (( p2_r0_s0 ),( p2_r0_s1 ),( p2_r0_s# )) |
| + | (( p2_r1_s0 ),( p2_r1_s1 ),( p2_r1_s# )) |
| + | (( p2_r2_s0 ),( p2_r2_s1 ),( p2_r2_s# )) |
| + | |
| + | Interaction Conditions: |
| + | |
| + | (( p0_r0 ) p0_r0_s0 ( p1_r0_s0 )) |
| + | (( p0_r0 ) p0_r0_s1 ( p1_r0_s1 )) |
| + | (( p0_r0 ) p0_r0_s# ( p1_r0_s# )) |
| + | |
| + | (( p0_r1 ) p0_r1_s0 ( p1_r1_s0 )) |
| + | (( p0_r1 ) p0_r1_s1 ( p1_r1_s1 )) |
| + | (( p0_r1 ) p0_r1_s# ( p1_r1_s# )) |
| + | |
| + | (( p0_r2 ) p0_r2_s0 ( p1_r2_s0 )) |
| + | (( p0_r2 ) p0_r2_s1 ( p1_r2_s1 )) |
| + | (( p0_r2 ) p0_r2_s# ( p1_r2_s# )) |
| + | |
| + | (( p1_r0 ) p1_r0_s0 ( p2_r0_s0 )) |
| + | (( p1_r0 ) p1_r0_s1 ( p2_r0_s1 )) |
| + | (( p1_r0 ) p1_r0_s# ( p2_r0_s# )) |
| + | |
| + | (( p1_r1 ) p1_r1_s0 ( p2_r1_s0 )) |
| + | (( p1_r1 ) p1_r1_s1 ( p2_r1_s1 )) |
| + | (( p1_r1 ) p1_r1_s# ( p2_r1_s# )) |
| + | |
| + | (( p1_r2 ) p1_r2_s0 ( p2_r2_s0 )) |
| + | (( p1_r2 ) p1_r2_s1 ( p2_r2_s1 )) |
| + | (( p1_r2 ) p1_r2_s# ( p2_r2_s# )) |
| + | |
| + | Transition Relations: |
| + | |
| + | ( p0_q0 p0_r1 p0_r1_s0 ( p1_q0 p1_r2 p1_r1_s0 )) |
| + | ( p0_q0 p0_r1 p0_r1_s1 ( p1_q1 p1_r2 p1_r1_s1 )) |
| + | ( p0_q0 p0_r1 p0_r1_s# ( p1_q# p1_r0 p1_r1_s# )) |
| + | ( p0_q0 p0_r2 p0_r2_s# ( p1_q# p1_r1 p1_r2_s# )) |
| + | |
| + | ( p0_q1 p0_r1 p0_r1_s0 ( p1_q1 p1_r2 p1_r1_s0 )) |
| + | ( p0_q1 p0_r1 p0_r1_s1 ( p1_q0 p1_r2 p1_r1_s1 )) |
| + | ( p0_q1 p0_r1 p0_r1_s# ( p1_q* p1_r0 p1_r1_s# )) |
| + | ( p0_q1 p0_r2 p0_r2_s# ( p1_q* p1_r1 p1_r2_s# )) |
| + | |
| + | ( p1_q0 p1_r1 p1_r1_s0 ( p2_q0 p2_r2 p2_r1_s0 )) |
| + | ( p1_q0 p1_r1 p1_r1_s1 ( p2_q1 p2_r2 p2_r1_s1 )) |
| + | ( p1_q0 p1_r1 p1_r1_s# ( p2_q# p2_r0 p2_r1_s# )) |
| + | ( p1_q0 p1_r2 p1_r2_s# ( p2_q# p2_r1 p2_r2_s# )) |
| + | |
| + | ( p1_q1 p1_r1 p1_r1_s0 ( p2_q1 p2_r2 p2_r1_s0 )) |
| + | ( p1_q1 p1_r1 p1_r1_s1 ( p2_q0 p2_r2 p2_r1_s1 )) |
| + | ( p1_q1 p1_r1 p1_r1_s# ( p2_q* p2_r0 p2_r1_s# )) |
| + | ( p1_q1 p1_r2 p1_r2_s# ( p2_q* p2_r1 p2_r2_s# )) |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~INTERPRETATION~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Interpretation of the Propositional Program: |
| + | |
| + | Mediate Conditions: |
| + | |
| + | ( p0_q# ( p1_q# )) |
| + | ( p0_q* ( p1_q* )) |
| + | |
| + | ( p1_q# ( p2_q# )) |
| + | ( p1_q* ( p2_q* )) |
| + | |
| + | In Ref Log, an expression of the form "( X ( Y ))" |
| + | expresses an implication or an if-then proposition: |
| + | "Not X without Y", "If X then Y", "X => Y", etc. |
| + | |
| + | A text string expression of the form "( X ( Y ))" |
| + | parses to a graphical data-structure of the form: |
| + | |
| + | X Y |
| + | o---o |
| + | | |
| + | @ |
| + | |
| + | All together, these Mediate Conditions state: |
| + | |
| + | | If at p_0 M is in state q_#, then at p_1 M is in state q_#, and |
| + | | If at p_0 M is in state q_*, then at p_1 M is in state q_*, and |
| + | | If at p_1 M is in state q_#, then at p_2 M is in state q_#, and |
| + | | If at p_1 M is in state q_*, then at p_2 M is in state q_*. |
| + | |
| + | Terminal Conditions: |
| + | |
| + | (( p2_q# )( p2_q* )) |
| + | |
| + | In Ref Log, an expression of the form "(( X )( Y ))" |
| + | expresses a disjunction "X or Y" and it parses into: |
| + | |
| + | X Y |
| + | o o |
| + | \ / |
| + | o |
| + | | |
| + | @ |
| + | |
| + | In effect, the Terminal Conditions state: |
| + | |
| + | | At p_2, M is in state q_#, or |
| + | | At p_2, M is in state q_*. |
| + | |
| + | State Partition: |
| + | |
| + | (( p0_q0 ),( p0_q1 ),( p0_q# ),( p0_q* )) |
| + | (( p1_q0 ),( p1_q1 ),( p1_q# ),( p1_q* )) |
| + | (( p2_q0 ),( p2_q1 ),( p2_q# ),( p2_q* )) |
| + | |
| + | In Ref Log, an expression of the form "(( e_1 ),( e_2 ),( ... ),( e_k ))" |
| + | expresses the fact that "exactly one of the e_j is true, for j = 1 to k". |
| + | Expressions of this form are called "universal partition" expressions, and |
| + | they parse into a type of graph called a "painted and rooted cactus" (PARC): |
| + | |
| + | e_1 e_2 ... e_k |
| + | o o o |
| + | | | | |
| + | o-----o--- ... ---o |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | \ / |
| + | @ |
| + | |
| + | The State Partition expresses the conditions that: |
| + | |
| + | | At each of the points-in-time p_i, for i = 0 to 2, |
| + | | M can be in exactly one state q_j, for j in the set {0, 1, #, *}. |
| + | |
| + | Register Partition: |
| + | |
| + | (( p0_r0 ),( p0_r1 ),( p0_r2 )) |
| + | (( p1_r0 ),( p1_r1 ),( p1_r2 )) |
| + | (( p2_r0 ),( p2_r1 ),( p2_r2 )) |
| + | |
| + | The Register Partition expresses the conditions that: |
| + | |
| + | | At each of the points-in-time p_i, for i = 0 to 2, |
| + | | H can be on exactly one cell r_j, for j = 0 to 2. |
| + | |
| + | Symbol Partition: |
| + | |
| + | (( p0_r0_s0 ),( p0_r0_s1 ),( p0_r0_s# )) |
| + | (( p0_r1_s0 ),( p0_r1_s1 ),( p0_r1_s# )) |
| + | (( p0_r2_s0 ),( p0_r2_s1 ),( p0_r2_s# )) |
| + | |
| + | (( p1_r0_s0 ),( p1_r0_s1 ),( p1_r0_s# )) |
| + | (( p1_r1_s0 ),( p1_r1_s1 ),( p1_r1_s# )) |
| + | (( p1_r2_s0 ),( p1_r2_s1 ),( p1_r2_s# )) |
| + | |
| + | (( p2_r0_s0 ),( p2_r0_s1 ),( p2_r0_s# )) |
| + | (( p2_r1_s0 ),( p2_r1_s1 ),( p2_r1_s# )) |
| + | (( p2_r2_s0 ),( p2_r2_s1 ),( p2_r2_s# )) |
| + | |
| + | The Symbol Partition expresses the conditions that: |
| + | |
| + | | At each of the points-in-time p_i, for i in {0, 1, 2}, |
| + | | in each of the tape-registers r_j, for j in {0, 1, 2}, |
| + | | there can be exactly one sign s_k, for k in {0, 1, #}. |
| + | |
| + | Interaction Conditions: |
| + | |
| + | (( p0_r0 ) p0_r0_s0 ( p1_r0_s0 )) |
| + | (( p0_r0 ) p0_r0_s1 ( p1_r0_s1 )) |
| + | (( p0_r0 ) p0_r0_s# ( p1_r0_s# )) |
| + | |
| + | (( p0_r1 ) p0_r1_s0 ( p1_r1_s0 )) |
| + | (( p0_r1 ) p0_r1_s1 ( p1_r1_s1 )) |
| + | (( p0_r1 ) p0_r1_s# ( p1_r1_s# )) |
| + | |
| + | (( p0_r2 ) p0_r2_s0 ( p1_r2_s0 )) |
| + | (( p0_r2 ) p0_r2_s1 ( p1_r2_s1 )) |
| + | (( p0_r2 ) p0_r2_s# ( p1_r2_s# )) |
| + | |
| + | (( p1_r0 ) p1_r0_s0 ( p2_r0_s0 )) |
| + | (( p1_r0 ) p1_r0_s1 ( p2_r0_s1 )) |
| + | (( p1_r0 ) p1_r0_s# ( p2_r0_s# )) |
| + | |
| + | (( p1_r1 ) p1_r1_s0 ( p2_r1_s0 )) |
| + | (( p1_r1 ) p1_r1_s1 ( p2_r1_s1 )) |
| + | (( p1_r1 ) p1_r1_s# ( p2_r1_s# )) |
| + | |
| + | (( p1_r2 ) p1_r2_s0 ( p2_r2_s0 )) |
| + | (( p1_r2 ) p1_r2_s1 ( p2_r2_s1 )) |
| + | (( p1_r2 ) p1_r2_s# ( p2_r2_s# )) |
| + | |
| + | In briefest terms, the Interaction Conditions merely express |
| + | the circumstance that the sign in a tape-cell cannot change |
| + | between two points-in-time unless the tape-head is over the |
| + | cell in question at the initial one of those points-in-time. |
| + | All that we have to do is to see how they manage to say this. |
| + | |
| + | In Ref Log, an expression of the following form: |
| + | |
| + | "(( p<i>_r<j> ) p<i>_r<j>_s<k> ( p<i+1>_r<j>_s<k> ))", |
| + | |
| + | and which parses as the graph: |
| + | |
| + | p<i>_r<j> o o p<i+1>_r<j>_s<k> |
| + | \ / |
| + | p<i>_r<j>_s<k> o |
| + | | |
| + | @ |
| + | |
| + | can be read in the form of the following implication: |
| + | |
| + | | If |
| + | | at the point-in-time p<i>, the tape-cell r<j> bears the mark s<k>, |
| + | | but it is not the case that |
| + | | at the point-in-time p<i>, the tape-head is on the tape-cell r<j>. |
| + | | then |
| + | | at the point-in-time p<i+1>, the tape-cell r<j> bears the mark s<k>. |
| + | |
| + | Folks among us of a certain age and a peculiar manner of acculturation will |
| + | recognize these as the "Frame Conditions" for the change of state of the TM. |
| + | |
| + | Transition Relations: |
| + | |
| + | ( p0_q0 p0_r1 p0_r1_s0 ( p1_q0 p1_r2 p1_r1_s0 )) |
| + | ( p0_q0 p0_r1 p0_r1_s1 ( p1_q1 p1_r2 p1_r1_s1 )) |
| + | ( p0_q0 p0_r1 p0_r1_s# ( p1_q# p1_r0 p1_r1_s# )) |
| + | ( p0_q0 p0_r2 p0_r2_s# ( p1_q# p1_r1 p1_r2_s# )) |
| + | |
| + | ( p0_q1 p0_r1 p0_r1_s0 ( p1_q1 p1_r2 p1_r1_s0 )) |
| + | ( p0_q1 p0_r1 p0_r1_s1 ( p1_q0 p1_r2 p1_r1_s1 )) |
| + | ( p0_q1 p0_r1 p0_r1_s# ( p1_q* p1_r0 p1_r1_s# )) |
| + | ( p0_q1 p0_r2 p0_r2_s# ( p1_q* p1_r1 p1_r2_s# )) |
| + | |
| + | ( p1_q0 p1_r1 p1_r1_s0 ( p2_q0 p2_r2 p2_r1_s0 )) |
| + | ( p1_q0 p1_r1 p1_r1_s1 ( p2_q1 p2_r2 p2_r1_s1 )) |
| + | ( p1_q0 p1_r1 p1_r1_s# ( p2_q# p2_r0 p2_r1_s# )) |
| + | ( p1_q0 p1_r2 p1_r2_s# ( p2_q# p2_r1 p2_r2_s# )) |
| + | |
| + | ( p1_q1 p1_r1 p1_r1_s0 ( p2_q1 p2_r2 p2_r1_s0 )) |
| + | ( p1_q1 p1_r1 p1_r1_s1 ( p2_q0 p2_r2 p2_r1_s1 )) |
| + | ( p1_q1 p1_r1 p1_r1_s# ( p2_q* p2_r0 p2_r1_s# )) |
| + | ( p1_q1 p1_r2 p1_r2_s# ( p2_q* p2_r1 p2_r2_s# )) |
| + | |
| + | The Transition Conditions merely serve to express, |
| + | by means of 16 complex implication expressions, |
| + | the data of the TM table that was given above. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~OUTPUTS~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | And here are the outputs of the computation, |
| + | as emulated by its propositional rendition, |
| + | and as actually generated within that form |
| + | of transmogrification by the program that |
| + | I wrote for finding all of the satisfying |
| + | interpretations (truth-value assignments) |
| + | of propositional expressions in Ref Log: |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~OUTPUT~0~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Output Conditions: |
| + | |
| + | p0_q0 |
| + | p0_r1 |
| + | p0_r0_s# |
| + | p0_r1_s0 |
| + | p0_r2_s# |
| + | p1_q0 |
| + | p1_r2 |
| + | p1_r2_s# |
| + | p1_r0_s# |
| + | p1_r1_s0 |
| + | p2_q# |
| + | p2_r1 |
| + | p2_r0_s# |
| + | p2_r1_s0 |
| + | p2_r2_s# |
| + | |
| + | The Output Conditions amount to the sole satisfying interpretation, |
| + | that is, a "sequence of truth-value assignments" (SOTVA) that make |
| + | the entire proposition come out true, and they state the following: |
| + | |
| + | | At the point-in-time p_0, M is in the state q_0, and |
| + | | At the point-in-time p_0, H is on the cell r_1, and |
| + | | At the point-in-time p_0, cell r_0 bears the mark "#", and |
| + | | At the point-in-time p_0, cell r_1 bears the mark "0", and |
| + | | At the point-in-time p_0, cell r_2 bears the mark "#", and |
| + | | |
| + | | At the point-in-time p_1, M is in the state q_0, and |
| + | | At the point-in-time p_1, H is on the cell r_2, and |
| + | | At the point-in-time p_1, cell r_0 bears the mark "#", and |
| + | | At the point-in-time p_1, cell r_1 bears the mark "0", and |
| + | | At the point-in-time p_1, cell r_2 bears the mark "#", and |
| + | | |
| + | | At the point-in-time p_2, M is in the state q_#, and |
| + | | At the point-in-time p_2, H is on the cell r_1, and |
| + | | At the point-in-time p_2, cell r_0 bears the mark "#", and |
| + | | At the point-in-time p_2, cell r_1 bears the mark "0", and |
| + | | At the point-in-time p_2, cell r_2 bears the mark "#". |
| + | |
| + | In brief, the output for our sake being the symbol that rests |
| + | under the tape-head H when the machine M gets to a rest state, |
| + | we are now amazed by the remarkable result that Parity(0) = 0. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~OUTPUT~1~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Output Conditions: |
| + | |
| + | p0_q0 |
| + | p0_r1 |
| + | p0_r0_s# |
| + | p0_r1_s1 |
| + | p0_r2_s# |
| + | p1_q1 |
| + | p1_r2 |
| + | p1_r2_s# |
| + | p1_r0_s# |
| + | p1_r1_s1 |
| + | p2_q* |
| + | p2_r1 |
| + | p2_r0_s# |
| + | p2_r1_s1 |
| + | p2_r2_s# |
| + | |
| + | The Output Conditions amount to the sole satisfying interpretation, |
| + | that is, a "sequence of truth-value assignments" (SOTVA) that make |
| + | the entire proposition come out true, and they state the following: |
| + | |
| + | | At the point-in-time p_0, M is in the state q_0, and |
| + | | At the point-in-time p_0, H is on the cell r_1, and |
| + | | At the point-in-time p_0, cell r_0 bears the mark "#", and |
| + | | At the point-in-time p_0, cell r_1 bears the mark "1", and |
| + | | At the point-in-time p_0, cell r_2 bears the mark "#", and |
| + | | |
| + | | At the point-in-time p_1, M is in the state q_1, and |
| + | | At the point-in-time p_1, H is on the cell r_2, and |
| + | | At the point-in-time p_1, cell r_0 bears the mark "#", and |
| + | | At the point-in-time p_1, cell r_1 bears the mark "1", and |
| + | | At the point-in-time p_1, cell r_2 bears the mark "#", and |
| + | | |
| + | | At the point-in-time p_2, M is in the state q_*, and |
| + | | At the point-in-time p_2, H is on the cell r_1, and |
| + | | At the point-in-time p_2, cell r_0 bears the mark "#", and |
| + | | At the point-in-time p_2, cell r_1 bears the mark "1", and |
| + | | At the point-in-time p_2, cell r_2 bears the mark "#". |
| + | |
| + | In brief, the output for our sake being the symbol that rests |
| + | under the tape-head H when the machine M gets to a rest state, |
| + | we are now amazed by the remarkable result that Parity(1) = 1. |
| + | |
| + | I realized after sending that last bunch of bits that there is room |
| + | for confusion about what is the input/output of the Study module of |
| + | the Theme One program as opposed to what is the input/output of the |
| + | "finitely approximated turing automaton" (FATA). So here is better |
| + | delineation of what's what. The input to Study is a text file that |
| + | is known as LogFile(Whatever) and the output of Study is a sequence |
| + | of text files that summarize the various canonical and normal forms |
| + | that it generates. For short, let us call these NormFile(Whatelse). |
| + | With that in mind, here are the actual IO's of Study, excluding the |
| + | glosses in square brackets: |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~INPUT~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | [Input To Study = FATA Initial Conditions + FATA Program Conditions] |
| + | |
| + | [FATA Initial Conditions For Input 0] |
| + | |
| + | p0_q0 |
| + | |
| + | p0_r1 |
| + | |
| + | p0_r0_s# |
| + | p0_r1_s0 |
| + | p0_r2_s# |
| + | |
| + | [FATA Program Conditions For Parity Machine] |
| + | |
| + | [Mediate Conditions] |
| + | |
| + | ( p0_q# ( p1_q# )) |
| + | ( p0_q* ( p1_q* )) |
| + | |
| + | ( p1_q# ( p2_q# )) |
| + | ( p1_q* ( p2_q* )) |
| + | |
| + | [Terminal Conditions] |
| + | |
| + | (( p2_q# )( p2_q* )) |
| + | |
| + | [State Partition] |
| + | |
| + | (( p0_q0 ),( p0_q1 ),( p0_q# ),( p0_q* )) |
| + | (( p1_q0 ),( p1_q1 ),( p1_q# ),( p1_q* )) |
| + | (( p2_q0 ),( p2_q1 ),( p2_q# ),( p2_q* )) |
| + | |
| + | [Register Partition] |
| + | |
| + | (( p0_r0 ),( p0_r1 ),( p0_r2 )) |
| + | (( p1_r0 ),( p1_r1 ),( p1_r2 )) |
| + | (( p2_r0 ),( p2_r1 ),( p2_r2 )) |
| + | |
| + | [Symbol Partition] |
| + | |
| + | (( p0_r0_s0 ),( p0_r0_s1 ),( p0_r0_s# )) |
| + | (( p0_r1_s0 ),( p0_r1_s1 ),( p0_r1_s# )) |
| + | (( p0_r2_s0 ),( p0_r2_s1 ),( p0_r2_s# )) |
| + | |
| + | (( p1_r0_s0 ),( p1_r0_s1 ),( p1_r0_s# )) |
| + | (( p1_r1_s0 ),( p1_r1_s1 ),( p1_r1_s# )) |
| + | (( p1_r2_s0 ),( p1_r2_s1 ),( p1_r2_s# )) |
| + | |
| + | (( p2_r0_s0 ),( p2_r0_s1 ),( p2_r0_s# )) |
| + | (( p2_r1_s0 ),( p2_r1_s1 ),( p2_r1_s# )) |
| + | (( p2_r2_s0 ),( p2_r2_s1 ),( p2_r2_s# )) |
| + | |
| + | [Interaction Conditions] |
| + | |
| + | (( p0_r0 ) p0_r0_s0 ( p1_r0_s0 )) |
| + | (( p0_r0 ) p0_r0_s1 ( p1_r0_s1 )) |
| + | (( p0_r0 ) p0_r0_s# ( p1_r0_s# )) |
| + | |
| + | (( p0_r1 ) p0_r1_s0 ( p1_r1_s0 )) |
| + | (( p0_r1 ) p0_r1_s1 ( p1_r1_s1 )) |
| + | (( p0_r1 ) p0_r1_s# ( p1_r1_s# )) |
| + | |
| + | (( p0_r2 ) p0_r2_s0 ( p1_r2_s0 )) |
| + | (( p0_r2 ) p0_r2_s1 ( p1_r2_s1 )) |
| + | (( p0_r2 ) p0_r2_s# ( p1_r2_s# )) |
| + | |
| + | (( p1_r0 ) p1_r0_s0 ( p2_r0_s0 )) |
| + | (( p1_r0 ) p1_r0_s1 ( p2_r0_s1 )) |
| + | (( p1_r0 ) p1_r0_s# ( p2_r0_s# )) |
| + | |
| + | (( p1_r1 ) p1_r1_s0 ( p2_r1_s0 )) |
| + | (( p1_r1 ) p1_r1_s1 ( p2_r1_s1 )) |
| + | (( p1_r1 ) p1_r1_s# ( p2_r1_s# )) |
| + | |
| + | (( p1_r2 ) p1_r2_s0 ( p2_r2_s0 )) |
| + | (( p1_r2 ) p1_r2_s1 ( p2_r2_s1 )) |
| + | (( p1_r2 ) p1_r2_s# ( p2_r2_s# )) |
| + | |
| + | [Transition Relations] |
| + | |
| + | ( p0_q0 p0_r1 p0_r1_s0 ( p1_q0 p1_r2 p1_r1_s0 )) |
| + | ( p0_q0 p0_r1 p0_r1_s1 ( p1_q1 p1_r2 p1_r1_s1 )) |
| + | ( p0_q0 p0_r1 p0_r1_s# ( p1_q# p1_r0 p1_r1_s# )) |
| + | ( p0_q0 p0_r2 p0_r2_s# ( p1_q# p1_r1 p1_r2_s# )) |
| + | |
| + | ( p0_q1 p0_r1 p0_r1_s0 ( p1_q1 p1_r2 p1_r1_s0 )) |
| + | ( p0_q1 p0_r1 p0_r1_s1 ( p1_q0 p1_r2 p1_r1_s1 )) |
| + | ( p0_q1 p0_r1 p0_r1_s# ( p1_q* p1_r0 p1_r1_s# )) |
| + | ( p0_q1 p0_r2 p0_r2_s# ( p1_q* p1_r1 p1_r2_s# )) |
| + | |
| + | ( p1_q0 p1_r1 p1_r1_s0 ( p2_q0 p2_r2 p2_r1_s0 )) |
| + | ( p1_q0 p1_r1 p1_r1_s1 ( p2_q1 p2_r2 p2_r1_s1 )) |
| + | ( p1_q0 p1_r1 p1_r1_s# ( p2_q# p2_r0 p2_r1_s# )) |
| + | ( p1_q0 p1_r2 p1_r2_s# ( p2_q# p2_r1 p2_r2_s# )) |
| + | |
| + | ( p1_q1 p1_r1 p1_r1_s0 ( p2_q1 p2_r2 p2_r1_s0 )) |
| + | ( p1_q1 p1_r1 p1_r1_s1 ( p2_q0 p2_r2 p2_r1_s1 )) |
| + | ( p1_q1 p1_r1 p1_r1_s# ( p2_q* p2_r0 p2_r1_s# )) |
| + | ( p1_q1 p1_r2 p1_r2_s# ( p2_q* p2_r1 p2_r2_s# )) |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~OUTPUT~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | [Output Of Study = FATA Output For Input 0] |
| + | |
| + | p0_q0 |
| + | p0_r1 |
| + | p0_r0_s# |
| + | p0_r1_s0 |
| + | p0_r2_s# |
| + | p1_q0 |
| + | p1_r2 |
| + | p1_r2_s# |
| + | p1_r0_s# |
| + | p1_r1_s0 |
| + | p2_q# |
| + | p2_r1 |
| + | p2_r0_s# |
| + | p2_r1_s0 |
| + | p2_r2_s# |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Turing automata, finitely approximated or not, make my head spin and |
| + | my tape go loopy, and I still believe 'twere a far better thing I do |
| + | if I work up to that level of complexity in a more gracile graduated |
| + | manner. So let us return to our Example in this gradual progress to |
| + | that vastly more well-guarded grail of our long-term pilgrim's quest: |
| + | |
| + | | boy male girl female |
| + | | o---o child o---o child |
