Wolfgang Schwarz

What Montague Pointed Out

Today I found Montague's paper, and it turns out that I was wrong. Well, Field's presentation was not entirely correct: We shouldn't take Robinson arithmetic itself as R, but some extension of it that contains an additional primitive predicate "True" (T, for short). The extension need not say anything about this predicate. This is why T needn't represent truth in R. (If R says nothing about T, T either represents nothing at all or the inconsistent property, depending on how precisely we define representation.) Montague then shows, very much like Field, that any theory that contains R -- no matter if it's axiomatizable or not --, as well as every instance of

i) T(A) A

iii) T(A) if A is logically true

and

iv') T(A) T(B) provided that it contains T(A B)

also contains

T(T(R N) (R N))

for some sentence N. (Lemma 4 in Montague's paper, the labels are his.) This is certainly an unpleasant consequence of accepting the left-to-right direction of the Tarski schema. Even more unpleasant I find that (by the argument of my previous posting) any such theory must contain

T(R).

So we should give up that direction of the Tarski schema. What about the other direction?

i*) A T(A).

By Löb's Theorem, any theory that contains R, (i*) and (iv') also contains every sentence A for which it contains T(A) A. So if such a theory is correct, it must not contain T(A) for any sentence A whatsoever. That is: we can have the right-to-left direction of the Tarski schema only if we are willing never to call any sentence false. I think I'd rather give up the schema.