[FOM] Truth theories and the conservativity argument

[henri galinon]

Dear fomers, in a previous post ( http://www.cs.nyu.edu/pipermail/fom/ 
2006-August/010741.html ) I've tried to sketch what a response to the  
"conservativity argument" of Shapiro and Ketland may look like.  
Thanks to conversations with R. Heck, I have somewhat clarified my idea.
  I think the following way to look at things have some clarificatiry  
power (I think for instance that it is a good basis to settle the  
Tennant-Ketland battle, besides answering the Shapiro-Ketland  
argument).  Before starting, it 's important to remind us of  
something : it makes sense to apply truth only to interpreted  
language. So formal systems of  arithmetic (for instance) we consider  
when truth-theorizing should better be called formalized system  
(rather than just "formal").
Now since we are speaking of formalized system, the question of the  
adequation of the system to its informal counterpart arises (this is  
not the case in formal system, stricto sensu, like group theory and  
the like, since for these later attributing truth simply makes no  
sense, as one may easily concvince himself).
  So a concept linking the informal theory we have of a domain (the  
basic understanding we have of our subject) and its formal  
counterparts may be  useful. I propose to use "faithfulness" (for  
lack of a better term), as briefly suggested in my first fom-post.  
The general question was, remember, wether non-conservativity results  
of some faithful truth-theory over PA show anything concerning the  
"substantiality" of our ordinary concept of truth or, to put it  
another way, wether the truth predicate has any explanatory power.

This being said, let me try to convince you that the conservativity  
argument is flawed.

-My main hypothesis is this, I believe :
  anybody grasping the natural numbers knows that the omega-rule is  
valid. (it is just trivial logical principle applied to an infinite  
set). *It is  known that it is truth-preserving*. I think this can be  
justified with some great plausibility.

- PA minus induction plus w-rule, that is "faithful arithmetic" as  
I'll call it, is complete.

- On the other hand, we may see the point of constructing formal  
systems as a mean to discover (that is to prove) some new truths from  
truths we already know about the domain under consideration (natural  
number, say). We achieve this by writing down axioms we know to be  
true and *applying* logical rules known to be truth-preserving (sorry  
for the platitudes). Of course, infinitary-rules, even known to be  
valid, are of no use in there. Only finitary rules can be used by us  
(finite being). Let me nonetheless insist on a point. It may be  
useful to contrast two kinds of inferences then : on the one hand  
those which are finitary, realy rule-like we may say, call them (for  
lack of a better term) discovery-rules. They are the useful rules to  
prove new things from old things, to learn more. On the other hand we  
have so called infinitary-rules,  which have as essential feature  
that they can be described but not applied. Whatever the importance  
of the difference may be between the two, it is important to  
recognize that both kind of rules may be valid (that is: truth- 
preserving), and *moreover known to be valid*. And basically, the w- 
inference is known to valid by someone who knows what natural numbers  
are.

-We may at first hope that it is possible to get (write down) all we  
know of the numbers in a finitary-rule setting (with r.e. set of  
axioms). But incompletness shows us precisely that this won't do. I  
take it, with many others I suppose, that this is the fundamental  
significance of the theorems (I don't think any "metaphysical" thesis  
can be extracted from them. We are no better than the machines : we  
can deal with infinity only under descriptions, names etc., as is  
clear too from the beginning of the discussion )

-  Thus we have to work in formal systems which only approximate  
faithful arithmetic. Systems like PRA might be sufficient to work out  
some theorems, but often we need more and are looking for stronger  
system which better approximate faithful arithmetic. And of course  
the strenghtening of approximations will often be done by some kind  
of formalization of the omega-rule since we know it to be valid (and  
in fact since it belongs to our very understanding of what numbers  
are). Induction axioms can be seen as one way to do this, reflection  
principles as others, formalized omega-rules, combination of them, etc .
We can also try what I here will take to be some kind of semantic  
ascent, by introducing talk about sets of natural numbers and axioms  
for them etc. All this are basically justified by our knowledge of  
the omega-rule to be valid.
But now, when using this tools to discover new theorems, what is  
basically explanatory ? The real explanatory work, it is plain now,  
is done by the w-rule, by our knowledge of this rule to be valid, or  
again our basic understanding of the natural numbers.
  *The other tools (induction, sets, etc.) basically adress an  
*expression* problem *: the problem to put (express) in a formal  
(that is finite) setting our knowledge of the structure of the  
natural numbers.
Among these tools, we can also introduce a truth-predicate, of course.
  But the non-conservativity result of T(PA) over PA does not show  
that truth is substantial, or does any real explanatory work (truth- 
talk is not like "electron" talk in physics, this is plain. I don't  
want to insist on this here) : on the contrary, we may (should, I  
think) see the results  concerning the respective strength of various  
truth-predicate (intertranslatability of T(PA) and ACA or RA for  
instance) as measurments of the *expressive* strength of the various  
predicates. But this is basically what the deflationist says : that  
truth is an expressive device ! (the "minimal" theories do not fulfil  
their purpose in this respect, at least in "typed"-truth framework.  
But note, a contrario, that the "minimal" theory of truth, when  
implemented in faithful arithmetics, in turn gives all the basic  
truth-theoretic principles).

-We may agree that conservativeness, in general, is a good measurment  
tool for relative substantiality (and for what is at issue, it may  
well be a good tool to show the substantiality of having concepts for  
dealing with the infinite, or potentiality). But that faithful  
theories of truth (that is : proving coherence and the like of the  
base theory, say)  are non-conservative over non-faithful arithmetic  
(formal systems) does not shows anything concerning  the  
substantiality of the concept of truth. On the other hand, truth- 
theories are, of course, for trivial reasons, conservative over  
faithful arithmetic (since this last is complete). Still, truth-like  
concept (set-theoretic concepts, semantic concepts etc) are of course  
indispensable, for we have seen that for the point of discovering new  
truths, there is a basic trade-off beetween faithful and useful  
theories (at least as long as we're interested in studying domain  
having as much "structure" as the natural numbers).


Then in a nutshell the response to the conservativity argument is  
this : the conservativity argument would show truth to be  
"substantial" only if we didn't knew the omega-rule to be valid. And  
our knowledge of the validity of the w-rule has nothing to do with  
our grasp of the concept of truth, but with our grasp of the  
structure of the natural numbers.



  As a concluding remark, I should note that it in quite another  
form, my argument is essentially the same as the one Field has  
developped in his own response to Shapiro (I 've realized this late).  
(approximate reference of Field's paper : Journal of Philosophy, 1999).

  I don't know what you'll think about this, but if you have a  
moment, I would be interested in hearing from you on this line of  
thought. Isn't it to simple to be true ?
Is this related to Carnap's work ? (a friend of mine suggested it  
might be)



Henri Galinon

NB: As Neil Tennant would put it, I'm not a deflationist, but ...