| + | | male female \ / \ / child human |
| + | | o---o o o o--o |
| + | | \ / | | | |
| + | | @ @ @ @ |
| + | | |
| + | | (male , female)((boy , male child))((girl , female child))(child (human)) |
| + | |
| + | One section of the Theme One program has a suite of utilities that fall |
| + | under the title "Theme One Study" ("To Study", or just "TOS" for short). |
| + | To Study is to read and to parse a so-called and a generally so-suffixed |
| + | "log" file, and then to conjoin what is called a "query", which is really |
| + | just an additional propositional expression that imposes a further logical |
| + | constraint on the input expression. |
| + | |
| + | The Figure roughly sketches the conjuncts of the graph-theoretic |
| + | data structure that the parser would commit to memory on reading |
| + | the appropriate log file that contains the text along the bottom. |
| + | |
| + | I will now explain the various sorts of things that the TOS utility |
| + | can do with the log file that describes the universe of discourse in |
| + | our present Example. |
| + | |
| + | Theme One Study is built around a suite of four successive generators |
| + | of "normal forms" for propositional expressions, just to use that term |
| + | in a very approximate way. The functions that compute these normal forms |
| + | are called "Model", "Tenor", "Canon", and "Sense", and so we may refer to |
| + | to their text-style outputs as the "mod", "ten", "can", and "sen" files. |
| + | |
| + | Though it could be any propositional expression on the same vocabulary |
| + | $A$ = {"boy", "child", "female", "girl", "human", "male"}, more usually |
| + | the query is a simple conjunction of one or more positive features that |
| + | we want to focus on or perhaps to filter out of the logical model space. |
| + | On our first run through this Example, we take the log file proposition |
| + | as it is, with no extra riders. |
| + | |
| + | | Procedural Note. TO Study Model displays a running tab of how much |
| + | | free memory space it has left. On some of the harder problems that |
| + | | you may think of to give it, Model may run out of free memory and |
| + | | terminate, abnormally exiting Theme One. Sometimes it helps to: |
| + | | |
| + | | 1. Rephrase the problem in logically equivalent |
| + | | but rhetorically increasedly felicitous ways. |
| + | | |
| + | | 2. Think of additional facts that are taken for granted but not |
| + | | made explicit and that cannot be logically inferred by Model. |
| + | |
| + | After Model has finished, it is ready to write out its mod file, |
| + | which you may choose to show on the screen or save to a named file. |
| + | Mod files are usually too long to see (or to care to see) all at once |
| + | on the screen, so it is very often best to save them for later replay. |
| + | In our Example the Model function yields a mod file that looks like so: |
| + | |
| + | Model Output and |
| + | Mod File Example |
| + | o-------------------o |
| + | | male | |
| + | | female - | 1 |
| + | | (female ) | |
| + | | girl - | 2 |
| + | | (girl ) | |
| + | | child | |
| + | | boy | |
| + | | human * | 3 * |
| + | | (human ) - | 4 |
| + | | (boy ) - | 5 |
| + | | (child ) | |
| + | | boy - | 6 |
| + | | (boy ) * | 7 * |
| + | | (male ) | |
| + | | female | |
| + | | boy - | 8 |
| + | | (boy ) | |
| + | | child | |
| + | | girl | |
| + | | human * | 9 * |
| + | | (human ) - | 10 |
| + | | (girl ) - | 11 |
| + | | (child ) | |
| + | | girl - | 12 |
| + | | (girl ) * | 13 * |
| + | | (female ) - | 14 |
| + | o-------------------o |
| + | |
| + | Counting the stars "*" that indicate true interpretations |
| + | and the bars "-" that indicate false interpretations of |
| + | the input formula, we can see that the Model function, |
| + | out of the 64 possible interpretations, has actually |
| + | gone through the work of making just 14 evaluations, |
| + | all in order to find the 4 models that are allowed |
| + | by the input definitions. |
| + | |
| + | To be clear about what this output means, the starred paths |
| + | indicate all of the complete specifications of objects in the |
| + | universe of discourse, that is, all of the consistent feature |
| + | conjunctions of maximum length, as permitted by the definitions |
| + | that are given in the log file. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Let's take a little break from the Example in progress |
| + | and look at where we are and what we have been doing from |
| + | computational, logical, and semiotic perspectives. Because, |
| + | after all, as is usually the case, we should not let our focus |
| + | and our fascination with this particular Example prevent us from |
| + | recognizing it, and all that we do with it, as just an Example of |
| + | much broader paradigms and predicaments and principles, not to say |
| + | but a glimmer of ultimately more concernful and fascinating objects. |
| + | |
| + | I chart the progression that we have just passed through in this way: |
| + | |
| + | | Parse |
| + | | Sign A o-------------->o Sign 1 |
| + | | ^ | |
| + | | / | |
| + | | / | |
| + | | / | |
| + | | Object o | Transform |
| + | | ^ | |
| + | | \ | |
| + | | \ | |
| + | | \ v |
| + | | Sign B o<--------------o Sign 2 |
| + | | Verse |
| + | | |
| + | | Figure. Computation As Sign Transformation |
| + | |
| + | In the present case, the Object is an objective situation |
| + | or a state of affairs, in effect, a particular pattern of |
| + | feature concurrences occurring to us in that world through |
| + | which we find ourselves most frequently faring, wily nily, |
| + | and the Signs are different tokens and different types of |
| + | data structures that we somehow or other find it useful |
| + | to devise or to discover for the sake of representing |
| + | current objects to ourselves on a recurring basis. |
| + | |
| + | But not all signs, not even signs of a single object, are alike |
| + | in every other respect that one might name, not even with respect |
| + | to their powers of relating, significantly, to that common object. |
| + | |
| + | And that is what our whole business of computation busies itself about, |
| + | when it minds its business best, that is, transmuting signs into signs |
| + | in ways that augment their powers of relating significantly to objects. |
| + | |
| + | We have seen how the Model function and the mod output format |
| + | indicate all of the complete specifications of objects in the |
| + | universe of discourse, that is, all of the consistent feature |
| + | conjunctions of maximal specificity that are permitted by the |
| + | constraints or the definitions that are given in the log file. |
| + | |
| + | To help identify these specifications of particular cells in |
| + | the universe of discourse, the next function and output format, |
| + | called "Tenor", edits the mod file to give only the true paths, |
| + | in effect, the "positive models", that are by default what we |
| + | usually mean when we say "models", and not the "anti-models" |
| + | or the "negative models" that fail to satisfy the formula |
| + | in question. |
| + | |
| + | In the present Example the Tenor function |
| + | generates a Ten file that looks like this: |
| + | |
| + | Tenor Output and |
| + | Ten File Example |
| + | o-------------------o |
| + | | male | |
| + | | (female ) | |
| + | | (girl ) | |
| + | | child | |
| + | | boy | |
| + | | human * | <1> |
| + | | (child ) | |
| + | | (boy ) * | <2> |
| + | | (male ) | |
| + | | female | |
| + | | (boy ) | |
| + | | child | |
| + | | girl | |
| + | | human * | <3> |
| + | | (child ) | |
| + | | (girl ) * | <4> |
| + | o-------------------o |
| + | |
| + | As I said, the Tenor function just abstracts a transcript of the models, |
| + | that is, the satisfying interpretations, that were already interspersed |
| + | throughout the complete Model output. These specifications, or feature |
| + | conjunctions, with the positive and the negative features listed in the |
| + | order of their actual budding on the "arboreal boolean expansion" twigs, |
| + | may be gathered and arranged in this antherypulogical flowering bouquet: |
| + | |
| + | 1. male (female ) (girl ) child boy human * |
| + | 2. male (female ) (girl ) (child ) (boy ) * |
| + | 3. (male ) female (boy ) child girl human * |
| + | 4. (male ) female (boy ) (child ) (girl ) * |
| + | |
| + | Notice that Model, as reflected in this abstract, did not consider |
| + | the six positive features in the same order along each path. This |
| + | is because the algorithm was designed to proceed opportunistically |
| + | in its attempt to reduce the original proposition through a series |
| + | of case-analytic considerations and the resulting simplifications. |
| + | |
| + | Notice, too, that Model is something of a lazy evaluator, quitting work |
| + | when and if a value is determined by less than the full set of variables. |
| + | This is the reason why paths <2> and <4> are not ostensibly of the maximum |
| + | length. According to this lazy mode of understanding, any path that is not |
| + | specified on a set of features really stands for the whole bundle of paths |
| + | that are derived by freely varying those features. Thus, specifications |
| + | <2> and <4> summarize four models altogether, with the logical choice |
| + | between "human" and "not human" being left open at the point where |
| + | they leave off their branches in the releavent deciduous tree. |
| + | |
| + | The last two functions in the Study section, "Canon" and "Sense", |
| + | extract further derivatives of the normal forms that are produced |
| + | by Model and Tenor. Both of these functions take the set of model |
| + | paths and simply throw away the negative labels. You may think of |
| + | these as the "rose colored glasses" or "job interview" normal forms, |
| + | in that they try to say everything that's true, so long as it can be |
| + | expressed in positive terms. Generally, this would mean losing a lot |
| + | of information, and the result could no longer be expected to have the |
| + | property of remaining logically equivalent to the original proposition. |
| + | |
| + | Fortunately, however, it seems that this type of positive projection of |
| + | the whole truth is just what is possible, most needed, and most clear in |
| + | many of the "natural" examples, that is, in examples that arise from the |
| + | domains of natural language and natural conceptual kinds. In these cases, |
| + | where most of the logical features are redundantly coded, for example, in |
| + | the way that "adult" = "not child" and "child" = "not adult", the positive |
| + | feature bearing redacts are often sufficiently expressive all by themselves. |
| + | |
| + | Canon merely censors its printing of the negative labels as it traverses the |
| + | model tree. This leaves the positive labels in their original columns of the |
| + | outline form, giving it a slightly skewed appearance. This can be misleading |
| + | unless you already know what you are looking for. However, this Canon format |
| + | is computationally quick, and frequently suffices, especially if you already |
| + | have a likely clue about what to expect in the way of a question's outcome. |
| + | |
| + | In the present Example the Canon function |
| + | generates a Can file that looks like this: |
| + | |
| + | Canon Output and |
| + | Can File Example |
| + | o-------------------o |
| + | | male | |
| + | | child | |
| + | | boy | |
| + | | human | |
| + | | female | |
| + | | child | |
| + | | girl | |
| + | | human | |
| + | o-------------------o |
| + | |
| + | The Sense function does the extra work that is required |
| + | to place the positive labels of the model tree at their |
| + | proper level in the outline. |
| + | |
| + | In the present Example the Sense function |
| + | generates a Sen file that looks like this: |
| + | |
| + | Sense Output and |
| + | Sen File Example |
| + | o-------------------o |
| + | | male | |
| + | | child | |
| + | | boy | |
| + | | human | |
| + | | female | |
| + | | child | |
| + | | girl | |
| + | | human | |
| + | o-------------------o |
| + | |
| + | The Canon and Sense outlines for this Example illustrate a certain |
| + | type of general circumstance that needs to be noted at this point. |
| + | Recall the model paths or the feature specifications that were |
| + | numbered <2> and <4> in the listing of the output for Tenor. |
| + | These paths, in effect, reflected Model's discovery that |
| + | the venn diagram cells for male or female non-children |
| + | and male or female non-humans were not excluded by |
| + | the definitions that were given in the Log file. |
| + | In the abstracts given by Canon and Sense, the |
| + | specifications <2> and <4> have been subsumed, |
| + | or absorbed unmarked, under the general topics |
| + | of their respective genders, male or female. |
| + | This happens because no purely positive |
| + | features were supplied to distinguish |
| + | the non-child and non-human cases. |
| + | |
| + | That completes the discussion of |
| + | this six-dimensional Example. |
| + | |
| + | Nota Bene, for possible future use. In the larger current of work |
| + | with respect to which this meander of a conduit was initially both |
| + | diversionary and tributary, before those high and dry regensquirm |
| + | years when it turned into an intellectual interglacial oxbow lake, |
| + | I once had in mind a scape in which expressions in a definitional |
| + | lattice were ordered according to their simplicity on some scale |
| + | or another, and in this setting the word "sense" was actually an |
| + | acronym for "semantically equivalent next-simplest expression". |
| + | |
| + | | If this is starting to sound a little bit familiar, |
| + | | it may be because the relationship between the two |
| + | | kinds of pictures of propositions, namely: |
| + | | |
| + | | 1. Propositions about things in general, here, |
| + | | about the times when certain facts are true, |
| + | | having the form of functions f : X -> B, |
| + | | |
| + | | 2. Propositions about binary codes, here, about |
| + | | the bit-vector labels on venn diagram cells, |
| + | | having the form of functions f' : B^k -> B, |
| + | | |
| + | | is an epically old story, one that I, myself, |
| + | | have related one or twice upon a time before, |
| + | | to wit, at least, at the following two cites: |
| + | | |
| + | | http://suo.ieee.org/email/msg01251.html |
| + | | http://suo.ieee.org/email/msg01293.html |
| + | | |
| + | | There, and now here, once more, and again, it may be observed |
| + | | that the relation is one whereby the proposition f : X -> B, |
| + | | the one about things and times and mores in general, factors |
| + | | into a coding function c : X -> B^k, followed by a derived |
| + | | proposition f' : B^k -> B that judges the resulting codes. |
| + | | |
| + | | f |
| + | | X o------>o B |
| + | | \ ^ |
| + | | c = <x_1, ..., x_k> \ / f' |
| + | | v / |
| + | | o |
| + | | B^k |
| + | | |
| + | | You may remember that this was supposed to illustrate |
| + | | the "factoring" of a proposition f : X -> B = {0, 1} |
| + | | into the composition f'(c(x)), where c : X -> B^k is |
| + | | the "coding" of each x in X as an k-bit string in B^k, |
| + | | and where f' is the mapping of codes into a co-domain |
| + | | that we interpret as t-f-values, B = {0, 1} = {F, T}. |
| + | |
| + | In short, there is the standard equivocation ("systematic ambiguity"?) as to |
| + | whether we are talking about the "applied" and concretely typed proposition |
| + | f : X -> B or the "pure" and abstractly typed proposition f' : B^k -> B. |
| + | Or we can think of the latter object as the approximate code icon of |
| + | the former object. |
| + | |
| + | Anyway, these types of formal objects are the sorts of things that |
| + | I take to be the denotational objects of propositional expressions. |
| + | These objects, along with their invarious and insundry mathematical |
| + | properties, are the orders of things that I am talking about when |
| + | I refer to the "invariant structures in these objects themselves". |
| + | |
| + | "Invariant" means "invariant under a suitable set of transformations", |
| + | in this case the translations between various languages that preserve |
| + | the objects and the structures in question. In extremest generality, |
| + | this is what universal constructions in category theory are all about. |
| + | |
| + | In summation, the functions f : X -> B and f' : B* -> B have invariant, formal, |
| + | mathematical, objective properties that any adequate language might eventually |
| + | evolve to express, only some languages express them more obscurely than others. |
| + | |
| + | To be perfectly honest, I continue to be surprised that anybody in this group |
| + | has trouble with this. There are perfectly apt and familiar examples in the |
| + | contrast between roman numerals and arabic numerals, or the contrast between |
| + | redundant syntaxes, like those that use the pentalphabet {~, &, v, =>, <=>}, |
| + | and trimmer syntaxes, like those used in existential and conceptual graphs. |
| + | Every time somebody says "Let's take {~, &, v, =>, <=>} as an operational |
| + | basis for logic" it's just like that old joke that mathematicians tell on |
| + | engineers where the ingenue in question says "1 is a prime, 2 is a prime, |
| + | 3 is a prime, 4 is a prime, ..." -- and I know you think that I'm being |
| + | hyperbolic, but I'm really only up to parabolas here ... |
| + | |
| + | I have already refined my criticism so that it does not apply to |
| + | the spirit of FOL or KIF or whatever, but only to the letters of |
