FOM: Second order logic

[John Mayberry]

	In his reply to Robert Black, Steve Simpson summarises his 
position by saying that (1) second order logic isn't logic at all, and 
(2) second order logic hides a lot of set-theoretic difficulties.

 Ad (1) ("Second order logic isn't logic at all"). This claim seems to 
rest on the following fact: since the set of universally valid formulas 
of 2nd order logic is not r.e., there cannot be a complete system of 
proof procedures for second order logic that anyone can actually use. 
No one disputes this, indeed, no one *can* dispute this: it is a 
mathematical fact and can be proved. What *is* in dispute is the 
conclusion that Steve Simpson draws from this fact, namely that, on its 
own, it precludes our regarding 2d order logic as a genuine logic. No 
doubt this position is defensible, but he has not defended it: he has 
merely asserted it. In particular, he has not addressed what I think is 
the central point, namely, that (a) the basic semantic definitions of 
satisfiability (in a particular structure - Tarski's definition), 
universal validity, logical consequence, and logical consistency are 
exactly parallel in the 2nd order case to the corresponding definitions 
in the 1st order case, and (b) these definitions are *logically prior* 
to the proof that, in the 1st order case, the notions of universal 
validity, logical consequence, and logical consistency can be 
characterised in terms of formal proof.

 Ad(2) ("Second order logic hides a lot of set theoretical 
difficulties") It seems to me that this is just wrong. Far from hiding 
set-theoretical difficulties, second order logic is precisely what 
reveals the extent of those difficulties to us. Surely this is obvious 
in the cases of natural number arithmetic and real analysis. In set 
theory itself, the contrast between the 1st and 2nd order versions 
raises acute difficulties about what axiomatic set theory is and how it 
can be interpreted. There is a good discussion of these difficulties in 
Shapiro's book. Kreisel has also discussed them at length in his 
classic paper "Informal rigour and completeness proofs". 

One further point: the Lowenheim number of second order logic is, as 
the name suggests, the smallest cardinal k such that any formula of 
that logic that has a model has a model of cardinality at most k. This 
definition is given and its relevance to the foundations of set theory 
is discussed with admirable clarity in Stuart Shapiro's book (Section 
6.4, p. 147ff).

John Mayberry
School of Mathematics
University of Bristol

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John Mayberry
J.P.Mayberry at bristol.ac.uk
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