On Feb 9, 2010, at 12:11 AM, John F. Sowa wrote:
> Chris,
>
> JFS>> Theory T1: (Ex)(Ay)(y=x).
>>>
>>> Theory T2: (Ex)(Ey)~(x=y).
>
> CM> I don't understand how the truth recursion will work for these
>> sentences.
>
> I don't understand what you are objecting to.
>
> Theory T1 is true of any model that contains exactly one individual
> in the domain. Theory T2 is true of any model that contains two or
> more individuals. There is no model that can make both theories true. (01)
Well, I'm certainly not objecting to *that*. My question is simply what the
actual clause for an identity statement looks like and how it functions in the
overall definition of the truth conditions for an arbitrary sentence. As I
understood you, you said that identity statements "x=y" involving variables x
and y are in fact about the variables x and y themselves. I simply don't
understand how any such account is to be woven into a standard Tarski-style
definition of truth in an interpretation since, in such a definition, variables
ultimately disappear from a sentence's truth conditions. (02)
> CM> The variables "x" and "y" drop out of the picture and ultimately
>> play no role whatever in the truth conditions of the sentence.
>
> I agree. That's what I was trying to say, ... (03)
Oh. I missed that. :-) (04)
> ...and what I interpreted Wittgenstein as saying:
>
> LW> 4.242 Expressions of the form "a=b" are therefore only expedients
>> in presentation (Behelfe der Darstellung).
>
> CM> According to your suggest, if I'm understanding, "x" and "y"
>> themselves occur in the truth conditions of "x=y" and, hence,
>> I assume in the truth conditions of "(Ex)(Ay)(y=x)".
>
> No. I meant that the variable names are artifacts of the notation,
> since they disappear in notations such as existential graphs. (05)
Well, that's good, but then I still don't quite understand what you are saying
when you say that "x=y" is about the variables x and y themselves. But pretty
clearly we're not arguing about anything of genuine substance here. (06)
-chris (07)
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