[FOM] Truth theories and the conservativity argument

[praatika@mappi.helsinki.fi]

henri galinon <henri.galinon at libertysurf.fr> wrote:

>   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.  

Well, I, for one, believe that whatever informal theory we have about 
natural numbers can be captured by some formal system (not necessarily any 
of those we have at hand now). Certainly we do not possess (informal or 
not) the complete theory of arithmetic. 

> -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.

This too can be disputed. I am pretty sure that all we know about natural 
numbers can be captured by some formal system.

> 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. 

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...

> 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))). (?) 

(can anyone confirm this?)


All the Best

Panu




Panu Raatikainen
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy 
University of Helsinki
Finland


E-mail: panu.raatikainen at helsinki.fi
 
http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm