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{{DISPLAYTITLE:Futures Of Logical Graphs}}
{{DISPLAYTITLE:Futures Of Logical Graphs}}
This article develops an extension of [[Charles Sanders Peirce]]'s [[Logical Graph]]s.
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
This article develops an extension of [[Charles Sanders Peirce]]'s [[Logical Graphs]].
==Introduction==
==Introduction==
I think I am finally ready to speculate on the futures of logical graphs that will be able to rise to the challenge of embodying the fundamental logical insights of Peirce.
I think I am finally ready to speculate on the futures of logical graphs that will be able to rise to the challenge of embodying the fundamental logical insights of Peirce.
For the sake of those who may be unfamiliar with it, let us first divert ourselves with an exposition of a standard way that graphs of the order that Peirce considered, those embedded in a continuous manifold like a plane sheet of paper, without or without the paper bridges that Peirce used to augment his genus, can be represented as parse-strings in Asciiish and sculpted into pointer-structures in computer memory.
For the sake of those who may be unfamiliar with it, let us first divert ourselves with an exposition of a standard way that graphs of the order that Peirce considered, those embedded in a continuous manifold like a plane sheet of paper, without or without the paper bridges that Peirce used to augment his genus, can be represented as parse-strings in Ascii and sculpted into pointer-structures in computer memory.
A blank sheet of paper can be represented as a blank space in a line of text, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display.
A blank sheet of paper can be represented as a blank space in a line of text, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display.
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This can be written inline as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}</math> or set off in a text display:
This can be written inline as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}\!</math> or set off in a text display:
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| width="33%" | <math>\texttt{(~(~)~)}</math>
| width="33%" | <math>\texttt{(~(~)~)}\!</math>
| width="34%" | <math>=\!</math>
| width="34%" | <math>=\!</math>
| width="33%" |
| width="33%" |
|}
|}
When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their topological duals. The planar regions of the original graph correspond to nodes (or points) of the dual graph, and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph.
When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their topological duals. The planar regions of the original graph correspond to nodes (or points) of the dual graph, and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph.
For example, overlaying the corresponding dual graphs on the plane-embedded graphs shown above, we get the following composite picture:
For example, overlaying the corresponding dual graphs on the plane-embedded graphs shown above, we get the following composite picture:
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The outermost region of the plane-embedded graph is singled out for special consideration and the corresponding node of the dual graph is referred to as its ''root node''. By way of graphical convention in the present text, the root node is indicated by means of a horizontal strike-through.
The outermost region of the plane-embedded graph is singled out for special consideration and the corresponding node of the dual graph is referred to as its ''root node''. By way of graphical convention in the present text, the root node is indicated by means of a horizontal strike-through.
Extracting the dual graph from its composite matrix, we get this picture:
Extracting the dual graph from its composite matrix, we get this picture:
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 graphs, that constitute the species of rooted trees 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 graphs, that constitute the species of rooted trees 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 case either <math>{}^{\backprime\backprime} \texttt{(} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \texttt{)} {}^{\prime\prime},</math> 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 case either <math>{}^{\backprime\backprime} \texttt{(} {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \texttt{)} {}^{\prime\prime},\!</math> that we happen to encounter in our travels.
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
|}
|}
This ritual is called ''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 graph'' of the string. I 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 ''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 graph'' of the string. I 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 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 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 different directions of applying the axioms.
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The ''parse graphs'' that we've been looking at so far are one step toward the ''pointer graphs'' that it takes to make trees live in computer memory, but they are still a couple of steps too abstract to properly suggest in much concrete detail the species of dynamic data structures that we need. I now proceed to flesh out the skeleton that I've drawn up to this point.
The ''parse graphs'' that we've been looking at so far are one step toward the ''pointer graphs'' that it takes to make trees live in computer memory, but they are still a couple of steps too abstract to properly suggest in much concrete detail the species of dynamic data structures that we need. I now proceed to flesh out the skeleton that I've drawn up to this point.
Nodes in a graph depict ''records'' in computer memory. A record is a collection of data that can be thought to reside at a specific ''address''. For semioticians, an address can be recognized as a type of index, and is commonly spoken of, on analogy with demonstrative pronouns, as a ''pointer'', even among computer programmers who are otherwise innocent of semiotics.
Nodes in a graph depict ''records'' in computer memory. A record is a collection of data that can be thought to reside at a specific ''address''. For semioticians, an address can be recognized as a type of index, and is commonly spoken of, on analogy with demonstrative pronouns, as a ''pointer'', even among computer programmers who are otherwise innocent of semiotics.
At the next level of concreteness, a pointer-record structure is represented as follows:
At the next level of concreteness, a pointer-record structure is represented as follows:
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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:
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
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, ''up'' being the same 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, ''up'' being the same as ''away from the root''.
We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.</math> For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.</math>
We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.\!</math> For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.\!</math>
First the plane-embedded maps:
First the plane-embedded maps:
Because I plan this time around a somewhat leisurely excursion through the primordial wilds of logics that were so intrepidly explored by C.S. Peirce and again in recent times revisited by George Spencer Brown, let me just give a few extra pointers to those who wish to run on ahead of this torturous tortoise pace:
Because I plan this time around a somewhat leisurely excursion through the primordial wilds of logics that were so intrepidly explored by C.S. Peirce and again in recent times revisited by George Spencer Brown, let me just give a few extra pointers to those who wish to run on ahead of this torturous tortoise pace:
:* Jon Awbrey, [http://forum.wolframscience.com/printthread.php?threadid=297&perpage=35 Propositional Equation Reasoning Systems].
:* Jon Awbrey, [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems].
:* Lou Kauffman, [http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form].
:* Lou Kauffman, [http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form].
When I call to mind a ''category of structured individuals'' (COSI), I get a picture of a certain form, with blanks to be filled in as the thought of it develops, that can be sketched at first like so:
When I call to mind a ''category of structured individuals'' (COSI), I get a picture of a certain form, with blanks to be filled in as the thought of it develops, that can be sketched at first like so:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
Category @
/ \
/ \
/ \
/ \
/ \
Individuals o ... o
/ \ / \
/ \ / \
/ \ / \
Structures o->-o->-o o->-o->-o
</pre>
</pre>
|}
|}
The various glyphs of this picturesque hierarchy serve to remind us that a COSI in general consists of many individuals, which in spite of their calling as such may have specific structures involving the ordering of their component parts. Of course, this generic picture may have degenerate realizations, as when we have a 1-adic relation, that may be viewed in most settings as nothing different than a set:
The various glyphs of this picturesque hierarchy serve to remind us that a COSI in general consists of many individuals, which in spite of their calling as such may have specific structures involving the ordering of their component parts. Of course, this generic picture may have degenerate realizations, as when we have a 1-adic relation, that may be viewed in most settings as nothing different than a set:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
Category @
/ \
/ \
/ \
/ \
/ \
Individuals o ... o
| |
| |
| |
Structures o ... o
</pre>
</pre>
|}
|}
The practical use of Peirce's categories is simply to organize our thoughts about what sorts of formal models are demanded by a material situation, for instance, a domain of phenomena from atoms to biology to culture. To say that "k-ness" is involved in a phenomenon is simply to say that we need k-adic relations to model it adequately, and that the phenomenon itself appears to demand nothing less. Aside from this, Peirce's realization that k-ness for k = 1, 2, 3 affords us with a sufficient basis for all that we need to model is a formal fact that depends on a particular theorem in the logic of relatives. If it weren't for that, there would hardly be any reason to single out three.
