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{{DISPLAYTITLE:Peirce's Logic Of Information}}
{{DISPLAYTITLE:Peirce's Logic Of Information}}
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
==Peirce's concept of information==
==Peirce's concept of information==
I've been meaning to get back to [[Charles Sanders Peirce|Peirce]]'s theory of information, because I believe that it contains a yet-to-be-tapped potential for many current issues, though it would take just a little bit of drilling to exploit its resources to the fullest that we can.
I've been meaning to get back to Peirce's theory of information, because I believe that it contains a yet-to-be-tapped potential for many current issues, though it would take just a little bit of drilling to exploit its resources to the fullest that we can.
In my own imagination, I tend to organize Peirce's ideas about information, along with its relationship to comprehension and extension, in what certain accidents of personal history lead me to think of as the ''light-cone picture'' — but it's really just the two branches of a geometric cone, or the pencil that is generated by a point in a lattice or partial order, with no real connection to physics intended, at least, not so directly as the picture at first suggests:
In my own imagination, I tend to organize Peirce's ideas about information, along with its relationship to comprehension and extension, in what certain accidents of personal history lead me to think of as the ''light-cone picture'' — but it's really just the two branches of a geometric cone, or the pencil that is generated by a point in a lattice or partial order, with no real connection to physics intended, at least, not so directly as the picture at first suggests:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
o.......o Properties
\ /
\ /
\ /
O Object
/ \
/ \
/ \
o.......o Instances
</pre>
</pre>
|}
Here, the Object could be the object of a generic concept, like ''human'', or it could be the object of a dynamic concept, like ''system in motion''. Then the Instances are the extension of that concept through some space, and the Properties are the comprehension of that concept, or what would count as the intensions of the Object in question.
Here, the Object could be the object of a generic concept, like ''human'', or it could be the object of a dynamic concept, like ''system in motion''. Then the Instances are the extension of that concept through some space, and the Properties are the comprehension of that concept, or what would count as the intensions of the Object in question.
I will start with some loose threads that were left hanging the last time that I tangled with this topic, but here are some bits of background reading for those who would like to thread the maze on their own:
I will start with some loose threads that were left hanging the last time that I tangled with this topic, but here are some bits of background reading for those who would like to thread the maze on their own:
* ICE. Information = Comprehension x Extension
* ICE. Information = Comprehension x Extension
* ICE. http://forum.wolframscience.com/showthread.php?threadid=609
* ICE. http://web.archive.org/web/20140330191602/http://forum.wolframscience.com/showthread.php?threadid=609
* ICE. http://stderr.org/pipermail/inquiry/2004-November/thread.html#1913
* ICE. http://web.archive.org/web/20150302042625/http://stderr.org/pipermail/inquiry/2004-November/thread.html#1913
In my current state of information I'm aware of Peirce having investigated at least two different measures of information, one that we may call ''multiplicative'' and the other that we may call ''exponential''. Though I'll take up with the multiplicative measure first, it will be one of my interests here to understand the possible relation between these two measures of information and also their relation to current concepts of information.
In my current state of information I'm aware of Peirce having investigated at least two different measures of information, one that we may call ''multiplicative'' and the other that we may call ''exponential''. Though I'll take up with the multiplicative measure first, it will be one of my interests here to understand the possible relation between these two measures of information and also their relation to current concepts of information.
The conical picture that I drew before illustrates the multiplicative case, if we are given that we have ''j'' properties in the [[comprehension]] of object ''x'' and ''k'' instances in the [[extension]] of object ''x'', then the information measure associated with object ''x'' is given by the product ''jk'', a product that also counts the number of edges in the complete bigraph between the two domains of properties and instances.
The conical picture that I drew before illustrates the multiplicative case, if we are given that we have ''j'' properties in the [[comprehension]] of object ''x'' and ''k'' instances in the [[extension]] of object ''x'', then the information measure associated with object ''x'' is given by the product ''jk'', a product that also counts the number of edges in the complete bigraph between the two domains of properties and instances.
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
o.......o Properties
\ /
\ /
\ /
O Object
/ \
/ \
/ \
o.......o Instances
</pre>
</pre>
|}
This ''area measure'' of information, given by Peirce's formula, "Information = Comprehension × Extension", is the background against which the change of information associated with all further constraints can be reckoned.
This ''area measure'' of information, given by Peirce's formula, “Information = Comprehension × Extension”, is the background against which the change of information associated with all further constraints can be reckoned.
That's a very rough sketch of just one aspect of Peirce's theory, but we can refine it as we go. The next order of business, though, is to balance these abstract speculations with a few well-posed concrete examples.
That's a very rough sketch of just one aspect of Peirce's theory, but we can refine it as we go. The next order of business, though, is to balance these abstract speculations with a few well-posed concrete examples.
