Re: Skolem Again


William of Ockham wrote:
> J.L. Perez-de-la-Cruz wrote:
> > William of Ockham wrote:
> >
> > > Chris Menzel wrote:
> > >
> > >>From the fact
> > >>that a given interpretation of a theory makes all of its axioms true it
> > >>does not follow that it can properly be considered the theory's meaning.
> > >
> > >
> > > But if grasp of meaning is grasp of truth-conditions, this is terribly
> > > difficult to avoid.
> > >
> >
> > May I irrupt with a little dialogue?
> >
> > Set-Semanticist: Ok, venerabilis inceptor,let us consider
> > the theory T1 given by just one axiom A1: \exists x P(x).
> > Pseudo-Ockham: Not a big one...
> > SS: Remember what posterity will attribute to you: 'Entia
> > non sunt multiplicanda', etc.
> > PO: Placet, go on.
> > SS: And now, would you be kind to give me a model for T1?
> > PO: Of course! For example, consider M1 as follows: the
> > domain of T1 is the real world; the interpretation of P(x)
> > is "x is a good philosopher", as understood by common
> > English speakers. Since there are at least one object in
> > real world (for example, me) that falls inside the extension
> > of P(x), A1 is true in M1, hence M1 is a model of T1.
> > SS: Terse exposition! Now I say that "Some people are not
> > good philosophers" is part of the meaning of T1.
> > PO: Excuse me?
> > SS: It is easily proven; in fact, is it true in M1, so
> > ...'it is terribly difficult to avoid' that conclusion.
> > PO: I haven't realized! Thanks!
> > SS: You are welcome. Good look with your Ph.D.
>
> Of course. One set of truth conditions can be included in another.
> The truth conditions of "for some x Px" are included in the truth
> conditions that correspond to M1. But where is there the analogy
> between this and the case of set theory? How are the truth conditions
> corresponding to a countable model for set theory, consistent with the
> truth conditions corresponding to an uncountable model?

Easy; just as there can be both good philosophers and bad philosophers,
there can be both countable and uncountable models.

> Don't the
> latter include the condition that there are not countably many things?

Yes, but not the condition that all things are not countable.

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