Functional Logic • Inquiry and Analogy • 6

Inquiry and AnalogyPeirce’s Formulation of AnalogyVersion 1

C.S. Peirce • “On the Natural Classification of Arguments” (1867)

The formula of analogy is as follows:

S^{\prime}, S^{\prime\prime}, \text{and}~ S^{\prime\prime\prime} are taken at random from such a class that their characters at random are such as {P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}}.

\begin{matrix}  T ~\text{is}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime},  \\[4pt]  S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\text{are}~ Q;  \\[4pt]  \therefore T ~\text{is}~ Q.  \end{matrix}

Such an argument is double.  It combines the two following:

\begin{matrix}  1.  \\[4pt]  S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\text{are taken as being}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime},  \\[4pt]  S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\text{are}~ Q;  \\[4pt]  \therefore ~(\text{By induction})~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime} ~\text{is}~ Q,  \\[4pt]  T ~\text{is}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime};  \\[4pt]  \therefore ~(\text{Deductively})~ T ~\text{is}~ Q.  \end{matrix}

\begin{matrix}  2.  \\[4pt]  S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\text{are, for instance,}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime},  \\[4pt]  T ~\text{is}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime};  \\[4pt]  \therefore ~(\text{By hypothesis})~ T ~\text{has the common characters of}~ S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime},  \\[4pt]  S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\text{are}~ Q;  \\[4pt]  \therefore ~(\text{Deductively})~ T ~\text{is}~ Q.  \end{matrix}

Owing to its double character, analogy is very strong with only a moderate number of instances.

(Peirce, CP 2.513, CE 2, 46–47)

Figure 7 shows the logical relationships involved in the above analysis.

Peirce's Formulation of Analogy (Version 1)
\text{Figure 7. Peirce's Formulation of Analogy (Version 1)}

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Functional Logic • Inquiry and Analogy • 5

Inquiry and AnalogyAristotle’s “Paradigm” • Reasoning by Analogy

Aristotle examines the subject of analogical inference or “reasoning by example” under the heading of the Greek word παραδειγμα, from which comes the English word paradigm.  In its original sense the word suggests a kind of “side-show”, or a parallel comparison of cases.

We have an Example (παραδειγμα, or analogy) when the major extreme is shown to be applicable to the middle term by means of a term similar to the third.  It must be known both that the middle applies to the third term and that the first applies to the term similar to the third.

E.g., let A be “bad”, B “to make war on neighbors”, C “Athens against Thebes”, and D “Thebes against Phocis”.  Then if we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad.  Evidence of this can be drawn from similar examples, e.g., that war by Thebes against Phocis is bad.  Then since war against neighbors is bad, and war against Thebes is against neighbors, it is evident that war against Thebes is bad.

(Aristotle, “Prior Analytics” 2.24)

Figure 6 shows the logical relationships involved in Aristotle’s example of analogy.

Aristotle's “Paradigm”
\text{Figure 6. Aristotle's ``Paradigm"}

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Functional Logic • Inquiry and Analogy • 4

Inquiry and AnalogyAristotle’s “Apagogy” • Abductive Reasoning

Peirce’s notion of abductive reasoning is derived from Aristotle’s treatment of it in the Prior Analytics.  Aristotle’s discussion begins with an example which may seem incidental but the question and its analysis are echoes of the investigation pursued in one of Plato’s Dialogue, the Meno.  It concerns nothing less than the possibility of knowledge and the relationship between knowledge and virtue, or between their objects, the true and the good.  It is not just because it forms a recurring question in philosophy, but because it preserves a close correspondence between its form and its content, that we shall find this example increasingly relevant to our study.

We have Reduction (απαγωγη, abduction):  (1) when it is obvious that the first term applies to the middle, but that the middle applies to the last term is not obvious, yet nevertheless is more probable or not less probable than the conclusion;  or (2) if there are not many intermediate terms between the last and the middle;  for in all such cases the effect is to bring us nearer to knowledge.

(1) E.g., let A stand for “that which can be taught”, B for “knowledge”, and C for “morality”.  Then that knowledge can be taught is evident;  but whether virtue is knowledge is not clear.  Then if BC is not less probable or is more probable than AC, we have reduction;  for we are nearer to knowledge for having introduced an additional term, whereas before we had no knowledge that AC is true.

(2) Or again we have reduction if there are not many intermediate terms between B and C;  for in this case too we are brought nearer to knowledge.  E.g., suppose that D is “to square”, E “rectilinear figure”, and F “circle”.  Assuming that between E and F there is only one intermediate term — that the circle becomes equal to a rectilinear figure by means of lunules — we should approximate to knowledge.

(Aristotle, “Prior Analytics” 2.25)

A few notes on the reading may be helpful.  The Greek text seems to imply a geometric diagram, in which directed line segments AB, BC, AC indicate logical relations between pairs of terms taken from A, B, C.  We have two options for reading the line labels, either as implications or as subsumptions, as in the following two paradigms for interpretation.

