Changing a semantics: opportunism or courage?

TL;DR

This paper systematically analyzes generalized model techniques in higher-order logics, exploring their conceptual foundations, technical roles, and potential for new insights in logic and set theory.

Contribution

It provides a comprehensive view of Henkin's generalized model approach, clarifying its mathematical and philosophical significance, and demonstrates its utility in various logical and set-theoretic contexts.

Findings

Clarifies the conceptual legitimacy of generalized models.

Shows how generalized models can lower complexity and calibrate provability.

Provides new results on fixed-point logics and set-theoretic absoluteness.

Abstract

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.