Re: Inconsistent = all sentences provable?

On Feb 9, 8:20 am, jackbat...@xxxxxxxxx wrote:
Hi,
is that true that "if a theory is inconsistent, then any sentence of
the theory turns out to be a theorem (i.e. can be proven
via inference rules)"?

"true" is not the best term here.
Math is not ABOUT what's true.
Math is about what follows FROM what.
In other words, what "inconsistent" means is a matter
of definition and context.

Most of the usually-used logics have some form of
reductio ad absurdum, indirect proof, or ex falso quodlibet
("from out of false, whatever") that guarantees that once
you have inferred a contradiction, you can thence infer
any and everything else.

The intuitive definition of "inconsistent" would involve the existence
of some P for which both P and ~P were derivable. However, in
the (usual) presence of the rules mentioned above, this ENTAILS
that you can derive everything. In that context, therefore, this
intuitive definition is EQUIVALENT to a DEFINITION of "inconsistent"
as "proving everything".

That has led a lot of people to (re)DEFINE being inconsistent as
proving everything. Once that re-definition has happened, people
need a new name for systems that prove a contradiction, but that,
because they (unusually) DON'T include RAA/EFQ, do NOT also
prove everything else. Those are called paraconsistent logics.

Standard classical propositional and first-order logic are NOT
paraconsistent (i.e., "proves a contradiction" and "proves
everything" are equivalent in them), so the question of "which
definition" of "inconsistent" to use ("proves a contradiction" or
"proves everything") is moot, as long as you are doing what's
usual.

What remains important, however, is that it is YOUR CHOICE
of which definition YOU use and which logic YOU use.
It is NOT like there is an overall FACT of the matter.
It is also important, however, IF you are not using the standard,
to acknowledge that and to alert people -- otherwise they will
infer that you are standard but mistaken (or even ignorant).


.

Relevant Pages

  • Re: Inconsistent = all sentences provable?
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  • Re: Consistent partial falsehood
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  • Re: Godels Incompleteness and Nonmonotonic Logic
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    (sci.logic)
  • Re: Godels Incompleteness and Nonmonotonic Logic
    ... > write a messed up set of inference rules that is inconsistent... ... I'm pretty sure asp doesn't make arithmetic inconsistent. ... > logics are used for computing things implies that there has to be some ... So, for a language strong enough to code arithmetic, only approximations ...
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  • Re: CLASS Terminology
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