Semiotics, Semiosis, Sign Relations • Discussion 2

Posted on October 26, 2019 by Jon Awbrey

Re: Richard Coyne • Recursion Again

It’s a common mistake to confound infinite with unbounded.  A process can continue without end and still be “bounded in a nutshell”.  So a sign process can pass from sign to interpretant sign to next interpretant sign ad infinitum without ever leaving a finite set of signs.

Resources

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Peirce’s 1870 “Logic Of Relatives” • Discussion 2

Posted on October 24, 2019 by Jon Awbrey

Re: Ecology of Systems Thinking • RS • TM

My previous comment summed up my observations of a general drift toward “absolutist and dyadic ways of thinking” in various communities of inquiry of interest to me over the past 20 years.  I traced its cause to “the stubborn pull of unchecked reductionism” and a corresponding failure to grasp the relational structures of complex phenomena.

A preference for simple models and theories is natural enough so long as the chosen models and theories are up to the task of explaining the phenomena at hand, but when a preference for a particular class of structures persists in the face of steadily mounting anomalies it becomes a hidebound and dysfunctional bias.

That is my description and my diagnosis of the situation as I see it.  I could be wrong about either or both.  But the reason for addressing the case in these terms is not simply to point out a dysfunctional state of affairs.  The purpose of a diagnosis is to indicate a remedy.

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Posted in C.S. Peirce, Category Theory, Differential Logic, Duality, Dyadic Relations, Graph Theory, Group Theory, Logic, Logic of Relatives, Logical Graphs, Logical Matrices, Mathematics, Peirce, Peirce's Categories, Predicate Calculus, Propositional Calculus, Relation Theory, Semiotics, Sign Relations, Teridentity, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , | 5 Comments

Peirce’s 1870 “Logic Of Relatives” • Comment 2

Posted on October 22, 2019 by Jon Awbrey

In a recent post on a related topic I gave this assessment of our present situation:

One of the more disconcerting developments, I might say “devolutions”, I’ve observed over the past 20 years has been the general slippage back to absolutist and dyadic ways of thinking, all of it due to the stubborn pull of unchecked reductionism and a failure to comprehend the relational paradigm, especially the basic facts about triadic relations, their irreducibility, and the consequences thereof.

For anyone who sees our situation this way, and who thinks it calls for a remedy, the question becomes:  How to remediate a persistent failure to comprehend the relational paradigm, especially the basic facts about triadic relations, their irreducibility, and the consequences thereof?

My answer to that naturally brings me back to this thread, so I’ll continue from here.

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Posted in C.S. Peirce, Category Theory, Differential Logic, Duality, Dyadic Relations, Graph Theory, Group Theory, Logic, Logic of Relatives, Logical Graphs, Logical Matrices, Mathematics, Peirce, Peirce's Categories, Predicate Calculus, Propositional Calculus, Relation Theory, Semiotics, Sign Relations, Teridentity, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , | 6 Comments

Peirce’s 1870 “Logic of Relatives” • Comment 1

Posted on October 22, 2019 by Jon Awbrey

Peirce often stressed his Logic of Relatives as the key to unlocking many puzzles.  As I read him, it was Peirce’s drive to understand the Logic of Science that required the grounding of logic in the mathematical forms of triadic sign relations and this in turn demanded a leap forward in the understanding of relations in general.

My long ago encounter with Peirce’s 1870 paper, “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, was one of the events precipitating my return from the hazier heights of philosophy to the solid plains of mathematics below.  Over the years I copied out various drafts of my study notes to the web, consisting of selections from Peirce’s paper along with my running commentary.  A few years back I serialized what progress I had made so far to this blog and this Overview consists of links to those installments.

Peirce’s 1870 “Logic of Relatives”

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 (CP 3.45–149), Chronological Edition (CE 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.  Cited as (CP volume.paragraph).
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Cited as (CE volume, page).

Resources

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The Difference That Makes A Difference That Peirce Makes : 31

Posted on October 21, 2019 by Jon Awbrey

One of the more disconcerting developments, I might say “devolutions”, I’ve observed over the last 20 years has been the general slippage back to absolutist and dyadic ways of thinking, all of it due to the stubborn pull of unchecked reductionism and a failure to comprehend the relational paradigm, especially the basic facts about triadic relations, their irreducibility, and the consequences thereof.

With all that in mind, I’ll return to a point in our earlier discussions, add a bit more on the concept of closure, and continue from there to its bearing on the pragmatic maxim.

The Difference That Makes A Difference That Peirce Makes : 23

A critical question in mathematical logic and its applications concerns the threshold of complexity between dyadic (binary) and triadic (ternary) relations, in essence, whether 2-place relations are universally adequate or whether 3-place relations are irreducible, minimally adequate, and even sufficient as a basis for all higher dimensions.

One of Peirce’s earliest arguments for the sufficiency of triadic relative terms occurs at the top of his 1870 “Logic of Relatives”.

