Re: If (P & ~P) -> Q is not derivable then Goedel's formula is not derivable


george wrote:
Newberry wrote:
If

(P & ~P) --> Q (1)

ought not to be derivable then Goedel's formula ought not to be
derivable either.

Well, obviously, since (P ^ ~P) --> Q
IS derivable, it ought to be.
It is derivable because its truth-table is all trues.

Here is why:


(1) is counter-intuitive and it ought not to be derivable.

No, (1) IS NOT counter-intuitive IF you understand what it MEANS.
Intuitively, YOU want --> to mean something like "implies"
(i.e. "causes to be true"), but that just ISN'T what "-->" MEANS
around here. Around here, P --> Q is just another way of saying ~P v
Q.

Exactly my point. Since (P & ~P) --> Q ought not to be derivable then
~(P & ~P) v Q ought not to be derivable either.

Another way of thinking of this is to note that everything whose main
connective is --> is equivalent to its contrapositive.
In this case, that would be ~Q --> (~P v P).

This ought not to be derivable either.

Now, obviously (or, at any rate, classically),
(~P v P) is true. Equally obviously, it HAS to be true.
Since the conclusion has to be true, anything that implies it
has to be true as well. (P v ~P) doesn't become doubtful just
because I alleged that "pigs can fly" implies it.

.

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