Definition and Determination • 21

Re: Definition and Determination • (18)(19)(20)
Re: FB | The Ecology of Systems ThinkingRichard Saunders

RS:
Don’t you think some little bit of unconscious knowledge and
logic comes preloaded, à priori, courtesy of our parents DNA? 
Is that simply experience one or more generations removed?

Dear Richard,

Excerpt 21 comes from a lecture on Kant in a series of lectures On the Logic of Science.  Peirce’s survey of conditions for the possibility of science reaches back through his time’s run of the mill dualism of deductive and inductive logic to encompass Aristotle’s notice of abductive reasoning.  This deeper perspective helps Peirce walk the line between empirical and rational sides of science without tumbling into either ism and it aids him in his quest for the questying beast of Kant’s synthetic à priori.  In this setting and under this sum of influences Peirce is led to his prescient theory of information, enabling him to integrate form and matter, intension and extension, into a unified whole.

With all that in mind, when Peirce says, “all our thought begins with experience, the mind furnishes no material for thought whatever”, we have to understand he is using “material” in the Aristotelian sense of matter versus form.  Saying the mind furnishes no material for thought still leaves room for the mind to furnish form for thought.  Much the same point is made in our contemporary literatures of cognitive psychology and linguistics under rubrics like “poverty of the stimulus” and “under-determination of theories by data”.

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Definition and Determination • 20

Re: Peirce ListRobert Marty

RM:
Thank you for this information.  I happen to have a work in progress (not yet written) on the question of determination.  I discovered that Peirce gave a quite remarkable definition in CP 8.361.

“We thus learn that the Object determines (i.e. renders definitely to be such as it will be,) the Sign in a particular manner.”  (CP 8.361, in CP 8.342–379, from M-20b, 1908).

It fits very well with what he writes in Excerpt 21.

“Hence universal and necessary elements of experience are not determined from without.  But are they, therefore, determined from within?  Are they determined at all?  Does not this very conception of determination imply causality and thus beg the whole question of causality at the very outset?  Not at all.  The determination here meant is not real determination but logical determination.  A cognition à priori is one which any experience contains reason for and therefore which no experience determines but which contains elements such as the mind introduces in working up the materials of sense, or rather as they are not new materials, they are the working up.”  (C.S. Peirce, Chronological Edition, CE 1, 246–247).

I have hosted this working paper on my personal website:  The Semiotics.Online, entitled DetermineWhat “Determine” Means.

I appreciate any suggestion or criticism, as usual.

Dear Robert,

Excerpt 21 comes from Peirce’s Harvard Lectures On the Logic of Science (1865).  It begins with a question about the possibility of knowledge à priori and draws conclusions about the grounds of validity for necessary and universal judgements.  For ease of discussion I copy the full excerpt below.

Is there any knowledge à priori?  All our thought begins with experience, the mind furnishes no material for thought whatever.  This is acknowledged by all the philosophers with whom we need concern ourselves at all.  The mind only works over the materials furnished by sense;  no dream is so strange but that all its elementary parts are reminiscences of appearance, the collocation of these alone are we capable of originating.

In one sense, therefore, everything may be said to be inferred from experience;  everything that we know, or think or guess or make up may be said to be inferred by some process valid or fallacious from the impressions of sense.  But though everything in this loose sense is inferred from experience, yet everything does not require experience to be as it is in order to afford data for the inference.  Give me the relations of any geometrical intuition you please and you give me the data for proving all the propositions of geometry.  In other words, everything is not determined by experience.

And this admits of proof.  For suppose there may be universal and necessary judgements;  as for example the moon must be made of green cheese.  But there is no element of necessity in an impression of sense for necessity implies that things would be the same as they are were certain accidental circumstances different from what they are.  I may here note that it is very common to misstate this point, as though the necessity here intended were a necessity of thinking.  But it is not meant to say that what we feel compelled to think we are absolutely compelled to think, as this would imply;  but that if we think a fact must be we cannot have observed that it must be.  The principle is thus reduced to an analytical one.  In the same way universality implies that the event would be the same were the things within certain limits different from what they are.

