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First-order friendliness

Guillermo Badia, David Clement Makinson

TL;DR

The paper extends Makinson's propositional notion of logical friendliness to first-order logic by defining a semantic relation using expansions and elementary embeddings to determine when a theory Γ can be friendly to a sentence φ. It shows that some propositional-continuity properties carry over, but fundamental first-order phenomena such as compactness and interpolation fail in this setting, while Beth definability remains true. Key contributions include a careful choice of the first-order option that preserves regularity, a consistency-based characterization and refinements, and explicit non-axiomatizability results that highlight a pronounced divergence from the propositional case. Overall, the work maps the fragility of first-order friendliness, clarifies its connections to model-theoretic notions, and outlines directions for future exploration across definability, resplendence, and alternative model-relations.

Abstract

In this note we study a counterpart in predicate logic of the notion of 'logical friendliness', introduced into propositional logic in Makinson (2007). The result is a new consequence relation for predicate languages using first-order models. Although compactness and interpolation fail dramatically, other properties are preserved from the propositional case.

First-order friendliness

TL;DR

The paper extends Makinson's propositional notion of logical friendliness to first-order logic by defining a semantic relation using expansions and elementary embeddings to determine when a theory Γ can be friendly to a sentence φ. It shows that some propositional-continuity properties carry over, but fundamental first-order phenomena such as compactness and interpolation fail in this setting, while Beth definability remains true. Key contributions include a careful choice of the first-order option that preserves regularity, a consistency-based characterization and refinements, and explicit non-axiomatizability results that highlight a pronounced divergence from the propositional case. Overall, the work maps the fragility of first-order friendliness, clarifies its connections to model-theoretic notions, and outlines directions for future exploration across definability, resplendence, and alternative model-relations.

Abstract

In this note we study a counterpart in predicate logic of the notion of 'logical friendliness', introduced into propositional logic in Makinson (2007). The result is a new consequence relation for predicate languages using first-order models. Although compactness and interpolation fail dramatically, other properties are preserved from the propositional case.
Paper Structure (4 sections, 14 theorems, 4 equations, 1 table)

This paper contains 4 sections, 14 theorems, 4 equations, 1 table.

Table of Contents

  1. Introduction and definition
  2. Continuities with the propositional case
  3. First-order discontinuities
  4. Non-axiomatizability

Key Result

Proposition 2

$\Gamma \vdash \phi$ only if $\Gamma \mathrel{\hbox{[}1.5]{$$}\mkern-3.1mu\raisebox{0.15ex}{$\sim$}} \phi$.

Theorems & Definitions (44)

  • Definition 1
  • Example 1
  • Proposition 2: Supraclassicality
  • proof
  • Proposition 3: First Reduction Case
  • proof
  • Proposition 4: Second Reduction Case
  • proof
  • Proposition 5: Third Reduction Case
  • proof
  • ...and 34 more