## All Liar, No Paradox • Discussion 2

Dear James, John, et al.

The questions arising in the present discussion take us back to the question of what we are using logical values like $\textsc{true}$ and $\textsc{false}$ for, which takes us back to the question of what we are using our logical systems for.

One of the things we use logical values like $\textsc{true}$ and $\textsc{false}$ for is to mark the sides of a distinction we have drawn, or noticed, or maybe just think we see in a logical universe of discourse or space $X.$

This leads us to speak of logical functions $f : X \to \mathbb{B},$ where $\mathbb{B}$ is the so-called boolean domain $\mathbb{B} = \{ \textsc{false}, \textsc{true} \}.$  But we are really using $\mathbb{B}$ only “up to isomorphism”, as they say in the trade, meaning we are using it as a generic 2-point set and any other 1-bit set will do as well, like $\mathbb{B} = \{ 0, 1 \}$ or $\mathbb{B} = \{ \textsc{white}, \textsc{blue} \},$ my favorite colors for painting the areas of a venn diagram.

A function like $f : X \to \mathbb{B} = \{ 0, 1 \}$ is called a “characteristic function” in set theory since it characterizes a subset $S$ of $X$ where the value of $f$ is $1.$  But I like the language they use in statistics, where $f : X \to \mathbb{B}$ is called an “indicator function” since it indicates a subset of $X$ where $f$ evaluates to $1.$

The indicator function of a subset $S$ of $X$ is notated as $f_S : X \to \mathbb{B}$ and defined as the function $f_S : X \to \mathbb{B}$ where $f_S (x) = 1$ if and only if $x \in S.$  I like this because it links up nicely with the sense of indication in the calculus of indications.

The indication in question is the subset $S$ of $X$ indicated by the function $f_S : X \to \mathbb{B}.$  Other names for it are the “fiber” or “pre-image“ of $1.$  It is computed by way of the “inverse function” $f_S^{-1}$ in the rather ugly but pre-eminently useful way as $S = f_S^{-1}(1).$

Regards,

Jon

## All Liar, No Paradox • Discussion 1

JM:
Several people have referred recently to the idea that Laws of Form, and particularly Chapter 11 with imaginary logical values, provides an answer to the problems Russell found in Principia Mathematica leading to the Theory of Logical Types, which essentially banned self-referential forms.

I am interested in this and wondered if anyone had done any work on it, or seen any work on it, which actually formulates self-referential forms such as “This sentence if false” into LoF notation?

If so I would be interested to work on it.

Dear John,

The problem with Russell, well, one of the problems with Russell, is not his having or wanting a theory of types but his lacking a theory of signs, a semiotics, which, being afflicted with the isms of logicism, nominalism, syntacticism, and their ilk, the need and utility of which he lacked the sense to know.  That is one of the reasons why I take up Spencer Brown’s calculus of indications and his Laws of Form within the sign-theoretic environment of Peirce’s theory of triadic sign relations.  I’ve written a few things about how the simpler so-called paradoxes look in that framework so I’ll post a sample of those later.

Regards,

Jon

## Logical Graphs, Iconicity, Interpretation • Discussion 2

JM:
The quote you have given does not match the standard Peircean trichotomy
of icon, index, symbol.  See this quote from CP 4.448 […]

Dear John,

I hesitate to call any sketch Peirce gave of the big three sign types a “standard Peircean trichotomy of icon, index, symbol”.  Several considerations give me pause on this point.

• Peirce gave so many instructive and useful characterizations of the main sign types over the years I’d be hard-pressed to declare any one text definitive.  It is not that we have a hermeneutic circle where every text is granted equal weight, only that it takes more analysis to define the terms as yet undefined and to sort all terms involved in order of their mutual and sole dependencies.
• A cursory inspection of Peirce’s sign types, from major to minor, shows we rarely if ever have true $k$-tomies, in the sense of mutually exclusive and exhaustive categories.  True, we often speak of dichotomies and trichotomies in loose terms, but now and again loose speech has led to sinking ontologies.

Oops, more to say but need to break for midday sustenance …

Regards,

Jon

## The Difference That Makes A Difference That Peirce Makes • 33

WF:
Are there any formal languages, such as Common Logic, that adequately represent statements about propositions — statements from which, in natural reasoning, one can draw conclusions about the elements of the embedded proposition?

