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. 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.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Algebraic Semantics for Functional Logic Programming with Polymorphic Order-Sorted Types

. In this paper we present the semantics of a functional logic language with parametric and order-sorted polymorphism. Typed programs consist of a polymorphic signature and a set of constructor-based conditional rewriting rules for which we define a semantic calculus. The denotational semantics of the language is based on Scott domains interpreting constructors and functions by monotonic and continuous mappings, respectively, in every instance of the declared type. We prove initiality results for the free ground term algebra. We also prove that the free term algebra with variables is freely generated in the category of models. The semantic calculus is proved to be sound and complete w.r.t. the denotational semantics. As in logic programming, we define the immediate consequence operator, proving that the Hebrand model is the least model of a program. 1 Introduction Type systems have been traditionally considered in functional languages and incorporated as an extension to logic program..