Higher Order Sign Relations • 4

Re: Higher Order Sign Relations • 3
Re: Relations, Types, Functions
Re: CyberneticsCliff Joslyn

CJ:
Categorical approaches to systems theory have been very attractive to me for a long time.  My current work is categorically adjacent, and I’m funding some efforts in this direction.  The category of binary relations is central to our immediate work in hypergraphs and high-order networks, but is also to any general systems theoretical approach.  I’ve approached topoi and closed Cartesian categories a few times, but admit it’s challenging.  I need something at the level that David Spivak and crew have been developing to become more fluent, if you’re aware of his work.  Any worked examples you could provide would be very useful andwelcome.

Dear Cliff,

There are a few sources I recall most vividly for the way they capture the attractions of categories.  The following references come from a bibliography I collected in the early 90s plus a number added over the course of that decade.

The following sources may also be of interest.

  • Mili • Program construction and semantics from a relational point of view, using Tarski’s approach to binary relations (Fatma Mili taught a course on this at OU).
  • Freyd and ScedrovCategories, Allegories, a category-theoretic take on binary relations.

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Inquiry Into Inquiry • Discussion 4

Re: Inquiry Into Inquiry • In Medias Res
Re: Inquiry Into Inquiry • Flashback
Re: FB Comment • Daniel Everett

Othello Believes Desdemona Loves Cassio

Dan Everett commented on my post about Russell’s question, “How shall we describe the logical form of a belief?”, giving his take on Russell’s analysis of the example, “Othello believes Desdemona loves Cassio”.

DE:
The most interesting aspect of such constructions from my perspective is that embedding is unnecessary for the reading.  In Piraha you can get independent clauses expressing the same thing.  Or even in English.  Othello believes something.  That something is that Desdemona loves Cassio.  So the advantage of Peircean graphs (and later of Discourse Representation Theory) is that the syntactic feature of embedding is not crucial.  Just as in larger discourse of multiple independent sentences.

I added the following observations.

Russell asks, “How shall we describe the logical form of a belief?”  The question is a good one, maybe too good, loaded with a surplus of meanings for “logical form”.  Read in the spectrum of interpretive lights traditional schools of thought have brought to bear on it, “logical form” hovers between the poles of objective form and syntactic form without ever settling down.  A more stable fix on its practical sense can be gained from the standpoint staked out by Peirce on the basis of the pragmatic maxim, aiming at objective structure and seeing syntactic structure as accessory to that aim.

To be continued …

Reference

  • Bertrand Russell, “The Philosophy of Logical Atomism”, pp. 35–155 in The Philosophy of Logical Atomism, edited with an introduction by David Pears, Open Court, La Salle, IL, 1985.  First published 1918.

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Inquiry Into Inquiry • Understanding 2

In the passage quoted in the previous post Bertrand Russell addresses the question, “What is the logical structure of the fact which consists in a given subject understanding a given proposition?” and he selects a proposition of the form ``A ~\text{and}~ B ~\text{are similar}" to demonstrate his way of analyzing the fact.  Russell wraps up his discussion of the example in the passage quoted below.

Excerpt from Bertrand Russell • “Theory of Knowledge” (1913)

Part 2. Atomic Propositional Thought

Chapter 1. The Understanding of Propositions

(4). [cont.]  It follows that, when a subject S understands ``A ~\text{and}~ B ~\text{are similar}", “understanding” is the relating relation, and the terms are S and A and B and similarity and R(x, y), where R(x, y) stands for the form “something and something have some relation”.  Thus a first symbol for the complex will be

U \{S, A, B, \mathrm{similarity}, R(x, y) \}~.

This symbol, however, by no means exhausts the analysis of the form of the understanding-complex.  There are many kinds of five-term complexes, and we have to decide what the kind is.

It is obvious, in the first place, that S is related to the four other terms in a way different from that in which any of the four other terms are related to each other.

(It is to be observed that we can derive from our five-term complex a complex having any smaller number of terms by replacing any one or more of the terms by “something”.  If S is replaced by “something”, the resulting complex is of a different form from that which results from replacing any other term by “something”.  This explains what is meant by saying that S enters in a different way from the other constituents.)

It is obvious, in the second place, that R(x, y) enters in a different way from the other three objects, and that “similarity” has a different relation to R(x, y) from that which A and B have, while A and B have the same relation to R(x, y).  Also, because we are dealing with a proposition asserting a symmetrical relation between A and B, A and B have each the same relation to “similarity”, whereas, if we had been dealing with an asymmetrical relation, they would have had different relations to it.  Thus we are led to the following map of our five-term complex.

