Handedness as Logical Primitiveness Re: [PEIRCE-L] Consequence as Logical Primitive (was Resending)

[Jerry LR Chandler]

Jon: 

> On Feb 10, 2021, at 7:44 PM, Jon Alan Schmidt  
> wrote:
> 
> JFS: In mathematics and logic, equivalence means freely interchangeable in 
> all contexts without any change in meaning.
> 
> No, it means freely interchangeable within a particular formal system. In the 
> context of classical logic, which treats the universe of discourse as 
> individual, a scroll is indeed equivalent to and interchangeable with nested 
> cuts/ovals. Nevertheless, as I demonstrated with Shafiei's example of ex 
> falso quodlibet 
> (https://list.iupui.edu/sympa/arc/peirce-l/2021-02/msg00019.html 
> ), the same 
> classical EG with nested cuts/ovals can have multiple natural-language 
> translations, which is not the case when scrolls are employed instead. After 
> all, does someone really mean exactly the same thing when saying "if A then 
> B" as when saying "not both A and not-B"? Peirce certainly does not think so.
> 
> 

While I agree with your assertion, Jon, in realism, the situation is more 
highly constrained than this paragraph alludes too.

CSP may have used the bedrock logic of chemical radicals to construct the 
logical distinctions among natural nominative objects (plural individuals) in 
order to construct the perplex predicates of meaningful abstract sentences.  
This is necessary to copulate the individual abstract symbols from the many to 
the one; that is, from atoms to a molecule, or, another words, from n 
individual distinctive atoms to a single polyatomic molecule. 

Within the possible patterns of “n” individual terms of a composite sentence, 
the natural logic may need to make specify a specific unique pattern for all n 
terms.

That is, “a" is here and not at any of the other n-1 locations, “b” is there 
and not at any of the other n-2 locations, “c” is over there and not at any of 
the other n-3 locations, “d” is opposite of “c” and not at any of the other n-4 
locations. And so forth to any arbitrary large n. 

 These logical assertions were well known to CSP under the guise of the 
handedness of molecules such as described by Pastuer and by van't Hoff and 
LaBel.

 Thus, such assertions are necessary but not sufficient to copulate the 
nominative forms of the numerical information as antecedent to the pragmatic 
consequences predicated by the indices of “n”. 

In the chemical sciences, these locative logics are organized into a scientific 
logic of valence and handedness and are components of the standard scientific 
methodology of today.

>From the perspective of CSP logical trichotomies, these forms of negations are 
>essential. And necessary to relate and to copulate the indices of the sin-sign 
>to the legi-sign via the dicisign. 

Copulation makes chemistry attractive, fun and creates natural sorts and kinds. 
Under thoughtful planning of contexts, repetition is habit forming and 
reproducible.

Cheers

Jerry _ _ _ _ _ _ _ _ _ _
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Sowa and the Meaning of Equivalence Relation. Was Re: [PEIRCE-L] Consequence as Logical Primitive (was Resending)

[Jerry LR Chandler]

List:

> On Feb 10, 2021, at 7:44 PM, Jon Alan Schmidt  
> wrote:
> 
> On Tue, Feb 9, 2021 at 11:25 PM John F. Sowa  > wrote:
>  and Jerry LRC,
> 
> JFS> In mathematics and logic, equivalence means freely interchangeable in 
> all contexts without any change in meaning.
> 
> JLRC> Really?
> 
> Yes, indeed.  As Casey Stengel used to say, "Ya could look it up." 
> 
> John
> 

The question of interchangeability of meaning of mathematical symbols is remote 
from the simplistic view you present, John.

In standard mathematics, that is, the usual mathematics of analysis and applied 
mathematics of physical analysis,
including such graphic structures as categories, topological spaces, Hilbert 
spaces, etc, 
the term equivalence relations is used to distinguish one form of mathematical 
structure from another.
The constraints used to separate the meanings of equivalence relations can be 
interpreted as a triad,
firstness (or transitivity), secondness (or symmetry) and thirdness (or 
reflexivity).


One very simple usage of the notion of “equivalence relations” is to structure 
the ordering of mathematical spaces such as (x,y,z), by inclusivity, that is, 
x is contained in y and
 y is contained in z and therefore
x is contained in z. 

With these antecedent definitions, one can order the meanings of the structural 
graphs for various spaces.
For example, a Hilbert space is logically contained in a metric space and a 
metric space is contained in a general topological space.

These distinctions are critical for interpreting the relationships between CSP 
usage of “topology” / graph theory
and the usage of Hilbert spaces in modern chemical graph theory.   (That is, 
perplex number theory of electrical relations.)
Note that the “handedness” of biomolecules are NOT symmetric as a consequence 
of the existential graphic patterns of internal electrical relations of 
chemical atoms in these compositions.

If you are not aware of these advanced mathematical concepts, to quote Casey 
Stengel, you could look it up.

But, I will save you the time and effort and suggest that the term 
“equivalence” as you used it in the sentence:

> JFS> In mathematics and logic, equivalence means freely interchangeable in 
> all contexts without any change in meaning.
> 

does not have a triadic meaning and does not imply an “equivalence relation”. 

