Differential Logic • 4

Posted on November 3, 2024 by Jon Awbrey

Differential Expansions of Propositions

Bird’s Eye View

An efficient calculus for the realm of logic represented by boolean functions and elementary propositions makes it feasible to compute the finite differences and the differentials of those functions and propositions.

For example, consider a proposition of the form {}^{\backprime\backprime} \, p ~\mathrm{and}~ q \, {}^{\prime\prime} graphed as two letters attached to a root node, as shown below.

Cactus Graph Existential p and q

Written as a string, this is just the concatenation p~q.

The proposition pq may be taken as a boolean function f(p, q) having the abstract type f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, where \mathbb{B} = \{ 0, 1 \} is read in such a way that 0 means \mathrm{false} and 1 means \mathrm{true}.

Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition pq is true, as shown in the following Figure.

Venn Diagram p and q

Now ask yourself:  What is the value of the proposition pq at a distance of \mathrm{d}p and \mathrm{d}q from the cell pq where you are standing?

Don’t think about it — just compute:

Cactus Graph (p,dp)(q,dq)

The cactus formula \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)} and its corresponding graph arise by replacing p with p + \mathrm{d}p and q with q + \mathrm{d}q in the boolean product or logical conjunction pq and writing the result in the two dialects of cactus syntax.  This follows because the boolean sum p + \mathrm{d}p is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form.

Cactus Graph (p,dp)

Next question:  What is the difference between the value of the proposition pq over there, at a distance of \mathrm{d}p and \mathrm{d}q from where you are standing, and the value of the proposition pq where you are, all expressed in the form of a general formula, of course?  The answer takes the following form.

Cactus Graph ((p,dp)(q,dq),pq)

There is one thing I ought to mention at this point:  Computed over \mathbb{B}, plus and minus are identical operations.  This will make the relation between the differential and the integral parts of the appropriate calculus slightly stranger than usual, but we will get into that later.

Last question, for now:  What is the value of this expression from your current standpoint, that is, evaluated at the point where pq is true?  Well, replacing p with 1 and q with 1 in the cactus graph amounts to erasing the labels p and q, as shown below.

Cactus Graph (( ,dp)( ,dq), )

And this is equivalent to the following graph.

Cactus Graph ((dp)(dq))

We have just met with the fact that the differential of the and is the or of the differentials.

\begin{matrix} p ~\mathrm{and}~ q & \quad & \xrightarrow{\quad\mathrm{Diff}\quad} & \quad & \mathrm{d}p ~\mathrm{or}~ \mathrm{d}q \end{matrix}

Cactus Graph pq → Diff → ((dp)(dq))

It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of De Morgan’s rule, it is no accident, as differentiation and negation turn out to be closely related operations.  Indeed, one can find discussion of logical difference calculus in the personal correspondence between Boole and De Morgan and Peirce, too, made use of differential operators in a logical context, but the exploration of those ideas has been hampered by a number of factors, not the least of which has been the lack of a syntax adequate to handle the complexity of expressions evolving in the process.

Resources

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Change, Cybernetics, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Inquiry Driven Systems, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , | 5 Comments

Differential Logic • 3

Posted on November 2, 2024 by Jon Awbrey

Cactus Language for Propositional Logic

Table 1 shows the cactus graphs, the corresponding cactus expressions, their logical meanings under the so‑called existential interpretation, and their translations into conventional notations for a sample of basic propositional forms.

Table 1. Syntax and Semantics of a Calculus for Propositional Logic

Table 1. Syntax and Semantics of a Calculus for Propositional Logic

The simplest expression for logical truth is the empty word, typically denoted by \boldsymbol\varepsilon or \lambda in formal languages, where it is the identity element for concatenation.  To make it visible in context, it may be denoted by the equivalent expression ``\texttt{(())}" or, especially if operating in an algebraic context, by a simple ``1".  Also when working in an algebraic mode, the plus sign ``+" may be used for exclusive disjunction.  Thus we have the following translations of algebraic expressions into cactus expressions.

\begin{matrix} a + b \quad = \quad \texttt{(} a \texttt{,} b \texttt{)} \\[8pt] a + b + c \quad = \quad \texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))} \quad = \quad \texttt{((} a \texttt{,} b \texttt{),} c \texttt{)} \end{matrix}

It is important to note the last expressions are not equivalent to the 3‑place form \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}.

Resources

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Change, Cybernetics, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Inquiry Driven Systems, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , | 4 Comments

Differential Logic • 2

Posted on October 31, 2024 by Jon Awbrey

Cactus Language for Propositional Logic

The development of differential logic is facilitated by having a moderately efficient calculus in place at the level of boolean-valued functions and elementary logical propositions.  One very efficient calculus on both conceptual and computational grounds is based on just two types of logical connectives, both of variable k-ary scope.  The syntactic formulas of that calculus map into a family of graph-theoretic structures called “painted and rooted cacti” which lend visual representation to the functional structures of propositions and smooth the path to efficient computation.

