Re: looking for a predicate hierarchy

A very interesting thread... It is completely off-topic,
I'm not following much of it, but it is giving me
the incentive to learn more. Thanks

V.J. Kumar a écrit :

Modus ponens does not hold. But weaker forms of do. Like:
1. (1=A and (A=>B)) => B

Yes, I am aware of that trick wrt to MP. It does not strike me as very
convincing because you lose ability to reason purely in syntactic terms,
you need to appeal to the current truth value and make sure it is
actually true and not T. With weakened MP, you do not have a pure
deductive system that would rely only on a system of axioms and inference
rule(s) any more, but some hybrid. Whether the hybrid is ugly or not,
is a matter of taste. At least, it makes one question the very idea of
a contradiction handling logic.

On the other hand, a purely syntactic system always seemed
to be a little miracle to me. Would it be possible that logics
whose inference rules never rely on the actual truth of
propositions are all unable to deal with inconsistencies without
explosion ?

--- Raoul

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