It's not just the genesis but the genius of inquiry that it begins in a muddle, that is to say, a muddled sort of experiential situation where the experiencer thereof may have a vague sense that this moment of experience has that mark of quality and this other moment of experience has that other mark of quality but can hardly say with any degree of certainty anything more than that. So let's define a ''muddle'' as a situation that can be represented in the following sort of bigraph model:
It's not just the genesis but the genius of inquiry that it begins in a muddle, that is to say, a muddled sort of experiential situation where the experiencer thereof may have a vague sense that this moment of experience has that mark of quality and this other moment of experience has that other mark of quality but can hardly say with any degree of certainty anything more than that. So let's define a ''muddle'' as a situation that can be represented in the following sort of bigraph model:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
q_1 q_2 q_3 q_4 q_5
o o o o o
\ |\ /|\ /| /
\ | \ / | \ / | /
\ | \ | \ | /
\ | / \ | / \ | /
\|/ \|/ \|/
o o o
m_1 m_2 m_3
</pre>
</pre>
|}
For concreteness, I have illustrated a muddle where the moment ''m''<sub>1</sub> has the qualities ''q''<sub>1</sub>, ''q''<sub>2</sub>, ''q''<sub>3</sub>, the moment ''m''<sub>2</sub> has the qualities ''q''<sub>2</sub>, ''q''<sub>3</sub>, ''q''<sub>4</sub>, and the moment ''m''<sub>3</sub> has the qualities ''q''<sub>3</sub>, ''q''<sub>4</sub>, ''q''<sub>5</sub>, but arbitrary muddles can be far more muddled than that and yet fall within the bounds of the bigraph model.
For concreteness, I have illustrated a muddle where the moment ''m''<sub>1</sub> has the qualities ''q''<sub>1</sub>, ''q''<sub>2</sub>, ''q''<sub>3</sub>, the moment ''m''<sub>2</sub> has the qualities ''q''<sub>2</sub>, ''q''<sub>3</sub>, ''q''<sub>4</sub>, and the moment ''m''<sub>3</sub> has the qualities ''q''<sub>3</sub>, ''q''<sub>4</sub>, ''q''<sub>5</sub>, but arbitrary muddles can be far more muddled than that and yet fall within the bounds of the bigraph model.
We have in fact returned to a sort of situation that precurses the constellation of the light-cone picture that I drew before:
We have in fact returned to a sort of situation that precurses the constellation of the light-cone picture that I drew before:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
o.......o Properties
\ /
\ /
\ /
O Object
/ \
/ \
/ \
o.......o Instances
</pre>
</pre>
|}
The progression from the muddle of experience to the spindle of necessity begins with a step of abductive reasoning or hypothesis formation, a step that integrates the manifold of sensuous impressions by factoring it over the medium of unitary concepts.
The progression from the muddle of experience to the spindle of necessity begins with a step of abductive reasoning or hypothesis formation, a step that integrates the manifold of sensuous impressions by factoring it over the medium of unitary concepts.
Here is a programme for keeping track of the players, that is, the various alternative and associated terms that might occur in the following discussion of Peirce's theory of information:
Here is a programme for keeping track of the players, that is, the various alternative and associated terms that might occur in the following discussion of Peirce's theory of information:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
Predicates Qualities Comprehension Connotation Content Depth
Properties o.......o
Characters \ /
\ /
\ / Concept
Object, Class O<----o Symbol
/ \ Term
/ \
Elements / \
Instances o.......o
Specimens Exemplars Extension Denotation Sphere Breadth
</pre>
</pre>
|}
I am deploying an ample plurality of rhetorical alternatives partly in order to emphasize the fact that the focus here is not on defining entities like exemplars and qualities but on the shape of the incidence relation between their rough ilks. Some of the terms lodged in the same stratum of the paradigm are close synonyms, in other cases, they are measures of the associated dimensions. I have placed ''class'' in accord with contemporary usage in set theory, as my sense of the reading tells me that Peirce is using ''class'' in a unitary sense, in contrast with his use of ''sphere'' in more distributive sense.
I am deploying an ample plurality of rhetorical alternatives partly in order to emphasize the fact that the focus here is not on defining entities like exemplars and qualities but on the shape of the incidence relation between their rough ilks. Some of the terms lodged in the same stratum of the paradigm are close synonyms, in other cases, they are measures of the associated dimensions. I have placed ''class'' in accord with contemporary usage in set theory, as my sense of the reading tells me that Peirce is using ''class'' in a unitary sense, in contrast with his use of ''sphere'' in more distributive sense.
A rough plot may serve to illustrate certain aspects of the situation. For convenience in graphing, let's indicate the measure of information by means of the symbols [1, ..., 9, a, b, c, d, e, f, *] to signify the range of integral values from 1 to 16. If we can imagine interpolating smooth curves through the symbols plotted for a given information value, then that would give us some idea of the ''constant information curves'' or the ''isoformal curves'' of this information topography.
A rough plot may serve to illustrate certain aspects of the situation. For convenience in graphing, let's indicate the measure of information by means of the symbols [1, ..., 9, a, b, c, d, e, f, *] to signify the range of integral values from 1 to 16. If we can imagine interpolating smooth curves through the symbols plotted for a given information value, then that would give us some idea of the ''constant information curves'' or the ''isoformal curves'' of this information topography.
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
Depth
^
|
| *
| f
| e
| d
| c
| b
| a
| 9
| 8 *
| 7 e
| 6 c
| 5 a f
| 4 8 c *
| 3 6 9 c f
| 2 4 6 8 a c e *
| 1 2 3 4 5 6 7 8 9 a b c d e f *
o----------------------------------> Breadth
</pre>
</pre>
|}
In struggling to sketch some of the Figures that I need for the next part of this discussion, I am finding the plaintext format so constraining that I think it has become almost indispensable to introduce a standard alternative form of representation for the requisite graphs, specifically, their representation in terms of ''adjacency matrices'' or ''incidence matrices''.