| + | specific syntactic proposals. There is a fact of the matter as |
| + | to whether a concrete language provides a clean or a cluttered |
| + | basis for representing the identified set of formal objects. |
| + | And it shows up in pragmatic realities like the efficiency |
| + | of real time concept formation, concept use, learnability, |
| + | reasoning power, and just plain good use of real time. |
| + | These are the dire consequences that I learned in my |
| + | very first tries at mathematically oriented theorem |
| + | automation, and the only factor that has obscured |
| + | them in mainstream work since then is the speed |
| + | with which folks can now do all of the same |
| + | old dumb things that they used to do on |
| + | their way to kludging out the answers. |
| + | |
| + | It seems to be darn near impossible to explain to the |
| + | centurion all of the neat stuff that he's missing by |
| + | sticking to his old roman numerals. He just keeps |
| + | on reckoning that what he can't count must be of |
| + | no account at all. There is way too much stuff |
| + | that these original syntaxes keep us from even |
| + | beginning to discuss, like differential logic, |
| + | just for starters. |
| + | |
| + | Our next Example illustrates the use of the Cactus Language |
| + | for representing "absolute" and "relative" partitions, also |
| + | known as "complete" and "contingent" classifications of the |
| + | universe of discourse, all of which amounts to divvying it |
| + | up into mutually exclusive regions, exhaustive or not, as |
| + | one frequently needs in situations involving a genus and |
| + | its sundry species, and frequently pictures in the form |
| + | of a venn diagram that looks just like a "pie chart". |
| + | |
| + | Example. Partition, Genus & Species |
| + | |
| + | The idea that one needs for expressing partitions |
| + | in cactus expressions can be summed up like this: |
| + | |
| + | | If the propositional expression |
| + | | |
| + | | "( p , q , r , ... )" |
| + | | |
| + | | means that just one of |
| + | | |
| + | | p, q, r, ... is false, |
| + | | |
| + | | then the propositional expression |
| + | | |
| + | | "((p),(q),(r), ... )" |
| + | | |
| + | | must mean that just one of |
| + | | |
| + | | (p), (q), (r), ... is false, |
| + | | |
| + | | in other words, that just one of |
| + | | |
| + | | p, q, r, ... is true. |
| + | |
| + | Thus we have an efficient means to express and to enforce |
| + | a partition of the space of models, in effect, to maintain |
| + | the condition that a number of features or propositions are |
| + | to be held in mutually exclusive and exhaustive disjunction. |
| + | This supplies a much needed bridge between the binary domain |
| + | of two values and any other domain with a finite number of |
| + | feature values. |
| + | |
| + | Another variation on this theme allows one to maintain the |
| + | subsumption of many separate species under an explicit genus. |
| + | To see this, let us examine the following form of expression: |
| + | |
| + | ( q , ( q_1 ) , ( q_2 ) , ( q_3 ) ). |
| + | |
| + | Now consider what it would mean for this to be true. We see two cases: |
| + | |
| + | 1. If the proposition q is true, then exactly one of the |
| + | propositions (q_1), (q_2), (q_3) must be false, and so |
| + | just one of the propositions q_1, q_2, q_3 must be true. |
| + | |
| + | 2. If the proposition q is false, then every one of the |
| + | propositions (q_1), (q_2), (q_2) must be true, and so |
| + | each one of the propositions q_1, q_2, q_3 must be false. |
| + | In short, if q is false then all of the other q's are also. |
| + | |
| + | Figures 1 and 2 illustrate this type of situation. |
| + | |
| + | Figure 1 is the venn diagram of a 4-dimensional universe of discourse |
| + | X = [q, q_1, q_2, q_3], conventionally named after the gang of four |
| + | logical features that generate it. Strictly speaking, X is made up |
| + | of two layers, the position space X of abstract type %B%^4, and the |
| + | proposition space X^ = (X -> %B%) of abstract type %B%^4 -> %B%, |
| + | but it is commonly lawful enough to sign the signature of both |
| + | spaces with the same X, and thus to give the power of attorney |
| + | for the propositions to the so-indicted position space thereof. |
| + | |
| + | Figure 1 also makes use of the convention whereby the regions |
| + | or the subsets of the universe of discourse that correspond |
| + | to the basic features q, q_1, q_2, q_3 are labelled with |
| + | the parallel set of upper case letters Q, Q_1, Q_2, Q_3. |
| + | |
| + | | o |
| + | | / \ |
| + | | / \ |
| + | | / \ |
| + | | / \ |
| + | | o o |
| + | | /%\ /%\ |
| + | | /%%%\ /%%%\ |
| + | | /%%%%%\ /%%%%%\ |
| + | | /%%%%%%%\ /%%%%%%%\ |
| + | | o%%%%%%%%%o%%%%%%%%%o |
| + | | / \%%%%%%%/ \%%%%%%%/ \ |
| + | | / \%%%%%/ \%%%%%/ \ |
| + | | / \%%%/ \%%%/ \ |
| + | | / \%/ \%/ \ |
| + | | o o o o |
| + | | / \ /%\ / \ / \ |
| + | | / \ /%%%\ / \ / \ |
| + | | / \ /%%%%%\ / \ / \ |
| + | | / \ /%%%%%%%\ / \ / \ |
| + | | o o%%%%%%%%%o o o |
| + | | ·\ / \%%%%%%%/ \ / \ /· |
| + | | · \ / \%%%%%/ \ / \ / · |
| + | | · \ / \%%%/ \ / \ / · |
| + | | · \ / \%/ \ / \ / · |
| + | | · o o o o · |
| + | | · ·\ / \ / \ /· · |
| + | | · · \ / \ / \ / · · |
| + | | · · \ / \ / \ / · · |
| + | | · Q · \ / \ / \ / ·Q_3 · |
| + | | ··········o o o·········· |
| + | | · \ /%\ / · |
| + | | · \ /%%%\ / · |
| + | | · \ /%%%%%\ / · |
| + | | · Q_1 \ /%%%%%%%\ / Q_2 · |
| + | | ··········o%%%%%%%%%o·········· |
| + | | \%%%%%%%/ |
| + | | \%%%%%/ |
| + | | \%%%/ |
| + | | \%/ |
| + | | o |
| + | | |
| + | | Figure 1. Genus Q and Species Q_1, Q_2, Q_3 |
| + | |
| + | Figure 2 is another form of venn diagram that one often uses, |
| + | where one collapses the unindited cells and leaves only the |
| + | models of the proposition in question. Some people would |
| + | call the transformation that changes from the first form |
| + | to the next form an operation of "taking the quotient", |
| + | but I tend to think of it as the "soap bubble picture" |
| + | or more exactly the "wire & thread & soap film" model |
| + | of the universe of discourse, where one pops out of |
| + | consideration the sections of the soap film that |
| + | stretch across the anti-model regions of space. |
| + | |
| + | o-------------------------------------------------o |
| + | | | |
| + | | X | |
| + | | | |
| + | | o | |
| + | | / \ | |
| + | | / \ | |
| + | | / \ | |
| + | | / \ | |
| + | | / \ | |
| + | | o Q_1 o | |
| + | | / \ / \ | |
| + | | / \ / \ | |
| + | | / \ / \ | |
| + | | / \ / \ | |
| + | | / \ / \ | |
| + | | / Q \ | |
| + | | / | \ | |
| + | | / | \ | |
| + | | / Q_2 | Q_3 \ | |
| + | | / | \ | |
| + | | / | \ | |
| + | | o-----------------o-----------------o | |
| + | | | |
| + | | | |
| + | | | |
| + | o-------------------------------------------------o |
| + | |
| + | Figure 2. Genus Q and Species Q_1, Q_2, Q_3 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Example. Partition, Genus & Species (cont.) |
| + | |
| + | Last time we considered in general terms how the forms |
| + | of complete partition and contingent partition operate |
| + | to maintain mutually disjoint and possibly exhaustive |
| + | categories of positions in a universe of discourse. |
| + | |
| + | This time we contemplate another concrete Example of |
| + | near minimal complexity, designed to demonstrate how |
| + | the forms of partition and subsumption can interact |
| + | in structuring a space of feature specifications. |
| + | |
| + | In this Example, we describe a universe of discourse |
| + | in terms of the following vocabulary of five features: |
| + | |
| + | | L. living_thing |
| + | | |
| + | | N. non_living |
| + | | |
| + | | A. animal |
| + | | |
| + | | V. vegetable |
| + | | |
| + | | M. mineral |
| + | |
| + | Let us construe these features as being subject to four constraints: |
| + | |
| + | | 1. Everything is either a living_thing or non_living, but not both. |
| + | | |
| + | | 2. Everything is either animal, vegetable, or mineral, |
| + | | but no two of these together. |
| + | | |
| + | | 3. A living_thing is either animal or vegetable, but not both, |
| + | | and everything animal or vegetable is a living_thing. |
| + | | |
| + | | 4. Everything mineral is non_living. |
| + | |
| + | These notions and constructions are expressed in the Log file shown below: |
| + | |
| + | Logical Input File |
| + | o-------------------------------------------------o |
| + | | | |
| + | | ( living_thing , non_living ) | |
| + | | | |
| + | | (( animal ),( vegetable ),( mineral )) | |
| + | | | |
| + | | ( living_thing ,( animal ),( vegetable )) | |
| + | | | |
| + | | ( mineral ( non_living )) | |
| + | | | |
| + | o-------------------------------------------------o |
| + | |
| + | The cactus expression in this file is the expression |
| + | of a "zeroth order theory" (ZOT), one that can be |
| + | paraphrased in more ordinary language to say: |
| + | |
| + | Translation |
| + | o-------------------------------------------------o |
| + | | | |
| + | | living_thing =/= non_living | |
| + | | | |
| + | | par : all -> {animal, vegetable, mineral} | |
| + | | | |
| + | | par : living_thing -> {animal, vegetable} | |
| + | | | |
| + | | mineral => non_living | |
| + | | | |
| + | o-------------------------------------------------o |
| + | |
| + | Here, "par : all -> {p, q, r}" is short for an assertion |
| + | that the universe as a whole is partitioned into subsets |
| + | that correspond to the features p, q, r. |
| + | |
| + | Also, "par : q -> {r, s}" asserts that "Q partitions into R and S. |
| + | |
| + | It is probably enough just to list the outputs of Model, Tenor, and Sense |
| + | when run on the preceding Log file. Using the same format and labeling as |
| + | before, we may note that Model has, from 2^5 = 32 possible interpretations, |
| + | made 11 evaluations, and found 3 models answering the generic descriptions |
| + | that were imposed by the logical input file. |
| + | |
| + | Model Outline |
| + | o------------------------o |
| + | | living_thing | |
| + | | non_living - | 1 |
| + | | (non_living ) | |
| + | | mineral - | 2 |
| + | | (mineral ) | |
| + | | animal | |
| + | | vegetable - | 3 |
| + | | (vegetable ) * | 4 * |
| + | | (animal ) | |
| + | | vegetable * | 5 * |
| + | | (vegetable ) - | 6 |
| + | | (living_thing ) | |
| + | | non_living | |
| + | | animal - | 7 |
| + | | (animal ) | |
| + | | vegetable - | 8 |
| + | | (vegetable ) | |
| + | | mineral * | 9 * |
| + | | (mineral ) - | 10 |
| + | | (non_living ) - | 11 |
| + | o------------------------o |
| + | |
| + | Tenor Outline |
| + | o------------------------o |
| + | | living_thing | |
| + | | (non_living ) | |
| + | | (mineral ) | |
| + | | animal | |
| + | | (vegetable ) * | <1> |
| + | | (animal ) | |
| + | | vegetable * | <2> |
| + | | (living_thing ) | |
| + | | non_living | |
| + | | (animal ) | |
| + | | (vegetable ) | |
| + | | mineral * | <3> |
| + | o------------------------o |
| + | |
| + | Sense Outline |
| + | o------------------------o |
| + | | living_thing | |
| + | | animal | |
| + | | vegetable | |
| + | | non_living | |
| + | | mineral | |
| + | o------------------------o |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Example. Molly's World |
| + | |
| + | I think that we are finally ready to tackle a more respectable example. |
| + | The Example known as "Molly's World" is borrowed from the literature on |
| + | computational learning theory, adapted with a few changes from the example |
| + | called "Molly’s Problem" in the paper "Learning With Hints" by Dana Angluin. |
| + | By way of setting up the problem, I quote Angluin's motivational description: |
| + | |
| + | | Imagine that you have become acquainted with an alien named Molly from the |
| + | | planet Ornot, who is currently employed in a day-care center. She is quite |
| + | | good at propositional logic, but a bit weak on knowledge of Earth. So you |
| + | | decide to formulate the beginnings of a propositional theory to help her |
| + | | label things in her immediate environment. |
| + | | |
| + | | Angluin, Dana, "Learning With Hints", pages 167-181, in: |
| + | | David Haussler & Leonard Pitt (eds.), 'Proceedings of the 1988 Workshop |
| + | | on Computational Learning Theory', Morgan Kaufmann, San Mateo, CA, 1989. |
| + | |
| + | The purpose of this quaint pretext is, of course, to make sure that the |
| + | reader appreciates the constraints of the problem: that no extra savvy |
| + | is fair, all facts must be presumed or deduced on the immediate premises. |
| + | |
| + | My use of this example is not directly relevant to the purposes of the |
| + | discussion from which it is taken, so I simply give my version of it |
| + | without comment on those issues. |
| + | |
| + | Here is my rendition of the initial knowledge base delimiting Molly’s World: |
| + | |
| + | Logical Input File: Molly.Log |
| + | o---------------------------------------------------------------------o |
| + | | | |
| + | | ( object ,( toy ),( vehicle )) | |
| + | | (( small_size ),( medium_size ),( large_size )) | |
| + | | (( two_wheels ),( three_wheels ),( four_wheels )) | |
| + | | (( no_seat ),( one_seat ),( few_seats ),( many_seats )) | |
| + | | ( object ,( scooter ),( bike ),( trike ),( car ),( bus ),( wagon )) | |
| + | | ( two_wheels no_seat ,( scooter )) | |
| + | | ( two_wheels one_seat pedals ,( bike )) | |
| + | | ( three_wheels one_seat pedals ,( trike )) | |
| + | | ( four_wheels few_seats doors ,( car )) | |
| + | | ( four_wheels many_seats doors ,( bus )) | |
| + | | ( four_wheels no_seat handle ,( wagon )) | |
| + | | ( scooter ( toy small_size )) | |
| + | | ( wagon ( toy small_size )) | |
| + | | ( trike ( toy small_size )) | |
| + | | ( bike small_size ( toy )) | |
| + | | ( bike medium_size ( vehicle )) | |
| + | | ( bike large_size ) | |
| + | | ( car ( vehicle large_size )) | |
| + | | ( bus ( vehicle large_size )) | |
| + | | ( toy ( object )) | |
| + | | ( vehicle ( object )) | |
| + | | | |
| + | o---------------------------------------------------------------------o |
| + | |
| + | All of the logical forms that are used in the preceding Log file |
| + | will probably be familiar from earlier discussions. The purpose |
| + | of one or two constructions may, however, be a little obscure, |
| + | so I will insert a few words of additional explanation here: |
| + | |
| + | The rule "( bike large_size )", for example, merely |
| + | says that nothing can be both a bike and large_size. |
| + | |
| + | The rule "( three_wheels one_seat pedals ,( trike ))" says that anything |
| + | with all the features of three_wheels, one_seat, and pedals is excluded |
| + | from being anything but a trike. In short, anything with just those |
| + | three features is equivalent to a trike. |
| + | |
| + | Recall that the form "( p , q )" may be interpreted to assert either |
| + | the exclusive disjunction or the logical inequivalence of p and q. |
| + | |
| + | The rules have been stated in this particular way simply |
| + | to imitate the style of rules in the reference example. |
| + | |
| + | This last point does bring up an important issue, the question |
| + | of "rhetorical" differences in expression and their potential |
| + | impact on the "pragmatics" of computation. Unfortunately, |
| + | I will have to abbreviate my discussion of this topic for |
| + | now, and only mention in passing the following facts. |
| + | |
| + | Logically equivalent expressions, even though they must lead |
| + | to logically equivalent normal forms, may have very different |
| + | characteristics when it comes to the efficiency of processing. |
| + | |
| + | For instance, consider the following four forms: |
| + | |
| + | | 1. (( p , q )) |
| + | | |
| + | | 2. ( p ,( q )) |
| + | | |
| + | | 3. (( p ), q ) |
| + | | |
| + | | 4. (( p , q )) |
| + | |
| + | All of these are equally succinct ways of maintaining that |
| + | p is logically equivalent to q, yet each can have different |
| + | effects on the route that Model takes to arrive at an answer. |
| + | Apparently, some equalities are more equal than others. |
| + | |
| + | These effects occur partly because the algorithm chooses to make cases |
| + | of variables on a basis of leftmost shallowest first, but their impact |
| + | can be complicated by the interactions that each expression has with |
| + | the context that it occupies. The main lesson to take away from all |
| + | of this, at least, for the time being, is that it is probably better |
| + | not to bother too much about these problems, but just to experiment |
| + | with different ways of expressing equivalent pieces of information |
| + | until you get a sense of what works best in various situations. |
| + | |
| + | I think that you will be happy to see only the |
| + | ultimate Sense of Molly’s World, so here it is: |
| + | |
| + | Sense Outline: Molly.Sen |
| + | o------------------------o |
| + | | object | |
| + | | two_wheels | |
| + | | no_seat | |
| + | | scooter | |
| + | | toy | |
| + | | small_size | |
| + | | one_seat | |
| + | | pedals | |
| + | | bike | |
| + | | small_size | |
| + | | toy | |
| + | | medium_size | |
| + | | vehicle | |
| + | | three_wheels | |
| + | | one_seat | |
| + | | pedals | |
| + | | trike | |
| + | | toy | |
| + | | small_size | |
| + | | four_wheels | |
| + | | few_seats | |
| + | | doors | |
| + | | car | |
| + | | vehicle | |
| + | | large_size | |
| + | | many_seats | |
| + | | doors | |
| + | | bus | |
| + | | vehicle | |
| + | | large_size | |
| + | | no_seat | |
| + | | handle | |
| + | | wagon | |