Table of Implications

Table of Subsumptions

In the latter case, P \geqslant Q is read as ``P ~\text{subsumes}~ Q", that is, ``P ~\text{applies to all}~ Q", or ``P ~\text{is predicated of all}~ Q".

The method of abductive reasoning bears a close relation to the sense of reduction in which we speak of one question reducing to another.  The question being asked is “Can virtue be taught?”  The type of answer which develops is as follows.

If virtue is a form of understanding, and if we are willing to grant that understanding can be taught, then virtue can be taught.  In this way of approaching the problem, by detour and indirection, the form of abductive reasoning is used to shift the attack from the original question, whether virtue can be taught, to the hopefully easier question, whether virtue is a form of understanding.

The logical structure of the process of hypothesis formation in the first example follows the pattern of “abduction to a case”, whose abstract form is diagrammed and schematized in Figure 5.

Teachability, Understanding, Virtue
\text{Figure 5. Teachability, Understanding, Virtue}

The sense of the Figure is explained by the following assignments.

Term, Position, Interpretation

Premiss, Predication, Inference Role

Abduction from a Fact to a Case proceeds according to the following schema.

\begin{array}{l}  ~ \text{Fact:}~ V \Rightarrow T?  \\  ~ \text{Rule:}~ U \Rightarrow T.  \\  \overline{~~~~~~~~~~~~~~~~~~~~~~} \\  ~ \text{Case:}~ V \Rightarrow U?  \end{array}

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Functional Logic • Inquiry and Analogy • 3

Inquiry and AnalogyComparison of the Analyses

The next two Figures will be of use when we turn to comparing the three types of inference as they appear in the respective analyses of Aristotle and Peirce.

Types of Reasoning in Transition

Types of Reasoning in Transition
\text{Figure 3. Types of Reasoning in Transition}

Types of Reasoning in Peirce

Types of Reasoning in Peirce
\text{Figure 4. Types of Reasoning in Peirce}

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Functional Logic • Inquiry and Analogy • 2

Inquiry and AnalogyThree Types of Reasoning

Types of Reasoning in C.S. Peirce

Peirce gives one of his earliest treatments of the three types of reasoning in his Harvard Lectures of 1865 “On the Logic of Science”.  There he shows how the same proposition may be reached from three directions, as the result of an inference in each of the three modes.

We have then three different kinds of inference.

  • Deduction or inference à priori,
  • Induction or inference à particularis,
  • Hypothesis or inference à posteriori.

(Peirce, CE 1, 267).

  • If I reason that certain conduct is wise because it has a character which belongs only to wise things, I reason à priori.
  • If I think it is wise because it once turned out to be wise, that is, if I infer that it is wise on this occasion because it was wise on that occasion, I reason inductively [à particularis].
  • But if I think it is wise because a wise man does it, I then make the pure hypothesis that he does it because he is wise, and I reason à posteriori.

(Peirce, CE 1, 180).

Suppose we make the following assignments.

\begin{array}{lll}  \mathrm{A} & = & \text{Wisdom}  \\  \mathrm{B} & = & \text{a certain character}  \\  \mathrm{C} & = & \text{a certain conduct}  \\  \mathrm{D} & = & \text{done by a wise man}  \\  \mathrm{E} & = & \text{a certain occasion}  \end{array}

Recognizing a little more concreteness will aid understanding, let us make the following substitutions in Peirce’s example.

\begin{array}{lllll}  \mathrm{B} & = & \text{Benevolence} & = & \text{a certain character}  \\  \mathrm{C} & = & \text{Contributes to Charity} & = & \text{a certain conduct}  \\  \mathrm{E} & = & \text{Earlier today} & = & \text{a certain occasion}  \end{array}

The converging operation of all three reasonings is shown in Figure 2.

A Triply Wise Act
\text{Figure 2. A Triply Wise Act}

The common proposition concluding each argument is AC, contributing to charity is wise.

  • Deduction could have obtained the Fact AC from the Rule AB, benevolence is wisdom, along with the Case BC, contributing to charity is benevolent.
  • Induction could have gathered the Rule AC, contributing to charity is exemplary of wisdom, from the Fact AE, the act of earlier today is wise, along with the Case CE, the act of earlier today was an instance of contributing to charity.
  • Abduction could have guessed the Case AC, contributing to charity is explained by wisdom, from the Fact DC, contributing to charity is done by this wise man, and the Rule DA, everything wise is done by this wise man.  Thus, a wise man, who does all the wise things there are to do, may nonetheless contribute to charity for no good reason and even be charitable to a fault.  But on seeing the wise man contribute to charity it is natural to think charity may well be the mark of his wisdom, in essence, that wisdom is the reason he contributes to charity.

