A Denotational Semantics for First-Order Logic

TL;DR

This paper introduces a denotational semantics for first-order logic that integrates ideas from programming language semantics, enabling a computational interpretation that aligns with logical truth and includes logical variables.

Contribution

It extends previous work by providing a denotational semantics for first-order logic, incorporating logical variables and combining concepts from programming language semantics.

Findings

Semantics aligns conjunction with sequential composition

Disjunction models 'don't know' nondeterminism

Existential quantification declares local variables

Abstract

In Apt and Bezem [AB99] (see cs.LO/9811017) we provided a computational interpretation of first-order formulas over arbitrary interpretations. Here we complement this work by introducing a denotational semantics for first-order logic. Additionally, by allowing an assignment of a non-ground term to a variable we introduce in this framework logical variables. The semantics combines a number of well-known ideas from the areas of semantics of imperative programming languages and logic programming. In the resulting computational view conjunction corresponds to sequential composition, disjunction to ``don't know'' nondeterminism, existential quantification to declaration of a local variable, and negation to the ``negation as finite failure'' rule. The soundness result shows correctness of the semantics with respect to the notion of truth. The proof resembles in some aspects the proof of the soundness of the SLDNF-resolution.