The practical use of Peirce's categories is simply to organize our thoughts about what sorts of formal models are demanded by a material situation, for instance, a domain of phenomena from atoms to biology to culture. To say that "k-ness" is involved in a phenomenon is simply to say that we need k-adic relations to model it adequately, and that the phenomenon itself appears to demand nothing less. Aside from this, Peirce's realization that k-ness for k = 1, 2, 3 affords us with a sufficient basis for all that we need to model is a formal fact that depends on a particular theorem in the logic of relatives. If it weren't for that, there would hardly be any reason to single out three.
In order to discuss the various forms of iconicity that might be involved in the application of Peirce's logical graphs and their kind to the object domain of logic itself, we will need to bring out two or three ''categories of structured individuals'' (COSIs), depending on how one counts. These are called the ''object domain'', the ''sign domain'', and the ''interpretant sign domain'', which may be written <math>O, S, I,\!</math> respectively, or <math>X, Y, Z,\!</math> respectively, depending on the style that fits the current frame of discussion.
In order to discuss the various forms of iconicity that might be involved in the application of Peirce's logical graphs and their kind to the object domain of logic itself, we will need to bring out two or three ''categories of structured individuals'' (COSIs), depending on how one counts. These are called the ''object domain'', the ''sign domain'', and the ''interpretant sign domain'', which may be written <math>{O, S, I},\!</math> respectively, or <math>{X, Y, Z},\!</math> respectively, depending on the style that fits the current frame of discussion.
For the time being, we will be considering systems where the sign domain and the interpretant domain are the same sets of entities, although, of course, their roles in a given ''[[sign relation]]'', say, <math>L \subseteq O \times S \times I</math> or <math>L \subseteq X \times Y \times Z,</math> remain as distinct as ever. We may use the term ''semiotic domain'' for the common set of elements that constitute the signs and the interpretant signs in any setting where the sign domain and the interpretant domain are equal as sets.
For the time being, we will be considering systems where the sign domain and the interpretant domain are the same sets of entities, although, of course, their roles in a given ''[[sign relation]]'', say, <math>L \subseteq O \times S \times I</math> or <math>L \subseteq X \times Y \times Z,</math> remain as distinct as ever. We may use the term ''semiotic domain'' for the common set of elements that constitute the signs and the interpretant signs in any setting where the sign domain and the interpretant domain are equal as sets.
Returning to <math>\operatorname{En}</math> and <math>\operatorname{Ex},</math> the two most popular interpretations of logical graphs, ones that happen to be dual to each other in a certain sense, let's see how they fly as ''hermeneutic arrows'' from the syntactic domain <math>S\!</math> to the object domain <math>O,\!</math> at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the ''primary arithmetic''.
Returning to <math>\operatorname{En}</math> and <math>\operatorname{Ex},</math> the two most popular interpretations of logical graphs, ones that happen to be dual to each other in a certain sense, let's see how they fly as ''hermeneutic arrows'' from the syntactic domain <math>S\!</math> to the object domain <math>O,\!</math> at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the ''primary arithmetic''.
Taking <math>\operatorname{En}</math> and <math>\operatorname{Ex}</math> as arrows of the form <math>\operatorname{En}, \operatorname{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}\!</math> and <math>O = \{ \operatorname{falsity}, \operatorname{truth} \},\!</math> it is possible to factor each arrow across the domain <math>S_0\!</math> that consists of a single rooted node plus a single rooted edge, in other words, the domain of formal constants <math>S_0 = \{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}.\!</math> This allows each arrow to be broken into a purely syntactic part <math>\operatorname{En}_\text{syn}, \operatorname{Ex}_\text{syn} : S \to S_0</math> and a purely semantic part <math>\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : S_0 \to O.</math>
Taking <math>\operatorname{En}\!</math> and <math>\operatorname{Ex}\!</math> as arrows of the form <math>\operatorname{En}, \operatorname{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}\!</math> and <math>O = \{ \operatorname{falsity}, \operatorname{truth} \},\!</math> it is possible to factor each arrow across the domain <math>S_0\!</math> that consists of a single rooted node plus a single rooted edge, in other words, the domain of formal constants <math>S_0 = \{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}.\!</math> This allows each arrow to be broken into a purely syntactic part <math>\operatorname{En}_\text{syn}, \operatorname{Ex}_\text{syn} : S \to S_0</math> and a purely semantic part <math>\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : S_0 \to O.</math>
As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone. Specifically, we have the following mappings:
As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone. Specifically, we have the following mappings:
Cast into the form of a 3-adic sign relation, the situation before us can now be given the following shape:
Cast into the form of a 3-adic sign relation, the situation before us can now be given the following shape:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
Y
Semiotic Domain
o-------------------o
| |
| Rooted Trees |
| |
o-------------------o
|
|
Syntactic
Reduction
|
|
v
o-----------o En_sem o-----------o
| |<--------------| o |
| F T | | O | |
| |<--------------| @ |
o-----------o Ex_sem o-----------o
X Y_0
Canonical
Object Domain Sign Domain
</pre>
</pre>
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|}
The more Peirce-sistent among you, on contemplating that last picture, will naturally ask, "What happened to the irreducible 3-adicity of sign relations in this portrayal of logical graphs?"
The more Peirce-sistent among you, on contemplating that last picture, will naturally ask, "What happened to the irreducible 3-adicity of sign relations in this portrayal of logical graphs?"
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
Y
Semiotic Domain
o-------------------o
| |
| Rooted Trees |
| |
o-------------------o
|
|
Syntactic
Reduction
|
|
v
o-----------o En_sem o-----------o
| |<--------------| o |
| F T | | O | |
| |<--------------| @ |
o-----------o Ex_sem o-----------o
X Y_0
Canonical
Object Domain Sign Domain
</pre>
</pre>
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|}
Thus, if you find youself in an argument with another interpreter who swears to the influence of some quality common to the object and the sign and that really does affect his or her conduct in regard to the two of them, then that argument is almost certainly bound to be utterly futile. I am sure we've all been there.