In struggling to sketch some of the Figures that I need for the next part of this discussion, I am finding the plaintext format so constraining that I think it has become almost indispensable to introduce a standard alternative form of representation for the requisite graphs, specifically, their representation in terms of ''adjacency matrices'' or ''incidence matrices''.
For example, consider the bigraph that we drew before:
For example, consider the bigraph that we drew before:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
q_1 q_2 q_3 q_4 q_5
o o o o o
\ |\ /|\ /| /
\ | \ / | \ / | /
\ | \ | \ | /
\ | / \ | / \ | /
\|/ \|/ \|/
o o o
e_1 e_2 e_3
</pre>
</pre>
|}
This can be represented by means of the following matrix:
This can be represented by means of the following matrix:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
e_1 e_2 e_3
o-------------------o
| |
q_1 | 1 0 0 |
| |
q_2 | 1 1 0 |
| |
q_3 | 1 1 1 |
| |
q_4 | 0 1 1 |
| |
q_5 | 0 0 1 |
| |
o-------------------o
</pre>
</pre>
|}
Here, the occurrence of a "1" at the intersection of ''e''<sub>''j''</sub> street and ''q''<sub>''k''</sub> avenue indicates that there is an edge connecting ''e''<sub>''j''</sub> and ''q''<sub>''k''</sub>, and the occurrence of a "0" says otherwise.
Here, the occurrence of a "1" at the intersection of ''e''<sub>''j''</sub> street and ''q''<sub>''k''</sub> avenue indicates that there is an edge connecting ''e''<sub>''j''</sub> and ''q''<sub>''k''</sub>, and the occurrence of a "0" says otherwise.
It had at first sight seemed to me that our protagonist's initial state of information could be represented like so:
It had at first sight seemed to me that our protagonist's initial state of information could be represented like so:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
Color o
|
|
|
|
|
Red o
/|\
/ | \
/ | \
/ | \
/ | \
o o o
A B C
</pre>
</pre>
|}
But on further reflection it seems to me that, in the absence of information to the contrary, our hero doesn't really know yet which of the terms ''color'' or ''red'', if either, to place above the other. In other words, as far as he can know at this stage of the game, red could just as well be the only color in town, making ''being red'' and ''having color'' into indiscernible predicates.
But on further reflection it seems to me that, in the absence of information to the contrary, our hero doesn't really know yet which of the terms ''color'' or ''red'', if either, to place above the other. In other words, as far as he can know at this stage of the game, red could just as well be the only color in town, making ''being red'' and ''having color'' into indiscernible predicates.
On this account, the agent's initial state of knowledge is more accurately represented by the following bigraph:
On this account, the agent's initial state of knowledge is more accurately represented by the following bigraph:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
A B C
o-------------------o
| |
Color | 1 1 1 |
| |
Red | 1 1 1 |
| |
o-------------------o
</pre>
</pre>
|}
I have no idea if this is the proper way to see things. But we can always backtrack to this point if necessary.
I have no idea if this is the proper way to see things. But we can always backtrack to this point if necessary.
Consider the graphical matrix that we last looked on:
Consider the graphical matrix that we last looked on:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
A B C
o-------------------o
| |
Color | 1 1 1 |
| |
Red | 1 1 1 |
| |
o-------------------o
</pre>
</pre>
|}
This is the incidence matrix of a ''concretely labeled graph''.
This is the incidence matrix of a ''concretely labeled graph''.
Taking one small step of abstraction up from there would give us the incidence matrix of an ''abstractly labeled graph'', like so:
Taking one small step of abstraction up from there would give us the incidence matrix of an ''abstractly labeled graph'', like so:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
1 2 3
o-------------------o
| |
4 | 1 1 1 |
| |
5 | 1 1 1 |
| |
o-------------------o
</pre>
</pre>
|}
This is the incidence matrix of a ''labeled bipartite graph'', or a ''labeled bigraph'', for short. The adjective ''bipartite'' means that its set of points (aka nodes or vertices) can be partitioned into just two parts, such that all the lines (aka edges or blocks) of the graph lie between the points of one part of the partition and the points of the other part of the partition, with no lines that lie between the points in any one part of the partition.
This is the incidence matrix of a ''labeled bipartite graph'', or a ''labeled bigraph'', for short. The adjective ''bipartite'' means that its set of points (aka nodes or vertices) can be partitioned into just two parts, such that all the lines (aka edges or blocks) of the graph lie between the points of one part of the partition and the points of the other part of the partition, with no lines that lie between the points in any one part of the partition.
Though the most careful among us will occasionally slip up and use the terms interchangeably, there is a slight but significant nuance of distinction between ''adjacency matrices'' and ''incidence matrices''. The incidence matrix of our labeled representative of ''K''<sub>2,3</sub> uses the underlying partition of points to economize the dimensions of matric materiel, tantamount to treating the two parts as if they contained different types of points, thus allowing the following compact form:
Though the most careful among us will occasionally slip up and use the terms interchangeably, there is a slight but significant nuance of distinction between ''adjacency matrices'' and ''incidence matrices''. The incidence matrix of our labeled representative of ''K''<sub>2,3</sub> uses the underlying partition of points to economize the dimensions of matric materiel, tantamount to treating the two parts as if they contained different types of points, thus allowing the following compact form:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
1 2 3
o-------------------o
| |
4 | 1 1 1 |
| |
5 | 1 1 1 |
| |
o-------------------o
</pre>
</pre>
|}
The economy of the incidence matrix, in those cases where it's available, will be evident if we compare it with the corresponding adjacency matrix:
The economy of the incidence matrix, in those cases where it's available, will be evident if we compare it with the corresponding adjacency matrix:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
1 2 3 4 5
o-------------------------------o
| |
1 | 0 0 0 1 1 |
| |
2 | 0 0 0 1 1 |
| |
3 | 0 0 0 1 1 |
| |
4 | 1 1 1 0 0 |
| |
5 | 1 1 1 0 0 |
| |
o-------------------------------o
</pre>
</pre>
|}
The adjacency matrix exhibits a 1 at the intersection of row ''j'' and column ''k'' if and only point ''j'' and point ''k'' are ''adjacent'', that is, connected by a line, and it exhibits a 0 otherwise. Since we are presently discussing graphs, where each line runs in two directions, that is, not directed graphs, or ''digraphs'', the adjacency matrix is symmetric about the main diagonal.
The adjacency matrix exhibits a 1 at the intersection of row ''j'' and column ''k'' if and only point ''j'' and point ''k'' are ''adjacent'', that is, connected by a line, and it exhibits a 0 otherwise. Since we are presently discussing graphs, where each line runs in two directions, that is, not directed graphs, or ''digraphs'', the adjacency matrix is symmetric about the main diagonal.
For instance, consider the graph that is pictured as follows:
For instance, consider the graph that is pictured as follows:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
o
/|\
/ | \
/ | \
o o o
\ | /
\ | /
\|/
o
</pre>
</pre>
|}
There is only one graph of this description. It is in fact our old friend ''K''<sub>2,3</sub>, as becomes manifest when the relevant partition of points is disclosed. There are, in comparison, 10 distinct labeled graphs that share this underlying graph:
There is only one graph of this description. It is in fact our old friend ''K''<sub>2,3</sub>, as becomes manifest when the relevant partition of points is disclosed. There are, in comparison, 10 distinct labeled graphs that share this underlying graph:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
1 1 1 1 2 2 2 3 3 4
/|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\
3 4 5 2 4 5 2 3 5 2 3 4 1 4 5 1 3 5 1 3 4 1 2 5 1 2 4 1 2 3
\|/ \|/ \|/ \|/ \|/ \|/ \|/ \|/ \|/ \|/
2 3 4 5 3 4 5 4 5 5
</pre>
</pre>
|}
Let's see if we have enough tackle to haul our first example into the light:
Let's see if we have enough tackle to haul our first example into the light:
Supposing for the moment that we begin with the minimal model, to wit, the "complete muddle" that has the form of the complete bigraph ''K''<sub>2,3</sub>, it's tantamount to preposing prior to the place where Peirce comes in a primal experience that is even more inchoate than the one where our hero takes the term ''red'' as denoting a class or an object of thought:
Supposing for the moment that we begin with the minimal model, to wit, the "complete muddle" that has the form of the complete bigraph ''K''<sub>2,3</sub>, it's tantamount to preposing prior to the place where Peirce comes in a primal experience that is even more inchoate than the one where our hero takes the term ''red'' as denoting a class or an object of thought:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
A B C
o-------------------o
| |
Color | 1 1 1 |
| |
Red | 1 1 1 |
| |
o-------------------o
</pre>
</pre>
|}
Initiated into this predicament of buzzing booming confusion, the most that we have is three moments of experience, ''A'', ''B'', ''C'', each of which has the same two marks of quality, ''Color'' and ''Red''. It doesn't make any real difference if we call them ''Exemplars'' and ''Qualities'', respectively, so for brevity we might as well.
Initiated into this predicament of buzzing booming confusion, the most that we have is three moments of experience, ''A'', ''B'', ''C'', each of which has the same two marks of quality, ''Color'' and ''Red''. It doesn't make any real difference if we call them ''Exemplars'' and ''Qualities'', respectively, so for brevity we might as well.
The odd thing about the maximal muddle, in so far as it's constrained to fill out the given frame, is that it's actually one of the easiest to bring to order through the simple act of introducing a middle term:
The odd thing about the maximal muddle, in so far as it's constrained to fill out the given frame, is that it's actually one of the easiest to bring to order through the simple act of introducing a middle term:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
Color Red
o o
\ /
\ /
\ /
\ /
\ /
Intermediate Object o<<<<<<<<<<<o Middle Term
/|\
/ | \
/ | \
/ | \
/ | \
o o o
A B C
</pre>
</pre>
|}
Except for the chevroned relation of denotation from the middle term to the intermediate object, this is cast as a lattice diagram, where transitive closure of lattice relations is understood to be in force. Accordingly, there is an implicit line between each point at the top and each point at the bottom, forming the same ''K''<sub>2,3</sub> on these points that we had before. With regard to the middling object that we used to factor the initial muddle, it has a breadth of 3 and a depth of 2. At least that's 1 way to look at it. I'm not saying it's the only 1.
Except for the chevroned relation of denotation from the middle term to the intermediate object, this is cast as a lattice diagram, where transitive closure of lattice relations is understood to be in force. Accordingly, there is an implicit line between each point at the top and each point at the bottom, forming the same ''K''<sub>2,3</sub> on these points that we had before. With regard to the middling object that we used to factor the initial muddle, it has a breadth of 3 and a depth of 2. At least that's 1 way to look at it. I'm not saying it's the only 1.
