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''In medias res'', as always, we nevertheless need a quantum of formal matter to keep the topical momentum going. A game try at supplying that least bit of motivation may be found in this duo of transformations between the indicated forms of enclosure:
''In medias res'', as always, we nevertheless need a quantum of formal matter to keep the topical momentum going. A game try at supplying that least bit of motivation may be found in this duo of transformations between the indicated forms of enclosure:
<pre>
o-----------------------------------------------------------o
| |
| ( ) ( ) = ( ) |
| |
o-----------------------------------------------------------o
</pre>
<pre>
o-----------------------------------------------------------o
| |
| (( )) = |
| |
o-----------------------------------------------------------o
</pre>
In lieu of better names, and in hope of a better reason to come in good time, we may for the moment refer to these two forms of transformation as ''[[axiom]]s'' or ''initials''.
In lieu of better names, and in hope of a better reason to come in good time, we may for the moment refer to these two forms of transformation as ''[[axiom]]s'' or ''initials''.
For example, consider the axiom drawn in box form below:
For example, consider the axiom drawn in box form below:
<pre>
o-----------o
o-----------o
o-----------o
o-----------o
</pre>
This can be written in linear text as "(( )) = ", or set off in the following way:
This can be written in linear text as "(( )) = ", or set off in the following way:
For example, overlaying the corresponding [[dual graph]]s on the plane-embedded graphs shown above, we get the following composite picture:
For example, overlaying the corresponding [[dual graph]]s on the plane-embedded graphs shown above, we get the following composite picture:
<pre>
o-----------o
o-----------o
@ = @
@ = @
</pre>
Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves almost compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''[[root node]]'' of the corresponding [[dual graph]], indicated in the above Figure by the ''amphora'' or ''at'' sign, "@".
Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves almost compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''[[root node]]'' of the corresponding [[dual graph]], indicated in the above Figure by the ''amphora'' or ''at'' sign, "@".
Extracting the dual graph from its composite matrix, we get this picture:
Extracting the dual graph from its composite matrix, we get this picture:
<pre>
o
o
@ = @
@ = @
</pre>
It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated [[dual graph]]s, that constitute the species of [[rooted tree]]s here to be described.
It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated [[dual graph]]s, that constitute the species of [[rooted tree]]s here to be described.
In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this particular case either "(" or ")", that we happen to encounter in our travels.
In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this particular case either "(" or ")", that we happen to encounter in our travels.
<pre>
o-----------------------------------------------------------o
| |
| o |
| ( | ) |
| o |
| ( | ) |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| (( )) = |
| |
o-----------------------------------------------------------o
</pre>
This ritual is called ''[[tree traversal|traversing]]'' the tree, and the string read off is often called the ''[[traversal string]]'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''[[parsing]]'' the string, and the tree constructed is often called the ''[[parse tree|parse graph]]'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''[[parse string]]'' to mean the string that gets parsed into the associated graph.
This ritual is called ''[[tree traversal|traversing]]'' the tree, and the string read off is often called the ''[[traversal string]]'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''[[parsing]]'' the string, and the tree constructed is often called the ''[[parse tree|parse graph]]'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''[[parse string]]'' to mean the string that gets parsed into the associated graph.
First the planar form:
First the planar form:
<pre>
o-------o o-------o o-------o
o-------o o-------o o-------o
o-------o o-------o o-------o
o-------o o-------o o-------o
</pre>
Next the planar and dual forms superimposed:
Next the planar and dual forms superimposed:
<pre>
o-------o o-------o o-------o
o-------o o-------o o-------o
@ = @
@ = @
</pre>
Finally the dual form by itself:
Finally the dual form by itself:
<pre>
o o o
o o o
@ = @
@ = @
</pre>
We have at this point enough material to begin thinking about the forms of [[analogy]], [[iconicity]], [[metaphor]], [[morphism]], whatever you want to call it, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''[[entitative graph]]s'' and ''[[existential graph]]s''.
