[FOM] truth axiomatizations (galinon)

[Leon Horsten]

Dear Henri:

You ask:
2) What  is the strenght of the following theory : PA + T-
biconditionals (restricted to the language of L(PA)) + the only above
tarskian clause for the universal quantifier (and full induction) ?
Does it prove Coh(PA) ? (other way : do the *other* tarskian clauses
have any part in the *arithmetical* strenght of T(PA) ?)
somewhere, but where ?)

Even though on its own, Tr(AxFx) iff  Ay Tr(F(y/x)) does not make a
truth theory non-conservative over Peano Arithmetic, this axiom is of
special significance in non-conservativeness phenomena. Michael Sheard
has investigated this in detail in his interesting article:

Sheard, M. Weak and strong theories of truth. Studia Logica 68(2001),
p. 89--91.

The (self-referential) truth theory FS, which can be regarded as the
natural  consistent classical self-referential extension of T(PA), is
arithmetically quite strong. It is "more non-conservative" over PA
than T(PA). What Sheard shows in his article is that a seemingly weak
subtheory of FS, which he calls S_1, is arithmetically just as
non-conservative as FS itself. The only *compositional* truth axiom
which S_1 contains is: Tr(AxFx) iff  Ay Tr(F(y/x)); the other truth
principles of S_1 are very weak when taken on their own. So there is
a sense in which this axiom 'Tr(AxFx) iff  Ay Tr(F(y/x))' makes an
extraordinary contribution to arithmetical non-conservativeness.  
(Sheard notes that *truth-theoretically*, S_1 is very weak: it fails  
to prove certain very basic laws of truth.)

Leon Horsten


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