| + | | toy | |
| + | | small_size | |
| + | o------------------------o |
| + | |
| + | This outline is not the Sense of the unconstrained Log file, |
| + | but the result of running Model with a query on the single |
| + | feature "object". Using this focus helps the Modeler |
| + | to make more relevant Sense of Molly’s World. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | DM = Douglas McDavid |
| + | |
| + | DM: This, again, is an example of how real issues of ontology are |
| + | so often trivialized at the expense of technicalities. I just |
| + | had a burger, some fries, and a Coke. I would say all that was |
| + | non-living and non-mineral. A virus, I believe is non-animal, |
| + | non-vegetable, but living (and non-mineral). Teeth, shells, |
| + | and bones are virtually pure mineral, but living. These are |
| + | the kinds of issues that are truly "ontological," in my |
| + | opinion. You are not the only one to push them into |
| + | the background as of lesser importance. See the |
| + | discussion of "18-wheelers" in John Sowa's book. |
| + | |
| + | it's not my example, and from you say, it's not your example either. |
| + | copied it out of a book or a paper somewhere, too long ago to remember. |
| + | i am assuming that the author or tardition from which it came must have |
| + | seen some kind of sense in it. tell you what, write out your own theory |
| + | of "what is" in so many variables, more or less, publish it in a book or |
| + | a paper, and then folks will tell you that they dispute each and every |
| + | thing that you have just said, and it won't really matter all that much |
| + | how complex it is or how subtle you are. that has been the way of all |
| + | ontology for about as long as anybody can remember or even read about. |
| + | me? i don't have sufficient arrogance to be an ontologist, and you |
| + | know that's saying a lot, as i can't even imagine a way to convince |
| + | myself that i believe i know "what is", really and truly for sure |
| + | like some folks just seem to do. so i am working to improve our |
| + | technical ability to do logic, which is mostly a job of shooting |
| + | down the more serious delusions that we often get ourselves into. |
| + | can i be of any use to ontologists? i dunno. i guess it depends |
| + | on how badly they are attached to some of the delusions of knowing |
| + | what their "common" sense tells them everybody ought to already know, |
| + | but that every attempt to check that out in detail tells them it just |
| + | ain't so. a problem for which denial was just begging to be invented, |
| + | and so it was. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Example. Molly's World (cont.) |
| + | |
| + | In preparation for a contingently possible future discussion, |
| + | I need to attach a few parting thoughts to the case workup |
| + | of Molly's World that may not seem terribly relevant to |
| + | the present setting, but whose pertinence I hope will |
| + | become clearer in time. |
| + | |
| + | The logical paradigm from which this Example was derived is that |
| + | of "Zeroth Order Horn Clause Theories". The clauses at issue |
| + | in these theories are allowed to be of just three kinds: |
| + | |
| + | | 1. p & q & r & ... => z |
| + | | |
| + | | 2. z |
| + | | |
| + | | 3. ~[p & q & r & ...] |
| + | |
| + | Here, the proposition letters "p", "q", "r", ..., "z" |
| + | are restricted to being single positive features, not |
| + | themselves negated or otherwise complex expressions. |
| + | |
| + | In the Cactus Language or Existential Graph syntax |
| + | these forms would take on the following appearances: |
| + | |
| + | | 1. ( p q r ... ( z )) |
| + | | |
| + | | 2. z |
| + | | |
| + | | 3. ( p q r ... ) |
| + | |
| + | The style of deduction in Horn clause logics is essentially |
| + | proof-theoretic in character, with the main burden of proof |
| + | falling on implication relations ("=>") and on "projective" |
| + | forms of inference, that is, information-losing inferences |
| + | like modus ponens and resolution. Cf. [Llo], [MaW]. |
| + | |
| + | In contrast, the method used here is substantially model-theoretic, |
| + | the stress being to start from more general forms of expression for |
| + | laying out facts (for example, distinctions, equations, partitions) |
| + | and to work toward results that maintain logical equivalence with |
| + | their origins. |
| + | |
| + | What all of this has to do with the output above is this: |
| + | >From the perspective that is adopted in the present work, |
| + | almost any theory, for example, the one that is founded |
| + | on the postulates of Molly's World, will have far more |
| + | models than the implicational and inferential mode of |
| + | reasoning is designed to discover. We will be forced |
| + | to confront them, however, if we try to run Model on |
| + | a large set of implications. |
| + | |
| + | The typical Horn clause interpreter gets around this |
| + | difficulty only by a stratagem that takes clauses to |
| + | mean something other than what they say, that is, by |
| + | distorting the principles of semantics in practice. |
| + | Our Model, on the other hand, has no such finesse. |
| + | |
| + | This explains why it was necessary to impose the |
| + | prerequisite "object" constraint on the Log file |
| + | for Molly's World. It supplied no more than what |
| + | we usually take for granted, in order to obtain |
| + | a set of models that we would normally think of |
| + | as being the intended import of the definitions. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Example. Jets & Sharks |
| + | |
| + | The propositional calculus based on the boundary operator, that is, |
| + | the multigrade logical connective of the form "( , , , ... )" can be |
| + | interpreted in a way that resembles the logic of activation states and |
| + | competition constraints in certain neural network models. One way to do |
| + | this is by interpreting the blank or unmarked state as the resting state |
| + | of a neural pool, the bound or marked state as its activated state, and |
| + | by representing a mutually inhibitory pool of neurons p, q, r by means |
| + | of the expression "( p , q , r )". To illustrate this possibility, |
| + | I transcribe into cactus language expressions a notorious example |
| + | from the "parallel distributed processing" (PDP) paradigm [McR] |
| + | and work through two of the associated exercises as portrayed |
| + | in this format. |
| + | |
| + | Logical Input File: JAS = ZOT(Jets And Sharks) |
| + | o----------------------------------------------------------------o |
| + | | | |
| + | | (( art ),( al ),( sam ),( clyde ),( mike ), | |
| + | | ( jim ),( greg ),( john ),( doug ),( lance ), | |
| + | | ( george ),( pete ),( fred ),( gene ),( ralph ), | |
| + | | ( phil ),( ike ),( nick ),( don ),( ned ),( karl ), | |
| + | | ( ken ),( earl ),( rick ),( ol ),( neal ),( dave )) | |
| + | | | |
| + | | ( jets , sharks ) | |
| + | | | |
| + | | ( jets , | |
| + | | ( art ),( al ),( sam ),( clyde ),( mike ), | |
| + | | ( jim ),( greg ),( john ),( doug ),( lance ), | |
| + | | ( george ),( pete ),( fred ),( gene ),( ralph )) | |
| + | | | |
| + | | ( sharks , | |
| + | | ( phil ),( ike ),( nick ),( don ),( ned ),( karl ), | |
| + | | ( ken ),( earl ),( rick ),( ol ),( neal ),( dave )) | |
| + | | | |
| + | | (( 20's ),( 30's ),( 40's )) | |
| + | | | |
| + | | ( 20's , | |
| + | | ( sam ),( jim ),( greg ),( john ),( lance ), | |
| + | | ( george ),( pete ),( fred ),( gene ),( ken )) | |
| + | | | |
| + | | ( 30's , | |
| + | | ( al ),( mike ),( doug ),( ralph ), | |
| + | | ( phil ),( ike ),( nick ),( don ), | |
| + | | ( ned ),( rick ),( ol ),( neal ),( dave )) | |
| + | | | |
| + | | ( 40's , | |
| + | | ( art ),( clyde ),( karl ),( earl )) | |
| + | | | |
| + | | (( junior_high ),( high_school ),( college )) | |
| + | | | |
| + | | ( junior_high , | |
| + | | ( art ),( al ),( clyde ),( mike ),( jim ), | |
| + | | ( john ),( lance ),( george ),( ralph ),( ike )) | |
| + | | | |
| + | | ( high_school , | |
| + | | ( greg ),( doug ),( pete ),( fred ),( nick ), | |
| + | | ( karl ),( ken ),( earl ),( rick ),( neal ),( dave )) | |
| + | | | |
| + | | ( college , | |
| + | | ( sam ),( gene ),( phil ),( don ),( ned ),( ol )) | |
| + | | | |
| + | | (( single ),( married ),( divorced )) | |
| + | | | |
| + | | ( single , | |
| + | | ( art ),( sam ),( clyde ),( mike ), | |
| + | | ( doug ),( pete ),( fred ),( gene ), | |
| + | | ( ralph ),( ike ),( nick ),( ken ),( neal )) | |
| + | | | |
| + | | ( married , | |
| + | | ( al ),( greg ),( john ),( lance ),( phil ), | |
| + | | ( don ),( ned ),( karl ),( earl ),( ol )) | |
| + | | | |
| + | | ( divorced , | |
| + | | ( jim ),( george ),( rick ),( dave )) | |
| + | | | |
| + | | (( bookie ),( burglar ),( pusher )) | |
| + | | | |
| + | | ( bookie , | |
| + | | ( sam ),( clyde ),( mike ),( doug ), | |
| + | | ( pete ),( ike ),( ned ),( karl ),( neal )) | |
| + | | | |
| + | | ( burglar , | |
| + | | ( al ),( jim ),( john ),( lance ), | |
| + | | ( george ),( don ),( ken ),( earl ),( rick )) | |
| + | | | |
| + | | ( pusher , | |
| + | | ( art ),( greg ),( fred ),( gene ), | |
| + | | ( ralph ),( phil ),( nick ),( ol ),( dave )) | |
| + | | | |
| + | o----------------------------------------------------------------o |
| + | |
| + | We now apply Study to the proposition that |
| + | defines the Jets and Sharks knowledge base, |
| + | that is to say, the knowledge that we are |
| + | given about the Jets and Sharks, not the |
| + | knowledge that the Jets and Sharks have. |
| + | |
| + | With a query on the name "ken" we obtain the following |
| + | output, giving all of the features associated with Ken: |
| + | |
| + | Sense Outline: JAS & Ken |
| + | o---------------------------------------o |
| + | | ken | |
| + | | sharks | |
| + | | 20's | |
| + | | high_school | |
| + | | single | |
| + | | burglar | |
| + | o---------------------------------------o |
| + | |
| + | With a query on the two features "college" and "sharks" |
| + | we obtain the following outline of all of the features |
| + | that satisfy these constraints: |
| + | |
| + | Sense Outline: JAS & College & Sharks |
| + | o---------------------------------------o |
| + | | college | |
| + | | sharks | |
| + | | 30's | |
| + | | married | |
| + | | bookie | |
| + | | ned | |
| + | | burglar | |
| + | | don | |
| + | | pusher | |
| + | | phil | |
| + | | ol | |
| + | o---------------------------------------o |
| + | |
| + | >From this we discover that all college Sharks |
| + | are 30-something and married. Furthermore, |
| + | we have a complete listing of their names |
| + | broken down by occupation, as I have no |
| + | doubt that all of them will be in time. |
| + | |
| + | | Reference: |
| + | | |
| + | | McClelland, James L. & Rumelhart, David E., |
| + | |'Explorations in Parallel Distributed Processing: |
| + | | A Handbook of Models, Programs, and Exercises', |
| + | | MIT Press, Cambridge, MA, 1988. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | One of the issues that my pondering weak and weary over |
| + | has caused me to burn not a few barrels of midnight oil |
| + | over the past elventeen years or so is the relationship |
| + | among divers and sundry "styles of inference", by which |
| + | I mean particular choices of inference paradigms, rules, |
| + | or schemata. The chief breakpoint seems to lie between |
| + | information-losing and information-maintaining modes of |
| + | inference, also called "implicational" and "equational", |
| + | or "projective" and "preservative" brands, respectively. |
| + | |
| + | Since it appears to be mostly the implicational and projective |
| + | styles of inference that are more familiar to folks hereabouts, |
| + | I will start off this subdiscussion by introducing a number of |
| + | risibly simple but reasonably manageable examples of the other |
| + | brand of inference, treated as equational reasoning approaches |
| + | to problems about satisfying "zeroth order constraints" (ZOC's). |
| + | |
| + | Applications of a Propositional Calculator: |
| + | Constraint Satisfaction Problems. |
| + | Jon Awbrey, April 24, 1995. |
| + | |
| + | The Four Houses Puzzle |
| + | |
| + | Constructed on the model of the "Five Houses Puzzle" in [VaH, 132-136]. |
| + | |
| + | Problem Statement. Four people with different nationalities live in the |
| + | first four houses of a street. They practice four distinct professions, |
| + | and each of them has a favorite animal, all of them different. The four |
| + | houses are painted different colors. The following facts are known: |
| + | |
| + | | 1. The Englander lives in the first house on the left. |
| + | | 2. The doctor lives in the second house. |
| + | | 3. The third house is painted red. |
| + | | 4. The zebra is a favorite in the fourth house. |
| + | | 5. The person in the first house has a dog. |
| + | | 6. The Japanese lives in the third house. |
| + | | 7. The red house is on the left of the yellow one. |
| + | | 8. They breed snails in the house to right of the doctor. |
| + | | 9. The Englander lives next to the green house. |
| + | | 10. The fox is in the house next to to the diplomat. |
| + | | 11. The Spaniard likes zebras. |
| + | | 12. The Japanese is a painter. |
| + | | 13. The Italian lives in the green house. |
| + | | 14. The violinist lives in the yellow house. |
| + | | 15. The dog is a pet in the blue house. |
| + | | 16. The doctor keeps a fox. |
| + | |
| + | The problem is to find all of the assignments of |
| + | features to houses that satisfy these requirements. |
| + | |
| + | Logical Input File: House^4.Log |
| + | o---------------------------------------------------------------------o |
| + | | | |
| + | | eng_1 doc_2 red_3 zeb_4 dog_1 jap_3 | |
| + | | | |
| + | | (( red_1 yel_2 ),( red_2 yel_3 ),( red_3 yel_4 )) | |
| + | | (( doc_1 sna_2 ),( doc_2 sna_3 ),( doc_3 sna_4 )) | |
| + | | | |
| + | | (( eng_1 gre_2 ), | |
| + | | ( eng_2 gre_3 ),( eng_2 gre_1 ), | |
| + | | ( eng_3 gre_4 ),( eng_3 gre_2 ), | |
| + | | ( eng_4 gre_3 )) | |
| + | | | |
| + | | (( dip_1 fox_2 ), | |
| + | | ( dip_2 fox_3 ),( dip_2 fox_1 ), | |
| + | | ( dip_3 fox_4 ),( dip_3 fox_2 ), | |
| + | | ( dip_4 fox_3 )) | |
| + | | | |
| + | | (( spa_1 zeb_1 ),( spa_2 zeb_2 ),( spa_3 zeb_3 ),( spa_4 zeb_4 )) | |
| + | | (( jap_1 pai_1 ),( jap_2 pai_2 ),( jap_3 pai_3 ),( jap_4 pai_4 )) | |
| + | | (( ita_1 gre_1 ),( ita_2 gre_2 ),( ita_3 gre_3 ),( ita_4 gre_4 )) | |
| + | | | |
| + | | (( yel_1 vio_1 ),( yel_2 vio_2 ),( yel_3 vio_3 ),( yel_4 vio_4 )) | |
| + | | (( blu_1 dog_1 ),( blu_2 dog_2 ),( blu_3 dog_3 ),( blu_4 dog_4 )) | |
| + | | | |
| + | | (( doc_1 fox_1 ),( doc_2 fox_2 ),( doc_3 fox_3 ),( doc_4 fox_4 )) | |
| + | | | |
| + | | (( | |
| + | | | |
| + | | (( eng_1 ),( eng_2 ),( eng_3 ),( eng_4 )) | |
| + | | (( spa_1 ),( spa_2 ),( spa_3 ),( spa_4 )) | |
| + | | (( jap_1 ),( jap_2 ),( jap_3 ),( jap_4 )) | |
| + | | (( ita_1 ),( ita_2 ),( ita_3 ),( ita_4 )) | |
| + | | | |
| + | | (( eng_1 ),( spa_1 ),( jap_1 ),( ita_1 )) | |
| + | | (( eng_2 ),( spa_2 ),( jap_2 ),( ita_2 )) | |
| + | | (( eng_3 ),( spa_3 ),( jap_3 ),( ita_3 )) | |
| + | | (( eng_4 ),( spa_4 ),( jap_4 ),( ita_4 )) | |
| + | | | |
| + | | (( gre_1 ),( gre_2 ),( gre_3 ),( gre_4 )) | |
| + | | (( red_1 ),( red_2 ),( red_3 ),( red_4 )) | |
| + | | (( yel_1 ),( yel_2 ),( yel_3 ),( yel_4 )) | |
| + | | (( blu_1 ),( blu_2 ),( blu_3 ),( blu_4 )) | |
| + | | | |
| + | | (( gre_1 ),( red_1 ),( yel_1 ),( blu_1 )) | |
| + | | (( gre_2 ),( red_2 ),( yel_2 ),( blu_2 )) | |
| + | | (( gre_3 ),( red_3 ),( yel_3 ),( blu_3 )) | |
| + | | (( gre_4 ),( red_4 ),( yel_4 ),( blu_4 )) | |
| + | | | |
| + | | (( pai_1 ),( pai_2 ),( pai_3 ),( pai_4 )) | |
| + | | (( dip_1 ),( dip_2 ),( dip_3 ),( dip_4 )) | |
| + | | (( vio_1 ),( vio_2 ),( vio_3 ),( vio_4 )) | |
| + | | (( doc_1 ),( doc_2 ),( doc_3 ),( doc_4 )) | |
| + | | | |
| + | | (( pai_1 ),( dip_1 ),( vio_1 ),( doc_1 )) | |
| + | | (( pai_2 ),( dip_2 ),( vio_2 ),( doc_2 )) | |
| + | | (( pai_3 ),( dip_3 ),( vio_3 ),( doc_3 )) | |
| + | | (( pai_4 ),( dip_4 ),( vio_4 ),( doc_4 )) | |
| + | | | |
| + | | (( dog_1 ),( dog_2 ),( dog_3 ),( dog_4 )) | |
| + | | (( zeb_1 ),( zeb_2 ),( zeb_3 ),( zeb_4 )) | |
| + | | (( fox_1 ),( fox_2 ),( fox_3 ),( fox_4 )) | |
| + | | (( sna_1 ),( sna_2 ),( sna_3 ),( sna_4 )) | |
| + | | | |
| + | | (( dog_1 ),( zeb_1 ),( fox_1 ),( sna_1 )) | |
| + | | (( dog_2 ),( zeb_2 ),( fox_2 ),( sna_2 )) | |
| + | | (( dog_3 ),( zeb_3 ),( fox_3 ),( sna_3 )) | |
| + | | (( dog_4 ),( zeb_4 ),( fox_4 ),( sna_4 )) | |
| + | | | |
| + | | )) | |
| + | | | |
| + | o---------------------------------------------------------------------o |
| + | |
| + | Sense Outline: House^4.Sen |
| + | o-----------------------------o |
| + | | eng_1 | |
| + | | doc_2 | |
| + | | red_3 | |
| + | | zeb_4 | |
| + | | dog_1 | |
| + | | jap_3 | |
| + | | yel_4 | |
| + | | sna_3 | |
| + | | gre_2 | |
| + | | dip_1 | |
| + | | fox_2 | |
| + | | spa_4 | |
| + | | pai_3 | |
| + | | ita_2 | |
| + | | vio_4 | |
| + | | blu_1 | |
| + | o-----------------------------o |
| + | |
| + | Table 1. Solution to the Four Houses Puzzle |
| + | o------------o------------o------------o------------o------------o |
| + | | | House 1 | House 2 | House 3 | House 4 | |
| + | o------------o------------o------------o------------o------------o |
| + | | Nation | England | Italy | Japan | Spain | |
| + | | Color | blue | green | red | yellow | |
| + | | Profession | diplomat | doctor | painter | violinist | |
| + | | Animal | dog | fox | snails | zebra | |