The first kind of connective is a parenthesized sequence of propositional expressions, written \texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)} to mean exactly one of the propositions e_1, e_2, \ldots, e_{k-1}, e_k is false, in short, their minimal negation is true.  An expression of that form is associated with a cactus structure called a lobe and is “painted” with the colors e_1, e_2, \ldots, e_{k-1}, e_k as shown below.

Lobe Connective

The second kind of connective is a concatenated sequence of propositional expressions, written e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k to mean all the propositions e_1, e_2, \ldots, e_{k-1}, e_k are true, in short, their logical conjunction is true.  An expression of that form is associated with a cactus structure called a node and is “painted” with the colors e_1, e_2, \ldots, e_{k-1}, e_k as shown below.

Node Connective

All other propositional connectives can be obtained through combinations of the above two forms.  As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it’s convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms.  While working with expressions solely in propositional calculus, it’s easiest to use plain parentheses for logical connectives.  In contexts where ordinary parentheses are needed for other purposes an alternate typeface \texttt{(} \ldots \texttt{)} may be used for the logical operators.

Resources

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Change, Cybernetics, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Inquiry Driven Systems, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , | 4 Comments

Differential Logic • 1

Posted on October 30, 2024 by Jon Awbrey

Introduction

Differential logic is the component of logic whose object is the description of variation — focusing on the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description.  A definition that broad naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.  To the extent a logical inquiry makes use of a formal system, its differential component governs the use of a differential logical calculus, that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

Simple examples of differential logical calculi are furnished by differential propositional calculi.  A differential propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe.  Such a calculus augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Resources

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Change, Cybernetics, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Inquiry Driven Systems, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , | 4 Comments

Information = Comprehension × Extension • Comment 7

Posted on October 28, 2024 by Jon Awbrey

Let’s stay with Peirce’s example of inductive inference a little longer and try to clear up the more troublesome confusions tending to arise.

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 2. Disjunctive Term u, Taken as Subject

\text{Figure 2. Disjunctive Term}~ u, \text{Taken as Subject}

Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign.

Figure 4. Disjunctive Subject u, Induction of Rule v ⇒ w

\text{Figure 4. Disjunctive Subject}~ u, \text{Induction of Rule}~ v \Rightarrow w

One final point needs to be stressed.  It is important to recognize the disjunctive term itself — the syntactic formula “neat, swine, sheep, deer” or any logically equivalent formula — is not an index but a symbol.  It has the character of an artificial symbol which is constructed to fill a place in a formal system of symbols, for example, a propositional calculus.  In that setting it would normally be interpreted as a logical disjunction of four elementary propositions, denoting anything in the universe of discourse which has any of the four corresponding properties.

The artificial symbol “neat, swine, sheep, deer” denotes objects which serve as indices of the genus herbivore by virtue of their belonging to one of the four named species of herbivore.  But there is in addition a natural symbol which serves to unify the manifold of given species, namely, the concept of a cloven‑hoofed animal.

As a symbol or general representation, the concept of a cloven‑hoofed animal connotes an attribute and connotes it in such a way as to determine what it denotes.  Thus we observe a natural expansion in the connotation of the symbol, amounting to what Peirce calls the “superfluous comprehension”, the information added by an “ampliative” or synthetic inference.

In sum we have sufficient information to motivate an inductive inference, from the Fact u \Rightarrow w and the Case u \Rightarrow v to the Rule v \Rightarrow w.

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Mathstodon • Academia.edu
cc: Research Gate

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information = Comprehension × Extension, Inquiry, Intension, Logic, Peirce's Categories, Pragmatic Semiotic Information, Pragmatism, Scientific Method, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , | 3 Comments

Information = Comprehension × Extension • Comment 6

Posted on October 23, 2024 by Jon Awbrey

Re: Information = Comprehension × Extension • Comment 2

Returning to Peirce’s example of inductive inference, let’s try to get a clearer picture of why he connects it with disjunctive terms and indicial signs.  At this point in time I can’t say I’m entirely satisfied with my understanding of the relationship between disjunctive terms, indicial signs, and inductive inferences as presented by Peirce in his early accounts.  What follows is just one of the simplest and least question‑begging attempts at rational reconstruction I’ve been able to devise.

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 2. Disjunctive Term u, Taken as Subject

\text{Figure 2. Disjunctive Term}~ u, \text{Taken as Subject}

Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign.

