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.
Overview
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Matteo Capucci explores extensions to first-order predicate logic by integrating quantitative semantics rooted in real numbers, identifying algebraic structures on $[0, fty]$ and presenting new logical connectives.
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The work proposes using means and harmonic means as quantifiers in a quantitative logic setting, linking these to integrals and softmax functions, and providing a coherent framework for quantitative predicate logic (QPL).
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This framework has implications for various fields including machine learning, statistics, and physics by offering theoretical backing for practices like softmax, and it opens potential for new interdisciplinary applications and enhanced proof systems.
On Quantifiers for Quantitative Reasoning
Matteo Capucci's paper "On Quantifiers for Quantitative Reasoning" explores innovative extensions to first-order predicate logic with a focus on quantitative semantics in the real numbers. This exploration is established on the multiplicative reals $[0,\infty]$ and identifies three generations of logical connectives: non-linear, linear additive, and linear multiplicative. The work presents means and harmonic means as natural candidates for bounded existential and universal quantifiers and demonstrates their coherence with other logical connectives.
Summary and Key Contributions
The paper's central thesis is that traditional predicate logic can be extended and enriched by adopting operations on the real numbers that have logical significance. This involves employing arithmetic and harmonic means as soft quantifiers. Here are the primary contributions:
Algebraic Structure on Reals:
- The paper identifies an algebraic structure in $[0,\infty]$, termed very linear logic, featuring non-linear, linear additive, and linear multiplicative connectives.
- It showcases a duality between additive and multiplicative structures, using Napierian duality $-\log \leftrightarrow 1/\exp$.
Quantifiers as Integrals:
- The paper proposes that quantification in a quantitative logic setting should correspond to integration, emphasizing means (normalized integrals) over sums (unbounded integrals).
- This yields $p$-means and $p$-integrals as soft versions of existential and universal quantifiers, seamlessly relating to min, max, and geometric means.
Syntax and Semantics of Quantitative Logic:
- An inductive definition of quantitative predicates over contexts, with semantics grounded in measure theory.
- A specific relationship between traditional logical quantifiers and their "soft" counterparts through $p$-means, solidifying how softmax functions can be understood as semantics of $\argmax$.
Relation to Existing Works:
- The framework illuminates connections to existing probability theories, many-valued logics, and category-theoretic approaches, referencing foundational work by Boole, Lawvere, and Bacci et al.
- The work draws connections with real-world applications such as softmax in machine learning and Renyi entropy in information theory.
Implications and Future Directions
Practical Implications:
- Machine Learning: The interpretation of softmax as a quantitative semantics of $\argmax$ provides deepened theoretical backing for practices in neural networks and probabilistic models.
- Statistics and Physics: Explaining entropy and diversity measures as quantitative logic constructs paves the way for novel interpretations and justifications in statistical mechanics and ecological modeling.
Theoretical Implications:
- Quantitative Predicate Logic (QPL): By framing quantifiers as integrative operations, this work enriches the classical logic landscape, aligning it with continuous semantics. This formulation suggests that quantitative reasoning can be formalized within a logical framework akin to qualitative reasoning in traditional logic.
- Category Theory: The paper hints at the potential for category-theoretic structures to be specified and reasoned about using this extended logical system, but also acknowledges existing challenges in aligning categorical semantics like enriched hyperdoctrines with the proposed extensions.
Speculation on Future Developments:
- Enhanced Proof Systems: Addressing the identified failure of current proof-theoretic frameworks to capture the nuances of QPL is crucial. This might entail developing new algebraic structures or proof rules to accommodate the properties and semantics elucidated in this work.
- Interdisciplinary Extensions: Further incorporating the principles of QPL into diverse fields beyond computer science, such as economics, cognitive science, and bioinformatics, where quantitative reasoning is paramount, could yield significant interdisciplinary advances.
- Refinement of Domain-Specific Logic: Exploring domain-specific adaptations of the proposed logic, such as censoring in survival analysis or stochastic processes in finance, will likely enhance the applicability and robustness of these ideas.
Conclusion
The paper by Matteo Capucci presents substantial progress towards a unified logic that incorporates both symbolic and quantitative reasoning. By defining a novel algebraic structure on the reals and connecting it with classical logical operations, this research sets the stage for a new perspective on quantitative reasoning in various scientific and practical domains. Emphasizing the role of means and integrals as bounded quantifiers opens up rich avenues for future exploration, both in theoretical advancements and practical applications. The challenge of establishing a cogent proof-theoretic foundation for QPL remains an open and stimulating area for ongoing research.
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