| + | o------------o------------o------------o------------o------------o |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | First off, I do not trivialize the "real issues of ontology", indeed, |
| + | it is precisely my estimate of the non-trivial difficulty of this task, |
| + | of formulating the types of "generic ontology" that we propose to do here, |
| + | that forces me to choose and to point out the inescapability of the approach |
| + | that I am currently taking, which is to enter on the necessary preliminary of |
| + | building up the logical tools that we need to tackle the ontology task proper. |
| + | And I would say, to the contrary, that it is those who think we can arrive at |
| + | a working general ontology by sitting on the porch shooting the breeze about |
| + | "what it is" until the cows come home -- that is, the method for which it |
| + | has become cliche to indict the Ancient Greeks, though, if truth be told, |
| + | we'd have to look to the pre-socratics and the pre-stoics to find a good |
| + | match for the kinds of revelation that are common hereabouts -- I would |
| + | say that it's those folks who trivialize the "real issues of ontology". |
| + | |
| + | A person, living in our times, who is serious about knowing the being of things, |
| + | really only has one choice -- to pick what tiny domain of things he or she just |
| + | has to know about the most, thence to hie away to the adept gurus of the matter |
| + | in question, forgeting the rest, cause "general ontology" is a no-go these days. |
| + | It is presently in a state like astronomy before telescopes, and that means not |
| + | entirely able to discern itself from astrology and other psychically projective |
| + | exercises of wishful and dreadful thinking like that. |
| + | |
| + | So I am busy grinding lenses ... |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | DM = Douglas McDavid |
| + | |
| + | DM: Thanks for both the original and additional response. I'm not trying to |
| + | single you out, as I have been picking on various postings in a similar |
| + | manner ever since I started contributing to this discussion. I agree with |
| + | you that the task of this working group is non-trivially difficult. In fact, |
| + | I believe we are still a long way from a clear and useful agreement about what |
| + | constitutes "upper" ontology, and what it would mean to standardize it. However, |
| + | I don't agree that the only place to make progress is in tiny domains of things. |
| + | I've contributed the thought that a fundamental, upper-level concept is the |
| + | concept of system, and that that would be a good place to begin. And I'll |
| + | never be able to refrain from evaluating the content as well as the form |
| + | of any examples presented for consideration here. Probably should |
| + | accompany these comments with a ;-) |
| + | |
| + | There will never be a standard universal ontology |
| + | of the absolute essential impertubable monolithic |
| + | variety that some people still dream of in their |
| + | fantasies of spectating on and speculating about |
| + | a pre-relativistically non-participatory universe |
| + | from their singular but isolated gods'eye'views. |
| + | The bells tolled for that one many years ago, |
| + | but some of the more blithe of the blissful |
| + | islanders have just not gotten the news yet. |
| + | |
| + | But there is still a lot to do that would be useful |
| + | under the banner of a "standard upper ontology", |
| + | if only we stay loose in our interpretation |
| + | of what that implies in practical terms. |
| + | |
| + | One likely approach to the problem would be to take |
| + | a hint from the afore-allusioned history of physics -- |
| + | to inquire for whom, else, the bell tolls -- and to |
| + | see if there are any bits of wisdom from that prior |
| + | round of collective experience that can be adapted |
| + | by dint of analogy to our present predicament. |
| + | I happen to think that there are. |
| + | |
| + | And there the answer was, not to try and force a return, |
| + | though lord knows they all gave it their very best shot, |
| + | to an absolute and imperturbable framework of existence, |
| + | but to see the reciprocal participant relation that all |
| + | partakers have to the constitution of that framing, yes, |
| + | even unto those who would abdictators and abstainees be. |
| + | |
| + | But what does that imply about some shred of a standard? |
| + | It means that we are better off seeking, not a standard, |
| + | one-size-fits-all ontology, but more standard resources |
| + | for trying to interrelate diverse points of view and to |
| + | transform the data that's gathered from one perspective |
| + | in ways that it can most appropriately be compared with |
| + | the data that is gathered from other standpoints on the |
| + | splendorous observational scenes and theorematic stages. |
| + | |
| + | That is what I am working on. |
| + | And it hasn't been merely |
| + | for a couple of years. |
| + | |
| + | As to this bit: |
| + | |
| + | o-------------------------------------------------o |
| + | | | |
| + | | ( living_thing , non_living ) | |
| + | | | |
| + | | (( animal ),( vegetable ),( mineral )) | |
| + | | | |
| + | | ( living_thing ,( animal ),( vegetable )) | |
| + | | | |
| + | | ( mineral ( non_living )) | |
| + | | | |
| + | o-------------------------------------------------o |
| + | |
| + | My 5-dimensional Example, that I borrowed from some indifferent source |
| + | of what is commonly recognized as "common sense" -- and I think rather |
| + | obviously designed more for the classification of pre-modern species |
| + | of whole critters and pure matters of natural substance than the |
| + | motley mixture of un/natural and in/organic conglouterites that |
| + | we find served up on the menu of modernity -- was not intended |
| + | even so much as a toy ontology, but simply as an expository |
| + | example, concocted for the sake of illustrating the sorts |
| + | of logical interaction that occur among four different |
| + | patterns of logical constraint, all of which types |
| + | arise all the time no matter what the domain, and |
| + | which I believe that my novel forms of expression, |
| + | syntactically speaking, express quite succinctly, |
| + | especially when you contemplate the complexities |
| + | of the computation that may flow and must follow |
| + | from even these meagre propositional expressions. |
| + | |
| + | Yes, systems -- but -- even here usage differs in significant ways. |
| + | I have spent ten years now trying to integrate my earlier efforts |
| + | under an explicit systems banner, but even within the bounds of |
| + | a systems engineering programme at one site there is a wide |
| + | semantic dispersion that issues from this word "system". |
| + | I am committed, and in writing, to taking what we so |
| + | glibly and prospectively call "intelligent systems" |
| + | seriously as dynamical systems. That has many |
| + | consequences, and I have to pick and choose |
| + | which of those I may be suited to follow. |
| + | |
| + | But that is too long a story for now ... |
| + | |
| + | ";-)"? |
| + | |
| + | Somehow that has always looked like |
| + | the Chesshire Cat's grin to me ... |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | By way of catering to popular demand, I have decided to |
| + | render this symposium a bit more à la carte, and thus to |
| + | serve up as faster food than heretofore a choice selection |
| + | of the more sumptuous bits that I have in my logical larder, |
| + | not yet full fare, by any means, but a sample of what might |
| + | one day approach to being an abundantly moveable feast of |
| + | ontological contents and general metaphysical delights. |
| + | I'll leave it to you to name your poison, as it were. |
| + | |
| + | Applications of a Propositional Calculator: |
| + | Constraint Satisfaction Problems. |
| + | Jon Awbrey, April 24, 1995. |
| + | |
| + | Fabric Knowledge Base |
| + | Based on the example in [MaW, pages 8-16]. |
| + | |
| + | Logical Input File: Fab.Log |
| + | o---------------------------------------------------------------------o |
| + | | | |
| + | | (has_floats , plain_weave ) | |
| + | | (has_floats ,(twill_weave ),(satin_weave )) | |
| + | | | |
| + | | (plain_weave , | |
| + | | (plain_weave one_color ), | |
| + | | (color_groups ), | |
| + | | (grouped_warps ), | |
| + | | (some_thicker ), | |
| + | | (crossed_warps ), | |
| + | | (loop_threads ), | |
| + | | (plain_weave flannel )) | |
| + | | | |
| + | | (plain_weave one_color cotton balanced smooth ,(percale )) | |
| + | | (plain_weave one_color cotton sheer ,(organdy )) | |
| + | | (plain_weave one_color silk sheer ,(organza )) | |
| + | | | |
| + | | (plain_weave color_groups warp_stripe fill_stripe ,(plaid )) | |
| + | | (plaid equal_stripe ,(gingham )) | |
| + | | | |
| + | | (plain_weave grouped_warps ,(basket_weave )) | |
| + | | | |
| + | | (basket_weave typed , | |
| + | | (type_2_to_1 ), | |
| + | | (type_2_to_2 ), | |
| + | | (type_4_to_4 )) | |
| + | | | |
| + | | (basket_weave typed type_2_to_1 thicker_fill ,(oxford )) | |
| + | | (basket_weave typed (type_2_to_2 , | |
| + | | type_4_to_4 ) same_thickness ,(monks_cloth )) | |
| + | | (basket_weave (typed ) rough open ,(hopsacking )) | |
| + | | | |
| + | | (typed (basket_weave )) | |
| + | | | |
| + | | (basket_weave ,(oxford ),(monks_cloth ),(hopsacking )) | |
| + | | | |
| + | | (plain_weave some_thicker ,(ribbed_weave )) | |
| + | | | |
| + | | (ribbed_weave ,(small_rib ),(medium_rib ),(heavy_rib )) | |
| + | | (ribbed_weave ,(flat_rib ),(round_rib )) | |
| + | | | |
| + | | (ribbed_weave thicker_fill ,(cross_ribbed )) | |
| + | | (cross_ribbed small_rib flat_rib ,(faille )) | |
| + | | (cross_ribbed small_rib round_rib ,(grosgrain )) | |
| + | | (cross_ribbed medium_rib round_rib ,(bengaline )) | |
| + | | (cross_ribbed heavy_rib round_rib ,(ottoman )) | |
| + | | | |
| + | | (cross_ribbed ,(faille ),(grosgrain ),(bengaline ),(ottoman )) | |
| + | | | |
| + | | (plain_weave crossed_warps ,(leno_weave )) | |
| + | | (leno_weave open ,(marquisette )) | |
| + | | (plain_weave loop_threads ,(pile_weave )) | |
| + | | | |
| + | | (pile_weave ,(fill_pile ),(warp_pile )) | |
| + | | (pile_weave ,(cut ),(uncut )) | |
| + | | | |
| + | | (pile_weave warp_pile cut ,(velvet )) | |
| + | | (pile_weave fill_pile cut aligned_pile ,(corduroy )) | |
| + | | (pile_weave fill_pile cut staggered_pile ,(velveteen )) | |
| + | | (pile_weave fill_pile uncut reversible ,(terry )) | |
| + | | | |
| + | | (pile_weave fill_pile cut ( (aligned_pile , staggered_pile ) )) | |
| + | | | |
| + | | (pile_weave ,(velvet ),(corduroy ),(velveteen ),(terry )) | |
| + | | | |
| + | | (plain_weave , | |
| + | | (percale ),(organdy ),(organza ),(plaid ), | |
| + | | (oxford ),(monks_cloth ),(hopsacking ), | |
| + | | (faille ),(grosgrain ),(bengaline ),(ottoman ), | |
| + | | (leno_weave ),(pile_weave ),(plain_weave flannel )) | |
| + | | | |
| + | | (twill_weave , | |
| + | | (warp_faced ), | |
| + | | (filling_faced ), | |
| + | | (even_twill ), | |
| + | | (twill_weave flannel )) | |
| + | | | |
| + | | (twill_weave warp_faced colored_warp white_fill ,(denim )) | |
| + | | (twill_weave warp_faced one_color ,(drill )) | |
| + | | (twill_weave even_twill diagonal_rib ,(serge )) | |
| + | | | |
| + | | (twill_weave warp_faced ( | |
| + | | (one_color , | |
| + | | ((colored_warp )(white_fill )) ) | |
| + | | )) | |
| + | | | |
| + | | (twill_weave warp_faced ,(denim ),(drill )) | |
| + | | (twill_weave even_twill ,(serge )) | |
| + | | | |
| + | | (( | |
| + | | ( ((plain_weave )(twill_weave )) | |
| + | | ((cotton )(wool )) napped ,(flannel )) | |
| + | | )) | |
| + | | | |
| + | | (satin_weave ,(warp_floats ),(fill_floats )) | |
| + | | | |
| + | | (satin_weave ,(satin_weave smooth ),(satin_weave napped )) | |
| + | | (satin_weave ,(satin_weave cotton ),(satin_weave silk )) | |
| + | | | |
| + | | (satin_weave warp_floats smooth ,(satin )) | |
| + | | (satin_weave fill_floats smooth ,(sateen )) | |
| + | | (satin_weave napped cotton ,(moleskin )) | |
| + | | | |
| + | | (satin_weave ,(satin ),(sateen ),(moleskin )) | |
| + | | | |
| + | o---------------------------------------------------------------------o |
| + | |
| + | | Reference [MaW] |
| + | | |
| + | | Maier, David & Warren, David S., |
| + | |'Computing with Logic: Logic Programming with Prolog', |
| + | | Benjamin/Cummings, Menlo Park, CA, 1988. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | I think that it might be a good idea to go back to a simpler example |
| + | of a constraint satisfaction problem, and to discuss the elements of |
| + | its expression as a ZOT in a less cluttered setting before advancing |
| + | onward once again to problems on the order of the Four Houses Puzzle. |
| + | |
| + | | Applications of a Propositional Calculator: |
| + | | Constraint Satisfaction Problems. |
| + | | Jon Awbrey, April 24, 1995. |
| + | |
| + | Graph Coloring |
| + | |
| + | Based on the discussion in [Wil, page 196]. |
| + | |
| + | One is given three colors, say, orange, silver, indigo, |
| + | and a graph on four nodes that has the following shape: |
| + | |
| + | | 1 |
| + | | o |
| + | | / \ |
| + | | / \ |
| + | | 4 o-----o 2 |
| + | | \ / |
| + | | \ / |
| + | | o |
| + | | 3 |
| + | |
| + | The problem is to color the nodes of the graph |
| + | in such a way that no pair of nodes that are |
| + | adjacent in the graph, that is, linked by |
| + | an edge, get the same color. |
| + | |
| + | The objective situation that is to be achieved can be represented |
| + | in a so-called "declarative" fashion, in effect, by employing the |
| + | cactus language as a very simple sort of declarative programming |
| + | language, and depicting the prospective solution to the problem |
| + | as a ZOT. |
| + | |
| + | To do this, begin by declaring the following set of |
| + | twelve boolean variables or "zeroth order features": |
| + | |
| + | {1_orange, 1_silver, 1_indigo, |
| + | 2_orange, 2_silver, 2_indigo, |
| + | 3_orange, 3_silver, 3_indigo, |
| + | 4_orange, 4_silver, 4_indigo} |
| + | |
| + | The interpretation to keep in mind will be such that |
| + | the feature name of the form "<node i>_<color j>" |
| + | says that the node i is assigned the color j. |
| + | |
| + | Logical Input File: Color.Log |
| + | o----------------------------------------------------------------------o |
| + | | | |
| + | | (( 1_orange ),( 1_silver ),( 1_indigo )) | |
| + | | (( 2_orange ),( 2_silver ),( 2_indigo )) | |
| + | | (( 3_orange ),( 3_silver ),( 3_indigo )) | |
| + | | (( 4_orange ),( 4_silver ),( 4_indigo )) | |
| + | | | |
| + | | ( 1_orange 2_orange )( 1_silver 2_silver )( 1_indigo 2_indigo ) | |
| + | | ( 1_orange 4_orange )( 1_silver 4_silver )( 1_indigo 4_indigo ) | |
| + | | ( 2_orange 3_orange )( 2_silver 3_silver )( 2_indigo 3_indigo ) | |
| + | | ( 2_orange 4_orange )( 2_silver 4_silver )( 2_indigo 4_indigo ) | |
| + | | ( 3_orange 4_orange )( 3_silver 4_silver )( 3_indigo 4_indigo ) | |
| + | | | |
| + | o----------------------------------------------------------------------o |
| + | |
| + | The first stanza of verses declares that |
| + | every node is assigned exactly one color. |
| + | |
| + | The second stanza of verses declares that |
| + | no adjacent nodes get the very same color. |
| + | |
| + | Each satisfying interpretation of this ZOT |
| + | that is also a program corresponds to what |
| + | graffitists call a "coloring" of the graph. |
| + | |
| + | Theme One's Model interpreter, when we set |
| + | it to work on this ZOT, will array before |
| + | our eyes all of the colorings of the graph. |
| + | |
| + | Sense Outline: Color.Sen |
| + | o-----------------------------o |
| + | | 1_orange | |
| + | | 2_silver | |
| + | | 3_orange | |
| + | | 4_indigo | |
| + | | 2_indigo | |
| + | | 3_orange | |
| + | | 4_silver | |
| + | | 1_silver | |
| + | | 2_orange | |
| + | | 3_silver | |
| + | | 4_indigo | |
| + | | 2_indigo | |
| + | | 3_silver | |
| + | | 4_orange | |
| + | | 1_indigo | |
| + | | 2_orange | |
| + | | 3_indigo | |
| + | | 4_silver | |
| + | | 2_silver | |
| + | | 3_indigo | |
| + | | 4_orange | |
| + | o-----------------------------o |
| + | |
| + | | Reference [Wil] |
| + | | |
| + | | Wilf, Herbert S., |
| + | |'Algorithms and Complexity', |
| + | | Prentice-Hall, Englewood Cliffs, NJ, 1986. |
| + | | |
| + | | Nota Bene. There is a wrong Figure in some |
| + | | printings of the book, that does not match |
| + | | the description of the Example that is |
| + | | given in the text. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Let us continue to examine the properties of the cactus language |
| + | as a minimal style of declarative programming language. Even in |
| + | the likes of this zeroth order microcosm one can observe, and on |
| + | a good day still more clearly for the lack of other distractions, |
| + | many of the buzz worlds that will spring into full bloom, almost |
| + | as if from nowhere, to become the first order of business in the |
| + | latter day logical organa, plus combinators, plus lambda calculi. |
| + | |
| + | By way of homage to the classics of the art, I can hardly pass |
| + | this way without paying my dues to the next sample of examples. |
| + | |
| + | N Queens Problem |
| + | |
| + | I will give the ZOT that describes the N Queens Problem for N = 5, |
| + | since that is the most that I and my old 286 could do when last I |
| + | wrote up this Example. |
| + | |
| + | The problem is now to write a "zeroth order program" (ZOP) that |
| + | describes the following objective: To place 5 chess queens on |
