Directed First-Order Logic

2025-04-15

AI Summary

This paper addresses the inherent symmetry of equality in traditional first-order logic by introducing a novel logical framework based on “directed equality,” interpreting equality as a directional term conversion (i.e., rewriting), thereby enabling types to be modeled as preordered sets. Methodologically, it establishes for the first time introduction and elimination rules for directed equality under polarity constraints, explicitly forbidding symmetric derivations. Semantically, it introduces the categorical model of “directed doctrines,” wherein directed equality is characterized as a relative left adjoint—generalizing Lawvere’s doctrine-theoretic account of equality. The main contributions are: (i) a sound and complete correspondence between syntax and semantics; (ii) the realization of genuinely directional reasoning; and (iii) precise tracking of variable variance behavior via a polarity system.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We present a first-order logic equipped with an"asymmetric"directed notion of equality, which can be thought of as transitions/rewrites between terms, allowing for types to be interpreted as preorders. We then provide a universal property to such"directed equalities"by describing introduction and elimination rules that allows them to be contracted only with certain syntactic restrictions, based on polarity, which do not allow for symmetry to be derived. We give a characterization of such directed equality as a relative left adjoint, generalizing the idea by Lawvere of equality as left adjoint. The logic is equipped with a precise syntactic system of polarities, inspired by dinaturality, that keeps track of the occurrence of variables (positive/negative/both). The semantics of this logic and its system of variances is then captured categorically using the notion of directed doctrine, which we prove sound and complete with respect to the syntax.

Problem

• Develops first-order logic with directed equality for preorder interpretations
• Provides universal property for directed equalities with polarity restrictions
• Characterizes directed equality as relative left adjoint generalizing Lawvere

Innovation

• Directed equality as asymmetric first-order logic transitions
• Polarity-based syntactic restrictions for equality contraction
• Categorical semantics via sound, complete directed doctrines