Hence universal and necessary elements of experience are not determined from without.  But are they, therefore, determined from within?  Are they determined at all?  Does not this very conception of determination imply causality and thus beg the whole question of causality at the very outset?  Not at all.  The determination here meant is not real determination but logical determination.  A cognition à priori is one which any experience contains reason for and therefore which no experience determines but which contains elements such as the mind introduces in working up the materials of sense, or rather as they are not new materials, they are the working up.  (C.S. Peirce, Chronological Edition, CE 1, 246–247).

Reference

  • Charles Sanders Peirce, “Harvard Lectures On the Logic of Science” (1865), Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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Definition and Determination • 19

Re: Peirce ListEdwina Taborsky

JA:
I needed to start thinking of semiotics and sign relations in the light of cybernetics and systems theory.  That required me to convert from understanding a sign relation as a relation among three sets, the Object, Sign, and Interpretant domains, to thinking of a sign relation as involving three active systems, Object, Sign, and Interpretant systems, respectively.
ET:
Sounds very interesting — that concept of semiosis as involving three active systems.

Dear Edwina,

Thanks for that.  As it happens, the transition from sets to systems is smoother than it may seem at first, since the first thing we need to know about a system evolving through time is its state space, which is a set among other things, and the next thing we need to know is the law governing its transition from one state to the next.  A mite steeper is the passage from dyadic, cause-effect, stimulus-response species of determination to more general orders of constraint, law, or rule-governed trajectory through state space.  And that is where Peirce’s anticipation of information theory comes into play.

Regards,

Jon

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Definition and Determination • 18

Various discussions in sundry places led me by a commodius vicus of recirculation back to the environs of old notes I gathered on the topics of Definition and Determination, that being on the occasion of a return to graduate school in a system engineering program where I needed to start thinking of semiotics and sign relations in the light of cybernetics and systems theory.

That required me to convert from understanding a sign relation as a relation among three sets, the Object, Sign, and Interpretant domains, to thinking of a sign relation as involving three active systems, Object, Sign, and Interpretant systems, respectively.

It will take a few posts to recover my notes from the Internet Archive (Wayback Machine), after which I’ll present the excerpts in a more digestible fashion and discuss more fully their implications.  Just for starters, a link-repaired version of the anchor post for this series is copied below.

Definition and Determination • 1

It looks like we might be due for one of our recurring reviews on the closely related subjects of definition and determination, with special reference to what Peirce himself wrote on the topics.

Arisbe List Archive

Here is a collection of excerpts on the subject of determination, mostly from Peirce but with a sampling of thoughts from other thinkers before and after him, on account of the larger questions of determinacy I was pursuing at the time

Collection Of Source Materials

One naturally looks to the Baldwin and Century dictionaries for Peirce-connected definitions of definition but I’d like to start with a series of texts I think are closer to Peirce’s own thoughts on definition, where he is not duty-bound to give a compendious account of every significant thinker’s point of view.  It may be a while before I get all the excerpts copied out.

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Survey of Animated Logical Graphs • 4

This is one of several Survey posts I’ll be drafting from time to time, starting with minimal stubs and collecting links to the better variations on persistent themes I’ve worked on over the years.  After that I’ll look to organizing and revising the assembled material with an eye toward developing more polished articles.

Beginnings

Elements

Examples

Excursions

Applications

Blog Dialogs

Anamnesis

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Animated Logical Graphs • 74

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30)(45)(57)(58)(59)(60)(61)(62)(63)(64)(65)(66)(69)(70)(71)(72)(73)

After the four orbits of self-dual logical graphs we come to six orbits of dual pairs.  In no particular order of importance, we may start by considering the following two.

  • The logical graphs for the constant functions f_{15} and f_{0} are dual to each other.
  • The logical graphs for the ampheck functions f_{7} and f_{1} are dual to each other.

The values of the constant and ampheck functions for each (x, y) \in \mathbb{B} \times \mathbb{B} and the text expressions for their logical graphs are given in the following Table.

