On Quantifiers for Quantitative Reasoning

TL;DR

The paper develops a quantitative predicate logic with real-valued semantics realized on the extended positive reals $[0,\infty]_{\otimes}$, revealing three generations of connectives and a dual additive world connected by Napierian duality. It defines $p$-sums and $p$-means as semantic quantifiers, and shows that softmax and argmax, as well as entropy measures, naturally arise within this framework. It provides syntax and semantics for the logic, demonstrates concrete links to information theory and machine learning concepts, and analyzes categorical semantics attempts via enriched hyperdoctrines, which fail, highlighting the need for new metatheory. The work aims to formalize how quantitative reasoning can be integrated into logical systems, with potential implications for statistics, ML, and theoretical computer science, while acknowledging unresolved proof theory and semantic foundations.

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 Rényi 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.

Figures (1)

• Figure 1: A comparison of $p$-sums (solid lines) and $p$-means (dashed lines) in their behaviour as $p$ changes. There is a clear tendency towards max/min (dotted) as $p \to \infty$, and the convergence of the $p$-means to the geometric one as $p \to 0$.

Theorems & Definitions (12)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Lemma 3.1
• Definition 3.2: $p$-sum
• Remark 3.3
• Remark 3.4
• Definition 3.5
• Definition 3.6