[FOM] Concerning definition of formulas
B. Sidney Smith
bsmith7 at radford.edu
Sun Oct 7 22:59:57 EDT 2007
Alex Blum wrote:
>It would be helpful to have a problematic example. After all, the rules
>of English are formulated in English.
>Alex
I don't know of a problematic example, if what you mean is a case of a
mathematical theory being on shaky ground owing to doubts about the
semantics of first-order logic. This may be a testament to the robustness
of Aristotelian categorical logic, but it is not necessarily a vindication
of our presumptions about the foundations of mathematics. Every author I've
checked seems to follow Tarski in taking the categorical notions of "every"
and "some" for granted when laying out the semantics of predicate logic.
And perhaps that's fair enough.
It does suggest that (1) a better answer to students who notice the apparent
circularity in the standard (ala Tarski) definitions of the quantifiers is
that we take the quantifiers to be primitive notions (like "set" itself, for
instance) that satisfy the given axioms -- unless formal set theory is
already part of the language -- and (2) that there is considerable fun to be
had in trying to analyze these primitive notions if you're inclined that
way. I've just spent a rewarding half-hour with Mary Tiles' "The Philosophy
of Set Theory," reviewing the early sections of Chapter 2 on "Classes and
Aristotelian Logic."
Another approach to quantification not yet mentioned during this exchange is
game-theoretic semantics (GTS), ala Hintikka. This too reduces the
categorical notion to one of "choice" in the form of the existence or
otherwise of strategies in a game. I find this approach very appealing.
P.s. I'd be interested in the rule-book for English if you can point me to
it. I'd just assumed linguists were still trying to figure it out. (The
point perhaps has merit even if the rule-book isn't written yet -- Lisp
interpreters are generally written in Lisp.)
Sid
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