Dear William,

Propositional attitudes and presuppositions were hot topics in the ’80s —
scanning an old bib I see:

• Salmon, N.U., and Soames, S. (eds., 1988), Propositions and Attitudes,
Oxford University Press, New York, NY.

But the roots of the problem go way back, and of course it can’t be rooted out till
more people read and comprehend and apply Peirce’s theory of triadic sign relations.

At this point in time, however, the gravitational pull of Russell’s Planet and its inconstant satellite Quine continue to weigh against any real progress being made.

But even Russell almost, barely, just not completely broke orbit at one of those critical branch points of intellectual history — it appears it was only Wittgenstein who pulled him back from the brink of 3-adicity and back to the 2-folds of dyadic relations.

I discussed all these issues in some detail in the old Standard Upper Ontology (SUO) group and its kin.  Here’s a few pertinent fragments I archived at my current haunts:

Regards,

Jon

## Theme One Program • Discussion 5

JC:
This post [Theme One Program • Motivation 1] is so muddled
that I gave up on a meaningful scientific interpretation of it.

Dear Jerry,

Thanks for the response.

I heartily agree with the sentiment we need to pay more attention to
Mathematical And Scientific Substance (MASS) and boggle our brains
less about Purity Of Orthodox Faith (POOF).

That is what I hope to do here, on the one hand giving a realistic account of real‑world problems I encountered through the years and along the way describing how the logic of science and the tools of mathematics, especially as articulated by C.S. Peirce and tested in computational experiment, helped to address them or at least to clarify my understanding of their nature.

But that first post is only a preamble, so I hope you’ll stay tuned …

Regards,

Jon

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## Theme One Program • Discussion 4

I’ve been going back and looking again at the problems and questions which nudged me into the computational sphere as a way of building our human capacities for inquiry, learning, and reasoning.

One critical issue, you might even say bifurcation point, came up again on the Peirce List almost a decade ago in discussing the so-called “Symbol Grounding Problem”, a problem I thought had long been laid to rest, at least, among readers of Peirce, who ought to have no trouble grasping how the problem dissolves as soon as placed in the medium of Peirce’s sign relations.

Here is how the ghost of a problem returned to haunt us on that occasion …

All of which led me to recall the problems I worked on all through the ’80s …

I spent one of my parallel lives in the 1980s earning a Master’s degree in psychology, concentrating on the quantitative-statistical branch with courses in systems theory, simulation, and mathematical models, plus a healthy diet of courses and seminars in cognitive science and counseling psychology.  Instead of the usual thesis I submitted a computer program which integrated a module for multi-level sequential learning with a module for propositional constraint satisfaction, the latter based on an extension of Peirce’s logical graphs.

All the hottest topics of artificial intelligence and cognitive science from those days enjoy no end of periodic revivals, and though it brings me a twinge of nostalgia to see those old chestnuts being fired up again, those problems now seem to me as problems existing only for a peculiar tradition of thought, a tradition ever occupied with chasing will o’ th’ wisps Peirce dispersed long before the chase began.

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## Logical Graphs, Iconicity, Interpretation • Discussion 1

JM:
I’m impressed that you have read Ricoeur — my impression is that Americans don’t have much time for Continental philosophy (a huge generalisation of course).

Have you looked at Habermas?  He uses Peirce’s work as well as hermeneutics (mainly Gadamer) and critical theory to come up with what he calls a theory of communicative action.  He also called it “universal pragmatics” at one time as a nod to both Chomsky and semiotics.

Dear John,

That observation from Ricoeur’s Conflict of Interpretations comes from a time when Susan Awbrey and I were exploring the synergies of action research, critical thinking, classical and post-modern hermeneutics, and C.S. Peirce’s triadic relational semiotics.  We benefited greatly from our study of Gadamer, Habermas, Ricoeur and a little more from Derrida, Foucault, Lyotard, aided by the panoramic surveys of Richard J. Bernstein.  All that led to a paper we gave at a conference on Hermeneutics and the Human Sciences, subsequently published as “Interpretation as Action : The Risk of Inquiry” (doc) (pdf).