The use of the phrase "total fact" recalls the classical articulations of the three kinds of inference — abduction, deduction, induction — by way of Cases, Facts, and Rules, as prefigured in the following generic sketch:
The use of the phrase "total fact" recalls the classical articulations of the three kinds of inference — abduction, deduction, induction — by way of Cases, Facts, and Rules, as prefigured in the following generic sketch:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
o
|\
| \
| \
| \ Rule
| \
| \
| \
Fact | o
| /
| /
| /
| / Case
| /
| /
|/
o
</pre>
</pre>
|}
In trying to form a minimal model of Peirce's story, which we must be able to do if his story is the least bit consistent, we find ourselves engaged in two different types of reasoning in parallel with each other. At one level, we are thinking out the constitution of a particular model, also known as a ''situation'', a ''structure'', or a ''universe of discourse''. At another level, we are thinking about the structures of many alternative models, many different imaginary scenarios that might flesh out the script. The former is actually a ''zeroth order model'' (ZOM), and can be constructed within the bounds of propositional logic or partial orderings of propositions. Getting to the next level will take a few ''higher order propositions'' (HOP's), that is, propositions about propositions, or propositions about the structures of whole different universes of discourse. That is more or less tantamount to the ''first order predicate calculus'', or the ''logic of quantified propositions''. However, if we are really satisfied with finding a ''minimal'' model, any way that we can arrange one, and don't really need a complete analysis of the story, then it is possible to carry out almost all of the reasoning in and about a simple ZOM.
In trying to form a minimal model of Peirce's story, which we must be able to do if his story is the least bit consistent, we find ourselves engaged in two different types of reasoning in parallel with each other. At one level, we are thinking out the constitution of a particular model, also known as a ''situation'', a ''structure'', or a ''universe of discourse''. At another level, we are thinking about the structures of many alternative models, many different imaginary scenarios that might flesh out the script. The former is actually a ''zeroth order model'' (ZOM), and can be constructed within the bounds of propositional logic or partial orderings of propositions. Getting to the next level will take a few ''higher order propositions'' (HOP's), that is, propositions about propositions, or propositions about the structures of whole different universes of discourse. That is more or less tantamount to the ''first order predicate calculus'', or the ''logic of quantified propositions''. However, if we are really satisfied with finding a ''minimal'' model, any way that we can arrange one, and don't really need a complete analysis of the story, then it is possible to carry out almost all of the reasoning in and about a simple ZOM.
The same raw data might be given in lattice form as follows:
The same raw data might be given in lattice form as follows:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
Color Color Color Red Red Red
o o o o o o
| | | | | |
| | | | | |
| | | | | |
o o o o o o
A B C A B C
</pre>
</pre>
|}
Arbitrary congeries of empirical data or synthetic facts can be represented in just these ways. As such, the ''total fact'' embodied in them is perfectly good information, as far as information goes. But the particular collection of empirical data or synthetic facts embodied in the above representations is not arbitrary — it has a special structure that permits it to be ''factored'' through the medium of a hypothetical object ''M'' and a middle term as follows:
Arbitrary congeries of empirical data or synthetic facts can be represented in just these ways. As such, the ''total fact'' embodied in them is perfectly good information, as far as information goes. But the particular collection of empirical data or synthetic facts embodied in the above representations is not arbitrary — it has a special structure that permits it to be ''factored'' through the medium of a hypothetical object ''M'' and a middle term as follows:
Or, to capture the total fact in a single picture:
Or, to capture the total fact in a single picture:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
Color, Red
o
|\
| \
| \
| \ Rule
| \
| \
| \
Fact | o M
| /
| /
| /
| / Case
| /
| /
|/
o
A, B, C
</pre>
</pre>
|}
Not every state of information allows the interpolation of a compact object with its own comprehension and extension, but this epistemic situation does.
Not every state of information allows the interpolation of a compact object with its own comprehension and extension, but this epistemic situation does.
The following Figure will give us some hint of the situation:
The following Figure will give us some hint of the situation:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
Color
o
/ \
/ \
/ \
/ \
/ \
Red o \
/|\ \
/ | \ \
/ | \ \
/ | \ \
/ | \ \
o o o o
A B C D
</pre>
</pre>
|}
This seems to suggest that the term ''Red'' has increased in depth while remaining the same in breadth, but that the term ''Color'' has increased in breadth while remaining the same in depth.
This seems to suggest that the term ''Red'' has increased in depth while remaining the same in breadth, but that the term ''Color'' has increased in breadth while remaining the same in depth.
We have of course just opened up a whole new can of worms — like they say, a single picture is worth a thousand worms:
We have of course just opened up a whole new can of worms — like they say, a single picture is worth a thousand worms:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
Celestial Body
*
/ \
/ \
/ \
/ \
/ \
Morning Star * * Evening Star
. . . .
. . . .
. . . .
. . . .
. . . .
* * *
? ? ?
</pre>
</pre>
|}
Here's the thousand worms part. Peirce exploits the literary device of a blind man's knowledge of color in order to rule out knowledge by acquaintance, leaving open only the channel of knowledge by being told. The real reason for doing this is of course that it forces us observers of the information process involved to make all the information explicit. In a more reflective moment we might also think to question our customary supposition that the putatively distinct modes of information acquisition are really all that different, logically analyzed, but save that for later.
Here's the thousand worms part. Peirce exploits the literary device of a blind man's knowledge of color in order to rule out knowledge by acquaintance, leaving open only the channel of knowledge by being told. The real reason for doing this is of course that it forces us observers of the information process involved to make all the information explicit. In a more reflective moment we might also think to question our customary supposition that the putatively distinct modes of information acquisition are really all that different, logically analyzed, but save that for later.