We have at this point enough material to begin thinking about the forms of [[analogy]], [[iconicity]], [[metaphor]], [[morphism]], whatever you want to call it, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''[[entitative graph]]s'' and ''[[existential graph]]s''.
At the next level of concretion, a record/node can be represented as follows:
At the next level of concretion, a record/node can be represented as follows:
<pre>
o-----------------------------o
o-----------------------------o
|
|
</pre>
This depicts the circumstance that index<sub>0</sub> is the address of the record in question, which record contains the following data:
This depicts the circumstance that index<sub>0</sub> is the address of the record in question, which record contains the following data:
What makes it possible to represent graph-theoretical structures as data structures in computer memory is the fact that an address is just another datum, and so we can have a circumstance like this:
What makes it possible to represent graph-theoretical structures as data structures in computer memory is the fact that an address is just another datum, and so we can have a circumstance like this:
<pre>
o-----o o-----o
o-----o o-----o
|
|
</pre>
Back at the abstract level, it takes three nodes to represent the three data records, with a root node connected to two other nodes. The ordinary bits of data are then treated as labels on the nodes:
Back at the abstract level, it takes three nodes to represent the three data records, with a root node connected to two other nodes. The ordinary bits of data are then treated as labels on the nodes:
<pre>
o o
o o
@ datum_1 datum_2 ...
@ datum_1 datum_2 ...
</pre>
Notice that, with rooted trees like these, drawing the arrows is optional, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, with 'up' being the same direction as 'away from the root'.
Notice that, with rooted trees like these, drawing the arrows is optional, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, with 'up' being the same direction as 'away from the root'.
This much preparation allows us to present the two most basic axioms of logical graphs, shown in graph and string forms below, along with handy names for referring to the two different directions of applying the axioms.
This much preparation allows us to present the two most basic axioms of logical graphs, shown in graph and string forms below, along with handy names for referring to the two different directions of applying the axioms.
<pre>
o-----------------------------------------------------------o
| |
| o o o |
| \ / | |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| ( ) ( ) = ( ) |
| |
o-----------------------------------------------------------o
| Axiom I_1. Distract <--- | ---> Condense |
o-----------------------------------------------------------o
</pre>
<pre>
o-----------------------------------------------------------o
| |
| o |
| | |
| o |
| | |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| (( )) = |
| |
o-----------------------------------------------------------o
| Axiom I_2. Unfold <--- | ---> Refold |
o-----------------------------------------------------------o
</pre>
===Primary arithmetic as semiotic system===
===Primary arithmetic as semiotic system===
Here are the axioms of the primary arithmetic:
Here are the axioms of the primary arithmetic:
<pre>
o-----------------------------------------------------------o
| |
| o o o |
| \ / | |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| ( ) ( ) = ( ) |
| |
o-----------------------------------------------------------o
| Axiom I_1. Distract <--- | ---> Condense |
o-----------------------------------------------------------o
</pre>
<pre>
o-----------------------------------------------------------o
| |
| o |
| | |
| o |
| | |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| (( )) = |
| |
o-----------------------------------------------------------o
| Axiom I_2. Unfold <--- | ---> Refold |
o-----------------------------------------------------------o
</pre>
Taking '''S''' to be the set of rooted trees and '''S'''<sub>0</sub> to be the set that has the root node and the rooted edge as its only two elements, writing these facts more briefly as '''S''' = {rooted trees} and '''S'''<sub>0</sub> = {O, |}, simple intuition, or a simple inductive proof, will assure us that any rooted tree can be reduced by means of the axioms of the primary arithmetic either to a root node "@" or else to a rooted edge "|".
Taking '''S''' to be the set of rooted trees and '''S'''<sub>0</sub> to be the set that has the root node and the rooted edge as its only two elements, writing these facts more briefly as '''S''' = {rooted trees} and '''S'''<sub>0</sub> = {O, |}, simple intuition, or a simple inductive proof, will assure us that any rooted tree can be reduced by means of the axioms of the primary arithmetic either to a root node "@" or else to a rooted edge "|".