I found Ricoeur’s comment fitting in the present connection because it speaks to the way identical modulations of a medium may convey different messages to different cultures and contexts of communication.  Conversely, conveying the same message to different cultures and contexts of communication may require different modulations of the same medium.

That is precisely the situation we observe in the Table from Episode 1, for ease of reference repeated below.  The objects to be conveyed are the 16 boolean functions on 2 variables, whose venn diagrams appear in Column 1.  And we have the two cultures of interpreters, Entitative and Existential, whose graphical and parenthetical forms of expression for the boolean functions are shown in Column 2 and Column 3, respectively.

$\text{Boolean Functions and Logical Graphs on Two Variables}$

### References

• Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.  ArchiveJournal.  Online (doc) (pdf).
• Awbrey, J.L., and Awbrey, S.M. (June 1992), “Interpretation as Action : The Risk of Inquiry”, The Eleventh International Human Science Research Conference, Oakland University, Rochester, Michigan.

## Conceptual Barriers • 4

Re: Conceptual Barriers • (1)(2)(3)
Re: Peirce ListJohn SowaGary RichmondRobert Marty

Dear Robert,

I believe you have pointed to the crux of the matter.

When I arrived on the campus of my first university in the late ’60s there was already in progress some sort of year-long cross-campus “Colloquium” or “Dialogue” going on — with many invited speakers and representatives from all the colleges and departments, springing from the issues raised by C.P. Snow’s 1959 lecture on “The Two Cultures”.  About the same time the University instituted three new residential colleges in cross-cultural liberal arts, “statecraft” (a mix of history, law, and political science), and the bridge between science and society.

I confess it was all over the head of a first year student with his brain buried in math and physics with a minor in beer but I did get a whiff now and then of a sea change in the air.  A Möbius twist of fate and my room-mate’s hankering to find a dorm with a reputation for better food soon led me to relocate to the residence hall housing the “relevant science” people.  They all took courses with titles like “Third Cultural Rhetoric” and so I osmoted some of that synthesis from my brushes with the crew of that Enterprise.  Plus I met my future wife.

Poking around the web I see I already began this story two years ago, in connection with remarks John Sowa made in the Ontolog Forum as to “Why A Single Unified Ontology Is Impossible”.  I’ve also just realized how one of the main themes of the present discussion links up with what I wrote earlier about the “Immune System Metaphor” — so I can save myself the effort of crafting new syntax by continuing the story as I did before under the heading of Conceptual Barriers.

Long time passing, I found myself returning to these questions around the turn of the millennium, addressing the “problem of silos” and the “scholarship of integration” from the perspective of Peirce’s and Dewey’s pragmatism and semiotics.  Here’s a couple of contributions Susan Awbrey and I made to the area.

#### Conference Presentation

• Awbrey, S.M., and Awbrey, J.L. (1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal ‘Organization’, Re‑Organizing Knowledge, Trans‑Forming Institutions : Knowing, Knowledge, and the University in the 21st Century, University of Massachusetts, Amherst, MA.  Online.

#### Published Paper

• Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, 269–284.  AbstractOnline.

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## C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • 2

### Selection from C.S. Peirce, “On the Algebra of Logic : A Contribution to the Philosophy of Notation” (1885)

#### §1.  Three Kinds Of Signs (cont.)

I have taken pains to make my distinction of icons, indices, and tokens clear, in order to enunciate this proposition:  in a perfect system of logical notation signs of these several kinds must all be employed.  Without tokens there would be no generality in the statements, for they are the only general signs;  and generality is essential to reasoning.  Take, for example, the circles by which Euler represents the relations of terms.  They well fulfill the function of icons, but their want of generality and their incompetence to express propositions must have been felt by everybody who has used them.  Mr. Venn has, therefore, been led to add shading to them;  and this shading is a conventional sign of the nature of a token.  In algebra, the letters, both quantitative and functional, are of this nature.