Constants and Amphecks

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Animated Logical Graphs • 73

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30)(45)(57)(58)(59)(60)(61)(62)(63)(64)(65)(66)(69)(70)(71)(72)

Last time we took up the four singleton orbits in the action of T on X and saw each consists of a single logical graph which T fixes, preserves, or transforms into itself.  On that account these four logical graphs are said to be self-dual or T-invariant.

In general terms, it is useful to think of the entitative and existential interpretations as two formal languages which happen to use the same set of signs, each in its own way, to denote the same set of formal objects.  Then T defines the translation between languages and the self-dual logical graphs are the points where the languages coincide, where the same signs denote the same objects in both.  Such constellations of “fixed stars” are indispensable to navigation between languages, as every argot-naut discovers in time.

Returning to the case at hand, where T acts on a selection of 16 logical graphs for the 16 boolean functions on two variables, the following Table shows the values of the denoted boolean function f : \mathbb{B} \times \mathbb{B} \to \mathbb{B} for each of the self-dual logical graphs.

Self-Dual Logical Graphs

The functions indexed here as f_{12} and f_{10} are known as the coordinate projections (x, y) \mapsto x and (x, y) \mapsto y on the first and second coordinates, respectively, and the functions indexed as f_{3} and f_{5} are the negations (x, y) \mapsto \tilde{x} and (x, y) \mapsto \tilde{y} of those projections, respectively.

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Animated Logical Graphs • 72

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30)(45)(57)(58)(59)(60)(61)(62)(63)(64)(65)(66)(69)(70)(71)

Turning again to our Table of Orbits let’s see what we can learn about the structure of the sign relational system in view.

We saw in Episode 62 that the transformation group T = \{ 1, t \} partitions the set X of 16 logical graphs and also the set O of 16 boolean functions into 10 orbits:  4 orbits of size 1 each and 6 orbits of size 2 each.

Points in singleton orbits are called fixed points of the transformation group T : X \to X since they are left unchanged, or changed into themselves, by all group actions.  Viewed in the frame of the sign relation L \subseteq O \times X \times X, where the transformations in T are literally translations in the linguistic sense, these T-invariant graphs have the same denotations in O for both Existential Interpreters and Entitative Interpreters.

\text{Peirce Duality as Sign Relation} \stackrel{_\bullet}{} \text{Orbit Order}

Peirce Duality as Sign Relation • Orbit Order

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Logical Graphs, Truth Tables, Venn Diagrams • 1

Re: Peirce ListMauro BertaniHelmut Raulien

Dear Mauro, Helmut,

I’ll be focusing on logical graphs, especially the duality between entitative and existential interpretations, for quite a while longer, so this doesn’t address your questions about modal logic, but you might find it useful to compare the representations of logical operators by means of truth tables with those using logical graphs.

You could start with the top eight entries in the section headed “Logical Operators” on the following page.

There’s also a page bringing all eight of those Truth Tables together in one place.

I had been meaning to include the corresponding Logical Graphs and Venn Diagrams — I’ll spend some of my pandemic time working on that — It looks like it would be worth the candle reviewing their properties as representations of basic operations and going over their relative utilities for various logical purposes.

The following two pages also contain useful synopses of the boolean basics.

Resource

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Animated Logical Graphs • 71

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (57)(58)(59)(60)(61)(62)(63)(64)(65)(66)(69)(70)

Our investigation has brought us to the point of seeing both a transformation group and a triadic sign relation in the duality between entitative and existential interpretations of logical graphs.

Given the level of the above abstractions it helps to anchor them in concrete structural experience.  In that spirit we’ve been pursuing the case of a group action T : X \to X and a sign relation L \subseteq O \times X \times X where O is the set of boolean functions on two variables and X is a set of logical graphs denoting those functions.  We drew up a Table combining the aspects of both structures and sorted it according to the orbits T induces on X and consequently on O.

\text{Peirce Duality as Sign Relation} \stackrel{_\bullet}{} \text{Orbit Order}

Peirce Duality as Sign Relation • Orbit Order

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