[FOM] Re: The rule of generalization in FOL
aa at tau.ac.il
Wed Sep 8 18:01:05 EDT 2004
Tim Chow wrote:
> Let A(b) denote a formula with free
> variable b. Then we have the rule
> X -> Y, A(b)
> X -> Y, forall x: A(x)
> provided that b does not occur free in Y.
b should not occur free in X either (otherwise you can prove
invalid sequents), and this is the standard =>\forall rule, as originally
formulated by Gentzen.
> On the other hand, "phi is a
> logical consequence of psi" is defined to hold iff for all structures M
> and all interpretations sigma, if M |= psi[sigma] then M |= phi[sigma].
> Working with this system makes my head spin because I have to keep reining
> in my natural urge to interchange derivability and logical consequence.
> My question is, what is the compensating advantage of this system that
> makes this apparent mismatch worthwhile?
LK (or any other Gentzen-type calculus) is just a calculus. It can
(and should) be *used* to characterize consequence relations. Now
there are two major methods for doing this that can be found
in the literature:
1) Define that B follows from A_1,..., A_n iff the sequent A_1,..., A_n=>B
is a theorem of G.
If G is LK than the resulting consequence relation is the "truth"
consequence relation of classical logica (thus A=>\forall x A is in general
not a theorem of G)
2) Define that B follows from A_1,..., A_n iff the singleton sequent =>B
is derivable in G from the n singleton premises =>A_1 ... => A_n (using
the axioms and rules of G, *including cut*).
If G is LK than the resulting consequence relation is the "validity"
consequence relation of classical logica (thus =>\forall x A *is*
derivable from =>A in LK).
The source of the confusion is that from a "horizontal" point of view,
sequents reflect the truth consequence relation between *formulas*r.
However, viewed from a "vertical" point of view, the rules of LK
preserve validity of sequents in Structures, not their truth given
a structure and an assignment (by truth/validity of a sequent I mean the
truth/validity of its standard translation into a formula of the language).
Hence a *sequent* is provable from other *sequents* iff it follows
from them according to the validity consequence relation. In fact, I
know no Gentzen-type system for classical logic in which all the rules
preserve truth of sequents.
So the answer to your question is that only this method works (or so
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