Date: Thu, 10 Jul 2008 08:59:43 -0400
"Marshall" <marshall.spight_at_gmail.com> wrote in message
> On Jul 9, 10:03 pm, "Brian Selzer" <br..._at_selzer-software.com> wrote:
>> "Marshall" <marshall.spi..._at_gmail.com> wrote in message
>> >> 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.
>> is determined through interpretation. Therefore, axioms are sentences
>> are supposed to be true under an interpretation--the intended
>> interpretation. So a logical theory consists of a set of sentences that
>> 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.
Isn't it true that even the most primitive axioms range over a collection of arbitrary objects? Received on Thu Jul 10 2008 - 07:59:43 CDT