But tokens alone do not state what is the subject of discourse;  and this can, in fact, not be described in general terms;  it can only be indicated.  The actual world cannot be distinguished from a world of imagination by any description.  Hence the need of pronouns and indices, and the more complicated the subject the greater the need of them.  The introduction of indices into the algebra of logic is the greatest merit of Mr. Mitchell’s system.  He writes $F_1$ to mean that the proposition $F$ is true of every object in the universe, and $F_u$ to mean that the same is true of some object.  This distinction can only be made in some such way as this.  Indices are also required to show in what manner other signs are connected together.

With these two kinds of signs alone any proposition can be expressed;  but it cannot be reasoned upon, for reasoning consists in the observation that where certain relations subsist certain others are found, and it accordingly requires the exhibition of the relations reasoned with in an icon.  It has long been a puzzle how it could be that, on the one hand, mathematics is purely deductive in its nature, and draws its conclusions apodictically, while on the other hand, it presents as rich and apparently unending a series of surprising discoveries as any observational science.  Various have been the attempts to solve the paradox by breaking down one or other of these assertions, but without success.  The truth, however, appears to be that all deductive reasoning, even simple syllogism, involves an element of observation;  namely, deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts.  (3.363).

### References

• Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
• 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 3 : Exact Logic (Published Papers), 1933.  CP 3.359–403.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 5 (1884–1886), 1993.  Item 30, 162–190.

## C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • 1

### Selection from C.S. Peirce, “On the Algebra of Logic : A Contribution to the Philosophy of Notation” (1885)

#### §1.  Three Kinds Of Signs

Any character or proposition either concerns one subject, two subjects, or a plurality of subjects.  For example, one particle has mass, two particles attract one another, a particle revolves about the line joining two others.  A fact concerning two subjects is a dual character or relation;  but a relation which is a mere combination of two independent facts concerning the two subjects may be called degenerate, just as two lines are called a degenerate conic.  In like manner a plural character or conjoint relation is to be called degenerate if it is a mere compound of dual characters.  (3.359).

A sign is in a conjoint relation to the thing denoted and to the mind.  If this triple relation is not of a degenerate species, the sign is related to its object only in consequence of a mental association, and depends upon a habit.  Such signs are always abstract and general, because habits are general rules to which the organism has become subjected.  They are, for the most part, conventional or arbitrary.  They include all general words, the main body of speech, and any mode of conveying a judgment.  For the sake of brevity I will call them tokens.  (3.360).

But if the triple relation between the sign, its object, and the mind, is degenerate, then of the three pairs

$\begin{array}{lll} \text{sign} & & \text{object} \\ \text{sign} & & \text{mind} \\ \text{object} & & \text{mind} \end{array}$

two at least are in dual relations which constitute the triple relation.  One of the connected pairs must consist of the sign and its object, for if the sign were not related to its object except by the mind thinking of them separately, it would not fulfill the function of a sign at all.  Supposing, then, the relation of the sign to its object does not lie in a mental association, there must be a direct dual relation of the sign to its object independent of the mind using the sign.  In the second of the three cases just spoken of, this dual relation is not degenerate, and the sign signifies its object solely by virtue of being really connected with it.  Of this nature are all natural signs and physical symptoms.  I call such a sign an index, a pointing finger being the type of this class.

The index asserts nothing;  it only says “There!”  It takes hold of our eyes, as it were, and forcibly directs them to a particular object, and there it stops.  Demonstrative and relative pronouns are nearly pure indices, because they denote things without describing them;  so are the letters on a geometrical diagram, and the subscript numbers which in algebra distinguish one value from another without saying what those values are.  (3.361).

The third case is where the dual relation between the sign and its object is degenerate and consists in a mere resemblance between them.  I call a sign which stands for something merely because it resembles it, an icon.  Icons are so completely substituted for their objects as hardly to be distinguished from them.  Such are the diagrams of geometry.  A diagram, indeed, so far as it has a general signification, is not a pure icon;  but in the middle part of our reasonings we forget that abstractness in great measure, and the diagram is for us the very thing.  So in contemplating a painting, there is a moment when we lose consciousness that it is not the thing, the distinction of the real and the copy disappears, and it is for the moment a pure dream — not any particular existence, and yet not general.  At that moment we are contemplating an icon.  (3.362).

### References

• Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
• 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 3 : Exact Logic (Published Papers), 1933.  CP 3.359–403.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 5 (1884–1886), 1993.  Item 30, 162–190.