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Re: [ontolog-forum] Inconsistent Theories

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Tue, 9 Feb 2010 10:24:23 -0600
Message-id: <069F1022-07E1-462A-9FAB-51A03E78051D@xxxxxxxx>
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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