[FOM] truth axiomatizations

[henri galinon]

Dear Fomers,
I have some few questions about (typed) truth theories  :

1) in the theory T(PA)  (= PA + inductive clauses for truth+ full  
induction),  is the clause for the universal quantifier [something  
like : Tr(AxFx) iff  Ay Tr(F(y/x)) ]  a kind of formalized omega- 
rule ? (I'm not able to elaborate much more on my question...)

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) ?)

3) Why is it, intuitively, that the extension of the induction scheme  
to L(PA+Tr), when the truth predicate is asked to commute with  
logical constants (as in T(PA)), is a genuine arithmetical extension  
(ie non-conservative over PA), whereas when the truth predicate is  
just asked to give the T-biconditionals, the extension is  
conservative ? (after all, neither truth predicate is definable, so  
when added to the induction scheme, things may happen... I'm confused  
somewhere, but where ?)
[Of course I expect something more informative than the simple facts  
that in the first you can prove that all theorems are true  
and..and... and then the consistency of PA ..., whereas on the second  
you just cannot prove any universal truth-theoretic  statement, thus  
not even the first part of the argument... ]

4) Now I would like to submit a modest tentative philosophical point  
related to the "conservativity argument" against deflationism. This  
is *very* sketchy, but I would be interested in objections.
Call a formalized theory T *faithful* to a informal theory Tee when  
explanations of  phenomena phi in Tee have couterparts as a proof in  
T (that is: T proves phi).
    Consider informally your theoretical commitments when you  
attribute truth to PA (PA seen not as a formal theory but as a  
formalized interpreted theory, for the attributions of truth to it to  
make sense). It seems plain that in a faithful theory of the truth of  
PA, you should be able to prove that all the axioms of PA are true,  
that inference-rules preserve truth, and then  that all theorems of  
PA are true. Finaly, from this, you prove easily that PA is  
consistent. So any faithfull theory of truth (for PA) is non- 
conservative over PA. On the other hand, deflationists say that the  
concept of truth, roughly, has no explanatory power (is not  
substantial) : so it seems that deflationists are committed to the  
view that a theory of truth should be conservative over PA. Hence a  
theory of truth cannot both be deflationary  and faithfull. Thus  
deflationists have gone wrong.
The argument is well known (Shapiro, Ketland) and repeated here for  
convenience. Now I want to question this predicament in a very simple  
way :
a) conservativeness is a *relative* phenomena (conservativness over a  
given theory); consequently it withnesses only *relative*  
substantiality.
b) PA itself is unfaithful to informal arithmetic : it is this way  
that Godel's theorems have mostly been understood (a refutation of  
formalism, say). Its unfaithfulness may be seen, for example, this  
way : you can have that PA prove F(0), F(1).... and still extend it  
consistently with not-AxF(x) (in brief, PA is omega-incomplete).
c) That faithful theories of truth are substantial over unfaithfull  
arithmetic does not show anything *specifically* for the alledged  
substantiality of our concept of truth.
[d) Moreover, faithfull truth theories  won't be substantial over  
faithfull arithmetics (take semi-formal systems with infinitary  
rules, progressions of formal systems, unfolding systems,  etc.. Of  
course, with Shapiro we can take also all this to show that a  
deflationist (but not only) should acknowledge that second-order  
arithmetics is better at faithfulness than first-order. This is  
another question )]

In a word, the argument is :  it is not possible to take godelian  
phenomena to refute *both* formalism  (or a version of it) and  
deflationism.
[A last word : it might be that deflationists (eg H. Field) are often  
physicalists, or formalists, or nominalist etc and  godelian  
phenomena *might* be taken to refute these positions ( that was  
another discussion, twenty years ago, between the same). But  
deflationism about truth does not imply any of this positions, as far  
as I can see (and deflationists who take truth to apply to  
propositions are consistent, even if not defending the most  
interesting version of deflationism).As for *truth*, and our putative  
need for a substantial theory of it, the conservative argument is not  
conclusive, or so it seems to me.]



Henri Galinon
PhD student
IHPST
Paris