| + | a 5 by 5 chessboard so that no queen attacks any other queen. |
| + | |
| + | It is clear that there can be at most one queen on each row |
| + | of the board and so by dint of regal necessity, exactly one |
| + | queen in each row of the desired array. This gambit allows |
| + | us to reduce the problem to one of picking a permutation of |
| + | five things in fives places, and this affords us sufficient |
| + | clue to begin down a likely path toward the intended object, |
| + | by recruiting the following phalanx of 25 logical variables: |
| + | |
| + | Literal Input File: Q5.Lit |
| + | o---------------------------------------o |
| + | | | |
| + | | q1_r1, q1_r2, q1_r3, q1_r4, q1_r5, | |
| + | | q2_r1, q2_r2, q2_r3, q2_r4, q2_r5, | |
| + | | q3_r1, q3_r2, q3_r3, q3_r4, q3_r5, | |
| + | | q4_r1, q4_r2, q4_r3, q4_r4, q4_r5, | |
| + | | q5_r1, q5_r2, q5_r3, q5_r4, q5_r5. | |
| + | | | |
| + | o---------------------------------------o |
| + | |
| + | Thus we seek to define a function, of abstract type f : %B%^25 -> %B%, |
| + | whose fibre of truth f^(-1)(%1%) is a set of interpretations, each of |
| + | whose elements bears the abstract type of a point in the space %B%^25, |
| + | and whose reading will inform us of our desired set of configurations. |
| + | |
| + | Logical Input File: Q5.Log |
| + | o------------------------------------------------------------o |
| + | | | |
| + | | ((q1_r1 ),(q1_r2 ),(q1_r3 ),(q1_r4 ),(q1_r5 )) | |
| + | | ((q2_r1 ),(q2_r2 ),(q2_r3 ),(q2_r4 ),(q2_r5 )) | |
| + | | ((q3_r1 ),(q3_r2 ),(q3_r3 ),(q3_r4 ),(q3_r5 )) | |
| + | | ((q4_r1 ),(q4_r2 ),(q4_r3 ),(q4_r4 ),(q4_r5 )) | |
| + | | ((q5_r1 ),(q5_r2 ),(q5_r3 ),(q5_r4 ),(q5_r5 )) | |
| + | | | |
| + | | ((q1_r1 ),(q2_r1 ),(q3_r1 ),(q4_r1 ),(q5_r1 )) | |
| + | | ((q1_r2 ),(q2_r2 ),(q3_r2 ),(q4_r2 ),(q5_r2 )) | |
| + | | ((q1_r3 ),(q2_r3 ),(q3_r3 ),(q4_r3 ),(q5_r3 )) | |
| + | | ((q1_r4 ),(q2_r4 ),(q3_r4 ),(q4_r4 ),(q5_r4 )) | |
| + | | ((q1_r5 ),(q2_r5 ),(q3_r5 ),(q4_r5 ),(q5_r5 )) | |
| + | | | |
| + | | (( | |
| + | | | |
| + | | (q1_r1 q2_r2 )(q1_r1 q3_r3 )(q1_r1 q4_r4 )(q1_r1 q5_r5 ) | |
| + | | (q2_r2 q3_r3 )(q2_r2 q4_r4 )(q2_r2 q5_r5 ) | |
| + | | (q3_r3 q4_r4 )(q3_r3 q5_r5 ) | |
| + | | (q4_r4 q5_r5 ) | |
| + | | | |
| + | | (q1_r2 q2_r3 )(q1_r2 q3_r4 )(q1_r2 q4_r5 ) | |
| + | | (q2_r3 q3_r4 )(q2_r3 q4_r5 ) | |
| + | | (q3_r4 q4_r5 ) | |
| + | | | |
| + | | (q1_r3 q2_r4 )(q1_r3 q3_r5 ) | |
| + | | (q2_r4 q3_r5 ) | |
| + | | | |
| + | | (q1_r4 q2_r5 ) | |
| + | | | |
| + | | (q2_r1 q3_r2 )(q2_r1 q4_r3 )(q2_r1 q5_r4 ) | |
| + | | (q3_r2 q4_r3 )(q3_r2 q5_r4 ) | |
| + | | (q4_r3 q5_r4 ) | |
| + | | | |
| + | | (q3_r1 q4_r2 )(q3_r1 q5_r3 ) | |
| + | | (q4_r2 q5_r3 ) | |
| + | | | |
| + | | (q4_r1 q5_r2 ) | |
| + | | | |
| + | | (q1_r5 q2_r4 )(q1_r5 q3_r3 )(q1_r5 q4_r2 )(q1_r5 q5_r1 ) | |
| + | | (q2_r4 q3_r3 )(q2_r4 q4_r2 )(q2_r4 q5_r1 ) | |
| + | | (q3_r3 q4_r2 )(q3_r3 q5_r1 ) | |
| + | | (q4_r2 q5_r1 ) | |
| + | | | |
| + | | (q2_r5 q3_r4 )(q2_r5 q4_r3 )(q2_r5 q5_r2 ) | |
| + | | (q3_r4 q4_r3 )(q3_r4 q5_r2 ) | |
| + | | (q4_r3 q5_r2 ) | |
| + | | | |
| + | | (q3_r5 q4_r4 )(q3_r5 q5_r3 ) | |
| + | | (q4_r4 q5_r3 ) | |
| + | | | |
| + | | (q4_r5 q5_r4 ) | |
| + | | | |
| + | | (q1_r4 q2_r3 )(q1_r4 q3_r2 )(q1_r4 q4_r1 ) | |
| + | | (q2_r3 q3_r2 )(q2_r3 q4_r1 ) | |
| + | | (q3_r2 q4_r1 ) | |
| + | | | |
| + | | (q1_r3 q2_r2 )(q1_r3 q3_r1 ) | |
| + | | (q2_r2 q3_r1 ) | |
| + | | | |
| + | | (q1_r2 q2_r1 ) | |
| + | | | |
| + | | )) | |
| + | | | |
| + | o------------------------------------------------------------o |
| + | |
| + | The vanguard of this logical regiment consists of two |
| + | stock'a'block platoons, the pattern of whose features |
| + | is the usual sort of array for conveying permutations. |
| + | Between the stations of their respective offices they |
| + | serve to warrant that all of the interpretations that |
| + | are left standing on the field of valor at the end of |
| + | the day will be ones that tell of permutations 5 by 5. |
| + | The rest of the ruck and the runt of the mill in this |
| + | regimental logos are there to cover the diagonal bias |
| + | against attacking queens that is our protocol to suit. |
| + | |
| + | And here is the issue of the day: |
| + | |
| + | Sense Output: Q5.Sen |
| + | o-------------------o |
| + | | q1_r1 | |
| + | | q2_r3 | |
| + | | q3_r5 | |
| + | | q4_r2 | |
| + | | q5_r4 | <1> |
| + | | q2_r4 | |
| + | | q3_r2 | |
| + | | q4_r5 | |
| + | | q5_r3 | <2> |
| + | | q1_r2 | |
| + | | q2_r4 | |
| + | | q3_r1 | |
| + | | q4_r3 | |
| + | | q5_r5 | <3> |
| + | | q2_r5 | |
| + | | q3_r3 | |
| + | | q4_r1 | |
| + | | q5_r4 | <4> |
| + | | q1_r3 | |
| + | | q2_r1 | |
| + | | q3_r4 | |
| + | | q4_r2 | |
| + | | q5_r5 | <5> |
| + | | q2_r5 | |
| + | | q3_r2 | |
| + | | q4_r4 | |
| + | | q5_r1 | <6> |
| + | | q1_r4 | |
| + | | q2_r1 | |
| + | | q3_r3 | |
| + | | q4_r5 | |
| + | | q5_r2 | <7> |
| + | | q2_r2 | |
| + | | q3_r5 | |
| + | | q4_r3 | |
| + | | q5_r1 | <8> |
| + | | q1_r5 | |
| + | | q2_r2 | |
| + | | q3_r4 | |
| + | | q4_r1 | |
| + | | q5_r3 | <9> |
| + | | q2_r3 | |
| + | | q3_r1 | |
| + | | q4_r4 | |
| + | | q5_r2 | <A> |
| + | o-------------------o |
| + | |
| + | The number at least checks with all of the best authorities, |
| + | so I can breathe a sigh of relief on that account, at least. |
| + | I am sure that there just has to be a more clever way to do |
| + | this, that is to say, within the bounds of ZOT reason alone, |
| + | but the above is the best that I could figure out with the |
| + | time that I had at the time. |
| + | |
| + | References: [BaC, 166], [VaH, 122], [Wir, 143]. |
| + | |
| + | [BaC] Ball, W.W. Rouse, & Coxeter, H.S.M., |
| + | 'Mathematical Recreations and Essays', |
| + | 13th ed., Dover, New York, NY, 1987. |
| + | |
| + | [VaH] Van Hentenryck, Pascal, |
| + | 'Constraint Satisfaction in Logic Programming, |
| + | MIT Press, Cambridge, MA, 1989. |
| + | |
| + | [Wir] Wirth, Niklaus, |
| + | 'Algorithms + Data Structures = Programs', |
| + | Prentice-Hall, Englewood Cliffs, NJ, 1976. |
| + | |
| + | http://mathworld.wolfram.com/QueensProblem.html |
| + | http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000170 |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | I turn now to another golden oldie of a constraint satisfaction problem |
| + | that I would like to give here a slightly new spin, but not so much for |
| + | the sake of these trifling novelties as from a sense of old time's ache |
| + | and a duty to -- well, what's the opposite of novelty? |
| + | |
| + | Phobic Apollo |
| + | |
| + | | Suppose Peter, Paul, and Jane are musicians. One of them plays |
| + | | saxophone, another plays guitar, and the third plays drums. As |
| + | | it happens, one of them is afraid of things associated with the |
| + | | number 13, another of them is afraid of cats, and the third is |
| + | | afraid of heights. You also know that Peter and the guitarist |
| + | | skydive, that Paul and the saxophone player enjoy cats, and |
| + | | that the drummer lives in apartment 13 on the 13th floor. |
| + | | |
| + | | Soon we will want to use these facts to reason |
| + | | about whether or not certain identity relations |
| + | | hold or are excluded. Assume X(Peter, Guitarist) |
| + | | means "the person who is Peter is not the person who |
| + | | plays the guitar". In this notation, the facts become: |
| + | | |
| + | | 1. X(Peter, Guitarist) |
| + | | 2. X(Peter, Fears Heights) |
| + | | 3. X(Guitarist, Fears Heights) |
| + | | 4. X(Paul, Fears Cats) |
| + | | 5. X(Paul, Saxophonist) |
| + | | 6. X(Saxophonist, Fears Cats) |
| + | | 7. X(Drummer, Fears 13) |
| + | | 8. X(Drummer, Fears Heights) |
| + | | |
| + | | Exercise attributed to Kenneth D. Forbus, pages 449-450 in: |
| + | | Patrick Henry Winston, 'Artificial Intelligence', 2nd ed., |
| + | | Addison-Wesley, Reading, MA, 1984. |
| + | |
| + | Here is one way to represent these facts in the form of a ZOT |
| + | and use it as a logical program to draw a succinct conclusion: |
| + | |
| + | Logical Input File: ConSat.Log |
| + | o-----------------------------------------------------------------------o |
| + | | | |
| + | | (( pete_plays_guitar ),( pete_plays_sax ),( pete_plays_drums )) | |
| + | | (( paul_plays_guitar ),( paul_plays_sax ),( paul_plays_drums )) | |
| + | | (( jane_plays_guitar ),( jane_plays_sax ),( jane_plays_drums )) | |
| + | | | |
| + | | (( pete_plays_guitar ),( paul_plays_guitar ),( jane_plays_guitar )) | |
| + | | (( pete_plays_sax ),( paul_plays_sax ),( jane_plays_sax )) | |
| + | | (( pete_plays_drums ),( paul_plays_drums ),( jane_plays_drums )) | |
| + | | | |
| + | | (( pete_fears_13 ),( pete_fears_cats ),( pete_fears_height )) | |
| + | | (( paul_fears_13 ),( paul_fears_cats ),( paul_fears_height )) | |
| + | | (( jane_fears_13 ),( jane_fears_cats ),( jane_fears_height )) | |
| + | | | |
| + | | (( pete_fears_13 ),( paul_fears_13 ),( jane_fears_13 )) | |
| + | | (( pete_fears_cats ),( paul_fears_cats ),( jane_fears_cats )) | |
| + | | (( pete_fears_height ),( paul_fears_height ),( jane_fears_height )) | |
| + | | | |
| + | | (( | |
| + | | | |
| + | | ( pete_plays_guitar ) | |
| + | | ( pete_fears_height ) | |
| + | | | |
| + | | ( pete_plays_guitar pete_fears_height ) | |
| + | | ( paul_plays_guitar paul_fears_height ) | |
| + | | ( jane_plays_guitar jane_fears_height ) | |
| + | | | |
| + | | ( paul_fears_cats ) | |
| + | | ( paul_plays_sax ) | |
| + | | | |
| + | | ( pete_plays_sax pete_fears_cats ) | |
| + | | ( paul_plays_sax paul_fears_cats ) | |
| + | | ( jane_plays_sax jane_fears_cats ) | |
| + | | | |
| + | | ( pete_plays_drums pete_fears_13 ) | |
| + | | ( paul_plays_drums paul_fears_13 ) | |
| + | | ( jane_plays_drums jane_fears_13 ) | |
| + | | | |
| + | | ( pete_plays_drums pete_fears_height ) | |
| + | | ( paul_plays_drums paul_fears_height ) | |
| + | | ( jane_plays_drums jane_fears_height ) | |
| + | | | |
| + | | )) | |
| + | | | |
| + | o-----------------------------------------------------------------------o |
| + | |
| + | Sense Outline: ConSat.Sen |
| + | o-----------------------------o |
| + | | pete_plays_drums | |
| + | | paul_plays_guitar | |
| + | | jane_plays_sax | |
| + | | pete_fears_cats | |
| + | | paul_fears_13 | |
| + | | jane_fears_height | |
| + | o-----------------------------o |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Phobic Apollo (cont.) |
| + | |
| + | It might be instructive to review various aspects |
| + | of how the Theme One Study function actually went |
| + | about arriving at its answer to that last problem. |
| + | Just to prove that my program and I really did do |
| + | our homework on that Phobic Apollo ConSat problem, |
| + | and didn't just provoke some Oracle or other data |
| + | base server to give it away, here is the middling |
| + | output of the Model function as run on ConSat.Log: |
| + | |
| + | Model Outline: ConSat.Mod |
| + | o-------------------------------------------------o |
| + | | pete_plays_guitar - | |
| + | | (pete_plays_guitar ) | |
| + | | pete_plays_sax | |
| + | | pete_plays_drums - | |
| + | | (pete_plays_drums ) | |
| + | | paul_plays_sax - | |
| + | | (paul_plays_sax ) | |
| + | | jane_plays_sax - | |
| + | | (jane_plays_sax ) | |
| + | | paul_plays_guitar | |
| + | | paul_plays_drums - | |
| + | | (paul_plays_drums ) | |
| + | | jane_plays_guitar - | |
| + | | (jane_plays_guitar ) | |
| + | | jane_plays_drums | |
| + | | pete_fears_13 | |
| + | | pete_fears_cats - | |
| + | | (pete_fears_cats ) | |
| + | | pete_fears_height - | |
| + | | (pete_fears_height ) | |
| + | | paul_fears_13 - | |
| + | | (paul_fears_13 ) | |
| + | | jane_fears_13 - | |
| + | | (jane_fears_13 ) | |
| + | | paul_fears_cats - | |
| + | | (paul_fears_cats ) | |
| + | | paul_fears_height - | |
| + | | (paul_fears_height ) - | |
| + | | (pete_fears_13 ) | |
| + | | pete_fears_cats - | |
| + | | (pete_fears_cats ) | |
| + | | pete_fears_height - | |
| + | | (pete_fears_height ) - | |
| + | | (jane_plays_drums ) - | |
| + | | (paul_plays_guitar ) | |
| + | | paul_plays_drums | |
| + | | jane_plays_drums - | |
| + | | (jane_plays_drums ) | |
| + | | jane_plays_guitar | |
| + | | pete_fears_13 | |
| + | | pete_fears_cats - | |
| + | | (pete_fears_cats ) | |
| + | | pete_fears_height - | |
| + | | (pete_fears_height ) | |
| + | | paul_fears_13 - | |
| + | | (paul_fears_13 ) | |
| + | | jane_fears_13 - | |
| + | | (jane_fears_13 ) | |
| + | | paul_fears_cats - | |
| + | | (paul_fears_cats ) | |
| + | | paul_fears_height - | |
| + | | (paul_fears_height ) - | |
| + | | (pete_fears_13 ) | |
| + | | pete_fears_cats - | |
| + | | (pete_fears_cats ) | |
| + | | pete_fears_height - | |
| + | | (pete_fears_height ) - | |
| + | | (jane_plays_guitar ) - | |
| + | | (paul_plays_drums ) - | |
| + | | (pete_plays_sax ) | |
| + | | pete_plays_drums | |
| + | | paul_plays_drums - | |
| + | | (paul_plays_drums ) | |
| + | | jane_plays_drums - | |
| + | | (jane_plays_drums ) | |
| + | | paul_plays_guitar | |
| + | | paul_plays_sax - | |
| + | | (paul_plays_sax ) | |
| + | | jane_plays_guitar - | |
| + | | (jane_plays_guitar ) | |
| + | | jane_plays_sax | |
| + | | pete_fears_13 - | |
| + | | (pete_fears_13 ) | |
| + | | pete_fears_cats | |
| + | | pete_fears_height - | |
| + | | (pete_fears_height ) | |
| + | | paul_fears_cats - | |
| + | | (paul_fears_cats ) | |
| + | | jane_fears_cats - | |
| + | | (jane_fears_cats ) | |
| + | | paul_fears_13 | |
| + | | paul_fears_height - | |
| + | | (paul_fears_height ) | |
| + | | jane_fears_13 - | |
| + | | (jane_fears_13 ) | |
| + | | jane_fears_height * | |
| + | | (jane_fears_height ) - | |
| + | | (paul_fears_13 ) | |
| + | | paul_fears_height - | |
| + | | (paul_fears_height ) - | |
| + | | (pete_fears_cats ) | |
| + | | pete_fears_height - | |
| + | | (pete_fears_height ) - | |
| + | | (jane_plays_sax ) - | |
| + | | (paul_plays_guitar ) | |
| + | | paul_plays_sax - | |
| + | | (paul_plays_sax ) - | |
| + | | (pete_plays_drums ) - | |
| + | o-------------------------------------------------o |
| + | |
| + | This is just the traverse of the "arboreal boolean expansion" (ABE) tree |
| + | that Model function germinates from the propositional expression that we |
| + | planted in the file Consat.Log, which works to describe the facts of the |
| + | situation in question. Since there are 18 logical feature names in this |
| + | propositional expression, we are literally talking about a function that |
| + | enjoys the abstract type f : %B%^18 -> %B%. If I had wanted to evaluate |
| + | this function by expressly writing out its truth table, then it would've |
| + | required 2^18 = 262144 rows. Now I didn't bother to count, but I'm sure |
| + | that the above output does not have anywhere near that many lines, so it |
| + | must be that my program, and maybe even its author, has done a couple of |
| + | things along the way that are moderately intelligent. At least, we hope. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | AK = Antti Karttunen |
| + | JA = Jon Awbrey |
| + | |
| + | AK: Am I (and other SeqFanaticians) missing something from this thread? |
| + | |
| + | AK: Your previous message on seqfan (headers below) is a bit of the same topic, |
| + | but does it belong to the same thread? Where I could obtain the other |
| + | messages belonging to those two threads? (I'm just now starting to |
| + | study "mathematical logic", and its relations to combinatorics are |
| + | very interesting.) Is this "cactus" language documented anywhere? |
| + | |
| + | here i was just following a courtesy of copying people |
| + | when i reference their works, in this case neil's site: |
| + | |
| + | http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000170 |
| + | |
| + | but then i thought that the seqfantasians might be amused, too. |
| + | |
| + | the bit on higher order propositions, in particular, |
| + | those of type h : (B^2 -> B) -> B, i sent because |
| + | of the significance that 2^2^2^2 = 65536 took on |
| + | for us around that time. & the ho, ho, ho joke. |
| + | |
| + | "zeroth order logic" (zol) is just another name for |
| + | the propositional calculus or the sentential logic |
| + | that comes before "first order logic" (fol), aka |
| + | first intens/tional logic, quantificational logic, |
| + | or predicate calculus, depending on who you talk to. |
| + | |
| + | the line of work that i have been doing derives from |
| + | the ideas of c.s. peirce (1839-1914), who developed |
| + | a couple of systems of "logical graphs", actually, |
| + | two variant interpretations of the same abstract |
| + | structures, called "entitative" and "existential" |
| + | graphs. he organized his system into "alpha", |
| + | "beta", and "gamma" layers, roughly equivalent |
| + | to our propositional, quantificational, and |
| + | modal levels of logic today. |
| + | |
| + | on the more contemporary scene, peirce's entitative interpretation |
| + | of logical graphs was revived and extended by george spencer brown |
| + | in his book 'laws of form', while the existential interpretation |
| + | has flourished in the development of "conceptual graphs" by |
| + | john f sowa and a community of growing multitudes. |
| + | |
| + | a passel of links: |
| + | |
| + | http://members.door.net/arisbe/ |
| + | http://www.enolagaia.com/GSB.html |
| + | http://www.cs.uah.edu/~delugach/CG/ |
| + | http://www.jfsowa.com/ |
| + | http://www.jfsowa.com/cg/ |
| + | http://www.jfsowa.com/peirce/ms514w.htm |
| + | http://users.bestweb.net/~sowa/ |
| + | http://users.bestweb.net/~sowa/peirce/ms514.htm |
| + | |
| + | i have mostly focused on "alpha" (prop calc or zol) -- |
| + | though the "func conception of quant logic" thread was |
| + | a beginning try at saying how the same line of thought |
