# Re: Guessing?

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

Message-ID: <qJndk.11934$cW3.4343@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 - 07:59:43 CDT