Re: All Liar, No Paradox

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Jon Awbrey

Oct 23, 2021, 9:00:17 AM10/23/21
to Peirce List, Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: All Liar, No Paradox • Discussion 1

Re: Laws of Form
::: John Mingers

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, or 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.



On 8/1/2015 11:28 AM, Jon Awbrey wrote:
> Post : All Liar, No Paradox
> Date : August 1, 2015 at 10:30 am
> | A statement S_0 asserts that a statement S_1 is a statement that S_1 is false.
> |
> | The statement S_0 violates an axiom of logic and it doesn't really
> | matter whether the ostensible statement S_1, the so-called “liar”,
> | really is a statement or has a truth value.
> Peircers,
> When I endeavored some years ago to examine the so-called “liar paradox”
> from what I take to be a pragmatic, semiotic, sign relational standpoint,
> I arrived at a way of understanding it that dispelled, for me, every air
> of paradox about it.  I wrote out an articulation of that analysis under
> the same title I'm using here and shared it in several discussion groups.
> The couplet above is a maximally trimmed down rendering of that analysis.
> The more rambling version can be found at these locations:
> •
> •
> Regards,
> Jon

Jon Awbrey

Oct 23, 2021, 3:25:45 PM10/23/21
to Peirce List, Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: All Liar, No Paradox • Discussion 2
Note. Please follow the above link for a much better formatted version.
::: James Bowery
::: John Mingers

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 “true” and “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 “true” and “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 → B, where B is the
so-called boolean domain B = {false, true}. But we are really using 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
B = {0, 1} or B = {white, blue}, my favorite colors for painting the areas
of a venn diagram.

A function like f : X → 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 → 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 fₛ : X → B and defined as
the function fₛ : X → B such that fₛ(x) = 1 if and only if x ∈ S. I think this
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
fₛ : X → B. Other names for it are the “fiber” or “pre-image” of 1. It is
computed via the “inverse function” fₛ⁻¹ in the rather ugly but pre-eminently
useful way as S = fₛ⁻¹(1).



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