For example, consider the reduction that proceeds as follows:
For example, consider the reduction that proceeds as follows:
<pre>
o-----------------------------------------------------------o
| |
| o o o |
| \| | |
| o o o |
| \|/ |
| @ |
| |
o===========================================================o
| |
| o o |
| | | |
| o o o |
| \|/ |
| @ |
| |
o===========================================================o
| |
| o |
| | |
| o o |
| |/ |
| @ |
| |
o===========================================================o
| |
| o |
| / |
| @ |
| |
o-----------------------------------------------------------o
</pre>
Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first being the [[interpretant]] of its predecessor, ending in a sign that we may regard as the canonical sign for their common object, in the upshot, the result of the computation process. Simple as it is, this exhibits the main features of all computation, specifically, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, in its aim and effect, an action on behalf of clarification.
Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first being the [[interpretant]] of its predecessor, ending in a sign that we may regard as the canonical sign for their common object, in the upshot, the result of the computation process. Simple as it is, this exhibits the main features of all computation, specifically, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, in its aim and effect, an action on behalf of clarification.
It is probably best to illustrate this theme in the setting of a concrete case, which we can do by revisiting the previous example of reductive evaluation:
It is probably best to illustrate this theme in the setting of a concrete case, which we can do by revisiting the previous example of reductive evaluation:
<pre>
o-----------------------------------------------------------o
| |
| o o o |
| \| | |
| o o o |
| \|/ |
| @ |
| |
o===========================================================o
| |
| o o |
| | | |
| o o o |
| \|/ |
| @ |
| |
o===========================================================o
| |
| o |
| | |
| o o |
| |/ |
| @ |
| |
o===========================================================o
| |
| o |
| / |
| @ |
| |
o-----------------------------------------------------------o
</pre>
The observation of several semioses of roughly this shape will most probably lead an observer with any observational facility whatever to notice that it doesn't really matter what sorts of branches happen to sprout from the side of the root aside from the lone edge that also grows there — the end will all be one.
The observation of several semioses of roughly this shape will most probably lead an observer with any observational facility whatever to notice that it doesn't really matter what sorts of branches happen to sprout from the side of the root aside from the lone edge that also grows there — the end will all be one.
Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following:
Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following:
<pre>
o-----------------------------------------------------------o
| |
| o |
| a / |
| @ |
| |
o===========================================================o
| |
| o |
| / |
| @ |
| |
o-----------------------------------------------------------o
</pre>
Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra.
Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra.
Speaking of algebra, and having encountered already one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from the works of [[Charles Sanders Peirce]] and [[George Spencer Brown]], respectively.
Speaking of algebra, and having encountered already one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from the works of [[Charles Sanders Peirce]] and [[George Spencer Brown]], respectively.
<pre>
o-----------------------------------------------------------o
| |
| a o o |
| | | |
| a @ = @ |
| |
o-----------------------------------------------------------o
| |
| a(a) = ( ) |
| |
o-----------------------------------------------------------o
| Axiom J_1. Insert <--- | ---> Delete |
o-----------------------------------------------------------o
</pre>
<pre>
o-----------------------------------------------------------o
| |
| ab ac b c |
| o o o o |
| \ / \ / |
| o o |
| | | |
| @ = a @ |
| |
o-----------------------------------------------------------o
| |
| ((ab)(ac)) = a((b)(c)) |
| |
o-----------------------------------------------------------o
| Axiom J_2. Distribute <--- | ---> Collect |
o-----------------------------------------------------------o
</pre>
The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems. As it happens, the example of an algebraic law that we noticed first, ''a''( ) = ( ), as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.
The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems. As it happens, the example of an algebraic law that we noticed first, ''a''( ) = ( ), as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.
