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Oct 23, 2021, 9:00:15 AM10/23/21

to Peirce List, Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: All Liar, No Paradox • Discussion 1

https://inquiryintoinquiry.com/2021/10/23/all-liar-no-paradox-discussion-1/

Re: Laws of Form

https://groups.io/g/lawsofform/topic/86335306

::: John Mingers

https://groups.io/g/lawsofform/message/990

<QUOTE 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.

</QUOTE>

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.

Regards,

Jon

On 8/1/2015 11:28 AM, Jon Awbrey wrote:

> Post : All Liar, No Paradox

> http://inquiryintoinquiry.com/2015/08/01/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:

>

> • http://permalink.gmane.org/gmane.comp.misc.ontology.general/1094

> • http://forum.wolframscience.com/archive/topic/266.html

>

> Regards,

>

> Jon

>

https://inquiryintoinquiry.com/2021/10/23/all-liar-no-paradox-discussion-1/

Re: Laws of Form

https://groups.io/g/lawsofform/topic/86335306

::: John Mingers

https://groups.io/g/lawsofform/message/990

<QUOTE 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.

</QUOTE>

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.

Regards,

Jon

On 8/1/2015 11:28 AM, Jon Awbrey wrote:

> Post : All Liar, No Paradox

> http://inquiryintoinquiry.com/2015/08/01/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:

>

> • http://permalink.gmane.org/gmane.comp.misc.ontology.general/1094

> • http://forum.wolframscience.com/archive/topic/266.html

>

> Regards,

>

> Jon

>

Oct 23, 2021, 3:25:43 PM10/23/21

to Peirce List, Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: All Liar, No Paradox • Discussion 2

https://inquiryintoinquiry.com/2021/10/23/all-liar-no-paradox-discussion-2/

Note. Please follow the above link for a much better formatted version.

::: James Bowery

https://groups.io/g/lawsofform/message/1079

::: John Mingers

https://groups.io/g/lawsofform/message/1080

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).

Regards,

Jon

https://inquiryintoinquiry.com/2021/10/23/all-liar-no-paradox-discussion-2/

Note. Please follow the above link for a much better formatted version.

::: James Bowery

https://groups.io/g/lawsofform/message/1079

::: John Mingers

https://groups.io/g/lawsofform/message/1080

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).

Regards,

Jon

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