Let us now contemplate the proposition ''Red'' ⇒ ''Hued'', henceforth ''R'' ⇒ ''H'', and what it says in the context of a suitable universe of discourse ''X''. The import becomes strikingly evident in the existential graph syntax, where ''R'' ⇒ ''H'' takes the form (''R'' (''H'')), making manifest that it excludes the existence of anything in the universe ''X'' from the region indicated by the propositional specification ''R'' (''H''), that is, "Red and not Hued". The situation can be diagrammed in a rough lattice fashion as follows:
Let us now contemplate the proposition ''Red'' ⇒ ''Hued'', henceforth ''R'' ⇒ ''H'', and what it says in the context of a suitable universe of discourse ''X''. The import becomes strikingly evident in the existential graph syntax, where ''R'' ⇒ ''H'' takes the form (''R'' (''H'')), making manifest that it excludes the existence of anything in the universe ''X'' from the region indicated by the propositional specification ''R'' (''H''), that is, "Red and not Hued". The situation can be diagrammed in a rough lattice fashion as follows:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
X
o
/ \
/ \
/ \
/ \
/ \
H o o (H)
/ \ \
/ \ \
/ \ \
/ \ \
/ \ \
R o o (R) o (R)
</pre>
</pre>
|}
In short, the formal existence of things under the propositional descriptions ''H'' (''R''), Hued Non-Reds, and (''H'')(''R''), Non-Hued Non-Reds, provides the asymmetry needed for a proper order relation ''R'' ⇒ ''H''.
In short, the formal existence of things under the propositional descriptions ''H'' (''R''), Hued Non-Reds, and (''H'')(''R''), Non-Hued Non-Reds, provides the asymmetry needed for a proper order relation ''R'' ⇒ ''H''.
Cosmetically revising ''H'' for ''Hued'' for things that have Color and superficially adding ''X'' for our tale's encompassing cosmos, we have come to form the following picture of the overall scene:
Cosmetically revising ''H'' for ''Hued'' for things that have Color and superficially adding ''X'' for our tale's encompassing cosmos, we have come to form the following picture of the overall scene:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
X
o
/ \
/ \
/ \
/ \
/ \
H o o (H)
/ \ \
/ \ \
/ \ \
/ \ \
/ \ \
R o o (R) o (R)
/|\
/ | \
/ | \
/ | \
/ | \
o o o
A B C
</pre>
</pre>
|}
We must hold in suspension many questions about the comprehension of ''Red'', till better comprehending what makes portions of comprehension superfluous, but there doesn't seem to be the same order of difficulty here with saying that the extension of ''Red'' is {''A'', ''B'', ''C''}, so let us explore that devolving branch of our conical configuration a little further.
We must hold in suspension many questions about the comprehension of ''Red'', till better comprehending what makes portions of comprehension superfluous, but there doesn't seem to be the same order of difficulty here with saying that the extension of ''Red'' is {''A'', ''B'', ''C''}, so let us explore that devolving branch of our conical configuration a little further.
This much we can do in any logical calculus, grammar, language, or syntax that is adequate to express the expressions of ZOL, but there are certain pragmatic benefits that accrue to a more efficient form of representation.
This much we can do in any logical calculus, grammar, language, or syntax that is adequate to express the expressions of ZOL, but there are certain pragmatic benefits that accrue to a more efficient form of representation.
Let's draw the sub-universe (subverse?) of Red things in euler-venn style, the heavier shading (^) for being's rule, the lighter (`) for its absence:
Let's draw the sub-universe (subverse?) of Red things in euler-venn style, the heavier shading (^) for being's rule, the lighter ( ) for its absence:
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| ` ` ` ` ` ` ` ` ` ` `o-------------o` ` ` ` ` ` ` ` ` ` ` |
| o-------------o |
| ` ` ` ` ` ` ` ` ` ` / ^ ^ ^ ^ ^ ^ ^ \ ` ` ` ` ` ` ` ` ` ` |
| / ^ ^ ^ ^ ^ ^ ^ \ |
| ` ` ` ` ` ` ` ` ` `/^ ^ ^ ^ ^ ^ ^ ^ ^\` ` ` ` ` ` ` ` ` ` |
| /^ ^ ^ ^ ^ ^ ^ ^ ^\ |
| ` ` ` ` ` ` ` ` ` / ^ ^ ^ ^ ^ ^ ^ ^ ^ \ ` ` ` ` ` ` ` ` ` |
| / ^ ^ ^ ^ ^ ^ ^ ^ ^ \ |
| ` ` ` ` ` ` ` ` `/^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^\` ` ` ` ` ` ` ` ` |
| /^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^\ |
| ` ` ` ` ` ` ` ` / ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ \ ` ` ` ` ` ` ` ` |
| / ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ \ |
| ` ` ` ` ` ` ` `o^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^o` ` ` ` ` ` ` ` |
| o^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^o |
| ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` |
| |^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^| |
| ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ A ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` |
| |^ ^ ^ ^ ^ ^ A ^ ^ ^ ^ ^ ^| |
| ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` |
| |^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^| |
| ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` |
| |^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^| |
| ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` |
| |^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^| |
| ` ` ` ` ` ` 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 |
| ` ` ` | ^ ^ ^ ^ ^ ^ ^ ^ | ` ` ` | ^ ^ ^ ^ ^ ^ ^ ^ | ` ` ` |
| | ^ ^ ^ ^ ^ ^ ^ ^ | | ^ ^ ^ ^ ^ ^ ^ ^ | |
| ` ` ` | ^ ^ ^ ^ ^ ^ ^ ^ | ` ` ` | ^ ^ ^ ^ ^ ^ ^ ^ | ` ` ` |
| | ^ ^ ^ ^ ^ ^ ^ ^ | | ^ ^ ^ ^ ^ ^ ^ ^ | |
| ` ` ` | ^ ^ ^ ^ ^ ^ ^ ^ | ` ` ` | ^ ^ ^ ^ ^ ^ ^ ^ | ` ` ` |
| | ^ ^ ^ ^ ^ ^ ^ ^ | | ^ ^ ^ ^ ^ ^ ^ ^ | |
| ` ` ` | ^ ^ ^ B ^ ^ ^ ^ | ` ` ` | ^ ^ ^ ^ C ^ ^ ^ | ` ` ` |
| | ^ ^ ^ B ^ ^ ^ ^ | | ^ ^ ^ ^ C ^ ^ ^ | |
| ` ` ` | ^ ^ ^ ^ ^ ^ ^ ^ | ` ` ` | ^ ^ ^ ^ ^ ^ ^ ^ | ` ` ` |
| | ^ ^ ^ ^ ^ ^ ^ ^ | | ^ ^ ^ ^ ^ ^ ^ ^ | |
| ` ` ` o ^ ^ ^ ^ ^ ^ ^ ^ o ` ` ` o ^ ^ ^ ^ ^ ^ ^ ^ o ` ` ` |
| o ^ ^ ^ ^ ^ ^ ^ ^ o o ^ ^ ^ ^ ^ ^ ^ ^ o |
| ` ` ` `\^ ^ ^ ^ ^ ^ ^ ^ ^\` ` `/^ ^ ^ ^ ^ ^ ^ ^ ^/` ` ` ` |
| \^ ^ ^ ^ ^ ^ ^ ^ ^\ /^ ^ ^ ^ ^ ^ ^ ^ ^/ |
| ` ` ` ` \ ^ ^ ^ ^ ^ ^ ^ ^ \ ` / ^ ^ ^ ^ ^ ^ ^ ^ / ` ` ` ` |
| \ ^ ^ ^ ^ ^ ^ ^ ^ \ / ^ ^ ^ ^ ^ ^ ^ ^ / |
| ` ` ` ` `\^ ^ ^ ^ ^ ^ ^ ^ ^\`/^ ^ ^ ^ ^ ^ ^ ^ ^/` ` ` ` ` |
| \^ ^ ^ ^ ^ ^ ^ ^ ^\ /^ ^ ^ ^ ^ ^ ^ ^ ^/ |
| ` ` ` ` ` \ ^ ^ ^ ^ ^ ^ ^ ^ o ^ ^ ^ ^ ^ ^ ^ ^ / ` ` ` ` ` |
| \ ^ ^ ^ ^ ^ ^ ^ ^ o ^ ^ ^ ^ ^ ^ ^ ^ / |
| ` ` ` ` ` `\^ ^ ^ ^ ^ ^ ^ ^/`\^ ^ ^ ^ ^ ^ ^ ^/` ` ` ` ` ` |
| \^ ^ ^ ^ ^ ^ ^ ^/ \^ ^ ^ ^ ^ ^ ^ ^/ |
| ` ` ` ` ` ` o-------------o ` o-------------o ` ` ` ` ` ` |
| o-------------o o-------------o |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
|}
In their imports for this sub-universe, the three propositional inequalities, ''A'' ≠ ''B'', ''B'' ≠ ''C'', ''C'' ≠ ''A'', so constrain the qualities of being ''A'', ''B'', ''C'', respectively, that just one of the corresponding propositions can be true of any given thing. The whole region marked by circumflecks is the rule of ''Red''.
In their imports for this sub-universe, the three propositional inequalities, ''A'' ≠ ''B'', ''B'' ≠ ''C'', ''C'' ≠ ''A'', so constrain the qualities of being ''A'', ''B'', ''C'', respectively, that just one of the corresponding propositions can be true of any given thing. The whole region marked by circumflecks is the rule of ''Red''.