| + | might be extended to 1st, 2nd, & higher order logics -- |
| + | and i devised a particular graph & string syntax that |
| + | is based on a species of cacti, officially described as |
| + | the "reflective extension of logical graphs" (ref log), |
| + | but more lately just referred to as "cactus language". |
| + | |
| + | it turns out that one can do many interesting things |
| + | with prop calc if one has an efficient enough syntax |
| + | and a powerful enough interpreter for it, even using |
| + | it as a very minimal sort of declarative programming |
| + | language, hence, the current thread was directed to |
| + | applying "zeroth order theories" (zot's) as brands |
| + | of "zeroth order programs" (zop's) to a set of old |
| + | constraint satisfaction and knowledge rep examples. |
| + | |
| + | more recent expositions of the cactus language have been directed |
| + | toward what some people call "ontology engineering" -- it sounds |
| + | so much cooler than "taxonomy" -- and so these are found in the |
| + | ieee standard upper ontology working group discussion archives. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Let's now pause and reflect on the mix of abstract and concrete material |
| + | that we have cobbled together in spectacle of this "World Of Zero" (WOZ), |
| + | since I believe that we may have seen enough, if we look at it right, to |
| + | illustrate a few of the more salient phenomena that would normally begin |
| + | to weigh in as a major force only on a much larger scale. Now, it's not |
| + | exactly like this impoverished sample, all by itself, could determine us |
| + | to draw just the right generalizations, or force us to see the shape and |
| + | flow of its immanent law -- it is much too sparse a scattering of points |
| + | to tease out the lines of its up and coming generations quite so clearly -- |
| + | but it can be seen to exemplify many of the more significant themes that |
| + | we know evolve in more substantial environments, that is, On Beyond Zero, |
| + | since we have already seen them, "tho' obscur'd", in these higher realms. |
| + | |
| + | One the the themes that I want to to keep an eye on as this discussion |
| + | develops is the subject that might be called "computation as semiosis". |
| + | |
| + | In this light, any calculus worth its salt must be capable of helping |
| + | us do two things, calculation, of course, but also analysis. This is |
| + | probably one of the reasons why the ordinary sort of differential and |
| + | integral calculus over quantitative domains is frequently referred to |
| + | as "real analysis", or even just "analysis". It seems quite clear to |
| + | me that any adequate logical calculus, in many ways expected to serve |
| + | as a qualitative analogue of analytic geometry in the way that it can |
| + | be used to describe configurations in logically circumscribed domains, |
| + | ought to qualify in both dimensions, namely, analysis and computation. |
| + | |
| + | With all of these various features of the situation in mind, then, we come |
| + | to the point of viewing analysis and computation as just so many different |
| + | kinds of "sign transformations in respect of pragmata" (STIROP's). Taking |
| + | this insight to heart, let us next work to assemble a comprehension of our |
| + | concrete examples, set in the medium of the abstract calculi that allow us |
| + | to express their qualitative patterns, that may hope to be an increment or |
| + | two less inchoate than we have seen so far, and that may even permit us to |
| + | catch the action of these fading fleeting sign transformations on the wing. |
| + | |
| + | Here is how I picture our latest round of examples |
| + | as filling out the framework of this investigation: |
| + | |
| + | o-----------------------------o-----------------------------o |
| + | | Objective Framework | Interpretive Framework | |
| + | o-----------------------------o-----------------------------o |
| + | | | |
| + | | s_1 = Logue(o) | | |
| + | | / | | |
| + | | / | | |
| + | | @ | | |
| + | | · \ | | |
| + | | · \ | | |
| + | | · i_1 = Model(o) v | |
| + | | · s_2 = Model(o) | | |
| + | | · / | | |
| + | | · / | | |
| + | | Object = o · · · · · · @ | | |
| + | | · \ | | |
| + | | · \ | | |
| + | | · i_2 = Tenor(o) v | |
| + | | · s_3 = Tenor(o) | | |
| + | | · / | | |
| + | | · / | | |
| + | | @ | | |
| + | | \ | | |
| + | | \ | | |
| + | | i_3 = Sense(o) v | |
| + | | | |
| + | o-----------------------------------------------------------o |
| + | Figure. Computation As Semiotic Transformation |
| + | |
| + | The Figure shows three distinct sign triples of the form <o, s, i>, where |
| + | o = ostensible objective = the observed, indicated, or intended situation. |
| + | |
| + | | A. <o, Logue(o), Model(o)> |
| + | | |
| + | | B. <o, Model(o), Tenor(o)> |
| + | | |
| + | | C. <o, Tenor(o), Sense(o)> |
| + | |
| + | Let us bring these several signs together in one place, |
| + | to compare and contrast their common and their diverse |
| + | characters, and to think about why we make such a fuss |
| + | about passing from one to the other in the first place. |
| + | |
| + | 1. Logue(o) = Consat.Log |
| + | o-----------------------------------------------------------------------o |
| + | | | |
| + | | (( pete_plays_guitar ),( pete_plays_sax ),( pete_plays_drums )) | |
| + | | (( paul_plays_guitar ),( paul_plays_sax ),( paul_plays_drums )) | |
| + | | (( jane_plays_guitar ),( jane_plays_sax ),( jane_plays_drums )) | |
| + | | | |
| + | | (( pete_plays_guitar ),( paul_plays_guitar ),( jane_plays_guitar )) | |
| + | | (( pete_plays_sax ),( paul_plays_sax ),( jane_plays_sax )) | |
| + | | (( pete_plays_drums ),( paul_plays_drums ),( jane_plays_drums )) | |
| + | | | |
| + | | (( pete_fears_13 ),( pete_fears_cats ),( pete_fears_height )) | |
| + | | (( paul_fears_13 ),( paul_fears_cats ),( paul_fears_height )) | |
| + | | (( jane_fears_13 ),( jane_fears_cats ),( jane_fears_height )) | |
| + | | | |
| + | | (( pete_fears_13 ),( paul_fears_13 ),( jane_fears_13 )) | |
| + | | (( pete_fears_cats ),( paul_fears_cats ),( jane_fears_cats )) | |
| + | | (( pete_fears_height ),( paul_fears_height ),( jane_fears_height )) | |
| + | | | |
| + | | (( | |
| + | | | |
| + | | ( pete_plays_guitar ) | |
| + | | ( pete_fears_height ) | |
| + | | | |
| + | | ( pete_plays_guitar pete_fears_height ) | |
| + | | ( paul_plays_guitar paul_fears_height ) | |
| + | | ( jane_plays_guitar jane_fears_height ) | |
| + | | | |
| + | | ( paul_fears_cats ) | |
| + | | ( paul_plays_sax ) | |
| + | | | |
| + | | ( pete_plays_sax pete_fears_cats ) | |
| + | | ( paul_plays_sax paul_fears_cats ) | |
| + | | ( jane_plays_sax jane_fears_cats ) | |
| + | | | |
| + | | ( pete_plays_drums pete_fears_13 ) | |
| + | | ( paul_plays_drums paul_fears_13 ) | |
| + | | ( jane_plays_drums jane_fears_13 ) | |
| + | | | |
| + | | ( pete_plays_drums pete_fears_height ) | |
| + | | ( paul_plays_drums paul_fears_height ) | |
| + | | ( jane_plays_drums jane_fears_height ) | |
| + | | | |
| + | | )) | |
| + | | | |
| + | o-----------------------------------------------------------------------o |
| + | |
| + | 2. Model(o) = Consat.Mod ><> http://suo.ieee.org/ontology/msg03718.html |
| + | |
| + | 3. Tenor(o) = Consat.Ten (Just The Gist Of It) |
| + | o-------------------------------------------------o |
| + | | (pete_plays_guitar ) | <01> - |
| + | | (pete_plays_sax ) | <02> - |
| + | | pete_plays_drums | <03> + |
| + | | (paul_plays_drums ) | <04> - |
| + | | (jane_plays_drums ) | <05> - |
| + | | paul_plays_guitar | <06> + |
| + | | (paul_plays_sax ) | <07> - |
| + | | (jane_plays_guitar ) | <08> - |
| + | | jane_plays_sax | <09> + |
| + | | (pete_fears_13 ) | <10> - |
| + | | pete_fears_cats | <11> + |
| + | | (pete_fears_height ) | <12> - |
| + | | (paul_fears_cats ) | <13> - |
| + | | (jane_fears_cats ) | <14> - |
| + | | paul_fears_13 | <15> + |
| + | | (paul_fears_height ) | <16> - |
| + | | (jane_fears_13 ) | <17> - |
| + | | jane_fears_height * | <18> + |
| + | o-------------------------------------------------o |
| + | |
| + | 4. Sense(o) = Consat.Sen |
| + | o-------------------------------------------------o |
| + | | pete_plays_drums | <03> |
| + | | paul_plays_guitar | <06> |
| + | | jane_plays_sax | <09> |
| + | | pete_fears_cats | <11> |
| + | | paul_fears_13 | <15> |
| + | | jane_fears_height | <18> |
| + | o-------------------------------------------------o |
| + | |
| + | As one proceeds through the subsessions of the Theme One Study session, |
| + | the computation transforms its larger "signs", in this case text files, |
| + | from one to the next, in the sequence: Logue, Model, Tenor, and Sense. |
| + | |
| + | Let us see if we can pin down, on sign-theoretic grounds, |
| + | why this very sort of exercise is so routinely necessary. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | We were in the middle of pursuing several questions about |
| + | sign relational transformations in general, in particular, |
| + | the following Example of a sign transformation that arose |
| + | in the process of setting up and solving a classical sort |
| + | of constraint satisfaction problem. |
| + | |
| + | o-----------------------------o-----------------------------o |
| + | | Objective Framework | Interpretive Framework | |
| + | o-----------------------------o-----------------------------o |
| + | | | |
| + | | s_1 = Logue(o) | | |
| + | | / | | |
| + | | / | | |
| + | | @ | | |
| + | | · \ | | |
| + | | · \ | | |
| + | | · i_1 = Model(o) v | |
| + | | · s_2 = Model(o) | | |
| + | | · / | | |
| + | | · / | | |
| + | | Object = o · · · · · · @ | | |
| + | | · \ | | |
| + | | · \ | | |
| + | | · i_2 = Tenor(o) v | |
| + | | · s_3 = Tenor(o) | | |
| + | | · / | | |
| + | | · / | | |
| + | | @ | | |
| + | | \ | | |
| + | | \ | | |
| + | | i_3 = Sense(o) v | |
| + | | | |
| + | o-----------------------------------------------------------o |
| + | Figure. Computation As Semiotic Transformation |
| + | |
| + | 1. Logue(o) = Consat.Log |
| + | o-----------------------------------------------------------------------o |
| + | | | |
| + | | (( pete_plays_guitar ),( pete_plays_sax ),( pete_plays_drums )) | |
| + | | (( paul_plays_guitar ),( paul_plays_sax ),( paul_plays_drums )) | |
| + | | (( jane_plays_guitar ),( jane_plays_sax ),( jane_plays_drums )) | |
| + | | | |
| + | | (( pete_plays_guitar ),( paul_plays_guitar ),( jane_plays_guitar )) | |
| + | | (( pete_plays_sax ),( paul_plays_sax ),( jane_plays_sax )) | |
| + | | (( pete_plays_drums ),( paul_plays_drums ),( jane_plays_drums )) | |
| + | | | |
| + | | (( pete_fears_13 ),( pete_fears_cats ),( pete_fears_height )) | |
| + | | (( paul_fears_13 ),( paul_fears_cats ),( paul_fears_height )) | |
| + | | (( jane_fears_13 ),( jane_fears_cats ),( jane_fears_height )) | |
| + | | | |
| + | | (( pete_fears_13 ),( paul_fears_13 ),( jane_fears_13 )) | |
| + | | (( pete_fears_cats ),( paul_fears_cats ),( jane_fears_cats )) | |
| + | | (( pete_fears_height ),( paul_fears_height ),( jane_fears_height )) | |
| + | | | |
| + | | (( | |
| + | | | |
| + | | ( pete_plays_guitar ) | |
| + | | ( pete_fears_height ) | |
| + | | | |
| + | | ( pete_plays_guitar pete_fears_height ) | |
| + | | ( paul_plays_guitar paul_fears_height ) | |
| + | | ( jane_plays_guitar jane_fears_height ) | |
| + | | | |
| + | | ( paul_fears_cats ) | |
| + | | ( paul_plays_sax ) | |
| + | | | |
| + | | ( pete_plays_sax pete_fears_cats ) | |
| + | | ( paul_plays_sax paul_fears_cats ) | |
| + | | ( jane_plays_sax jane_fears_cats ) | |
| + | | | |
| + | | ( pete_plays_drums pete_fears_13 ) | |
| + | | ( paul_plays_drums paul_fears_13 ) | |
| + | | ( jane_plays_drums jane_fears_13 ) | |
| + | | | |
| + | | ( pete_plays_drums pete_fears_height ) | |
| + | | ( paul_plays_drums paul_fears_height ) | |
| + | | ( jane_plays_drums jane_fears_height ) | |
| + | | | |
| + | | )) | |
| + | | | |
| + | o-----------------------------------------------------------------------o |
| + | |
| + | 2. Model(o) = Consat.Mod ><> http://suo.ieee.org/ontology/msg03718.html |
| + | |
| + | 3. Tenor(o) = Consat.Ten (Just The Gist Of It) |
| + | o-------------------------------------------------o |
| + | | (pete_plays_guitar ) | <01> - |
| + | | (pete_plays_sax ) | <02> - |
| + | | pete_plays_drums | <03> + |
| + | | (paul_plays_drums ) | <04> - |
| + | | (jane_plays_drums ) | <05> - |
| + | | paul_plays_guitar | <06> + |
| + | | (paul_plays_sax ) | <07> - |
| + | | (jane_plays_guitar ) | <08> - |
| + | | jane_plays_sax | <09> + |
| + | | (pete_fears_13 ) | <10> - |
| + | | pete_fears_cats | <11> + |
| + | | (pete_fears_height ) | <12> - |
| + | | (paul_fears_cats ) | <13> - |
| + | | (jane_fears_cats ) | <14> - |
| + | | paul_fears_13 | <15> + |
| + | | (paul_fears_height ) | <16> - |
| + | | (jane_fears_13 ) | <17> - |
| + | | jane_fears_height * | <18> + |
| + | o-------------------------------------------------o |
| + | |
| + | 4. Sense(o) = Consat.Sen |
| + | o-------------------------------------------------o |
| + | | pete_plays_drums | <03> |
| + | | paul_plays_guitar | <06> |
| + | | jane_plays_sax | <09> |
| + | | pete_fears_cats | <11> |
| + | | paul_fears_13 | <15> |
| + | | jane_fears_height | <18> |
| + | o-------------------------------------------------o |
| + | |
| + | We can worry later about the proper use of quotation marks |
| + | in discussing such a case, where the file name "Yada.Yak" |
| + | denotes a piece of text that expresses a proposition that |
| + | describes an objective situation or an intentional object, |
| + | but whatever the case it is clear that we are knee & neck |
| + | deep in a sign relational situation of a modest complexity. |
| + | |
| + | I think that the right sort of analogy might help us |
| + | to sort it out, or at least to tell what's important |
| + | from the things that are less so. The paradigm that |
| + | comes to mind for me is the type of context in maths |
| + | where we talk about the "locus" or the "solution set" |
| + | of an equation, and here we think of the equation as |
| + | denoting its solution set or describing a locus, say, |
| + | a point or a curve or a surface or so on up the scale. |
| + | |
| + | In this figure of speech, we might say for instance: |
| + | |
| + | | o is |
| + | | what "x^3 - 3x^2 + 3x - 1 = 0" denotes is |
| + | | what "(x-1)^3 = 0" denotes is |
| + | | what "1" denotes |
| + | | is 1. |
| + | |
| + | Making explicit the assumptive interpretations |
| + | that the context probably enfolds in this case, |
| + | we assume this description of the solution set: |
| + | |
| + | {x in the Reals : x^3 - 3x^2 + 3x -1 = 0} = {1}. |
| + | |
| + | In sign relational terms, we have the 3-tuples: |
| + | |
| + | | <o, "x^3 - 3x^2 + 3x - 1 = 0", "(x-1)^3 = 0"> |
| + | | |
| + | | <o, "(x-1)^3 = 0", "1"> |
| + | | |
| + | | <o, "1", "1"> |
| + | |
| + | As it turns out we discover that the |
| + | object o was really just 1 all along. |
| + | |
| + | But why do we put ourselves through the rigors of these |
| + | transformations at all? If 1 is what we mean, why not |
| + | just say "1" in the first place and be done with it? |
| + | A person who asks a question like that has forgetten |
| + | how we keep getting ourselves into these quandaries, |
| + | and who it is that assigns the problems, for it is |
| + | Nature herself who is the taskmistress here and the |
| + | problems are set in the manner that she determines, |
| + | not in the style to which we would like to become |
| + | accustomed. The best that we can demand of our |
| + | various and sundry calculi is that they afford |
| + | us with the nets and the snares more readily |
| + | to catch the shape of the problematic game |
| + | as it flies up before us on its own wings, |
| + | and only then to tame it to the amenable |
| + | demeanors that we find to our liking. |
| + | |
| + | In sum, the first place is not ours to take. |
| + | We are but poor second players in this game. |
| + | |
| + | That understood, I can now lay out our present Example |
| + | along the lines of this familiar mathematical exercise. |
| + | |
| + | | o is |
| + | | what Consat.Log denotes is |
| + | | what Consat.Mod denotes is |
| + | | what Consat.Ten denotes is |
| + | | what Consat.Sen denotes. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | It will be good to keep this picture before us a while longer: |
| + | |
| + | o-----------------------------o-----------------------------o |
| + | | Objective Framework | Interpretive Framework | |
| + | o-----------------------------o-----------------------------o |
| + | | | |
| + | | s_1 = Logue(o) | | |
| + | | / | | |
| + | | / | | |
| + | | @ | | |
| + | | · \ | | |
| + | | · \ | | |
| + | | · i_1 = Model(o) v | |
| + | | · s_2 = Model(o) | | |
| + | | · / | | |
| + | | · / | | |
| + | | Object = o · · · · · · @ | | |
| + | | · \ | | |
| + | | · \ | | |
| + | | · i_2 = Tenor(o) v | |
| + | | · s_3 = Tenor(o) | | |
| + | | · / | | |
| + | | · / | | |
| + | | @ | | |
| + | | \ | | |
| + | | \ | | |
| + | | i_3 = Sense(o) v | |
| + | | | |
| + | o-----------------------------------------------------------o |
| + | Figure. Computation As Semiotic Transformation |
| + | |
| + | The labels that decorate the syntactic plane and indicate |
| + | the semiotic transitions in the interpretive panel of the |
| + | framework point us to text files whose contents rest here: |
| + | |
| + | http://suo.ieee.org/ontology/msg03722.html |
| + | |
| + | The reason that I am troubling myself -- and no doubt you -- |
| + | with the details of this Example is because it highlights |
| + | a number of the thistles that we will have to grasp if we |