Figure 4. Disjunctive Subject u, Induction of Rule v ⇒ w

\text{Figure 4. Disjunctive Subject}~ u, \text{Induction of Rule}~ v \Rightarrow w

If there is any distinguishing feature shared by all the instances under the disjunctive description “neat, swine, sheep, deer” then sign users may take that feature as a predictor of being herbivorous, precisely because all the things under the disjunctive description are herbivorous.  But everything under the disjunctive description is cloven‑hoofed, so the cases under the disjunctive description serve to indicate, support, or witness the utility of the induction from cloven‑hoofed to herbivorous.

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Mathstodon • Academia.edu
cc: Research Gate

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information = Comprehension × Extension, Inquiry, Intension, Logic, Peirce's Categories, Pragmatic Semiotic Information, Pragmatism, Scientific Method, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , | 4 Comments

Information = Comprehension × Extension • Comment 5

Posted on October 16, 2024 by Jon Awbrey

Let’s stay with Peirce’s example of abductive inference a little longer and try to clear up the more troublesome confusions tending to arise.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 1. Conjunctive Term z, Taken as Predicate

\text{Figure 1. Conjunctive Term}~ z, \text{Taken as Predicate}

Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign.

Figure 3. Conjunctive Predicate z, Abduction of Case x ⇒ y

\text{Figure 3. Conjunctive Predicate}~ z, \text{Abduction of Case}~ x \Rightarrow y

One thing needs to be stressed at this point.  It is important to recognize the conjunctive term itself — namely, the syntactic string “spherical bright fragrant juicy tropical fruit” — is not an icon but a symbol.  It has its place in a formal system of symbols, for example, a propositional calculus, where it would normally be interpreted as a logical conjunction of six elementary propositions, denoting anything in the universe of discourse with all six of the corresponding properties.  The symbol denotes objects which may be taken as icons of oranges by virtue of their bearing those six properties in common with oranges.  But there are no objects denoted by the symbol which aren’t already oranges themselves.  Thus we observe a natural reduction in the denotation of the symbol, consisting in the absence of cases outside of oranges which have all the properties indicated.

The above analysis provides another way to understand the abductive inference from the Fact x \Rightarrow z and the Rule y \Rightarrow z to the Case x \Rightarrow y.  The lack of any cases which are z and not y is expressed by the implication z \Rightarrow y.  Taking this together with the Rule y \Rightarrow z gives the logical equivalence y = z.  But this reduces the Case x \Rightarrow y to the Fact x \Rightarrow z and so the Case is justified.

Viewed in the light of the above analysis, Peirce’s example of abductive reasoning exhibits an especially strong form of inference, almost deductive in character.  Do all abductive arguments take this form, or may there be weaker styles of abductive reasoning which enjoy their own levels of plausibility?  That must remain an open question at this point.

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Mathstodon • Academia.edu
cc: Research Gate

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information = Comprehension × Extension, Inquiry, Intension, Logic, Peirce's Categories, Pragmatic Semiotic Information, Pragmatism, Scientific Method, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , | 4 Comments

Information = Comprehension × Extension • Comment 4

Posted on October 14, 2024 by Jon Awbrey

Re: Information = Comprehension × Extension • Comment 3

Many things still puzzle me about Peirce’s account at this point.  The question marks I added to the Figures of the previous post indicate the node labels I have remaining doubts about.  For example, in Figure 3, is z really an icon of object y?  Again, in Figure 4, is u really an index of object v?  There is nothing for it but returning to Peirce’s text and trying again to follow his reasoning.

Let’s go back to Peirce’s example of abductive inference and try to get a clearer picture of why he connects it with conjunctive terms and iconic signs.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 1. Conjunctive Term z, Taken as Predicate

\text{Figure 1. Conjunctive Term}~ z, \text{Taken as Predicate}

Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign.

Figure 3. Conjunctive Predicate z, Abduction of Case x ⇒ y

\text{Figure 3. Conjunctive Predicate}~ z, \text{Abduction of Case}~ x \Rightarrow y

The relationship between conjunctive terms and iconic signs may be understood along the following lines.  If there is anything with all the properties described by the conjunctive term “spherical bright fragrant juicy tropical fruit” then sign users may use that thing as an icon of an orange, precisely because it shares those properties with an orange.  But the only natural examples of things with all those properties are oranges themselves, so the only thing qualified to serve as a natural icon of an orange by virtue of those very properties is that orange itself or another orange.

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Mathstodon • Academia.edu
cc: Research Gate

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information = Comprehension × Extension, Inquiry, Intension, Logic, Peirce's Categories, Pragmatic Semiotic Information, Pragmatism, Scientific Method, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , | 4 Comments

Information = Comprehension × Extension • Comment 3

Posted on October 13, 2024 by Jon Awbrey

Peirce identifies inference with a process he describes as symbolization.  Let us consider what that might imply.

I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of information.  (467).

