Re: Question on logic and Godel



Chris Menzel wrote:
> On 9 Jul 2005 22:52:34 -0700, RichD <r_delaney2001@xxxxxxxxx> said:
> > Chris Menzel wrote:
> >> > If arithmetic is consistent, one cannot prove both S and ~S.
> >> >
> >> > We assume that if S is proved, it is true. But can we prove that...?
> >>
> >> Sure. Let's suppose your system is Peano Arithmetic and
> >> that S is provable in PA. Then S is true. Proof: All the
> >> axioms of PA are true
> >
> > Which we know/believe because...
>
> I'm not sure what kind of an answer you are looking for.
>
> >> and the rules of inference of first-order logic are truth
> >> preserving.
> >
> > Ditto...
>
> Ditto. Do you have doubts that B might be false even if A and "If A
> then B" are true? Do you think this principle needs proof in order for
> us to be able to say that we know it?

What could constitute a proof? We are defining truth in a specific way
by giving Tarskian truth definitions (for example). We do not need to
prove that "if A, A -> B, then B" is truth preserving because it is by
the definition of the connective -> and truth under an interpretation.

.

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