The axioms are just four in number, divided into the ''arithmetic initials'' I<sub>1</sub> and I<sub>2</sub>, and the ''algebraic initials'' J<sub>1</sub> and J<sub>2</sub>.
The axioms are just four in number, divided into the ''arithmetic initials'' I<sub>1</sub> and I<sub>2</sub>, and the ''algebraic initials'' J<sub>1</sub> and J<sub>2</sub>.
<pre>
o-----------------------------------------------------------o
| |
| o o o |
| \ / | |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| ( ) ( ) = ( ) |
| |
o-----------------------------------------------------------o
| Axiom I_1. Distract <--- | ---> Condense |
o-----------------------------------------------------------o
</pre>
<pre>
o-----------------------------------------------------------o
| |
| o |
| | |
| o |
| | |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| (( )) = |
| |
o-----------------------------------------------------------o
| Axiom I_2. Unfold <--- | ---> Refold |
o-----------------------------------------------------------o
</pre>
<pre>
o-----------------------------------------------------------o
| |
| a o o |
| | | |
| a @ = @ |
| |
o-----------------------------------------------------------o
| |
| a(a) = ( ) |
| |
o-----------------------------------------------------------o
| Axiom J_1. Insert <--- | ---> Delete |
o-----------------------------------------------------------o
</pre>
<pre>
o-----------------------------------------------------------o
| |
| ab ac b c |
| o o o o |
| \ / \ / |
| o o |
| | | |
| @ = a @ |
| |
o-----------------------------------------------------------o
| |
| ((ab)(ac)) = a((b)(c)) |
| |
o-----------------------------------------------------------o
| Axiom J_2. Distribute <--- | ---> Collect |
o-----------------------------------------------------------o
</pre>
Here is one way of reading the axioms under the entitative interpretation:
Here is one way of reading the axioms under the entitative interpretation:
The first theorem goes under the names of ''Consequence 1'' (C<sub>1</sub>), the ''double negation theorem'' (DNT), or ''Reflection''.
The first theorem goes under the names of ''Consequence 1'' (C<sub>1</sub>), the ''double negation theorem'' (DNT), or ''Reflection''.
<pre>
o-----------------------------------------------------------o
| C_1. Double Negation Theorem |
o-----------------------------------------------------------o
| |
| a |
| o |
| | |
| o |
| | a |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| ((a)) = a |
| |
o-----------------------------------------------------------o
| Reflect <---- | ----> Reflect |
o-----------------------------------------------------------o
</pre>
The proof that follows is adapted from the one that was given by [[George Spencer Brown]] in his book ''Laws of Form'' (LOF), and credited to two of his students, John Dawes and D.A. Utting.
The proof that follows is adapted from the one that was given by [[George Spencer Brown]] in his book ''Laws of Form'' (LOF), and credited to two of his students, John Dawes and D.A. Utting.
<pre>
o-----------------------------------------------------------o
| C_1. Double Negation Theorem. Proof. |
o-----------------------------------------------------------o
| |
| a o |
| \ |
| \ |
| o |
| \ |
| \ |
| @ |
| |
o=============================< I2. Unfold "(())" >=========o
| |
| a o o |
| \ / |
| \ / |
| o o |
| \ / |
| \ / |
| @ |
| |
o=============================< J1. Insert "(a)" >==========o
| |
| a o |
| / |
| / |
| a o a o o |
| \ \ / |
| \ \ / |
| o o |
| \ / |
| \ / |
| @ |
| |
o=============================< J2. Distribute "((a))" >====o
| |
| a o a o |
| \ \ |
| \ \ |
| o o a o |
| \ \ / |
| \ \ / |
| a o o |
| \ / |
| \ / |
| o |
| / |
| / |
| @ |
| |
o=============================< J1. Delete "(a)" >==========o
| |
| a o |
| \ |
| \ |
| o o |
| \ \ |
| \ \ |
| a o o |
| \ / |
| \ / |
| o |
| / |
| / |
| @ |
| |
o=============================< J1. Insert "a" >============o
| |
| a o |
| \ |
| \ |
| o o a |
| \ \ |
| \ \ |
| a o o a |
| \ / |
| \ / |
| o |
| / |
| / |
| @ |
| |
o=============================< J2. Collect "a" >===========o
| |
| a o |
| \ |
| \ |
| o o a |
| \ \ |
| \ \ |
| o o |
| \ / |
| \ / |
| o |
| / |
| / |
| a @ |
| |
o=============================< J1. Delete "((a))" >========o
| |
| o |
| \ |
| \ |
| o |
| / |
| / |
| a @ |
| |
o=============================< I2. Refold "(())" >=========o
| |
| a |
| @ |
| |
o=============================< QED >=======================o
</pre>
====C<sub>2</sub>. Generation theorem====
====C<sub>2</sub>. Generation theorem====
One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased (retracted or withdrawn) anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the name of ''Consequence 2'' (C<sub>2</sub>), or ''Generation''.