The conjugative term involves the conception of third, the relative that of second or other, the absolute term simply considers an object.  No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.  Whether this reason for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory or not, the fact itself is made perfectly evident by the study of the logic of relatives.  (Peirce, CP 3.63).

Peirce’s argument invokes what is known as a closure principle, as I remarked in the following comment:

What strikes me about the initial installment this time around is its use of a certain pattern of argument I can recognize as invoking a closure principle, and this is a figure of reasoning Peirce uses in three other places:  his discussion of continuous predicates, his definition of sign relations, and in the formulation of the pragmatic maxim itself.

In mathematics, a closure operator is one whose repeated application yields the same result as its first application.

If we consider an arbitrary operator \mathrm{A}, the result of applying \mathrm{A} to an operand x is \mathrm{A}x, the result of applying \mathrm{A} again is \mathrm{AA}x, the result of applying \mathrm{A} again is \mathrm{AAA}x, and so on.  In general, it is perfectly possible each application yields a novel result, distinct from all previous results.

But a closure operator \mathrm{C} is defined by the property \mathrm{CC} = \mathrm{C}, so nothing new results beyond the first application.

The Difference That Makes A Difference That Peirce Makes : 24

The concepts of closure and idempotence are closely related.

We usually speak of a closure operator in contexts where the objects acted on are the primary interest, as in topology, where the objects of interest are open sets, boundaries, closed sets, etc.  In contexts where we abstract away from the operand space, as in algebra, we tend to say idempotence for the detached application \mathrm{CC} = \mathrm{C}.  (If I recall right, it was actually Charles Peirce’s father Benjamin who coined the term idempotence.)

At any rate, I’ll have to mutate the principle a bit to cover the uses Peirce makes of it.

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Posted in Analogy, C.S. Peirce, Communication, Descriptive Science, Fixation of Belief, Formal Systems, Information, Inquiry, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Mathematics, Normative Science, Paradigms, Peirce, Pragmatic Maxim, Pragmatism, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

The Difference That Makes A Difference That Peirce Makes : 30

Posted on October 18, 2019 by Jon Awbrey

Re: Ontolog Forum • Mihai Nadin
Re: Peirce List • Helmut Raulien

I first encountered Peirce’s dimensions of generality and vagueness — two measures of determinacy on sign relations describing the extent to which objects are determined by signs and interpretant signs — while exploring the closely related subjects of definition and determination.

Lately I’ve noticed Peirce’s treatment of objectively indeterminate signs has a bearing on my approach to Category Theory through the Logic of Relatives, so it looks worth paying attention to their potential relationships.  To get things rolling, here’s a good entry point:

Accurate writers have apparently made a distinction between the definite and the determinate.  A subject is determinate in respect to any character which inheres in it or is (universally and affirmatively) predicated of it, as well as in respect to the negative of such character, these being the very same respect.  In all other respects it is indeterminate.  The definite shall be defined presently.

A sign (under which designation I place every kind of thought, and not alone external signs), that is in any respect objectively indeterminate (i.e., whose object is undetermined by the sign itself) is objectively general in so far as it extends to the interpreter the privilege of carrying its determination further.

Example:  “Man is mortal.”  To the question, What man?  the reply is that the proposition explicitly leaves it to you to apply its assertion to what man or men you will.

A sign that is objectively indeterminate in any respect is objectively vague in so far as it reserves further determination to be made in some other conceivable sign, or at least does not appoint the interpreter as its deputy in this office.

Example:  “A man whom I could mention seems to be a little conceited.”  The suggestion here is that the man in view is the person addressed;  but the utterer does not authorize such an interpretation or any other application of what she says.  She can still say, if she likes, that she does not mean the person addressed.

Every utterance naturally leaves the right of further exposition in the utterer;  and therefore, in so far as a sign is indeterminate, it is vague, unless it is expressly or by a well-understood convention rendered general.

C.S. Peirce, Collected Papers, CP 5.447

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The Difference That Makes A Difference That Peirce Makes : 29

Posted on October 17, 2019 by Jon Awbrey

Re: Ontolog Forum • JA • JFS

\begin{array}{l} \texttt{We are called to recall Peirce's deeper meanings}\\ \texttt{of ideas like Form (think Platonic Ideas and the}\\ \texttt{way Aristotle compounded Form and Matter). When}\\ \texttt{we come to Sentiment, by any other word, Feeling,}\\ \texttt{that is the medium of Aesthetics, which concerns}\\ \texttt{Beauty in no merely skin-deep sense but all that}\\ \texttt{embodies and manifests `the admirable in itself',}\\ \texttt{thus every form of life worth living. So Peirce}\\ \texttt{stands the normative science of Logic on grounds}\\ \texttt{within the pale of Ethics and fixes the sight of}\\ \texttt{Ethics on the prize Aesthetics picks to steer by.} \end{array}

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The Difference That Makes A Difference That Peirce Makes : 28

Posted on October 15, 2019 by Jon Awbrey

Re: The Difference That Makes A Difference That Peirce Makes : 20

I had to go back and remind myself why I took up this thread again, but at least it supplies a lot of material for future study on the difficulties of communicating across paradigms.

