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

From: Jeffrey Ketland (ketland_at_ketland.fsnet.co.uk)
Date: 06/22/04 

Date: Tue, 22 Jun 2004 18:15:53 +0100

Paul Holbach wrote in message
<881c8779.0406211820.43c78402@posting.google.com>...
>Let´s consider Tarski´s famous T-scheme:
>
>True("p") <-> p
>
>Now what about the statement "Nothing exists"?
>
>True("Nothing exists") <-> Nothing exists
>
>Truth is a property of statements, and if nothing exists, there aren´t
>any statements either. The point is that nonexistent statements are
>neither true nor false, and so it is not the case that "Nothing
>exists" is true iff nothing exists.

The T-scheme implies the existence of at least two things. In particular, a
syntactic item A must be distinct from its negation ~A. (For each item, the
T-scheme implies "~A is true if and only if A is not true", so "A = ~A"
would be inconsistent with the T-scheme.)
If one considers a model with at least two objects---and preferably one
where all syntactic items are elements of the domain---then the relevant
restricted T-scheme can be made *true* (including the instance using
"~Ex(x=x)").

But this merely tells us that the T-scheme itself implies that something
exists. This is no surprise, since its instances refer to syntactical items.
What is wrong with that? No one ever said the T-scheme was a tautology, and
it isn't a tautology.
So, this is not an "exception to the rule". It merely notes that the
T-scheme has some existential content. Actually, the unrestricted T-scheme
is in fact inconsistent with arithmetic (or syntax), as Alfred Tarski noted
in 1933. Tarski's theory of truth is based on *abandoning* the T-scheme.

--- Jeff

Relevant Pages

  • Re: Exception to the rule? (Tarski´s T-scheme)
    ... In Der Wahrheitsbegriff and "The Semantic Conception of Truth", Tarski ... referred to instances of the T-scheme as "partial definitions". ...
    (sci.logic)
  • Re: JSH: But what is truth?
    ... actual definition of truth was (a variant of) the usual inductive one. ... The T-scheme is an adequacy condition on any such definition; according to Tarski a definition of truth must yield all instances of the T-scheme. ...
    (sci.math)
  • 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: Exception to the rule? (Tarski´s T-scheme)
    ... >>Truth is a property of statements, and if nothing exists, there aren´t ... The point is that nonexistent statements are ... > But this merely tells us that the T-scheme itself implies that something ... > it isn't a tautology. ...
    (sci.logic)
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