Cheers

Jerry 

Research Professor
Krasnow Institute of Advanced Study

Headwater House
Little Falls, MN




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Re: Sowa and the Meaning of Equivalence Relation. Was Re: [PEIRCE-L] Consequence as Logical Primitive (was Resending)

[Auke van Breemen]

Jerry,

I think you did hit the weak spot: 

> But, I will save you the time and effort and suggest that the term 
> “equivalence” as you used it in the sentence:
> 
> 
> > > 
> > > > > 
> > > JFS> In mathematics and logic, equivalence means freely 
> > > interchangeable in all contexts without any change in meaning.
> > > 
> > > > > 
> > > does not have a triadic meaning and does not imply an 
> > “equivalence relation”. 
> 


I only would add 'real' to your remark. For logical space it is freely 
interchangable as Jon acknowledges.


Best,

Auke

> Cheers
> 
> Jerry 
> 
> Research Professor
> Krasnow Institute of Advanced Study
> 
> Headwater House
> Little Falls, MN
> 
> 
> 
> 
> 


 

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[PEIRCE-L] Re: Animated Logical Graphs

[Jon Awbrey]

Cf: Animated Logical Graphs • 58
http://inquiryintoinquiry.com/2021/02/11/animated-logical-graphs-58/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/animated_logical_graphs/79952098
:: Lyle Anderson
https://groups.io/g/lawsofform/message/109

Re: Brading, K., Castellani, E. and Teh, N, (2017),
“Symmetry and Symmetry Breaking”, The Stanford Encyclopedia
of Philosophy (Winter 2017), Edward N. Zalta (ed.).  Online
https://plato.stanford.edu/archives/win2017/entries/symmetry-breaking/

Dear Lyle,

Thanks for the link to the article on symmetry and its breaking.  I did once
take a Master's in Mathematics, specializing in combinatorics, graph theory,
and group theory.  As far as the applications to logical graphs and the
calculus of indications goes, it will take careful attention to the details
of the relationship between the two interpretations recognized by Peirce and
Spencer Brown.

Both Peirce and Spencer Brown recognized the relevant duality, if they differed
in what they found most convenient to use in their development and exposition,
and most of us will emphasize one interpretation or the other as a matter of
taste or facility in a chosen application, so it requires a bit of effort to
keep the underlying unity in focus.  I recently made another try at taking
a more balanced view, drawing up a series of tables in parallel columns the
way one commonly does with dual theorems in projective geometry, so I will
shortly share more of that work.

Regards,

Jon
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[PEIRCE-L] Re: Animated Logical Graphs

[Jon Awbrey]

Cf: Animated Logical Graphs • 57
http://inquiryintoinquiry.com/2021/02/11/animated-logical-graphs-57/

| All other sciences without exception depend upon
| the principles of mathematics;  and mathematics
| borrows nothing from them but hints.
|
| C.S. Peirce • “Logic of Number”

| A principal intention of this essay is to separate
| what are known as algebras of logic from the subject
| of logic, and to re-align them with mathematics.
|
| G. Spencer Brown • Laws of Form

The duality between entitative and existential interpretations
of logical graphs tells us something important about the relation
between logic and mathematics.  It tells us that the mathematical
forms giving structure to reasoning are deeper and more abstract
at once than their logical interpretations.

A formal duality points to a more encompassing unity, founding
a calculus of forms whose expressions can be read in alternate
ways by switching the meanings assigned to a pair of primitive
terms.  Spencer Brown’s mathematical approach to Laws of Form
and the whole of Peirce’s work on the mathematics of logic
shows both thinkers were deeply aware of this principle.

Peirce explored a variety of dualities in logic which he treated on
analogy with the dualities in projective geometry.  This gave rise to
formal systems where the initial constants, and thus their geometric and
graph-theoretic representations, had no uniquely fixed meanings but could be
given dual interpretations in logic.

It was in this context that Peirce’s systems of logical graphs developed,
issuing in dual interpretations of the same formal axioms which Peirce
referred to as “entitative graphs” and “existential graphs”, respectively.
He developed only the existential interpretation to any great extent, since
the extension from propositional to relational calculus appeared more natural
in that case, but whether there is any logical or mathematical reason for the
symmetry to break at that point is a good question for further research.

Resources
=

• Duality Indicating Unity
https://inquiryintoinquiry.com/2013/01/31/duality-indicating-unity-1/

• C.S. Peirce • Logic of Number
https://inquiryintoinquiry.com/2012/09/01/c-s-peirce-logic-of-number-ms-229/

• C.S. Peirce • Syllabus • Selection 1
https://inquiryintoinquiry.com/2014/08/24/c-s-peirce-syllabus-selection-1/

References
==

• Peirce, C.S., [Logic of Number — Le Fevre] (MS 229),
  in Carolyn Eisele (ed., 1976), The New Elements of
  Mathematics by Charles S. Peirce, vol. 2, 592–595.

• Spencer Brown, G. (1969), Laws of Form,
  George Allen and Unwin, London, UK.

Regards,

Jon
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