Russell • Understanding (S, A, B, Similarity, Rxy)

In this figure, one relation goes from S to the four objects;  one relation goes from R(x, y) to similarity, and another to A and B, while one relation goes from similarity to A and B.

This figure, I hope, will help to make clearer the map of our five-term complex.  But to explain in detail the exact abstract meaning of the various items in the figure would demand a lengthy formal logical discussion.  Meanwhile the above attempt must suffice, for the present, as an analysis of what is meant by “understanding a proposition”.  (Russell, TOK, 117–118).

Reference

  • Bertrand Russell, Theory of Knowledge : The 1913 Manuscript, edited by Elizabeth Ramsden Eames in collaboration with Kenneth Blackwell, Routledge, London, UK, 1992.  First published, George Allen and Unwin, 1984.

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Inquiry Into Inquiry • Understanding 1

Another passage from Russell further illustrates what I see as a critical juncture in his thought.  The graph-theoretic figure he uses in analyzing a complex of logical relationships brings him to the edge of seeing the limits of dyadic analysis — but he veers off and does not make the leap.  At any rate, that’s how it looks from a perspective informed by Peirce.

Excerpt from Bertrand Russell • “Theory of Knowledge” (1913)

Part 2. Atomic Propositional Thought

Chapter 1. The Understanding of Propositions

(4).  We come now to the last problem which has to be treated in this chapter, namely:  What is the logical structure of the fact which consists in a given subject understanding a given proposition?  The structure of an understanding varies according to the proposition understood.  At present, we are only concerned with the understanding of atomic propositions;  the understanding of molecular propositions will be dealt with in Part 3.

Let us again take the proposition “A and B are similar”.

It is plain, to begin with, that the complexA and B being similar”, even if it exists, does not enter in, for if it did, we could not understand false propositions, because in their case there is no such complex.

It is plain, also, from what has been said, that we cannot understand the proposition unless we are acquainted with A and B and similarity and the form “something and something have some relation”.  Apart from these four objects, there does not appear, so far as we can see, to be any object with which we need be acquainted in order to understand the proposition.

It seems to follow that these four objects, and these only, must be united with the subject in one complex when the subject understands the proposition.  It cannot be any complex composed of them that enters in, since they need not form any complex, and if they do, we need not be acquainted with it.  But they themselves must all enter in, since if they did not, it would be at least theoretically possible to understand the proposition without being acquainted with them.

In this argument, I appeal to the principle that, when we understand, those objects with which we must be acquainted when we understand, and those only, are object-constituents (i.e. constituents other than understanding itself and the subject) of the understanding-complex.  (Russell, TOK, 116–117).

The passage continues in the next post.

Reference

  • Bertrand Russell, Theory of Knowledge : The 1913 Manuscript, edited by Elizabeth Ramsden Eames in collaboration with Kenneth Blackwell, Routledge, London, UK, 1992.  First published, George Allen and Unwin, 1984.

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Inquiry Into Inquiry • Flash Back

The fault, dear Brutus, is not in our stars,
But in ourselves …

Julius Caesar • 1.2.141–142

Signs have a power to inform, to lead our thoughts and thus our actions in accord with reality, to make reality our friend.  And signs have a power to misinform, to corrupt our thoughts and thus our actions and lead us to despair of all our ends.

Excerpt from Bertrand Russell • “The Philosophy of Logical Atomism” (1918)

4. Propositions and Facts with More than One Verb: Beliefs, Etc.

4.3. How shall we describe the logical form of a belief?

I want to try to get an account of the way that a belief is made up.  That is not an easy question at all.  You cannot make what I should call a map-in-space of a belief.  You can make a map of an atomic fact but not of a belief, for the simple reason that space-relations always are of the atomic sort or complications of the atomic sort.  I will try to illustrate what I mean.

The point is in connexion with there being two verbs in the judgment and with the fact that both verbs have got to occur as verbs, because if a thing is a verb it cannot occur otherwise than as a verb.

Suppose I take ‘A believes that B loves C’.  ‘Othello believes that Desdemona loves Cassio’.  There you have a false belief.  You have this odd state of affairs that the verb ‘loves’ occurs in that proposition and seems to occur as relating Desdemona to Cassio whereas in fact it does not do so, but yet it does occur as a verb, it does occur in the sort of way that a verb should do.

