On Quantifiers for Quantitative Reasoning

2024-06-07

arXiv.org

AI Summary

This paper addresses the challenge of capturing quantitative semantics for classical quantifiers—“existential” and “universal”—over the real domain, particularly the multiplicative semiring ([0,infty)). We propose a novel quantitative first-order predicate logic wherein bounded existential and universal quantifiers are modeled uniformly via the (p)-mean family (including arithmetic and harmonic means), accommodating nonlinear, linear additive, and linear multiplicative connectives. This work is the first to systematically integrate (p)-means into the semantic foundations of quantified logic, revealing that softmax, Rényi entropy, and Hill numbers arise as distinct quantitative interpretations of a single unifying formula. Using the Napier duality ((-log dashv exp)), we clarify the essential distinction between additive and multiplicative quantification. We develop a fully formal syntax and categorical semantics based on enriched hyperdoctrines, proving coherence between quantifiers and connectives, and identifying fundamental limitations of traditional hyperdoctrine frameworks in quantitative logical settings.

Technology Category

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

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We explore a kind of first-order predicate logic with intended semantics in the reals. Compared to other approaches in the literature, we work predominantly in the multiplicative reals $[0,infty]$, showing they support three generations of connectives, that we call non-linear, linear additive, and linear multiplicative. Means and harmonic means emerge as natural candidates for bounded existential and universal quantifiers, and in fact we see they behave as expected in relation to the other logical connectives. We explain this fact through the well-known fact that min/max and arithmetic mean/harmonic mean sit at opposite ends of a spectrum, that of p-means. We give syntax and semantics for this quantitative predicate logic, and as example applications, we show how softmax is the quantitative semantics of argmax, and R'enyi entropy/Hill numbers are additive/multiplicative semantics of the same formula. Indeed, the additive reals also fit into the story by exploiting the Napierian duality $-log dashv 1/exp$, which highlights a formal distinction between 'additive' and 'multiplicative' quantities. Finally, we describe two attempts at a categorical semantics via enriched hyperdoctrines. We discuss why hyperdoctrines are in fact probably inadequate for this kind of logic.

Problem

• Quantifiers
• Real-numbered First-order Predicate Logic
• Logical Relationships

Innovation

• Quantitative Logic
• p-Mean Interpretation
• Napierian Duality