First-Order Implication-Space Semantics

TL;DR

This paper extends implication-space semantics to first-order logic, enabling a nonmonotonic, inferentialist framework that captures classical logic features while allowing for nonclassical inference patterns.

Contribution

It introduces a first-order implication-space semantics that maintains classical inference properties while accommodating nonmonotonic and supraclassical consequence relations.

Findings

Achieves a nonmonotonic first-order consequence relation

Preserves deduction-detachment and disjunction rules

Supports multiplicative conjunctions and counterexamples

Abstract

This paper extends implication-space semantics to include first-order quantification. Implication-space semantics has recently been introduced as an inferentialist formal semantics that can capture nonmonotonic and nontransitive material inferences. Extant versions, however, include only propositional logic. This paper extends the framework so as to recover classical first-order logic. The goal is to formulate a theory in which consequence relations can be nonmonotonic and supraclassical, while obeying the deduction-detachment theorem and disjunction simplification, while also including conjunctions that behave multiplicatively as premises and counterexamples to the usual quantifier rules. The paper explains these constraints and shows how they can be met jointly. The result is a first-order version of implication-space semantics that has all the virtues for which inferentialists and inferential expressivists praise propositional implication-space semantics.