Changing a Semantics: Opportunism or Courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin’s approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. 1. General models for second-order logic In Henkin [46] general models were introduced for second-order logic and type theory that restrict the ranges of available predicates to a designated family in the model. This enlarged model class for higher-order logics supported perspicuous completeness proofs for natural axiom systems on the pattern of the famous method in Henkin [45] for proving completeness for first-order logic. To fix the historical background for this paper, we review the basic notions here, loosely following the exposition in van Benthem and Doets [17]. The language of second-order logic has the usual vocabulary and syntactic constructions of first-order logic, including quantifiers over individual objects, plus variables for n-ary predicates, and a formation rule for existential and universal second-order quantifiers ∃Xφ,∀Xφ. Standard models M = (D, I) for this language have a domain of objects D, and an interpretation map I for constant predicate symbols. 1 Next, assignments s send individual variables to objects in D, and predicate variables to real predicates over D, viewed as sets of tuples of objects. Then we have the following standard truth condition: M, s |= ∃Xφ iff there is some predicate P of suitable arity with M, s[X := P ] |= φ Now we take the control of available predicates in our own hands, instead of leaving their supply to set theory. A general model is a tuple (M,V ) where M is a standard model and V is some non-empty family of predicates on the domain D of M . There may be some further constraints on what needs to be in the family, but we will look into this below. Now interpretation on standard models is generalized as follows: This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend. The authors wish to thank several colleagues for their helpful comments, including Solomon Feferman, Maria Manzano, Vaughan Pratt, Gabriel Sandu, and, especially, Jouko Väänänen. The first and fourth authors would like to acknowledge the support of the Hungarian National Grant for Basic Research No T81188. The third author would like to acknowledge the support of the Rustaveli Foundation of Georgia grant FR/489/5-105/11. 1For simplicity only, we mostly omit individual constants and function terms in what follows.