Thus, if you find youself in an argument with another interpreter who swears to the influence of some quality common to the object and the sign and that really does affect his or her conduct in regard to the two of them, then that argument is almost certainly bound to be utterly futile. I am sure we've all been there.
When I first became acquainted with the Entish and Extish hermenautics of logical graphs, back in the late great 1960's, I was struck in the spirit of those times by what I imagined to be their Zen and Zenoic sensibilities, the ''tao is silent'' wit of the Zen mind being the empty mind, that seems to go along with the <math>\operatorname{Ex}</math> interpretation, and the way from ''the way that's marked is not the true way'' to ''the mark that's marked is not the remarkable mark'' and to ''the sign that's signed is not the significant sign'' of the <math>\operatorname{En}</math> interpretation, reminding us that the sign is not the object, no matter how apt the image. And later, when my discovery of the cactus graph extension of logical graphs led to the leimons of neural pools, where <math>\operatorname{En}</math> says that truth is an active condition, while <math>\operatorname{Ex}</math> says that sooth is a quiescent mind, all these themes got reinforced more still.
When I first became acquainted with the Entish and Extish hermenautics of logical graphs, back in the late great 1960s, I was struck in the spirit of those times by what I imagined to be their Zen and Zenoic sensibilities, the ''tao is silent'' wit of the Zen mind being the empty mind, that seems to go along with the <math>\operatorname{Ex}\!</math> interpretation, and the way from ''the way that's marked is not the true way'' to ''the mark that's marked is not the remarkable mark'' and to ''the sign that's signed is not the significant sign'' of the <math>\operatorname{En}\!</math> interpretation, reminding us that the sign is not the object, no matter how apt the image. And later, when my discovery of the cactus graph extension of logical graphs led to the leimons of neural pools, where <math>\operatorname{En}\!</math> says that truth is an active condition, while <math>\operatorname{Ex}\!</math> says that sooth is a quiescent mind, all these themes got reinforced more still.
We hold these truths to be self-iconic, but they come in complementary couples, in consort to the flip-side of the tao.
We hold these truths to be self-iconic, but they come in complementary couples, in consort to the flip-side of the tao.
At the level of the ''primary arithmetic'', we have a set-up like this:
At the level of the ''primary arithmetic'', we have a set-up like this:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
Categories` ` ` ` `!O!` ` ` ` ` ` ` ` ` ` ` ` `!S!` ` ` ` ` `
Categories !O! !S!
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ Denotes / \
Individuals `{F}` ` ` ` `{T}` <------ ` {...} ` ` ` {...} ` `
Individuals {F} {T} <------ {...} {...}
| | /|\ /|\
| | / | \ / | \
| | / | \ / | \
Structures` ` F ` ` ` ` ` T ` ` ` ` ` o ` o ` o ` o ` o ` o `
Structures F T o o o o o o
o
|
o o o o
| \ / |
o o o o o o
| | | \ / |
F T @ @ @ @ @ @
</pre>
</pre>
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|}
In thinking about mappings between categories of structured individuals, we can take each mapping in two parts. At the first level of analysis, there is the part that maps individuals to individuals. At the second level of analysis, there is the part that maps the structural parts of each individual to the structural parts of the individual that forms its counterpart under the first part of the mapping in question.
In thinking about mappings between categories of structured individuals, we can take each mapping in two parts. At the first level of analysis, there is the part that maps individuals to individuals. At the second level of analysis, there is the part that maps the structural parts of each individual to the structural parts of the individual that forms its counterpart under the first part of the mapping in question.
The general scheme of things is suggested by the following Figure, where the mapping <math>f\!</math> from COSI <math>U\!</math> to COSI <math>V\!</math> is analyzed in terms of a mapping <math>g\!</math> that takes individuals to individuals, ignoring their inner structures, and a set of mappings <math>h_j,\!</math> where <math>j\!</math> ranges over the individuals of COSI <math>U,\!</math> and where <math>h_j\!</math> specifies just how the parts of <math>j\!</math> map to the parts of <math>g(j),\!</math> its counterpart under <math>g.\!</math>
The general scheme of things is suggested by the following Figure, where the mapping <math>f\!</math> from COSI <math>U\!</math> to COSI <math>V\!</math> is analyzed in terms of a mapping <math>g\!</math> that takes individuals to individuals, ignoring their inner structures, and a set of mappings <math>h_j,\!</math> where <math>j\!</math> ranges over the individuals of COSI <math>U,\!</math> and where <math>h_j\!</math> specifies just how the parts of <math>j\!</math> map to the parts of <math>{g(j)},\!</math> its counterpart under <math>{g}.\!</math>
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
U f V
@ --------------------> @
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ g / \
o ... o --------> o ... o
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
o->-o->-o o->-o->-o o->-o->-o o->-o->-o
\ /
\ /
\ /
\ /
\ h_j /
o-------->----------o
</pre>
</pre>
|}
|}
Next time we'll apply this general scheme to the <math>\operatorname{En}</math> and <math>\operatorname{Ex}</math> interpretations of logical graphs, and see how it helps us to sort out the varieties of iconic mapping that are involved in that setting.
Next time we'll apply this general scheme to the <math>\operatorname{En}\!</math> and <math>\operatorname{Ex}\!</math> interpretations of logical graphs, and see how it helps us to sort out the varieties of iconic mapping that are involved in that setting.
Corresponding to the Entitative and Existential interpretations of the primary arithmetic, there are two distinct mappings from the sign domain <math>S,\!</math> containing the topological equivalents of bare and rooted trees, onto the object domain <math>O,\!</math> containing the two objects whose conventional, ordinary, or meta-language names are ''falsity'' and ''truth'', respectively.
Corresponding to the Entitative and Existential interpretations of the primary arithmetic, there are two distinct mappings from the sign domain <math>S,\!</math> containing the topological equivalents of bare and rooted trees, onto the object domain <math>O,\!</math> containing the two objects whose conventional, ordinary, or meta-language names are ''falsity'' and ''truth'', respectively.
Here is the Figure for the Entitative interpretation:
Here is the Figure for the Entitative interpretation:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
o
|
o o o o
| | \ /
Structures` ` F ` ` ` ` ` T ` ` ` ` ` @ ` @ `...` @ ` @ `...`
Structures F T @ @ ... @ @ ...
| | \ | / \ | /
| | \ | / \ | /
| | En \|/ \|/
Individuals ` o ` ` ` ` ` o ` <-----` ` ` o ` ` ` ` ` o ` ` `
Individuals o o <----- o o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
Categories` ` ` ` `!O!` ` ` ` ` ` ` ` ` ` ` ` `!S!` ` ` ` ` `
Categories !O! !S!
