Re: Exception to the rule? (Tarski´s T-scheme)

From: Andrew Boucher (Helene.Boucher_at_wanadoo.fr)
Date: 06/26/04 

Date: 26 Jun 2004 00:43:49 -0700

"Jeffrey Ketland" <ketland@ketland.fsnet.co.uk> wrote in message news:<cbiesd$kmu$1@news5.svr.pol.co.uk>...
> Andrew Boucher wrote:
>
> >> Do you agree that
> >>
> >> (i) "Snow" contains four letters
> >> implies
> >> (ii) There is something which contains four letters
> >
> >Yes. But, unless I'm missing something, that doesn't seem to be
> >relevant to what I'm saying.
>
> OK. Maybe I'm confused about what you're saying. I took the criticism to
> concern whether existential generalization on quotation names or
> structural-descriptive terms is legitimate.
>
> Let's consider this criticism first. Presumably (unless we insist on free
> logic for names), we have,
>
> (i) "Snow is white" is true iff snow is white
> implies
> (ii) There is something x such that x is true iff snow is white

I agree (of course) with this. My point (maybe it's not much of a
point, but it's my point) is that (i) is an application of the
T-schema. And you can only apply the schema if you know that "Snow is
white" exists and can be substituted in for A.

>
> If that's OK (it's classical logic), then the following is OK:
>
> (iii) "Snow is white" is true iff snow is white
> (iv) "Snow is not white" is true iff snow is not white
> jointly imply
> (v) "Snow is not white" is distinct from "Snow is not white"
>
> Finally, from (v), we have:
> (vi) There are x, y such that x is distinct from y
>
> This is a classically valid argument from two T-sentences, (iii) and (iv).
> The only step which is not valid in free logic is the step which you seem to
> accept (i.e., existential generalization on quotation names).
>
> So, it seems to me that this first criticism, that the T-scheme doesn't
> imply the existence of at least two objects, depends upon rejecting EG on
> quotation names.

I think this is the source of the confusion - probably my fault. My
claim is: the T-schema *plus* other assumptions imply the existence of
at least two objects. By itself the T-schema implies only that the
negation of a sentence is not the same as a sentence.

> There is also a second criticism, which might be your point. This is based
> on a different response: namely, that just which set of sentences count as
> the relevant instances of the T-scheme depends upon the object language.
> This is right, of course. If we consider a language L with just one sentence
> A (so that we don't even have ~A), there is just one T-sentence, namely,
>
> Tr(t) iff A
>
> where Tr, t, and "iff" are expressions in the meta-language. If we consider
> a language L with just one sentence, then I agree that the relevant T-scheme
> for that language doesn't imply the existence of two objects.

Yes that's it. And I would go even further: if L doesn't have any
sentence at all then the T-schema doesn't imply the existence of two
objects either.
>
> But if our object language has just A and ~A, then

Yes, with this added assumption, the T-schema implies that there are
at least two objects.

>
> At the start of this thread, I was simply assuming a language with the usual
> properties. I.e., a non-trivial language, which necessarily has denumerably
> many sentences. The T-scheme doesn't imply the distinctness of all of these,
> but it does imply that there are at least two of them.

But if you're assuming a non-trivial language (say with two letters, 0
and 1), it would seem you can prove the existence of two things
without even invoking the T-scheme.

In the systems I study, you would be able to prove there are three
types of language (without invocation of the T-scheme): the empty
language (no sentences), the language with only one sentence composed
of one letter, and other languages. And, for these other languages,
you would be able to prove that there are more than two things, again
without invocation of the T-scheme. That is, if you're outlawing
non-trivial languages, then you get the third case and you can prove
the existence of two objects, without using the T-scheme. I think you
probably are looking at the subject from a different (and probably
more standard) way, and this probably doesn't happen for you.

>
> I've discussed what I take to be the two criticisms:
> (a) We are applying EG to quotation names. I replied that not doing this
> would be weird.
> (b) We might consider a language L with just one sentence (or, e.g., without
> a negation connective). Then there is just one T-sentence, and it doesn't
> imply the existence of two distinct objects. Agreed. But modulo (a), it does
> imply the existence of at least one: namely, the sentence in question.

Agreed. This was my point about going the "Cartesian" route. It's
not the content of the T-scheme which implies the existence of a
sentence but the T-scheme itself (which is stated in a sentence).
Similarly, the content of "There are no unicorns" doesn't imply the
existence of anything; but the sentence "There are no unicorns"
implies the existence of one thing. Still, your argument needs the
existence of a sentence with a negation.

I have to leave it at that. My brother (from Cambridge actually) is
here in Paris for the day, and he's ready to go!

Relevant Pages

  • Re: Exception to the rule? (Tarski´s T-scheme)
    ... Let's consider this criticism first. ... So, it seems to me that this first criticism, that the T-scheme doesn't ... And similarly through all the other sentences of the language in question. ... the existence of both y and y*). ...
    (sci.logic)
  • Re: Exception to the rule? (Tarski´s T-scheme)
    ... >sentence but the T-scheme itself. ... >implies the existence of one thing. ... My brother (from Cambridge actually) is ...
    (sci.logic)
  • Re: Exception to the rule? (Tarski´s T-scheme)
    ... > You cannot compress the T-scheme to a single sentence, since it is a scheme, ... > language of arithmetic L_extended with a primitive predicate Tr, ... And the existence of at least two things certainly implies the ... > truth predicate symbol Tr. ...
    (sci.logic)
  • Re: Qualia Question
    ... > define words in language any way you want. ... sense happening in our head - our thoughts etc), ... exist in an alternate universe (the universe of everything ... existence, when in fact, there is only one in our universe. ...
    (comp.ai.philosophy)
  • Re: Creating a "toy" OO/AO language...
    ... since when calculus became a proof of existence of a result? ... I asked you for the language we are speaking. ... language of mathematical analysis indeed defines sine. ... Clearly operations are defined on the values of R. In mathematics we cannot ...
    (comp.object)
Loading