Date: Wed, 9 Jul 2008 23:37:15 -0700 (PDT)
> >> Logical propositions without an intended interpretation are when written
> >> just squiggles--something akin to doodles--with no significance or
> >> utility
> >> whatsoever, and are when spoken just noise--they do not rise even to the
> >> level of being a tale told by an idiot: they're just noise.
> > That turns out not to be the case. Axioms are just sentences in
> > a language, for example. A first order theory is just a bunch
> > of syntactic statements. There may be a variety of different
> > possible interpretations, or models. There may be exactly
> > one, or there may be none at all.
> > Marshall
> I don't think so: axioms are sentences that are suppposed to be true. Truth
> is determined through interpretation. Therefore, axioms are sentences that
> are supposed to be true under an interpretation--the intended
> interpretation. So a logical theory consists of a set of sentences that are
> supposed to be true under an interpretation along with that which can be
> derived from those sentences.
You're in the right ballpark, but you're overstating things to the point of being wrong. The interpretation is not a requirement. It may not be unique; it may not even exist.
More relevantly, any interpretation is *not* part of the theory.
Axioms are sentences that a theory assumes. Within that theory, axioms are the definition of truth. We may have an associated model for the theory. Or we might have five, or there might not be any model possible. So it does not make sense to speak of "_the_ intended interpretation."
And for most proofs, (and this is my big objection to your position) we don't even need to consider any model, or even whether there is a model. For example, consider the group axioms. It is a theorem of group theory that the identity element is unique. Why can we say such a thing without specifying *which* interpretation of the group axioms we mean? Because it *doesn't matter* which interpretation we are talking about: the naturals under addition, the nonzero rationals under multiplication, translations in the Cartesian plane, whatever. It's true for *all* interpretations.
Marshall Received on Thu Jul 10 2008 - 08:37:15 CEST