# Re: Guessing?

Date: Thu, 10 Jul 2008 08:59:43 -0400

Message-ID: <qJndk.11934$cW3.4343_at_nlpi064.nbdc.sbc.com>

"Marshall" <marshall.spight_at_gmail.com> wrote in message
news:56409430-6e29-4ac4-be4d-84b2eae5bc67_at_d45g2000hsc.googlegroups.com...

> 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.
**>>
**>> >http://en.wikipedia.org/wiki/First_order_theory
**>>
**>> > 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
*

Isn't it true that even the most primitive axioms range over a collection of arbitrary objects? Received on Thu Jul 10 2008 - 14:59:43 CEST