</pre>
</pre>
|}
|}
Here is the Figure for the Existential interpretation:
Here is the Figure for the Existential interpretation:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
o
|
o o o o
| \ / |
Structures` ` F ` ` ` ` ` T ` ` ` ` ` @ ` @ `...` @ ` @ `...`
Structures F T @ @ ... @ @ ...
| | \ | / \ | /
| | \ | / \ | /
| | Ex \|/ \|/
Individuals ` o ` ` ` ` ` o ` <-----` ` ` o ` ` ` ` ` o ` ` `
Individuals o o <----- o o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
Categories` ` ` ` `!O!` ` ` ` ` ` ` ` ` ` ` ` `!S!` ` ` ` ` `
Categories !O! !S!
</pre>
</pre>
|}
|}
{| align="center" cellpadding="10" width="90%"
{| align="center" cellpadding="10" width="90%"
|
|
<p><math>\operatorname{En}</math> maps every tree on the left side of <math>S\!</math> to the left side of <math>O.\!</math></p>
<p><math>\operatorname{En}\!</math> maps every tree on the left side of <math>S\!</math> to the left side of <math>O.\!</math></p>
<p><math>\operatorname{En}</math> maps every tree on the right side of <math>S\!</math> to the right side of <math>O.\!</math></p>
<p><math>\operatorname{En}\!</math> maps every tree on the right side of <math>S\!</math> to the right side of <math>O.\!</math></p>
|-
|-
|
|
<p><math>\operatorname{Ex}</math> maps every tree on the left side of <math>S\!</math> to the right side of <math>O.\!</math></p>
<p><math>\operatorname{Ex}\!</math> maps every tree on the left side of <math>S\!</math> to the right side of <math>O.\!</math></p>
<p><math>\operatorname{Ex}</math> maps every tree on the right side of <math>S\!</math> to the left side of <math>O.\!</math></p>
<p><math>\operatorname{Ex}\!</math> maps every tree on the right side of <math>S\!</math> to the left side of <math>O.\!</math></p>
|}
|}
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
o
|
o o o o
| | \ /
Structures` ` F ` ` ` ` ` T ` ` ` ` ` @ ` @ `...` @ ` @ `...`
Structures F T @ @ ... @ @ ...
| | \ | / \ | /
| | \ | / \ | /
| | En \|/ \|/
Individuals ` o ` ` ` ` ` o ` <-----` ` ` o ` ` ` ` ` o ` ` `
Individuals o o <----- o o
\ / Ex \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
Categories` ` ` ` `!O!` ` ` ` ` ` ` ` ` ` ` ` `!S!` ` ` ` ` `
Categories !O! !S!
</pre>
</pre>
|}
|}
: In: [http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104 FOLG]
: In: [http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104 FOLG]
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
In <math>\operatorname{Ex},</math> all four of the listed signs are expressions of Falsity, and, viewed within the special type of semiotic procedure that is being considered here, each sign interprets its predecessor in the sequence. Thus we might begin by drawing up this Table:
In <math>\operatorname{Ex},</math> all four of the listed signs are expressions of Falsity, and, viewed within the special type of semiotic procedure that is being considered here, each sign interprets its predecessor in the sequence. Thus we might begin by drawing up this Table:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
Let's take another look at the semiotic sequence associated with a logical evaluation and the corresponding sample of a sign relation that we were looking at last time.
Let's take another look at the semiotic sequence associated with a logical evaluation and the corresponding sample of a sign relation that we were looking at last time.
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
|}
|}
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
The sign of equality "=", interpreted as logical equivalence "⇔", that marked our steps in the process of conducting the evaluation, is evidently intended to denote an equivalence relation, and this is a 2-adic relation that is reflexive, symmetric, and transitive. If we then pass to the reflexive, symmetric, transitive closure of the <''s'', ''i''> pairs that occur in our initial sample, attaching the constant reference to Falsity in the object domain, we will sweep out a more complete selection of the sign relation that inheres in the definition of the primary logical arithmetic.
The sign of equality "=", interpreted as logical equivalence "⇔", that marked our steps in the process of conducting the evaluation, is evidently intended to denote an equivalence relation, and this is a 2-adic relation that is reflexive, symmetric, and transitive. If we then pass to the reflexive, symmetric, transitive closure of the <''s'', ''i''> pairs that occur in our initial sample, attaching the constant reference to Falsity in the object domain, we will sweep out a more complete selection of the sign relation that inheres in the definition of the primary logical arithmetic.
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
Before we leave it for richer coasts — not to say we won't find ourselves returning eternally — let's note one other feature of our randomly chosen microcosm, one I suspect we'll see many echoes of in the macrocosm of our future wanderings.
Before we leave it for richer coasts — not to say we won't find ourselves returning eternally — let's note one other feature of our randomly chosen microcosm, one I suspect we'll see many echoes of in the macrocosm of our future wanderings.
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
For example, consider the following expression:
For example, consider the following expression:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
a a
o-----o
|
|
@
</pre>
</pre>
|}
|}
We may regard this algebraic expression as a general expression for an infinite set of arithmetic expressions, starting like so:
We may regard this algebraic expression as a general expression for an infinite set of arithmetic expressions, starting like so:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<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-----o o-----o o-----o o-----o o-----o
| | | | | |
| | | | | |
@ @ @ @ @ @
</pre>
</pre>
|}
|}
Now consider what this says about the following algebraic law:
Now consider what this says about the following algebraic law:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
a a
o-----o
|
|
@ = @
</pre>
</pre>
|}
|}
To begin with a concrete case that's as easy as possible, let's examine this extremely simple algebraic expression:
To begin with a concrete case that's as easy as possible, let's examine this extremely simple algebraic expression:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
a
o
|
@
</pre>
</pre>
|}
|}
As we've already discussed, the algebraic variable name indicates the contemplated absence or presence of any arithmetic expression taking its place in the surrounding template, which expression is proxied well enough by its value, of which values we know but two. Thus, the given algebraic expression varies between these choices:
As we've already discussed, the algebraic variable name indicates the contemplated absence or presence of any arithmetic expression taking its place in the surrounding template, which expression is proxied well enough by its value, of which values we know but two. Thus, the given algebraic expression varies between these choices:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
o
|
o o
| |
@ , @
</pre>
</pre>
|}
|}
We had been contemplating the penultimately simple algebraic expression "(a)" as a name for a set of arithmetic expressions, namely, (a) = {(), (())}, taking the equality sign in the appropriate sense.
We had been contemplating the penultimately simple algebraic expression "(a)" as a name for a set of arithmetic expressions, namely, (a) = {(), (())}, taking the equality sign in the appropriate sense.
