Re: Existential, Universal Quantifiers, Generalize Booleans

David C. Ullrich wrote:
On Mon, 10 Jul 2006 12:27:14 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> wrote:
[snip]

Exactly correct. You didn't quote the definition of logical
equivalence, you quoted the definition of the truth value
of a formula in an interpretation. Yes, there _is_ such
an assumption in _that_ definition. No, there is no such
assumption in the definition of logical equivalence.

See, if we assume we have a "context", as you're using the
term, then we could in fact give names to all the individuals
and consider the truth values of P(c) for all individuals c
in determining the truth value of Ax P(x). But that doesn't
say that Ax P(x) is equivalent to the conjunction of all those
formulas P(c), precisely because we are considering only
one "context".

But it does make sense to write the infinite conjunction, within this single "context", as: /\_{x} P(x).
With that syntax, there's no need to enumerate any constants in the formula.
And it's perfectly ok to hold that for all models, -that- formula is equivalent to \forall x P(x).

Perhaps it becomes clearer if we allow ourselves also to write
/\_{x \in D} P(x)
where D is any set, in a fixed model of ZFC.

--
Cheers,
Herman Jurjus





************************

David C. Ullrich
.

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