From Newsgroup: sci.math

On 12/15/2023 12:03 PM, Ross Finlayson wrote:

On Friday, December 15, 2023 at 4:49:41rC>AM UTC-8, Richard Damon wrote:

On 12/14/23 11:20 PM, olcott wrote:

On 12/14/2023 9:56 PM, Andr|- G. Isaak wrote:

On 2023-12-14 17:14, olcott wrote:

On 12/14/2023 9:58 AM, olcott wrote:

"from a contradiction, any proposition (including its negation)
can be inferred from it; this is known as deductive explosion." >>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion

Here is a contradiction as a syllogism that integrates the full
semantics of the contradiction as defined sets.
(a) All Cats are dogs
(b) Some Cats are not dogs // AKA Not(All Cats are dogs)
(c) therefore NULL (the empty set)

The principle of explosion would says that (a) and (b)
proves that the Moon is made from green cheese.

No. It doesn't say that. Given a contradiction (I'll use A & -4A), the >>>> principle of explosion says that for any statement X, "A & -4A
therefore X" is a *valid* argument.

*Which is itself conventionally defined incorrectly*
The correct way that valid should be defined is that the
conclusion is a necessary consequence of all of its premises.

Which it is, according to the rules of the logic system. You are just
showing your lack of understanding.

Any system which claims to be non-contradictory in logc form, that has a
pair of statements that are contradictory, is just broken. The Principle
of Explosion just makes the breakage total,

This eliminates the Principle of Explosion before it
even gets started.

Nope, it proves that you don't understand what you are talking about.

Truth is established by having a set (possibly infinite) of valid steps
from the initial truthmakers of the system to the statement.

A Proof is just a finite listing of one possible set of those links,
thus anything that can be proven, must be true.

Yes, if you limit the forms of links that can be used as steps, you can
make some things not provable, but this MIGHT also reduce what is
actually true in the system.

To *prove* a statement, the statement needs to appear as the
conclusion to a *sound* argument (being valid is necessary but not
sufficient), and the principle of explosion does *not* claim that your >>>> hypothetical argument is sound.

Andr|-

You mean that doesn't have all sorts compounding theorems?

It's pretty easy to show someone following linearly that
some proof-by-contradiction implies something unrelated
in terms, that it never was, and that it's separated into what
is entirely its own independent, logically, stipulation.

This is one reason why all orders of elements must be considered,
because there is the entire context and complement of both
what _is_ and what _not is_, that de Morgan requests you make
it so that your data structures or tables, get put in a box and
rolled like dice, and come out same.

It's like, the person who reads your developments, that any
entry point to reading, is as good as any other. If that's not so,
it all goes on one line.

Here the notion of Tarski truth is also contingent, it's
always so conditioned, "unless otherwise, ..., the stipulations
that went into it", where then one might wonder: is there
a Tarski truth of a complete and consistent theory?

You can show a given ordering of inference of statements
under connectives so follows, but not necessarily certify
"and that's the whole truth", when for example reviewing
a summary of "C or D, C, D", that it's also "C, D, C or D".

I.e. the completeness of the evaluation of the inferences,
is under the inferences under any ordering of whatever
otherwise their terms are, "independent".

So, "right" is a bit stronger than "not necessarily wrong".

"Not ultimately untrue", ....


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