Re: Formulas & Sentences of PL...



William Elliot wrote:
> On Tue, 21 Jun 2005, 1st Semester Logic Student wrote:
>
> > The following is the recursive definition of FORMULA of PL
> >
> > 1. Every atomic formula of PL is a formula of PL.
> > 2. If P is a formula of PL, so id ~P.
> > 3. If P and Q are formulas of PL, so are (P&Q), (PvQ), (P->Q), and
> > (P<->Q).
>
> > 4. If P is a formula of PL that contains at least one occurance of x
> > and no x-quantifier, then (Allx)P and (Ex)P are both formulas of PL.
>
> This is weird requiring a theorem like
> (Ax)(P -> Q) -> ((Ax)P -> (Ax)Q)
>
> to have three different versions for when there's
> free x in P and Q
> free x in P, no free x in Q
> no free x in Q, free x in Q
> and the outlawed fourth version.
>
>
> What about (Ax)P(x) & Q(x)? It can't be quantified?
> Picky picky. Use theorem
> (Ax)P(x) <-> (Ay)P(y)
> to get
> (Ax)P(x) & Q(x) <-> (Ay)P(y) & Q(x)
> So we have two equivalent formulas, one which can be
> quantified and one which can't. "Weirder and Weirder"
>
> > 5. Nothing is a formula of PL unless it can be formed by repeated
> > applications of clause 1-4.

What's so weird? There are a lot of systems whose formation rules
disallow vacuous and duplicate quantification, free variables, and even
sequences of quantifiers and variables that are not in alphabetical
order in the prefix and matrix. Why would you want a duplicate
quantification (x)(x)P to be well-formed?

.

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