|
|
| o-------------------o-------------------o
| o-------------------o-------------------o
| | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| | | |
| | dotted` ` ` ` ` ` | 4 circles ` ` ` ` |
| | dotted | 4 circles |
| | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| | | |
| o-------------------o-------------------o
| o-------------------o-------------------o
| | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| | | |
| | dotted & crossed` | 2 circles ` ` ` ` |
| | dotted & crossed | 2 circles |
| | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| | | |
| o-------------------o-------------------o
| o-------------------o-------------------o
|
|
==Document history==
==Document history==
===Peirce's Logic Of Information (2005)===
===2005 • Inquiry List • Peirce's Logic Of Information===
* http://stderr.org/pipermail/inquiry/2005-November/thread.html#3254
* http://web.archive.org/web/20140930152003/http://stderr.org/pipermail/inquiry/2005-November/thread.html#3254
* http://stderr.org/pipermail/inquiry/2005-December/thread.html#3273
* http://web.archive.org/web/20120512004315/http://stderr.org/pipermail/inquiry/2005-December/thread.html#3273
# http://stderr.org/pipermail/inquiry/2005-November/003254.html
# http://web.archive.org/web/20140930152801/http://stderr.org/pipermail/inquiry/2005-November/003254.html
# http://stderr.org/pipermail/inquiry/2005-November/003255.html
# http://web.archive.org/web/20140927150048/http://stderr.org/pipermail/inquiry/2005-November/003255.html
# http://stderr.org/pipermail/inquiry/2005-November/003259.html
# http://web.archive.org/web/20140927150049/http://stderr.org/pipermail/inquiry/2005-November/003259.html
# http://stderr.org/pipermail/inquiry/2005-November/003260.html
# http://web.archive.org/web/20120210072641/http://stderr.org/pipermail/inquiry/2005-November/003260.html
# http://stderr.org/pipermail/inquiry/2005-November/003262.html
# http://web.archive.org/web/20081007063200/http://stderr.org/pipermail/inquiry/2005-November/003262.html
# http://stderr.org/pipermail/inquiry/2005-November/003266.html
# http://web.archive.org/web/20120210073319/http://stderr.org/pipermail/inquiry/2005-November/003266.html
# http://stderr.org/pipermail/inquiry/2005-November/003267.html
# http://web.archive.org/web/20120210073101/http://stderr.org/pipermail/inquiry/2005-November/003267.html
# http://stderr.org/pipermail/inquiry/2005-November/003268.html
# http://web.archive.org/web/20120229142008/http://stderr.org/pipermail/inquiry/2005-November/003268.html
# http://stderr.org/pipermail/inquiry/2005-November/003270.html
# http://web.archive.org/web/20120210073017/http://stderr.org/pipermail/inquiry/2005-November/003270.html
# http://stderr.org/pipermail/inquiry/2005-November/003271.html
# http://web.archive.org/web/20120229142645/http://stderr.org/pipermail/inquiry/2005-November/003271.html
# http://stderr.org/pipermail/inquiry/2005-December/003273.html
# http://web.archive.org/web/20081012073104/http://stderr.org/pipermail/inquiry/2005-December/003273.html
# http://stderr.org/pipermail/inquiry/2005-December/003275.html
# http://web.archive.org/web/20080828012728/http://stderr.org/pipermail/inquiry/2005-December/003275.html
# http://stderr.org/pipermail/inquiry/2005-December/003285.html
# http://web.archive.org/web/20080829073702/http://stderr.org/pipermail/inquiry/2005-December/003285.html
# http://stderr.org/pipermail/inquiry/2005-December/003286.html
# http://web.archive.org/web/20080828012439/http://stderr.org/pipermail/inquiry/2005-December/003286.html
# http://stderr.org/pipermail/inquiry/2005-December/003287.html
# http://web.archive.org/web/20081006121225/http://stderr.org/pipermail/inquiry/2005-December/003287.html
# http://stderr.org/pipermail/inquiry/2005-December/003288.html
# http://web.archive.org/web/20081012105240/http://stderr.org/pipermail/inquiry/2005-December/003288.html
# http://stderr.org/pipermail/inquiry/2005-December/003289.html
# http://web.archive.org/web/20081007044212/http://stderr.org/pipermail/inquiry/2005-December/003289.html
# http://stderr.org/pipermail/inquiry/2005-December/003290.html
# http://web.archive.org/web/20061013233710/http://stderr.org/pipermail/inquiry/2005-December/003290.html
# http://stderr.org/pipermail/inquiry/2005-December/003291.html
# http://web.archive.org/web/20081007064627/http://stderr.org/pipermail/inquiry/2005-December/003291.html
# http://stderr.org/pipermail/inquiry/2005-December/003292.html
# http://web.archive.org/web/20081007045426/http://stderr.org/pipermail/inquiry/2005-December/003292.html
# http://stderr.org/pipermail/inquiry/2005-December/003305.html
# http://web.archive.org/web/20081006113912/http://stderr.org/pipermail/inquiry/2005-December/003305.html
# http://stderr.org/pipermail/inquiry/2005-December/003306.html
# http://web.archive.org/web/20080908071532/http://stderr.org/pipermail/inquiry/2005-December/003306.html
===Peirce's Logic Of Information : Discussion (2005)===
===2005 • Inquiry List • Peirce's Logic Of Information • Discussion===
* http://stderr.org/pipermail/inquiry/2005-December/thread.html#3280
* http://web.archive.org/web/20120512004315/http://stderr.org/pipermail/inquiry/2005-December/thread.html#3280
# http://stderr.org/pipermail/inquiry/2005-December/003280.html
# http://web.archive.org/web/20081007062911/http://stderr.org/pipermail/inquiry/2005-December/003280.html
# http://stderr.org/pipermail/inquiry/2005-December/003284.html
# http://web.archive.org/web/20081012073556/http://stderr.org/pipermail/inquiry/2005-December/003284.html
# http://stderr.org/pipermail/inquiry/2005-December/003293.html
# http://web.archive.org/web/20061013233835/http://stderr.org/pipermail/inquiry/2005-December/003293.html
# http://stderr.org/pipermail/inquiry/2005-December/003294.html
# http://web.archive.org/web/20080908075216/http://stderr.org/pipermail/inquiry/2005-December/003294.html
# http://stderr.org/pipermail/inquiry/2005-December/003295.html
# http://web.archive.org/web/20081012073814/http://stderr.org/pipermail/inquiry/2005-December/003295.html
[[Category:Charles Sanders Peirce]]
[[Category:Charles Sanders Peirce]]