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
o
a |
o o o
| | |
@ = { @ , @ }
</pre>
</pre>
|}
|}
Clearly, a variation between the absence and the presence of the operator "(_)" in the algebraic expression "(a)" refers to a variation between the algebraic expressions "a" and "(a)", respectively, somewhat as pictured here:
Clearly, a variation between the absence and the presence of the operator "(_)" in the algebraic expression "(a)" refers to a variation between the algebraic expressions "a" and "(a)", respectively, somewhat as pictured here:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
a a
o o
? a |
@ = { @ , @ }
</pre>
</pre>
|}
|}
Here is how we might suggest an algebraic expression of the form "(q)" where the absence or presence of the operator "(_)" depends on the value of the algebraic expression "p", the operator "(_)" being absent whenever p is unmarked and present when whenever p is marked.
Here is how we might suggest an algebraic expression of the form "(q)" where the absence or presence of the operator "(_)" depends on the value of the algebraic expression "p", the operator "(_)" being absent whenever p is unmarked and present when whenever p is marked.
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
o---o
| q |
o---o = { q , (q) }
|
|
o
</pre>
</pre>
|}
|}
One of the other tactics of syntax that I tried at this time — somewhere in the 70's ... when did we quit using punchcards? — by way of porting operator variables into logical graphs and the laws of form, was to hollow out a leg of Spencer-Brown's crosses, gnomons, markers, whatever you call them, like this:
One of the other tactics of syntax that I tried at this time — somewhere in the 70's ... when did we quit using punchcards? — by way of porting operator variables into logical graphs and the laws of form, was to hollow out a leg of Spencer-Brown's crosses, gnomons, markers, whatever you call them, like this:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
----------o---o
| |
q | p | = { q , (q) }
| |
</pre>
</pre>
|}
|}
A funny thing just happened. Let's see if we can tell where. We started with the algebraic expression "(a)", in which the operand "a" suggests the contemplated absence or presence of any arithmetic expression or its value, then we contemplated the absence of presence of the operator "(_)" in "(a)" to be indicated by a cross or a space, respectively, for the value of a newly introduced variable, "b", placed in a new slot of a newly extended operator form, as suggested by this picture:
A funny thing just happened. Let's see if we can tell where. We started with the algebraic expression "(a)", in which the operand "a" suggests the contemplated absence or presence of any arithmetic expression or its value, then we contemplated the absence of presence of the operator "(_)" in "(a)" to be indicated by a cross or a space, respectively, for the value of a newly introduced variable, "b", placed in a new slot of a newly extended operator form, as suggested by this picture:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
----------o---o
| |
a | b |
| |
</pre>
</pre>
|}
|}
Writing out a formal operation table yields the following summary:
Writing out a formal operation table yields the following summary:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
The step of controlled reflection that we just took can be iterated just as far as we wish to take it, as suggested by the following set:
The step of controlled reflection that we just took can be iterated just as far as we wish to take it, as suggested by the following set:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
----------o
|
a |
|
----------o---o
| |
a | b |
| |
----------o---o---o
| | |
a | b | c |
| | |
----------o---o---o---o
| | | |
a | b | c | d |
| | | |
</pre>
</pre>
|}
|}
The following Table will suffice to suggest the syntactic correspondences among the "streamer-cross" forms that Peirce used in his essay on "Qualitative Logic" and Spencer Brown used in his book ''Laws of Form'', as they become extended by successive steps of controlled reflection, the plaintext string syntax, and the rooted cactus graphs:
The following Table will suffice to suggest the syntactic correspondences among the "streamer-cross" forms that Peirce used in his essay on "Qualitative Logic" and Spencer Brown used in his book ''Laws of Form'', as they become extended by successive steps of controlled reflection, the plaintext string syntax, and the rooted cactus graphs:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
Let us examine the formal operation table for the next in our series of reflective operations to see if we can elicit the general pattern.
Let us examine the formal operation table for the next in our series of reflective operations to see if we can elicit the general pattern.
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
Or, thinking in terms of the graphic equivalents, writing "o" for a blank node and "|" for an edge:
Or, thinking in terms of the graphic equivalents, writing "o" for a blank node and "|" for an edge:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
The formal rule of evaluation for a "''k''-lobe" or "''k''-operator" is:
The formal rule of evaluation for a "''k''-lobe" or "''k''-operator" is:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:
The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
I think that I am at long last ready to state the following:
I think that I am at long last ready to state the following:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
In order to think of tackling even the roughest sketch toward a proof of this theorem, we need to add a number of axioms and axiom schemata. Because I abandoned proof-theoretic purity somewhere in the middle of grinding this calculus into computational form, I never got around to finding the most elegant and minimal, or anything near a complete set of axioms for the ''cactus language'', so what I list here are just the slimmest rudiments of the hodge-podge of rules of thumb that I have found over time to be necessary and useful in most working settings. Some of these special precepts are probably provable from genuine axioms, but I have yet to go looking for a more proper formulation.
In order to think of tackling even the roughest sketch toward a proof of this theorem, we need to add a number of axioms and axiom schemata. Because I abandoned proof-theoretic purity somewhere in the middle of grinding this calculus into computational form, I never got around to finding the most elegant and minimal, or anything near a complete set of axioms for the ''cactus language'', so what I list here are just the slimmest rudiments of the hodge-podge of rules of thumb that I have found over time to be necessary and useful in most working settings. Some of these special precepts are probably provable from genuine axioms, but I have yet to go looking for a more proper formulation.
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
Here is a proof sketch for the ''Case Analysis-Synthesis Theorem'' (CAST):
Here is a proof sketch for the ''Case Analysis-Synthesis Theorem'' (CAST):
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
| [[Image:Proof Praeclarum Theorema CAST 02.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 02.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 03.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 03.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 04.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 04.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 05.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 05.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 07.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 07.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 08.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 08.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 09.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 09.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 10.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 10.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 13.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 13.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 14.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 14.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 15.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 15.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Cast C.jpg|500px]]
| [[Image:Equational Inference Bar -- Cast C ISW.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 16.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 16.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- QED.jpg|500px]]
| [[Image:Equational Inference Bar -- QED.jpg|500px]]
|}
|}
The following Figure provides an animated recap of the graphical transformations that occur in the above proof:
{| align="center" cellpadding="8"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
|-
| [[Image:Praeclarum Theorema CAST 500 x 389 Animation.gif]]
|}
|}
|}
|}
Remembering that a blank node is the graphical equivalent of a logical value <math>\operatorname{true},</math> the resulting DNF may be read as follows:
Remembering that a blank node is the graphical equivalent of a logical value <math>\operatorname{true},</math> the resulting DNF may be read as follows:
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" cellpadding="10" style="text-align:center; width:60%"
|
|
<pre>
<pre>
{| align="center" cellpadding="10" width="90%"
{| align="center" cellpadding="10" width="90%"
| [[Directory:Jon_Awbrey/Papers/Futures_Of_Logical_Graphs#Praeclarum_theorema|The first way of transforming the expression]] that appears on the left hand side of the equation can be described as ''proof-theoretic'' in character.