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Functional Logic • Inquiry and Analogy • 1

Inquiry and AnalogyThree Types of Reasoning

Types of Reasoning in Aristotle

Figure 1 gives a quick overview of traditional terminology I’ll have occasion to refer to as discussion proceeds.

Types of Reasoning in Aristotle
\text{Figure 1. Types of Reasoning in Aristotle}

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Peirce’s 1870 “Logic of Relatives” • Selection 13

I continue with my Selections and Comments examining Peirce’s 1870 Logic of Relatives, one of those works which convinced me from my earliest grapplings I would need to learn a lot more mathematics before I’d have any hope of understanding what Peirce was up to.  What I’ve put on the Web so far is linked in this Overview.

We continue with §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives”Selection 13

The Sign of Involution (cont.)

A servant of every man and woman will be denoted by \mathit{s}^{\mathrm{m} \;+\!\!,~ \mathrm{w}} and \mathit{s}^\mathrm{m}\!,\!\mathit{s}^\mathrm{w} will denote a servant of every man that is a servant of every woman.  So that

s(m+,w) = (s^m),(s^w)

(Peirce, CP 3.77)

References

  • Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.  Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429.  Online (1) (2) (3).
  • Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.

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Sign Relations, Triadic Relations, Relation Theory • Discussion 10

Re: FB | Dan EverettOn the Origin of Symbols and the Descent of Signs

Continuing a discussion on the generative power of symbols (1) (2) (3).

If it’s true what I say about symbols being the genus of all signs then it must be possible to say what differentia are added to the genus in order to generate every subtended species, beginning with icons and indices.

Turning first to icons, we have the following from Peirce.

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.  (Peirce 1866, Lecture 7, 467).

Let’s say we look inside a triadic sign relation L \subseteq O \times S \times I and we notice a triple (o, s, i) where o and s have a character \chi in common.  We may quite naturally be tempted to make a further leap and suppose the sign s receives the interpretant sign i precisely by virtue of the character \chi shared by o and s.  I know that looks like a lot of supposing but the fact is we do the like all the time without hardly giving it a second thought.  But critical reflection demands we bat an i and give it second and third thoughts.

The catch is tucked away in Peirce’s last sentence.  “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.”

There may be a lot of characters shared by o and s in a given environment or universe of discourse, any selection of which may account for the linking of o and s to i.  As long as we remain content to operate in a theoretical vacuum devoid of empirical grounding, who’s to say any number of them do not qualify?

But a question arises when we use a sign relation L to model an empirical system of interpretive practice, whether its agent is a single individual or a whole community of interpretation.  The question is — Do the characters we mark as effective in our model actually do the job for the agent?

An icon denotes its objects by virtue of qualities it shares with its objects.  But icons are icons solely because they are interpreted as icons, by dint of particular qualities chosen from many by the very process of interpretation in view.  This gives us a glimmer of the interpretive character of sign typing, that sign typologies are not absolute but relative to the sign relation at hand.  To paraphrase William James — The trail of the hermeneutic serpent is over all.

Reference

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

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Sign Relations, Triadic Relations, Relation Theory • Discussion 9

Once, there was nothing there, nothing moving on its own, just data and people shuffling it around.  Then something happened, and it … it knew itself.

William Gibson • Count Zero (1) (2)

Re: FB | Dan EverettOn the Origin of Symbols and the Descent of Signs

Continuing a discussion on the generative power of symbols (1) (2).

Here’s the skinny on the big three types of signs.  Despite its simplicity, or maybe because of it, the larger implications for the interpretive character of sign typing still go widely missed.

SemeioticTypes of Signs

There are three principal ways a sign may denote its objects.  The modes of representation are often referred to as kinds, species, or types of signs but it is important to recognize they are not ontological species, that is, they are not mutually exclusive features of description, since the same thing can be a sign in several different ways.

Beginning very roughly, the three main ways of being a sign can be described as follows.

  • An icon denotes its objects by virtue of a quality it shares with its objects.
  • An index denotes its objects by virtue of an existential connection it has to its objects.
  • A symbol denotes its objects solely by virtue of being interpreted to do so.

One of Peirce’s early delineations of the three types of signs affords a useful first approach to understanding their differences and their relationships to each other.

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.  (Peirce 1866, Lecture 7, 467–468).

Reference

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

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Sign Relations, Triadic Relations, Relation Theory • Discussion 8

Re: FB | Dan EverettOn the Origin of Symbols and the Descent of Signs

Continuing a discussion on the primal character of symbols.

There are a few passages from Peirce going most quickly to the root of the matter and working to keep the main ideas in mind — before one gets too bogged down and bewildered by the full‑blown classification mania so common in the literature.

The following statement is key.

Thought is not necessarily connected with a brain.  It appears in the work of bees, of crystals, and throughout the purely physical world;  and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there.

C.S. Peirce, Collected Papers (CP 4.551)

I know that is a Golden Oldie, but as the years go by I find many people have taken away different messages from even the most familiar tunes, making it fruitful every now and again to accord old themes another turn.

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