Wolfgang Schwarz

Truth, Field, Montague, and Robinson Arithmetic

So I've started to actually read Field's papers. Unfortunately I already got stuck on page 4 of "The Semantic Paradoxes and the Paradoxes of Truth". Field there discusses the following restriction of the naive truth schema:

T**) If True(p) then p.

He notes that this is rather weak, since it doesn't even imply that there are any truths at all. Hence, he says, one would presumably add principles like

TPMP) If True(p q) then (True(p) True(q))

and

TP) For any theorem p of first-order logic, True(p).

Then comes the passage where I got stuck:

Montague [16] pointed out that (T**) plus (TPMP) plus [TP] yields a proof of the untruth of some instances of (T**): that is, there is a sentence M [...] such that we can prove:

True(If True(M) then M)

I've spent most of this afternoon searching for Montague [16], which is his "Syntactic Treatment of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability", but, as I should have expected, I couldn't find it in the horrible libraries of Humboldt University. So all I've got is Field's own explanation of Montague's interesting result, in his footnote 6:

Let R be the conjunction of the axioms of Robinson arithmetic (which is adequate to construct self-referential sentences). Standard techniques of self-reference allow the construction of a sentence N that is provably equivalent, in Robinson arithmetic, to True(R N); M will be R N.

Field then uses (TPMP) to derive this fact, closely resembling Löb's theorem:

True(True(R N) (R N)) True(R N).

Together with True(R N), which he derives from (T**), R and again R (N (True(R N))), this yields the promised result.

What I don't understand is step 1: Why must there be a sentence N that is provably equivalent, in Robinson arithmetic, to True(R N)? Why must Truth be representable in Robinson arithmetic (or in any other finitely axiomatizable theory)? If it isn't, "standard techniques of self-reference" cannot be applied.

So whether what Montague pointed out really holds depends on whether Truth is representable in Robinson arithmetic (or other finitely axiomatizable systems of arithmetic), which in turn depends on what further principles we lay down for Truth. -- Certainly (T**), (TPMP), and (TP) cannot be the entire account: they still leave it open whether anything besides first-order tautologies is True. For instance, one could simply add a principle stating that every arithmetical sentence that holds in the standard model of arithmetic is True. Together with (T**), this would imply, by Tarski's theorem, that Truth is not representable in arithmetic. Even less controversially, one could add a principle stating that Robinson arithmetic is true. This, together with (T**), (TPMP) and (TP), suffices to rule out the representability of Truth in Robinson arithmetic:

1.	R (G True(G))	(Diagonal Lemma)
2.	True(G) G	(T**)
3.	R G	(1,2, truth-functional)
4.	R True(G)	(1,2, truth-functional)
5.	True(R G)	(3, TP)
6.	True(R) True(G)	(5, TPMP)
7.	True(R) R	(4,6)
8.	True(R) R	(T**)
9.	True(R)	(7,8)