Re: What isn't a tautology?


Chris Menzel wrote:
> Well, that is just so not true on the (completely standard and common)
> notion of "tautology of predicate logic" that I clarified in my previous
> post (quoted below), and which you appeared to agreed with. On that
> notion, it is trivially provable that no universally quantified sentence
> is a tautology. Period. There's absolutely no way to replace
> subformulas with propositional constants in a quantified sentence to
> yield a PC tautology. It just can't be done.

No, there isn't, but maybe George is looking for something like the
following. Whatever is implied by a valid formula is valid. Hence if |-
Ax(p -> q), then |- Axp -> Axq (by |- Ax(p -> q) -> (Axp -> Axq) and
MP). In some cases, e.g. in the case of Ax(Px -> Px), an equivalent
tautology is implied--i.e. AxPx -> AxPx. (That was my point before but
I didn't state it properly and "loss of information" got thrown in as a
red herring or a practical joke.) So certain valid quantified
expressions can be equivalently rewritten as tautologies. I understand
this has only little bearing on your discussion as a whole.

.

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