One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased (retracted or withdrawn) anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the name of ''Consequence 2'' (C<sub>2</sub>), or ''Generation''.
<pre>
o-----------------------------------------------------------o
| C_2. Generation Theorem |
o-----------------------------------------------------------o
| |
| b o a o b |
| | | |
| a @ = a @ |
| |
o-----------------------------------------------------------o
| |
| a(b) = a(ab) |
| |
o-----------------------------------------------------------o
| Degenerate <---- | ----> Regenerate |
o-----------------------------------------------------------o
</pre>
Here is a proof of the Generation Theorem.
Here is a proof of the Generation Theorem.
<pre>
o-----------------------------------------------------------o
| C_2. Generation Theorem. Proof. |
o-----------------------------------------------------------o
| |
| b o |
| | |
| a @ |
| |
o=============================< C1. Reflect "a(b)" >========o
| |
| b o |
| | |
| a o |
| | |
| o |
| | |
| @ |
| |
o=============================< I2. Unfold "(())" >=========o
| |
| b o o |
| | / |
| a o o |
| |/ |
| o |
| | |
| @ |
| |
o=============================< J1. Insert "a" >============o
| |
| b o o a |
| | / |
| a o o a |
| |/ |
| o |
| | |
| @ |
| |
o=============================< J2. Collect "a" >===========o
| |
| b o o a |
| | / |
| o o |
| |/ |
| o |
| | |
| a @ |
| |
o=============================< C1. Reflect "a", "b" >======o
| |
| a o b |
| | |
| a @ |
| |
o=============================< QED >=======================o
</pre>
====C<sub>3</sub>. Dominant form theorem====
====C<sub>3</sub>. Dominant form theorem====
The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence 3'' (C<sub>3</sub>), or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs.
The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence 3'' (C<sub>3</sub>), or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs.
<pre>
o-----------------------------------------------------------o
| C_3. Dominant Form Theorem |
o-----------------------------------------------------------o
| |
| o o |
| | | |
| a @ = @ |
| |
o-----------------------------------------------------------o
| |
| a( ) = ( ) |
| |
o-----------------------------------------------------------o
| Remark <---- | ----> Recess |
o-----------------------------------------------------------o
</pre>
Here is a proof of the Dominant Form Theorem.
Here is a proof of the Dominant Form Theorem.
