[FOM] truth axiomatizations

[Richard Heck]

henri galinon wrote:
> 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...)
>   
Two options: (i) You formalize the theory in terms of satisfaction 
rather than truth, in which case the clause for the universal quantifier 
is stated in terms of satisfaction and sequences, as in Tarski; (ii) you 
treat quantification substitutionally, in which case the clause is: 
T('Ax.B(x)') iff An.T('B(n)'), where 'B(n)' here is the result of 
substituting the standard numeral for n for 'x' in 'B(x)'. So the 
latter, in a way, is a formalized omega-rule.
> 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 Con(PA) ? (other way : do the *other* tarskian clauses have any part in the *arithmetical* strenght of T(PA) ?)
>   
I don't think this theory will be strong at all. Can we even prove that 
modus ponens preserves truth? I don't see how we could, since we don't 
know anything about conditionals, in general. Or again: I don't see how 
we could prove that all induction axioms are true. Certainly, for each 
axiom, we can prove that it is true. But how would we prove the 
generalization that every instance of induction is true?

I think I see the connection between the two questions: If the answer to 
(1) were "yes", then, since we can prove '~Bew(n,'0=1')' for each n, it 
might seem we could somehow use the clause for the universal quantifier 
to prove Con(PA). But we can't. To do that, we'd need to be able to 
prove the generalization that, for every n, the sentence '~Bew(n,'0=1')' 
was true. But all we can do is prove, for each n, that '~Bew(n,'0=1')' 
is true.
> 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... ]
>   
Well, one way to think of it is that, when you just add the 
T-biconditionals, you are only adding T-sentences for sentences denoted 
by *standard* Goedel numbers. So it isn't just that you can't prove 
universal statements such as that, if a conjunction is true, then its 
first conjunct is true; that statement can be made false in non-standard 
models of the theory, since we can do whatever we like to the extension 
of T on non-standard Goedel numbers.  (Of course, by completeness, if 
it's not a theorem, there's a countermodel. I mean to be explaining why 
there's a countermodel and so why the generalizations aren't theorems.) 
Adding the Tarskian clauses, on the other hand, constrains the 
interpretation of T throughout the model: What you can do on the 
non-standard Goedel numbers is then constrained just as what you can do 
on the standard ones is.

So here's a question. If T* is not conservative over T, then there are 
models of T that cannot be extended to models of T*. What is an example 
of a model of PA that cannot be extended to a model of T(PA)? Surely no 
model of PA+~Con(PA) can be extended to a model of PA+Tr(PA), since the 
latter proves Con(PA). What precisely goes wrong when we try to extend 
such a model to a model of T(PA)? I haven't worked this out, but I'll 
guess there is some kind of insight into your question to be found here.

I'll leave it to others to defend the conservativity argument against 
deflationism. But the suggestion you make seems one worth developing. I 
take it to be, in short, that one shouldn't expect a deflationary 
truth-theory to be conservative over *every* base theory but only over 
those that are "faithful". Developing the suggestion will involve making 
it a lot more clear what it is for a theory to be "faithful". Here are a 
couple worries you may want to address. It seems, first, that you are 
committed to the view that no formal theory could be "faithful" to 
informal arithmetic. That seems a very strong claim, one that is 
worryingly close, to me, to the Lucas-Penrose line. Second, you suggest 
that Tr(T) will be conservative so long as T is faithful. I don't see 
any argument for this. Perhaps you have in mind some very specific view 
about what a faithful theory would be, but I can't see how we get it 
from the kind of informal gloss you give on "faithful".

Richard Heck