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First-Order Implication-Space Semantics

Ulf Hlobil

TL;DR

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.

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.

First-Order Implication-Space Semantics

TL;DR

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.

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.
Paper Structure (11 sections, 13 theorems, 1 equation)

This paper contains 11 sections, 13 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Background and Aim
  3. First-Order Implication-Space Semantics
  4. First-Order Implication Spaces
  5. Interpreting First-Order Languages
  6. Are the Desiderata Met?
  7. Recovering Classical First-Order Logic
  8. Recasting Truth in a Tarskian Model
  9. Mapping Tarskian Models into Implication-Space Models
  10. One Model is Enough for FOL
  11. Conclusion

Key Result

Proposition 12

Let $\mathsf{G}\cup\mathsf{D}\subseteq\mathbb{B}$ and $\mathtt{G}=\{\mathcal{R}(\mathsf{g})\mid\mathsf{g}\in\mathsf{G}\}$$=\{\mathtt{g}_{0},...,\mathtt{g}_{n}\}$ and $\mathtt{D}=\{\mathcal{R}(\mathsf{d})\mid\mathsf{d}\in\mathsf{D}\}$$=\{\mathtt{d}_{0},...,\mathtt{d}_{n}\}$, then: $\left\langle \math

Theorems & Definitions (48)

  • Definition 1: Bearers, $\mathbb{B}$
  • Definition 2: Implication space, $\mathbb{S}$
  • Definition 3: Implication and implication frames, $\left\langle \mathbb{B},\mathbb{I}\right\rangle$
  • Definition 5: Range of subjunctive robustness, $\mathsf{RSR}(\cdotp)$
  • Definition 6: Implicational Role, $\mathcal{R}(\cdot)$
  • Definition 7: Conceptual Content
  • Definition 8: Adjunction, $\sqcup$
  • Definition 9: Symjunction, $\sqcap$
  • Definition 10: Power-Symjunction, $\APLdown$
  • Definition 11: Content entailment in an implication frame, $\dttstile{}{}$
  • ...and 38 more