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george wrote:
> The point IS that HE was using variables
> while insisting that he wasn't.
Change example for a moment.
Propositional logic, I take it we can agree, deals with sentences, not
quantifiers and variables. It is, in a trivial sense, variable free.
And that trivial sense isn't of course undermined by the fact that
when, metalinguistically, we say e.g. for every wff A, B, the
expression Q(A --> (B -->A))Q is an axiom -- where the Q's are
quine-quotes. For here the A, B do not belong to PL the language of
propositional logic: they are entirely dispensable augmentations of
English. But the fact that we use such metalinguistic quantifiers and
variables talking about propositional logic doesn't mean that
propositional logic involves quantifiers and variables.
Likewise, a described another theory (involving numerals, function
expressions for p.r. functions, identity and the connectives). This
theory too doesn't involve quantifiers and variables (or even just free
variables): in the same trivial sense, it is variable free. The fact
that we use metalinguistic quantifiers and variables in giving its
axioms doesn't mean that the theory contains variables. Which was the
point at issue, and all I claimed.
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