<pre>
o-----------------------------------------------------------o
| C_3. Dominant Form Theorem. Proof. |
o-----------------------------------------------------------o
| |
| o |
| | |
| a @ |
| |
o=============================< C2. Regenerate "a" >========o
| |
| a o |
| | |
| a @ |
| |
o=============================< J1. Delete "a" >============o
| |
| o |
| | |
| @ |
| |
o=============================< QED >=======================o
</pre>
===Exemplary proofs===
===Exemplary proofs===
The first order of business is present the statement as it appears in the so-called ''existential interpretation'' of Peirce's own ''logical graphs''. Here is the statement of Peirce's law, as rendered under the existential interpretation into (the topological dual forms of) Peirce's logical graphs:
The first order of business is present the statement as it appears in the so-called ''existential interpretation'' of Peirce's own ''logical graphs''. Here is the statement of Peirce's law, as rendered under the existential interpretation into (the topological dual forms of) Peirce's logical graphs:
<pre>
o-----------------------------------------------------------o
| Peirce's Law |
o-----------------------------------------------------------o
| |
| p o---o q |
| | |
| o---o p |
| | |
| o---o p |
| | |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| (((p (q)) (p)) (p))) = |
| |
o-----------------------------------------------------------o
</pre>
Finally, here's the promised proof of Peirce's law:
Finally, here's the promised proof of Peirce's law:
<pre>
o-----------------------------------------------------------o
| Peirce's Law. Proof |
o-----------------------------------------------------------o
| |
| p o---o q |
| | |
| o---o p |
| | |
| o---o p |
| | |
| @ |
| |
o==================================< Collect >==============o
| |
| o---o q |
| | |
| o---o |
| | |
| p o---o p |
| | |
| @ |
| |
o==================================< Recess >===============o
| |
| o---o |
| | |
| p o---o p |
| | |
| @ |
| |
o==================================< Refold >===============o
| |
| p o---o p |
| | |
| @ |
| |
o==================================< Delete >===============o
| |
| o---o |
| | |
| @ |
| |
o==================================< Refold >===============o
| |
| @ |
| |
o==================================< QED >==================o
</pre>
====Praeclarum theorema====
====Praeclarum theorema====
Under the existential interpretation, the praeclarum theorema is represented by means of the following logical graph.
Under the existential interpretation, the praeclarum theorema is represented by means of the following logical graph.
<pre>
o-----------------------------------------------------------o
| Praeclarum Theorema (Leibniz) |
o-----------------------------------------------------------o
| |
| b o o c o bc |
| | | | |
| a o o d o ad |
| \ / | |
| o---------o |
| | |
| | |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| ((a(b))(d(c))((ad(bc)))) = |
| |
o-----------------------------------------------------------o
</pre>
And here's a neat proof of that nice theorem.
And here's a neat proof of that nice theorem.
<pre>
o-----------------------------------------------------------o
| Praeclarum Theorema (Leibniz). Proof. |
o-----------------------------------------------------------o
| |
| b o o c o bc |
| | | | |
| a o o d o ad |
| \ / | |
| o---------o |
| | |
| | |
| @ |
| |
o=============================< C1. Reflect "ad(bc)" >======o
| |
| b o o c |
| | | |
| a o o d |
| \ / |
| ad o---------o bc |
| | |
| | |
| @ |
| |
o=============================< Weed "a", "d" >=============o
| |
| b o o c |
| | | |
| o o |
| \ / |
| ad o---------o bc |
| | |
| | |
| @ |
| |
o=============================< C1. Reflect "b", "c" >======o
| |
| abcd o---------o bc |
| | |
| | |
| @ |
| |
o=============================< Weed "bc" >=================o
| |
| abcd o---------o |
| | |
| | |
| @ |
| |
o=============================< C3. Recess "abcd" >=========o
| |
| o---------o |
| | |
| | |
| @ |
| |
o=============================< I2. Refold "(())" >=========o
| |
| @ |
| |
o=============================< QED >=======================o
</pre>
==References==
==References==
* [http://www.lawsofform.org/aum/session1.html Spencer-Brown's talks at Esalen 1973] — Self-referential forms are introduced in the section entitled "Degree of Equations and the Theory of Types"
* [http://www.lawsofform.org/aum/session1.html Spencer-Brown's talks at Esalen 1973] — Self-referential forms are introduced in the section entitled "Degree of Equations and the Theory of Types"
* [http://www.math.uic.edu/~kauffman/ Louis H. Kauffman] — ''[http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form]''
* [http://www.math.uic.edu/~kauffman/ Louis H. Kauffman] — ''[http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form]''
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