At this point it seems worth adding to the record a few exhibits on Peirce’s definition of logic as “formal semiotic” and his variant description of logic as “semiotic, the quasi-necessary, or formal, doctrine of signs”.

Here are two variants of a paragraph where Peirce defines logic as “formal semiotic”.

Selections from C.S. Peirce, “Carnegie Application” (1902)

No. 12.  On the Definition of Logic

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.  It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has virtually been quite generally held, though not generally recognized.  (NEM 4, 20–21).

No. 12.  On the Definition of Logic [Earlier Draft]

Logic is formal semiotic.  A sign is something, A, which brings something, B, its interpretant sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, C, its object, as that in which itself stands to C.  This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time.  It is from this definition that I deduce the principles of logic by mathematical reasoning, and by mathematical reasoning that, I aver, will support criticism of Weierstrassian severity, and that is perfectly evident.  The word “formal” in the definition is also defined.  (NEM 4, 54).

Reference

  • Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75), published in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.

In the following passage Peirce explains what he means by calling logic “the quasi-necessary, or formal, doctrine of signs”.

Selection from C.S. Peirce, “Ground, Object, and Interpretant” (c. 1897)

Logic, in its general sense, is, as I believe I have shown, only another name for semiotic (σημειωτική), the quasi-necessary, or formal, doctrine of signs.  By describing the doctrine as “quasi-necessary”, or formal, I mean that we observe the characters of such signs as we know, and from such an observation, by a process which I will not object to naming Abstraction, we are led to statements, eminently fallible, and therefore in one sense by no means necessary, as to what must be the characters of all signs used by a “scientific” intelligence, that is to say, by an intelligence capable of learning by experience.  As to that process of abstraction, it is itself a sort of observation.

The faculty which I call abstractive observation is one which ordinary people perfectly recognize, but for which the theories of philosophers sometimes hardly leave room.  It is a familiar experience to every human being to wish for something quite beyond his present means, and to follow that wish by the question, “Should I wish for that thing just the same, if I had ample means to gratify it?”  To answer that question, he searches his heart, and in doing so makes what I term an abstractive observation.  He makes in his imagination a sort of skeleton diagram, or outline sketch, of himself, considers what modifications the hypothetical state of things would require to be made in that picture, and then examines it, that is, observes what he has imagined, to see whether the same ardent desire is there to be discerned.  By such a process, which is at bottom very much like mathematical reasoning, we can reach conclusions as to what would be true of signs in all cases, so long as the intelligence using them was scientific.

C.S. Peirce, Collected Papers, CP 2.227
From an unidentified fragment, c. 1897

Reference

  • 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. Volume 2 : Elements of Logic, 1932.

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The Difference That Makes A Difference That Peirce Makes • 27

Posted on October 14, 2019 by Jon Awbrey

Re: Peirce List • John Sowa

It’s been my observation over many decades that people invoke the “ethics of terminology” mainly to inveigh against everyone’s innovations but their own, so these days I’ve shifted my attention to the “pragmatics of communication”, the critical case being communication across the boundaries and through the filters of diverse communities of usage.  In that spirit, I’ll copy here my last best attempt to construct a bridge between Peirce’s special sense of “formal” and the more generic construals we likely know.

The most general meaning of formal is concerned with form, but the Latin forma can mean beauty in addition to form, so perhaps a normative goodness of form enters at this root.

The Latin word norma literally means a carpenter’s square.  The Greek gnomon is a sundial pointer taking a similar form.  The most general meaning of normative is “having to do with what a person ought to do”, but a pragmatic interpretation of ethical imperatives tends to treat that as “having to do with what a person ought to do in order to achieve a given object”, so another formula might be “relating to the good that befits a being of our kind, and what must be done in order to bring that good into being, and how to tell the signs that show the way”.

Defining logic as formal or normative semiotic differentiates logic from other species of semiotic under the general theory of signs, leaving a niche open for descriptive semiotic, just to mention the obvious branch.  This brings us to the question:

How does a concern with form, or goodness of form, along with the question of what is required to achieve an object, modify our perspective on sign relations in a way that duly marks it as a logical point of view?

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The Difference That Makes A Difference That Peirce Makes • 26

Posted on October 14, 2019 by Jon Awbrey

Re: Peirce List • John Sowa

Questions about Peirce’s use of “formal” and “normative” in relation to logic and semiotic have arisen on the Peirce List once again, but I have to run off to another appointment, so for now I’ll just post a link to a relevant previous discussion.

In other recurring discussions, as far as my personal usage goes, I’ve always suggested there is a place for descriptive semiotics, whether of not that was Peirce’s way of drawing the distinctions.

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Posted in C.S. Peirce, Descriptive Science, Fixation of Belief, Formal Systems, Information, Inquiry, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Mathematics, Normative Science, Pragmatic Maxim, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , , , , | Leave a comment