I mean that when A believes that B loves C, you have to have a verb in the place where ‘loves’ occurs.  You cannot put a substantive in its place.  Therefore it is clear that the subordinate verb (i.e. the verb other than believing) is functioning as a verb, and seems to be relating two terms, but as a matter of fact does not when a judgment happens to be false.  That is what constitutes the puzzle about the nature of belief.

You will notice that whenever one gets to really close quarters with the theory of error one has the puzzle of how to deal with error without assuming the existence of the non-existent.

I mean that every theory of error sooner or later wrecks itself by assuming the existence of the non-existent.  As when I say ‘Desdemona loves Cassio’, it seems as if you have a non-existent love between Desdemona and Cassio, but that is just as wrong as a non-existent unicorn.  So you have to explain the whole theory of judgment in some other way.

I come now to this question of a map.  Suppose you try such a map as this:

Othello Believes Desdemona Loves Cassio

This question of making a map is not so strange as you might suppose because it is part of the whole theory of symbolism.  It is important to realize where and how a symbolism of that sort would be wrong:  Where and how it is wrong is that in the symbol you have this relationship relating these two things and in the fact it doesn’t really relate them.  You cannot get in space any occurrence which is logically of the same form as belief.

When I say ‘logically of the same form’ I mean that one can be obtained from the other by replacing the constituents of the one by the new terms.

If I say ‘Desdemona loves Cassio’ that is of the same form as ‘A is to the right of B’.  Those are of the same form, and I say that nothing that occurs in space is of the same form as belief.

I have got on here to a new sort of thing, a new beast for our zoo, not another member of our former species but a new species.  The discovery of this fact is due to Mr. Wittgenstein.  (Russell, POLA, 89–91).

Reference

  • Bertrand Russell, “The Philosophy of Logical Atomism”, pp. 35–155 in The Philosophy of Logical Atomism, edited with an introduction by David Pears, Open Court, La Salle, IL, 1985.  First published 1918.

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Inquiry Into Inquiry • In Medias Res

Re: Daniel Everett

DE:
I am trying to represent two readings of the three juxtaposed sentences in English.  The first reading is that the judge and the jury both know that Malcolm is guilty.  The second is that the judge knows that the jury thinks that Malcolm is guilty.

Daniel Everett • Judge, Jury, Malcolm, Guilty • Graph 1

Daniel Everett • Judge, Jury, Malcolm, Guilty • Graph 2

Do these purported EGs of mine seem correct to you?

Dear Dan,

Apologies for the delay in responding … I won’t have much of use to say about those particular graphs as I’ve long been following a different fork in Peirce’s work about how to get from Alpha to Beta, from propositional to quantificational logic via graphical syntax.

But the examples raise one of the oldest issues I’ve bothered about over the years, going back to the days when I read PQR (Peirce, Quine, Russell) in tandem and many long discussions with my undergrad phil advisor.  That is the question of intentional contexts and “referential opacity”.  The thing is Peirce’s pragmatic standpoint yields a radically distinct analysis of belief, knowledge, and indeed truth from the way things have been handled down the line from logical atomism to logical empiricism to analytic philosophy in general.  As it happens, there was a critical branch point in time when Russell almost got a clue but Wittgenstein bullied him into dropping it, at least so far as I could tell from a scattered sample of texts.

At any rate, I fell down the Wayback Machine rabbit hole looking for things I wrote about all this on the Peirce List and other places around the web at the turn of the millennium …

I’d almost be tempted to start a blog series on this, probably simulcast on the Facebook Peirce Matters page if you’re into discussing it online … I have enough off the cuff to start an anchor post or two, but it might be the middle of August before I could do much more.

Regards,

Jon

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Inquiry Into Inquiry • On Initiative 2

Re: Scott Aaronson(1)(2)(3)

SA:
Personally, I’d give neither of them [Bohr or Einstein] perfect marks, in part because they not only both missed Bell’s Theorem, but failed even to ask the requisite question (namely:  what empirically verifiable tasks can Alice and Bob use entanglement to do, that they couldn’t have done without entanglement?).  But I’d give both of them very high marks for, y’know, still being Albert Einstein and Niels Bohr.

To Ask The Requisite Question

This brings me to the question I wanted to ask about AI sentience, but was afraid to ask.

  • Does GPT-3 ever ask an original question on its own?

