Chimp

The tautologies of classical 2v-logic have symmetry. Can the
tautologies of logic be categorized as symmetry groups?

The Law of Excluded Middle:

X|~X|X V~X
T| F| T
F| T| T

(T|F) = (F|T) = T

Generalizing:

T = A

F = B

? = C

? = D

MV-logic = {A = T, B = F, C = ?, ...Z = ?, ...n = ? }


1valued logic

A = A

2valued logic

(A|B) = (B|A) = A

3valued logic

(A|B|C) = (B|C|A) = (C|A|B)

= (C|B|A) = (B|A|C) = (A|B|C) = A


4valued logic

[A|B|C|D]=[B|C|D|A]=[C|D|A|B]=[D|A|B|C]

=[D|C|B|A]=[C|B|A|D]=[B|A|D|C]=[A|D|C|D]

=[A|C|B|D]=[C|B|D|A]=[B|D|A|C]=[D|A|C|B]

=[B|A|C|D]=[A|C|D|B]=[C|D|B|A]=[D|B|A|C]

=[A|B|D|C]=[B|D|C|A]=[D|C|A|B]=[C|A|B|D]

=[D|B|C|A]=[B|C|A|D]=[C|A|D|B]=[A|D|B|C]

= A

Tautologies of *generalized* logic are "invariant" under choice of
truth value since they are always true.

Can these logical-tautology groups, be used in the mathematics of QM?

Russell E. Rierson
analog57@yahoo.com

Feather

That seems rather clumsy, to be honest.

One uses the rules of logic and the given set of axioms that define what a group is to determine whether a system of relations can be classified as a group. Alternatively, one defines a system of relations that conforms with the requirements for it to be a group and then determines whether said group has any meaning/applicability.

Would not classifying logic as a group unto itself be somewhat circular, then? Or are you classifying the specific true-false, a-b-c-d-n, rule sets as groups?

Maybe I'm just not getting what you're saying here, possibly due to confusion/ambiguity of the symbology and/or inadequate background in the fields from which you are drawing. In the case of the latter, that'll take time to remedy. I've only studied group theory to any extent; formal logic is something with which I'm only passingly familiar.

For example, is "(A|B)" supposed to be some sort of product between elements of the group (defined in whatever way the product for the group is to be defined, obviously)? Or is it something from formal logic?

Aside from all this, group theory is an essential tool for advancing in Quantum Mechanics, or at least it seems so. So I imagine any new theorems pertaining to group theory may have an impact in the study of the subject. But you must keep in mind that logic is only a tool, as far as science is concerned. A given set of perfectly valid logic statements can be completely meaningless when compared with the real world. Particularly sets of statements that are so abstract that they cannot be applied to real-world observations would be useless.

Chimp

All mathematical proofs are ...circular?

3x+1 = 4

3x = 3

x = 1

1 = 1



Interesting question.

Does T*F = F*T ?

Does T+F = F+T ?

In this instance the symmetry is "T or F"

(T or F) = (F or T)

A logical tautology appears to be an invariance principle. So symmetry could be a higher level of generalization?

For every set A there is a choice function, f, such that for any non empty set B of A, f(B) is a member of B. There may be infinitely many sets B within A. Yet this axiom of choice is independent of the other axioms of "set theory". Are the other axioms of set theory independent of this "axiom of choice"?

True, but a given "set?" of abstract relations could sit on the shelf
until a correspondence between them and the real world is discovered.

Russ

wade-w

In logic, tautologies are generally ignored because they are ultimately not very interesting. What can you say about a statement whose truth value is always T?