| [[#Praeclarum theorema|The first way of transforming the expression]] that appears on the left hand side of the equation can be described as ''proof-theoretic'' in character.
|-
|-
| [[Directory:Jon_Awbrey/Papers/Futures_Of_Logical_Graphs#Example|The second way of transforming the expression]] that appears on the left hand side of the equation can be described as ''model-theoretic'' in character.
| [[#Praeclarum theorema : Proof by CAST|The second way of transforming the expression]] that appears on the left hand side of the equation can be described as ''model-theoretic'' in character.
|}
|}
For example, consider the existential graph for <math>p \Leftrightarrow q</math> that is shown below:
For example, consider the existential graph for <math>p \Leftrightarrow q</math> that is shown below:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
o-----------------o o-----------------o
| o-------o | | o-------o |
| | | | | | | |
| p | q | | | q | p | |
| | | | | | | |
| o-------o | | o-------o |
o-----------------o o-----------------o
</pre>
</pre>
|}
|}
Graphing the topological dual form, one obtains the following rooted tree:
Graphing the topological dual form, one obtains the following rooted tree:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
q o o p
| |
p o o q
\ /
@
(p (q)) (q (p))
</pre>
</pre>
|}
|}
However, the cactus graph expression for equivalence works much better:
However, the cactus graph expression for equivalence works much better:
{| align="center" style="text-align:center; width:90%"
{| align="center" cellpadding="10"
|
|
<pre>
<pre>
p o---o q
\ /
o
|
@
((p , q))
</pre>
</pre>
|}
|}
|}
|}
For the sake of simplicity in discussing this example, let's stick with the existential interpretation (<math>\operatorname{Ex}</math>) of logical graphs and their corresponding parse strings. Under <math>\operatorname{Ex}</math> the formal expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> translates into the vernacular expression <math>{}^{\backprime\backprime} p ~\operatorname{implies}~ q ~\operatorname{and}~ p ~\operatorname{implies}~ r {}^{\prime\prime},</math> in symbols, <math>(p \Rightarrow q) \land (p \Rightarrow r),</math> so this is the reading that we'll want to keep in mind for the present. Where brevity is required, we may refer to the propositional expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> under the name <math>f\!</math> by making use of the following definition:
For the sake of simplicity in discussing this example, let's stick with the existential interpretation (<math>\operatorname{Ex}\!</math>) of logical graphs and their corresponding parse strings. Under <math>\operatorname{Ex}\!</math> the formal expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> translates into the vernacular expression <math>{}^{\backprime\backprime} p ~\operatorname{implies}~ q ~\operatorname{and}~ p ~\operatorname{implies}~ r {}^{\prime\prime},</math> in symbols, <math>(p \Rightarrow q) \land (p \Rightarrow r),</math> so this is the reading that we'll want to keep in mind for the present. Where brevity is required, we may refer to the propositional expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> under the name <math>f\!</math> by making use of the following definition:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>f ~=~ \texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math>
| <math>f ~=~ \texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}\!</math>
|}
|}
Since the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation. There are in fact two different ways to execute the picture.
Since the expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation. There are in fact two different ways to execute the picture.
Figure 27 indicates the points of the universe of discourse <math>X\!</math> for which the proposition <math>f : X \to \mathbb{B}</math> has the value 1, here interpreted as the logical value <math>\operatorname{true}.</math> In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe <math>X,\!</math> going according to some previously adopted convention, for instance: Let the cells that get the value 0 under <math>f\!</math> remain untinted and let the cells that get the value 1 under <math>f\!</math> be painted or shaded. In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the <math>0, 1\!</math> in <math>\mathbb{B},</math> but in the pattern of regions that they indicate.
Figure 27 indicates the points of the universe of discourse <math>X\!</math> for which the proposition <math>f : X \to \mathbb{B}</math> has the value 1, here interpreted as the logical value <math>\operatorname{true}.</math> In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe <math>X,\!</math> going according to some previously adopted convention, for instance: Let the cells that get the value 0 under <math>f\!</math> remain untinted and let the cells that get the value 1 under <math>f\!</math> be painted or shaded. In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the <math>0, 1\!</math> in <math>\mathbb{B},</math> but in the pattern of regions that they indicate.
While we go through each of these ways let us keep one eye out for the character and the conduct of each type of proceeding as a semiotic process, that is, as an orbit, in this case discrete, through a locus of signs, in this case propositional expressions, and as it happens in this case, a sequence of transformations that perseveres in the denotative objective of each expression, that is, in the abstract proposition that it expresses, while it preserves the informed constraint on the universe of discourse that gives us one viable candidate for the informational content of each expression in the interpretive chain of sign metamorphoses.
While we go through each of these ways let us keep one eye out for the character and the conduct of each type of proceeding as a semiotic process, that is, as an orbit, in this case discrete, through a locus of signs, in this case propositional expressions, and as it happens in this case, a sequence of transformations that perseveres in the denotative objective of each expression, that is, in the abstract proposition that it expresses, while it preserves the informed constraint on the universe of discourse that gives us one viable candidate for the informational content of each expression in the interpretive chain of sign metamorphoses.
A ''sign relation'' <math>L\!</math> is a subset of a cartesian product <math>O \times S \times I,</math> where <math>O, S, I\!</math> are sets known as the ''object'', ''sign'', and ''interpretant sign'' domains, respectively. These facts are symbolized by writing <math>L \subseteq O \times S \times I.</math> Accordingly, a sign relation <math>L\!</math> consists of ordered triples of the form <math>(o, s, i),\!</math> where <math>o, s, i\!</math> belong to the domains <math>O, S, I,\!</math> respectively. An ordered triple of the form <math>(o, s, i) \in L</math> is referred to as a ''sign triple'' or an ''elementary sign relation''.
A ''sign relation'' <math>L\!</math> is a subset of a cartesian product <math>O \times S \times I,</math> where <math>{O, S, I}\!</math> are sets known as the ''object'', ''sign'', and ''interpretant sign'' domains, respectively. These facts are symbolized by writing <math>L \subseteq O \times S \times I.</math> Accordingly, a sign relation <math>L\!</math> consists of ordered triples of the form <math>(o, s, i),\!</math> where <math>o, s, i\!</math> belong to the domains <math>{O, S, I},\!</math> respectively. An ordered triple of the form <math>(o, s, i) \in L</math> is referred to as a ''sign triple'' or an ''elementary sign relation''.