| + | ever want to escape from the traps of YARNBOL and YARWARS |
| + | in which so many of our fairweather fiends are seeking to |
| + | ensnare us, and not just us -- the whole web of the world. |
| + | |
| + | YARNBOL = Yet Another Roman Numeral Based Ontology Language. |
| + | YARWARS = Yet Another Representation Without A Reasoning System. |
| + | |
| + | In order to avoid this, or to reverse the trend once it gets started, |
| + | we just have to remember what a dynamic living process a computation |
| + | really is, precisely because it is meant to serve as an iconic image |
| + | of dynamic, deliberate, purposeful transformations that we are bound |
| + | to go through and to carry out in a hopeful pursuit of the solutions |
| + | to the many real live problems that life and society place before us. |
| + | So I take it rather seriously. |
| + | |
| + | Okay, back to the grindstone. |
| + | |
| + | The question is: "Why are these trips necessary?" |
| + | |
| + | How come we don't just have one proper expression |
| + | for each situation under the sun, or all possible |
| + | suns, I guess, for some, and just use that on any |
| + | appearance, instance, occasion of that situation? |
| + | |
| + | Why is it ever necessary to begin with an obscure description |
| + | of a situation? -- for that is exactly what the propositional |
| + | expression caled "Logue(o)", for Example, the Consat.Log file, |
| + | really is. |
| + | |
| + | Maybe I need to explain that first. |
| + | |
| + | The first three items of syntax -- Logue(o), Model(o), Tenor(o) -- |
| + | are all just so many different propositional expressions that |
| + | denote one and the same logical-valued function p : X -> %B%, |
| + | and one whose abstract image we may well enough describe as |
| + | a boolean function of the abstract type q : %B%^k -> %B%, |
| + | where k happens to be 18 in the present Consat Example. |
| + | |
| + | If we were to write out the truth table for q : %B%^18 -> %B% |
| + | it would take 2^18 = 262144 rows. Using the bold letter #x# |
| + | for a coordinate tuple, writing #x# = <x_1, ..., x_18>, each |
| + | row of the table would have the form <x_1, ..., x_18, q(#x#)>. |
| + | And the function q is such that all rows evalue to %0% save 1. |
| + | |
| + | Each of the four different formats expresses this fact about q |
| + | in its own way. The first three are logically equivalent, and |
| + | the last one is the maximally determinate positive implication |
| + | of what the others all say. |
| + | |
| + | From this point of view, the logical computation that we went through, |
| + | in the sequence Logue, Model, Tenor, Sense, was a process of changing |
| + | from an obscure sign of the objective proposition to a more organized |
| + | arrangement of its satisfying or unsatisfying interpretations, to the |
| + | most succinct possible expression of the same meaning, to an adequate |
| + | positive projection of it that is useful enough in the proper context. |
| + | |
| + | This is the sort of mill -- it's called "computation" -- that we have |
| + | to be able to put our representations through on a recurrent, regular, |
| + | routine basis, that is, if we expect them to have any utility at all. |
| + | And it is only when we have started to do that in genuinely effective |
| + | and efficient ways, that we can even begin to think about facilitating |
| + | any bit of qualitative conceptual analysis through computational means. |
| + | |
| + | And as far as the qualitative side of logical computation |
| + | and conceptual analysis goes, we have barely even started. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | We are contemplating the sequence of initial and normal forms |
| + | for the Consat problem and we have noted the following system |
| + | of logical relations, taking the enchained expressions of the |
| + | objective situation o in a pairwise associated way, of course: |
| + | |
| + | Logue(o) <=> Model(o) <=> Tenor(o) => Sense(o). |
| + | |
| + | The specifics of the propositional expressions are cited here: |
| + | |
| + | http://suo.ieee.org/ontology/msg03722.html |
| + | |
| + | If we continue to pursue the analogy that we made with the form |
| + | of mathematical activity commonly known as "solving equations", |
| + | then there are many salient features of this type of logical |
| + | problem solving endeavor that suddenly leap into the light. |
| + | |
| + | First of all, we notice the importance of "equational reasoning" |
| + | in mathematics, by which I mean, not just the quantitative type |
| + | of equation that forms the matter of the process, but also the |
| + | qualitative type of equation, or the "logical equivalence", |
| + | that connects each expression along the way, right up to |
| + | the penultimate stage, when we are satisfied in a given |
| + | context to take a projective implication of the total |
| + | knowledge of the situation that we have been taking |
| + | some pains to preserve at every intermediate stage |
| + | of the game. |
| + | |
| + | This general pattern or strategy of inference, working its way through |
| + | phases of "equational" or "total information preserving" inference and |
| + | phases of "implicational" or "selective information losing" inference, |
| + | is actually very common throughout mathematics, and I have in mind to |
| + | examine its character in greater detail and in a more general setting. |
| + | |
| + | Just as the barest hint of things to come along these lines, you might |
| + | consider the question of what would constitute the equational analogue |
| + | of modus ponens, in other words the scheme of inference that goes from |
| + | x and x=>y to y. Well the answer is a scheme of inference that passes |
| + | from x and x=>y to x&y, and then being reversible, back again. I will |
| + | explore the rationale and the utility of this gambit in future reports. |
| + | |
| + | One observation that we can make already at this point, |
| + | however, is that these schemes of equational reasoning, |
| + | or reversible inference, remain poorly developed among |
| + | our currently prevailing styles of inference in logic, |
| + | their potentials for applied logical software hardly |
| + | being broached in our presently available systems. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Extra Examples |
| + | |
| + | 1. Propositional logic example. |
| + | Files: Alpha.lex + Prop.log |
| + | Ref: [Cha, 20, Example 2.12] |
| + | |
| + | 2. Chemical synthesis problem. |
| + | Files: Chem.* |
| + | Ref: [Cha, 21, Example 2.13] |
| + | |
| + | 3. N Queens problem. |
| + | Files: Queen*.*, Q8.*, Q5.* |
| + | Refs: [BaC, 166], [VaH, 122], [Wir, 143]. |
| + | Notes: Only the 5 Queens example will run in 640K memory. |
| + | Use the "Queen.lex" file to load the "Q5.eg*" log files. |
| + | |
| + | 4. Five Houses puzzle. |
| + | Files: House.* |
| + | Ref: [VaH, 132]. |
| + | Notes: Will not run in 640K memory. |
| + | |
| + | 5. Graph coloring example. |
| + | Files: Color.* |
| + | Ref: [Wil, 196]. |
| + | |
| + | 6. Examples of Cook's Theorem in computational complexity, |
| + | that propositional satisfiability is NP-complete. |
| + | |
| + | Files: StiltN.* = "Space and Time Limited Turing Machine", |
| + | with N units of space and N units of time. |
| + | StuntN.* = "Space and Time Limited Turing Machine", |
| + | for computing the parity of a bit string, |
| + | with Number of Tape cells of input equal to N. |
| + | Ref: [Wil, 188-201]. |
| + | Notes: Can only run Turing machine example for input of size 2. |
| + | Since the last tape cell is used for an end-of-file marker, |
| + | this amounts to only one significant digit of computation. |
| + | Use the "Stilt3.lex" file to load the "Stunt2.egN" files. |
| + | Their Sense file outputs appear on the "Stunt2.seN" files. |
| + | |
| + | 7. Fabric knowledge base. |
| + | Files: Fabric.*, Fab.* |
| + | Ref: [MaW, 8-16]. |
| + | |
| + | 8. Constraint Satisfaction example. |
| + | Files: Consat1.*, Consat2.* |
| + | Ref: [Win, 449, Exercise 3-9]. |
| + | Notes: Attributed to Kenneth D. Forbus. |
| + | |
| + | References |
| + | |
| + | | Angluin, Dana, |
| + | |"Learning with Hints", in |
| + | |'Proceedings of the 1988 Workshop on Computational Learning Theory', |
| + | | edited by D. Haussler & L. Pitt, Morgan Kaufmann, San Mateo, CA, 1989. |
| + | |
| + | | Ball, W.W. Rouse, & Coxeter, H.S.M., |
| + | |'Mathematical Recreations and Essays', 13th ed., |
| + | | Dover, New York, NY, 1987. |
| + | |
| + | | Chang, Chin-Liang & Lee, Richard Char-Tung, |
| + | |'Symbolic Logic and Mechanical Theorem Proving', |
| + | | Academic Press, New York, NY, 1973. |
| + | |
| + | | Denning, Peter J., Dennis, Jack B., and Qualitz, Joseph E., |
| + | |'Machines, Languages, and Computation', |
| + | | Prentice-Hall, Englewood Cliffs, NJ, 1978. |
| + | |
| + | | Edelman, Gerald M., |
| + | |'Topobiology: An Introduction to Molecular Embryology', |
| + | | Basic Books, New York, NY, 1988. |
| + | |
| + | | Lloyd, J.W., |
| + | |'Foundations of Logic Programming', |
| + | | Springer-Verlag, Berlin, 1984. |
| + | |
| + | | Maier, David & Warren, David S., |
| + | |'Computing with Logic: Logic Programming with Prolog', |
| + | | Benjamin/Cummings, Menlo Park, CA, 1988. |
| + | |
| + | | McClelland, James L. and Rumelhart, David E., |
| + | |'Explorations in Parallel Distributed Processing: |
| + | | A Handbook of Models, Programs, and Exercises', |
| + | | MIT Press, Cambridge, MA, 1988. |
| + | |
| + | | Peirce, Charles Sanders, |
| + | |'Collected Papers of Charles Sanders Peirce', |
| + | | edited by Charles Hartshorne, Paul Weiss, & Arthur W. Burks, |
| + | | Harvard University Press, Cambridge, MA, 1931-1960. |
| + | |
| + | | Peirce, Charles Sanders, |
| + | |'The New Elements of Mathematics', |
| + | | edited by Carolyn Eisele, Mouton, The Hague, 1976. |
| + | |
| + | |'Charles S. Peirce: Selected Writings; Values in a Universe of Chance', |
| + | | edited by Philip P. Wiener, Dover, New York, NY, 1966. |
| + | |
| + | | Spencer Brown, George, |
| + | |'Laws of Form', |
| + | | George Allen & Unwin, London, UK, 1969. |
| + | |
| + | | Van Hentenryck, Pascal, |
| + | |'Constraint Satisfaction in Logic Programming', |
| + | | MIT Press, Cambridge, MA, 1989. |
| + | |
| + | | Wilf, Herbert S., |
| + | |'Algorithms and Complexity', |
| + | | Prentice-Hall, Englewood Cliffs, NJ, 1986. |
| + | |
| + | | Winston, Patrick Henry, |
| + | |'Artificial Intelligence, 2nd ed., |
| + | | Addison-Wesley, Reading, MA, 1984. |
| + | |
| + | | Wirth, Niklaus, |
| + | |'Algorithms + Data Structures = Programs', |
| + | | Prentice-Hall, Englewood Cliffs, NJ, 1976. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Cactus Town Cartoons |
| + | |
| + | 01. http://suo.ieee.org/ontology/msg03567.html |
| + | 02. http://suo.ieee.org/ontology/msg03571.html |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Differential Analytic Turing Automata (DATA) |
| + | |
| + | 01. http://suo.ieee.org/ontology/msg00596.html |
| + | 02. http://suo.ieee.org/ontology/msg00618.html |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Differential Logic |
| + | |
| + | 01. http://suo.ieee.org/ontology/msg04040.html |
| + | 02. http://suo.ieee.org/ontology/msg04041.html |
| + | 03. http://suo.ieee.org/ontology/msg04045.html |
| + | 04. http://suo.ieee.org/ontology/msg04046.html |
| + | 05. http://suo.ieee.org/ontology/msg04047.html |
| + | 06. http://suo.ieee.org/ontology/msg04048.html |
| + | 07. http://suo.ieee.org/ontology/msg04052.html |
| + | 08. http://suo.ieee.org/ontology/msg04054.html |
| + | 09. http://suo.ieee.org/ontology/msg04055.html |
| + | 10. http://suo.ieee.org/ontology/msg04067.html |
| + | 11. http://suo.ieee.org/ontology/msg04068.html |
| + | 12. http://suo.ieee.org/ontology/msg04069.html |
| + | 13. http://suo.ieee.org/ontology/msg04070.html |
| + | 14. http://suo.ieee.org/ontology/msg04072.html |
| + | 15. http://suo.ieee.org/ontology/msg04073.html |
| + | 16. http://suo.ieee.org/ontology/msg04074.html |
| + | 17. http://suo.ieee.org/ontology/msg04077.html |
| + | 18. http://suo.ieee.org/ontology/msg04079.html |
| + | 19. http://suo.ieee.org/ontology/msg04080.html |
| + | 20. |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Extensions Of Logical Graphs |
| + | |
| + | 01. http://www.virtual-earth.de/CG/cg-list/old/msg03351.html |
| + | 02. http://www.virtual-earth.de/CG/cg-list/old/msg03352.html |
| + | 03. http://www.virtual-earth.de/CG/cg-list/old/msg03353.html |
| + | 04. http://www.virtual-earth.de/CG/cg-list/old/msg03354.html |
| + | 05. http://www.virtual-earth.de/CG/cg-list/old/msg03376.html |
| + | 06. http://www.virtual-earth.de/CG/cg-list/old/msg03379.html |
| + | 07. http://www.virtual-earth.de/CG/cg-list/old/msg03381.html |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Functional Conception Of Quantificational Logic |
| + | |
| + | 01. http://suo.ieee.org/ontology/msg03562.html |
| + | 02. http://suo.ieee.org/ontology/msg03563.html |
| + | 03. http://suo.ieee.org/ontology/msg03577.html |
| + | 04. http://suo.ieee.org/ontology/msg03578.html |
| + | 05. http://suo.ieee.org/ontology/msg03579.html |
| + | 06. http://suo.ieee.org/ontology/msg03580.html |
| + | 07. http://suo.ieee.org/ontology/msg03581.html |
| + | 08. http://suo.ieee.org/ontology/msg03582.html |
| + | 09. http://suo.ieee.org/ontology/msg03583.html |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Propositional Equation Reasoning Systems (PERS) |
| + | |
| + | 01. http://suo.ieee.org/email/msg04187.html |
| + | 02. http://suo.ieee.org/email/msg04305.html |
| + | 03. http://suo.ieee.org/email/msg04413.html |
| + | 04. http://suo.ieee.org/email/msg04419.html |
| + | 05. http://suo.ieee.org/email/msg04422.html |
| + | 06. http://suo.ieee.org/email/msg04423.html |
| + | 07. http://suo.ieee.org/email/msg04432.html |
| + | 08. http://suo.ieee.org/email/msg04454.html |
| + | 09. http://suo.ieee.org/email/msg04455.html |
| + | 10. http://suo.ieee.org/email/msg04476.html |
| + | 11. http://suo.ieee.org/email/msg04510.html |
| + | 12. http://suo.ieee.org/email/msg04517.html |
| + | 13. http://suo.ieee.org/email/msg04525.html |
| + | 14. http://suo.ieee.org/email/msg04533.html |
| + | 15. http://suo.ieee.org/email/msg04536.html |
| + | 16. http://suo.ieee.org/email/msg04542.html |
| + | 17. http://suo.ieee.org/email/msg04546.html |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Reflective Extension Of Logical Graphs (RefLog) |
| + | |
| + | 01. http://suo.ieee.org/email/msg05694.html |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Sequential Interactions Generating Hypotheses |
| + | |
| + | 01. http://suo.ieee.org/email/msg02607.html |
| + | 02. http://suo.ieee.org/email/msg02608.html |
| + | 03. http://suo.ieee.org/email/msg03183.html |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Sowa's Top Level Categories |
| + | |
| + | 01. http://suo.ieee.org/email/msg01949.html |
| + | 02. http://suo.ieee.org/email/msg01956.html |
| + | 03. http://suo.ieee.org/email/msg01966.html |
| + | |
| + | 04. http://suo.ieee.org/ontology/msg00048.html |
| + | 05. http://suo.ieee.org/ontology/msg00051.html |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Zeroth Order Logic (ZOL) |
| + | |
| + | 01. http://suo.ieee.org/email/msg01246.html |
| + | 02. http://suo.ieee.org/email/msg01406.html |
| + | 03. http://suo.ieee.org/email/msg01546.html |
| + | 04. http://suo.ieee.org/email/msg01561.html |
| + | 05. http://suo.ieee.org/email/msg01670.html |
| + | 06. http://suo.ieee.org/email/msg01739.html |
| + | 07. http://suo.ieee.org/email/msg01966.html |
| + | 08. http://suo.ieee.org/email/msg01985.html |
| + | 09. http://suo.ieee.org/email/msg01988.html |
| + | |
| + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| + | |
| + | Zeroth Order Theories (ZOT's) |
| + | |
| + | 01. http://suo.ieee.org/ontology/msg03680.html |
| + | 02. http://suo.ieee.org/ontology/msg03681.html |
| + | 03. http://suo.ieee.org/ontology/msg03682.html |
| + | 04. http://suo.ieee.org/ontology/msg03683.html |
| + | 05. http://suo.ieee.org/ontology/msg03685.html |
| + | 06. http://suo.ieee.org/ontology/msg03687.html |
| + | 07. http://suo.ieee.org/ontology/msg03689.html |
| + | 08. http://suo.ieee.org/ontology/msg03691.html |
| + | 09. http://suo.ieee.org/ontology/msg03693.html |
| + | 10. http://suo.ieee.org/ontology/msg03694.html |
| + | 11. http://suo.ieee.org/ontology/msg03695.html |
| + | 12. http://suo.ieee.org/ontology/msg03696.html |
| + | 13. http://suo.ieee.org/ontology/msg03700.html |
| + | 14. http://suo.ieee.org/ontology/msg03701.html |
| + | 15. http://suo.ieee.org/ontology/msg03702.html |
| + | 16. http://suo.ieee.org/ontology/msg03703.html |
| + | 17. http://suo.ieee.org/ontology/msg03705.html |
| + | 18. http://suo.ieee.org/ontology/msg03706.html |
| + | 19. http://suo.ieee.org/ontology/msg03707.html |
| + | 20. http://suo.ieee.org/ontology/msg03708.html |
| + | 21. http://suo.ieee.org/ontology/msg03709.html |
| + | 22. http://suo.ieee.org/ontology/msg03711.html |
| + | 23. http://suo.ieee.org/ontology/msg03712.html |
| + | 24. http://suo.ieee.org/ontology/msg03715.html |
| + | 25. http://suo.ieee.org/ontology/msg03716.html |
| + | 26. http://suo.ieee.org/ontology/msg03717.html |
| + | 27. http://suo.ieee.org/ontology/msg03718.html |
| + | 28. http://suo.ieee.org/ontology/msg03720.html |
| + | 29. http://suo.ieee.org/ontology/msg03721.html |
| + | 30. http://suo.ieee.org/ontology/msg03722.html |
| + | 31. http://suo.ieee.org/ontology/msg03723.html |
| + | 32. http://suo.ieee.org/ontology/msg03724.html |
| | | |
| o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| </pre> | | </pre> |