[FOM] Truth theories and the conservativity argument

[henri galinon]

Well, for starting, I know that the "argument" I've presented is  
flawed and confused as it stands, so I apologize for this. But I have  
hard time getting things clear and I'm interested in objections.

Panu wrote :
>  Certainly we do not possess (informal or
> not) the complete theory of arithmetic.

Well, I think your right, and it seems that incompletness says that  
it will never happen.  On the other hand it seems that we have a  
clear concept of what natural numbers are (this is not to say that we  
know all there is to know about them), and it seems also that PA (and  
extensions) is not faithfull or complete (in a sense) relative to  
that knowledge.
My tentative remark was : there is something important we now about  
the natural numbers, namely that the w-rule is valid : had we proved  
for each number n that phi(n), w e would have proved that for all  
natural numbers phi.   This knowledge (the validity the w-rule) does  
not help in proving new theorems from old ones, but it seems most  
important in guiding us when we want to know wether the theory we're  
working in/on is a theory of the natural numbers or not.
  For example, consider again the difference between formal theories  
and arithmetic. Groups are just whatever is specified by the axioms  
of group-theory. Add some axioms to it and you have another theory,  
ring-theory, say. Things are really different for arithmetic. We may  
write down PA as a theory of arithmetic, but still we can add some  
axioms to it and still be doing arithmetic, while adding some others,  
this is no more arithmetic in any straighforward sense : when working  
in PA+non-Con(PA), that is in an w-inconsistent theory, you're not  
doing arithmetic anymore.( We can't anymore read "Ax phi(x)" as "for  
every natural numbers phi", or so it seems).
This being said, this seemingly important principle (the validity of  
the w-rule) is not completely implemented in PA. Induction gives  
something like this in implying that, had you proved phi(o) and the  
hereditarity of phi, you would have a proof of the fact that all  
natural numbers are phi. But it seems it's not enough (see again the  
example above), and that we know more.
Take whatever finitary-ruled r.e. system S of arithmetic, and you'll  
have that  for some phi, S proves phi(0), phi(1), etc. and does not  
prove that for all natural numbers phi(0), while *we*, looking at the  
system from outside and through a representation of it (an emulation  
of the so-called "semantic" point of view I suppose ?), we can see  
that the conclusion is implied. Thus it seems that from outside, the  
finite-proving-system will always seems defective. Is it right ? And  
if it is, what is the significance of this ?

  Panu :
> I vaguely recall that Charles Parsons once suggested that there is  
> a close
> connection between concepts of truth and class  - sorry, I don't  
> have the
> reference at hand (1974 paper?). But his might be relevant issue  
> here...

ok, thanks. And by the way, let me add that it is a paper of Parsons  
( "Informal axiomatization, Formalization and the concept of truth",  
maybe the one you're talking about ?) that reminded me of  
formalization being a process. I've been perplexed since.


Panu :

>
>> But note, a contrario, that the "minimal" theory of truth, when
>> implemented in faithful arithmetics, in turn gives all the basic
>> truth-theoretic principles).
>
> I am not sure this is true. Even if we take the complete theory of
> arithmetic, and add T-sentences, I don't think we get  
> generalizations such
> as (For all x)(x sentence ->  (True(x) or False(x))). (?)

By faithfull arithmetic, I had in mind something like PA, minus  
induction, plus w-rule (not just complete arithmetic with usual rules).
There, since for each natural number we can prove Sent(x)---> (true 
(x) or true(not-x), it seems that the w-rule gives you the  
generalization.


When willing to assess the alledged arithmetical explanatory power of  
our concept of truth, where do we have to look at, given that,  
intuitively, truth applies only to interpreted theory ? In such  
theory, of arithmetic say , "Ax phi(x)" means (and thus implies !)  
"for all natural numbers phi".
If, on the one hand, we look at the effect of truth-theorizing on a  
theory like PA, we face the problem that, as an interpreted theory,  
PA is not faithful, or even is "quasi-logically" defective. For in PA  
having a proof of phi(n) for all natural numbers n does not implie  
that for all natural number phi. And when adding whatever purely  
expressive device to PA, you make it proof-theoretically stronger  
"automaticaly" : the induction scheme is reinforced, and you allow  
that because you know that improving the induction scheme will not  
bring you out of arithmetic and in fact quite the contrary (you know  
that your induction scheme could be improved on). It seems that it is  
the base system which is somewhat deficient, leading to "unexpected"  
results when combined with some new expressive tools, not the added  
concept that is substantial.
On the other hand adding truth to an w-ruled arithmetic sytem adds  
nothing (arithmetically speaking). But  this "faithful" system, we  
can have some meta-description of, but not working in. It has no  
procedural value. Knowing that this w-arithmetic is faithfull is not  
knowing much arithmetic, but just knowing what arithmetic is about.

A friend of mine (Denis Bonnay) suggested to me the following  
comparison, inspired by a example from Dummett  :
Take a very simple logical language, with only the implication  
connector  --->, ruled by the usual in- and out- natural deduction  
rules. Then add the following deviant connector " v' " governed by  
the same rules as the usual " or " except that the elimination rule  
has an artificial limitation: " v' " can be eliminated only if the  
derivation from A to C and from B to C are one-step.

                 [A]       [B]
                 ___    ___ (One step)
A v' B       C        C
_________________
              C

Then add the usual "or" with its usual rules.  The system (---> + v'  
+ or)   is not conservative over the (----> + v') system. Does it  
mean that "or" is substantial ?I think we don't want to say that.  It  
is the (---> + v') system which somehow is defective. More plausibly,  
the non-substantiality of "or" is accounted by the conservativity of  
the (----> + or) system over the (---->)-system.


Henri