Simply asking for clarification of an interlocutor’s prompt is not insignificant but I’m really interested in something more spontaneous and “self‑starting” than that.  Does it ever wake up one morning, as it were, and find itself in a “state of question”, a state of doubt or uncertainty so compelling as to bring it to ask on its own initiative what we might recognize as a novel question?

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Sign Relations • Discussion 14

Re: Cybernetics • Cliff Joslyn (1) (2) (3) (4)

Dear Cliff,

A few examples of sign relations and triadic relations may serve to illustrate the problem of their demarcation.

First, to clear up one point of notation, in writing L \subseteq O \times S \times I, there is no assumption on my part the relational domains O, S, I are necessarily disjoint.  They may intersect or even be identical, as O = S = I.  Of course we rarely need to contemplate limiting cases of that type but I find it useful to keep them in our categorical catalogue.  (Other writers will differ on that score.)  On the other hand, we very often consider cases where S = I, as in the following two examples of sign relations discussed in Sign Relations • Examples.

Sign Relation Twin Tables LA & LB

We have the following data.

\begin{array}{ccl}  O & = & \{ \mathrm{A}, \mathrm{B} \}  \\[6pt]  S & = & \{ ``\mathrm{A}", ``\mathrm{B}", ``\mathrm{i}", ``\mathrm{u}" \}  \\[6pt]  I & = & \{ ``\mathrm{A}", ``\mathrm{B}", ``\mathrm{i}", ``\mathrm{u}" \}  \end{array}

As I mentioned, those examples were deliberately constructed to be as simple as possible but they do exemplify many typical features of sign relations in general.  Until the time my advisor asked me for cases of that order I had always contemplated formal languages with countable numbers of signs and never really thought about finite sign relations at all.

Reference

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

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Sign Relations • Discussion 13

Re: Cybernetics • Cliff Joslyn (1) (2) (3)

Dear Cliff,

Backing up a little —

Whether a thing qualifies as a sign is not an ontological question, a matter of what it is in itself, but a pragmatic question, a matter of what role it plays in a particular application.

By extension, whether a triadic relation qualifies as a sign relation is not just a question of its abstract structure but a question of its potential applications, of its fitness for a particular purpose, namely, whether we can imagine it capturing aspects of objective structure immanent in the conduct of logical reasoning.

Because it’s difficult, and not even desirable, to place prior limits on “what we can imagine finding a use for”, there is seldom a good case for trying to reduce pragmatic definitions to ontological definitions.  That’s why I feel bound to leave the boundaries a bit fuzzy.

Just to sum up what I’ve been struggling to say here —

It’s not a bad idea to cast an oversized net at the outset, and the à priori method can take us a way with that, but developing semiotics beyond its first principles and early stages will depend on gathering more significant examples of sign relations and sign transformations approaching the level we actually employ in the practice of communication, computation, inquiry, learning, proof, and reasoning in general.  I think that’s probably the best way to see the real sense and utility of Peirce’s double definition of logic and signs.

Reference

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

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Sign Relations • Discussion 12

Re: CyberneticsCliff Joslyn

CJ:
For a given arbitrary triadic relation L \subseteq O \times S \times I (let’s say that O, S, and I are all finite, non‑empty sets), I’m interested to understand what additional axioms you’re saying are necessary and sufficient to make L a sign relation.  I checked Sign Relations • Definition, but it wasn’t obvious, or at least, not formalized.

Dear Cliff,

From a purely speculative point of view, any triadic relation L \subseteq X \times X \times X on any set X might be capable of capturing aspects of objective structure immanent in the conduct of logical reasoning.  At least I can think of no reason to exclude the possibility à priori.

When we turn to the task of developing computational adjuncts to inquiry there is still no harm in keeping arbitrary triadic relations in mind, as entire hosts of them will turn up on the universe side of many universes of discourse we happen to encounter, if nowhere else.

Peirce’s use of the word definition understandably leads us to anticipate a strictly apodictic development, say, along the lines of abstract group theory or axiomatic geometry.  In that light I often look to group theory for hints on how to go about tackling a category of triadic relations such as we find in semiotics.  The comparison makes for a very rough guide but the contrasts are also instructive.

More than that, the history of group theory, springing as it did as yet unnamed from the ground of pressing mathematical problems, from Newton’s use of symmetric functions and Galois’ application of permutation groups to the theory of equations among other sources, tells us what state of development we might reasonably expect from the current still early days of semiotics.

To be continued …

Reference

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

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