On Quantifiers for Quantitative Reasoning
Matteo Capucci

TL;DR
This paper develops a first-order predicate logic over the reals with novel bounded quantifiers based on means, exploring their semantics and applications in quantitative reasoning.
Contribution
It introduces a new quantitative predicate logic using multiplicative reals and means as quantifiers, with semantics and categorical approaches, advancing formal reasoning with quantities.
Findings
Means and harmonic means serve as natural bounded quantifiers.
Softmax functions as semantics of argmax in this logic.
Rényi entropy and Hill numbers are modeled as additive/multiplicative semantics.
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 , 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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsSemantic Web and Ontologies · AI-based Problem Solving and Planning · Fuzzy Logic and Control Systems
