Do Not Claim Too Much: Second-order Logic and First-order Logic

STEWART SHAPIRO
1999 Philosophia Mathematica  
I once heard a story about a museum that claimed to have the skull of Christopher Columbus. In fact, they claimed to have two such skulls, one of Columbus when he was a small boy and one when he was a grown man. Whether there was such a museum or not, the clear moral is that one should not claim too much. The purpose of this paper is to apply the moral to the contrast between first-order logic and second-order logic, as articulated in my Foundations without foundationalism: A case for
more » ... er logic (Shapiro [1991] ; see also Shapiro [1985] ). Important philosophical issues concerning the nature of logic and logical theory lie in the vicinity. In a review of my book, John Burgess [1993] wrote: ... there is a tendency, signaled by the use of the word 'case' in the subtitle and the phrase 'the competition' as the title for the last chapter, for the author to step into the role of a lawyer or salesman for Second-Order, Inc., and this approach leads to some exaggerated and tendentious formulations. Thus Burgess thinks that in my enthusiastic defense of second-order logic, I claim too much. So do a few other commentators. There is little point to an exegetical study of my own book, to see whether it contains the exaggerated claims in question, but a study of the critical remarks will reveal what should and should not be claimed for second-order logic. The focus in my book, and here, is on second-order languages with standard model-theoretic semantics. In each interpretation, the property or set variables range over the entire powerset of the domain d, the binary relation variables range over the powerset of d 2 , etc. I do not insist on extensionality. If one takes the higher-order variables to range over intensional items, like concepts, then the issue of standard semantics is whether, for each subset s of d, there is a concept whose extension is s, and similarly for relation and function variables. Let AR be the conjunction of the standard Peano axioms, including the
doi:10.1093/philmat/7.1.42 fatcat:m2spad6v3rb4hnt6jzpqo4jscy