The ''syntactic domain'' of a sign relation <math>L \subseteq O \times S \times I</math> is defined as the set-theoretic union <math>S \cup I</math> of its sign domain <math>S\!</math> and its interpretant domain <math>I.\!</math> It is not uncommon, especially in formal examples, for the sign domain and the interpretant domain to be equal as sets, in short, to have <math>S = I.\!</math>
The ''syntactic domain'' of a sign relation <math>L \subseteq O \times S \times I</math> is defined as the set-theoretic union <math>S \cup I</math> of its sign domain <math>S\!</math> and its interpretant domain <math>I.\!</math> It is not uncommon, especially in formal examples, for the sign domain and the interpretant domain to be equal as sets, in short, to have <math>S = I.\!</math>
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-2.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-2.jpg|500px]]
|-
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| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-3.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-3.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-6.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-6.jpg|500px]]
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| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-7.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-7.jpg|500px]]
Remembering that a blank node is the graphical equivalent of a logical value <math>\operatorname{true},</math> the resulting DNF may be read as follows:
Remembering that a blank node is the graphical equivalent of a logical value <math>\operatorname{true},</math> the resulting DNF may be read as follows:
{| align="center" cellpadding="8" style="text-align:center; width:90%"
{| align="center" cellpadding="8" style="text-align:center; width:60%"
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<pre>
<pre>
| [[Image:Equational Inference Bar -- Cast P.jpg|500px]]
| [[Image:Equational Inference Bar -- Cast P.jpg|500px]]
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-2.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-2 ISW.jpg|500px]]
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| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-3.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-3.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-5.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-5.jpg|500px]]
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| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-6.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-6 ISW.jpg|500px]]
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| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
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|}
This can be read to say <math>{}^{\backprime\backprime} \operatorname{either}~ p q r ~\operatorname{or}~ \operatorname{not}~ p {}^{\prime\prime},</math> which gives us yet another equivalent for the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> and the expression <math>\texttt{(} p \texttt{(} q r \texttt{))}.</math> Still another way of writing the same thing would be as follows:
This can be read to say <math>{}^{\backprime\backprime} \operatorname{either}~ p q r ~\operatorname{or}~ \operatorname{not}~ p {}^{\prime\prime},</math> which gives us yet another equivalent for the expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> and the expression <math>\texttt{(} p \texttt{(} q r \texttt{))}.</math> Still another way of writing the same thing would be as follows:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
In other words, <math>{}^{\backprime\backprime} p ~\operatorname{is~equivalent~to}~ p ~\operatorname{and}~ q ~\operatorname{and}~ r {}^{\prime\prime}.</math>
In other words, <math>{}^{\backprime\backprime} p ~\operatorname{is~equivalent~to}~ p ~\operatorname{and}~ q ~\operatorname{and}~ r {}^{\prime\prime}.</math>
One lemma that suggests itself at this point is a principle that may be canonized as the ''Emptiness Rule''. It says that a bare lobe expression like <math>\texttt{( \_, \_, \ldots )},</math> with any number of places for arguments but nothing but blanks as filler, is logically tantamount to the proto-typical expression of its type, namely, the constant expression <math>\texttt{(~)}</math> that <math>\operatorname{Ex}</math> interprets as denoting the logical value <math>\operatorname{false}.</math> To depict the rule in graphical form, we have the continuing sequence of equations:
One lemma that suggests itself at this point is a principle that may be canonized as the ''Emptiness Rule''. It says that a bare lobe expression like <math>\texttt{( \_, \_, \ldots )},</math> with any number of places for arguments but nothing but blanks as filler, is logically tantamount to the proto-typical expression of its type, namely, the constant expression <math>\texttt{(~)}</math> that <math>\operatorname{Ex}\!</math> interprets as denoting the logical value <math>\operatorname{false}.</math> To depict the rule in graphical form, we have the continuing sequence of equations:
{| align="center" cellpadding="8" style="text-align:center; width:90%"
{| align="center" cellpadding="8" style="text-align:center; width:60%"
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<pre>
<pre>
Yet another rule that we'll need is the following:
Yet another rule that we'll need is the following:
{| align="center" cellpadding="8" style="text-align:center; width:90%"
{| align="center" cellpadding="8" style="text-align:center; width:60%"
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<pre>
<pre>
This one is easy enough to derive from rules that are already known, but just for the sake of ready reference it is useful to canonize it as the ''Indistinctness Rule''. Finally, let me introduce a rule-of-thumb that is a bit more suited to routine computation, and that serves to replace the indistinctness rule in many cases where we actually have to call on it. This is actually just a special case of the evaluation rule listed above:
This one is easy enough to derive from rules that are already known, but just for the sake of ready reference it is useful to canonize it as the ''Indistinctness Rule''. Finally, let me introduce a rule-of-thumb that is a bit more suited to routine computation, and that serves to replace the indistinctness rule in many cases where we actually have to call on it. This is actually just a special case of the evaluation rule listed above:
{| align="center" cellpadding="8" style="text-align:center; width:90%"
{| align="center" cellpadding="8" style="text-align:center; width:60%"
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<pre>
<pre>
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-02.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-02.jpg|500px]]
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| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-03.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-03.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-08.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-08.jpg|500px]]
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| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-09.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-09.jpg|500px]]
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
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| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-14.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-14 ISW.jpg|500px]]
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| [[Image:Equational Inference Bar -- Emptiness.jpg|500px]]
| [[Image:Equational Inference Bar -- Emptiness.jpg|500px]]
: ''e''<sub>5</sub> = "(( (p (q))(p (r)) , (p (q r)) ))"
: ''e''<sub>5</sub> = "(( (p (q))(p (r)) , (p (q r)) ))"
Under <math>\operatorname{Ex}</math> we have the following interpretations:
Under <math>\operatorname{Ex}\!</math> we have the following interpretations:
: ''e''<sub>0</sub> expresses the logical constant "false"
: ''e''<sub>0</sub> expresses the logical constant "false"
Proof 2 lit on by burning the candle at both ends, changing ''e''<sub>2</sub> into a normal form that reduced to ''e''<sub>4</sub>, changing ''e''<sub>3</sub> into a normal form that reduced to e_4, in this way tethering ''e''<sub>2</sub> and ''e''<sub>3</sub> to a common point. We got that (p (q))(p (r)) is equal to (p q r, (p)), then we got that (p (q r)) is equal to (p q r, (p)), so we got that (p (q))(p (r)) is equal to (p (q r)).