Even if it were only a rough analogy between inference and symbolization, a principle of logical continuity, what is known in physics as a correspondence principle, would suggest parallels between steps of reasoning in the neighborhood of exact inferences and signs in the vicinity of genuine symbols.  This would lead us to expect a correspondence between degrees of inference and degrees of symbolization extending from exact to approximate (non‑demonstrative) inferences and from genuine to approximate (degenerate) symbols.

For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.

In the first place there are likenesses or copies — such as statues, pictures, emblems, hieroglyphics, and the like.  Such representations stand for their objects only so far as they have an actual resemblance to them — that is agree with them in some characters.  The peculiarity of such representations is that they do not determine their objects — they stand for anything more or less;  for they stand for whatever they resemble and they resemble everything more or less.

The second kind of representations are such as are set up by a convention of men or a decree of God.  Such are tallies, proper names, &c.  The peculiarity of these conventional signs is that they represent no character of their objects.

Likenesses denote nothing in particular;  conventional signs connote nothing in particular.

The third and last kind of representations are symbols or general representations.  They connote attributes and so connote them as to determine what they denote.  To this class belong all words and all conceptions.  Most combinations of words are also symbols.  A proposition, an argument, even a whole book may be, and should be, a single symbol.  (467–468).

In addition to Aristotle, the influence of Kant on Peirce is very strongly marked in these earliest expositions.  The invocations of “conceptions of the understanding”, the “use of concepts” and thus of symbols in reducing the manifold of extension, and the not so subtle hint of the synthetic à priori in Peirce’s discussion, not only of natural kinds but also of the kinds of signs leading up to genuine symbols, can all be recognized as pervasive Kantian themes.

In order to draw out these themes and see how Peirce was led to develop their leading ideas, let us bring together our previous Figures, abstracting from their concrete details, and see if we can figure out what is going on.

Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign.

Figure 3. Conjunctive Predicate z, Abduction of Case x ⇒ y

\text{Figure 3. Conjunctive Predicate}~ z, \text{Abduction of Case}~ x \Rightarrow y

Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign.

Figure 4. Disjunctive Subject u, Induction of Rule v ⇒ w

\text{Figure 4. Disjunctive Subject}~ u, \text{Induction of Rule}~ v \Rightarrow w

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Mathstodon • Academia.edu
cc: Research Gate

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information = Comprehension × Extension, Inquiry, Intension, Logic, Peirce's Categories, Pragmatic Semiotic Information, Pragmatism, Scientific Method, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , | 5 Comments

Information = Comprehension × Extension • Comment 2

Posted on October 12, 2024 by Jon Awbrey

Let’s examine Peirce’s second example of a disjunctive term — neat, swine, sheep, deer — within the style of lattice framework we used before.

Hence if we find out that neat are herbivorous, swine are herbivorous, sheep are herbivorous, and deer are herbivorous;  we may be sure that there is some class of animals which covers all these, all the members of which are herbivorous.  (468–469).

Accordingly, if we are engaged in symbolizing and we come to such a proposition as “Neat, swine, sheep, and deer are herbivorous”, we know firstly that the disjunctive term may be replaced by a true symbol.  But suppose we know of no symbol for neat, swine, sheep, and deer except cloven‑hoofed animals.  (469).

This is apparently a stock example of inductive reasoning Peirce is borrowing from traditional discussions, so let us pass over the circumstance that modern taxonomies may classify swine as omnivores.

In view of the analogical symmetries the disjunctive term shares with the conjunctive case, we can run through this example in fairly short order.  We have the following four terms.

\begin{array}{lll} s_1 & = & \mathrm{neat} \\ s_2 & = & \mathrm{swine} \\ s_3 & = & \mathrm{sheep} \\ s_4 & = & \mathrm{deer} \end{array}

Suppose u is the logical disjunction of the above four terms.

\begin{array}{lll} u & = & \texttt{((} s_1 \texttt{)(} s_2 \texttt{)(} s_3 \texttt{)(} s_4 \texttt{))} \end{array}

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 2. Disjunctive Term u, Taken as Subject

\text{Figure 2. Disjunctive Term}~ u, \text{Taken as Subject}

Here we have a situation which is dual to the structure of the conjunctive example.  There is a gap between the logical disjunction u, in lattice terminology, the least upper bound of the disjoined terms, u = \mathrm{lub} \{ s_1, s_2, s_3, s_4 \}, and what we might regard as the natural disjunction or natural lub of those terms, namely, v, cloven‑hoofed.

Once again, the sheer implausibility of imagining the disjunctive term u would ever be embedded exactly as such in a lattice of natural kinds leads to the evident naturalness of the induction to the implication v \Rightarrow w, namely, the rule that cloven‑hoofed animals are herbivorous.

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Mathstodon • Academia.edu
cc: Research Gate

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information = Comprehension × Extension, Inquiry, Intension, Logic, Peirce's Categories, Pragmatic Semiotic Information, Pragmatism, Scientific Method, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , | 5 Comments