Proof 2 lit on by burning the candle at both ends, changing ''e''<sub>2</sub> into a normal form that reduced to ''e''<sub>4</sub>, changing ''e''<sub>3</sub> into a normal form that reduced to e_4, in this way tethering ''e''<sub>2</sub> and ''e''<sub>3</sub> to a common point. We got that (p (q))(p (r)) is equal to (p q r, (p)), then we got that (p (q r)) is equal to (p q r, (p)), so we got that (p (q))(p (r)) is equal to (p (q r)).
Proof 3 took the path of reflection, expressing the meta-equation between ''e''<sub>2</sub> and ''e''<sub>3</sub> via the object equation ''e''<sub>5</sub>, then taking ''e''<sub>5</sub> as ''s''<sub>1</sub> and exchanging it by dint of value preserving steps for ''e''<sub>1</sub> as ''s''<sub>''n''</sub>. Thus we went from "(( (p (q))(p (r)) , (p (q r)) ))" to the blank expression that <math>\operatorname{Ex}</math> recognizes as true.
Proof 3 took the path of reflection, expressing the meta-equation between ''e''<sub>2</sub> and ''e''<sub>3</sub> via the object equation ''e''<sub>5</sub>, then taking ''e''<sub>5</sub> as ''s''<sub>1</sub> and exchanging it by dint of value preserving steps for ''e''<sub>1</sub> as ''s''<sub>''n''</sub>. Thus we went from "(( (p (q))(p (r)) , (p (q r)) ))" to the blank expression that <math>\operatorname{Ex}\!</math> recognizes as true.
I need to say something about the concept of ''reflection'' that I've been using according to my informal intuitions about it at numerous points in this discussion. This is, of course, distinct from the use of the word "reflection" to license an application of the double negation theorem.
I need to say something about the concept of ''reflection'' that I've been using according to my informal intuitions about it at numerous points in this discussion. This is, of course, distinct from the use of the word "reflection" to license an application of the double negation theorem.
* [[Charles Sanders Peirce|Peirce, C.S.]], [[Charles Sanders Peirce (Bibliography)|Bibliography]].
* [[Charles Sanders Peirce|Peirce, C.S.]], [[Charles Sanders Peirce (Bibliography)|Bibliography]].
* Peirce, C.S. (1902), [Application to the Carnegie Institution] (L 75), pp. 13–73 in ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Carolyn Eisele (ed.), Mouton, The Hague, 1976. [http://www.cspeirce.com/menu/library/bycsp/l75/l75.htm Eprint].
* Peirce, C.S. (1902), [Application to the Carnegie Institution] (L 75), pp. 13–73 in ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Carolyn Eisele (ed.), Mouton, The Hague, 1976. [http://www.cspeirce.com/menu/library/bycsp/l75/l75.htm Online].
* Peirce, C.S. (c. 1903), "Logical Tracts, No. 2", in ''Collected Papers'', CP 4.418–509. [http://www.existentialgraphs.com/peirceoneg/existentialgraphs4.418-529.htm Eprint].
* Peirce, C.S. (c. 1903), “Logical Tracts, No. 2”, in ''Collected Papers'', CP 4.418–509. [http://www.existentialgraphs.com/peirceoneg/existentialgraphs4.418-529.htm Online].
==See also==
==See also==
===Related essays and projects===
===Related essays and projects===
* [[Information Equals Comprehension Times Extension|Information = Comprehension × Extension]]
{{col-begin}}
* [[Inquiry Driven Systems]]
{{col-break}}
* [[Cactus Language]]
* [[Semiotic Information]]
* [[Peirce's Logic Of Information]]
* [[Peirce's 1870 Logic Of Relatives]]
* [[Information = Comprehension × Extension]]
{{col-break}}
* [[Introduction to Inquiry Driven Systems]]
* [[Introduction to Inquiry Driven Systems]]
* [[Peirce’s Logic Of Information]]
* [[Prospects for Inquiry Driven Systems]]
* [[Inquiry Driven Systems|Inquiry Driven Systems : Inquiry Into Inquiry]]
* [[Propositional Equation Reasoning Systems]]
* [[Propositional Equation Reasoning Systems]]
* [[Semiotic Theory Of Information]]
* [[Differential Logic : Introduction]]
{{col-end}}
===Related concepts and topics===
===Related concepts and topics===
==Document history==
==Document history==
===Inquiry List : Exposition (Oct–Nov 2005)===
===2005 • Inquiry List • Futures Of Logical Graphs • Exposition===
* http://web.archive.org/web/20150224133200/http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104
* http://web.archive.org/web/20140930152003/http://stderr.org/pipermail/inquiry/2005-November/thread.html#3165
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===2005 • Inquiry List • Futures Of Logical Graphs • Discussion===
* http://web.archive.org/web/20150224133200/http://stderr.org/pipermail/inquiry/2005-October/thread.html#3135
* http://web.archive.org/web/20140930152003/http://stderr.org/pipermail/inquiry/2005-November/thread.html#3167
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[[Category:Adaptive Systems]]
[[Category:Artificial Intelligence]]
[[Category:Automated Reasoning]]
[[Category:Boolean Algebra]]
[[Category:Boolean Functions]]
[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Combinatorics]]
[[Category:Computational Complexity]]
[[Category:Computer Science]]
[[Category:Computer Science]]
[[Category:Constraint Satisfaction]]
[[Category:Cybernetics]]
[[Category:Cybernetics]]
[[Category:Declarative Programming]]
[[Category:Differential Logic]]
[[Category:Equational Reasoning]]
[[Category:Equational Reasoning]]
[[Category:Formal Languages]]
[[Category:Formal Languages]]
[[Category:Formal Systems]]
[[Category:Formal Systems]]
[[Category:Functional Logic]]
[[Category:George Spencer Brown]]
[[Category:Graph Theory]]
[[Category:Graph Theory]]
[[Category:History of Logic]]
[[Category:History of Logic]]
[[Category:History of Mathematics]]
[[Category:History of Mathematics]]
[[Category:Inquiry]]
[[Category:Inquiry Driven Systems]]
[[Category:Intelligent Systems]]
[[Category:Knowledge Representation]]
[[Category:Laws Of Form]]
[[Category:Learning Systems]]
[[Category:Learning Theory]]
[[Category:Logic]]
[[Category:Logic]]
[[Category:Logical Graphs]]
[[Category:Logical Modeling]]
[[Category:Machine Learning]]
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Model Theory]]
[[Category:Philosophy]]
[[Category:Philosophy]]
[[Category:Programming]]
[[Category:Proof Theory]]
[[Category:Propositional Calculus]]
[[Category:Scientific Method]]
[[Category:Semiotics]]
[[Category:Semiotics]]
[[Category:Sign Relations]]
[[Category:Systems Engineering]]
[[Category:Systems Theory]]
[[Category:Visualization]]
[[Category:Visualization]]
