Axiomatization in the Meaning Sciences

Holliday, Wesley H.; Icard, Thomas

doi:10.1093/oso/9780198739548.003.0002

Cited by 4 publications

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References 37 publications

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“…Notably, such constructions come up in the context of probabilistic reasoning as well [54], in a way that reverberates elsewhere in natural language, witness expressions for probability and likelihood [62].…”

Section: Fact 92 Lmentioning

confidence: 99%

Interleaving Logic and Counting

VAN BENTHEM,

ICARD

2023

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Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus 'grassroots mathematics'.We begin with a brief review of FO(#), first-order logic with counting operators and cardinality comparisons. This system is known to be of very high complexity, and drowns out finer aspects of the combination of logic and counting. We therefore move to a small fragment that can represent numerical syllogisms and basic reasoning about comparative size: monadic first-order logic with counting, MFO(#). We provide normal forms that allow for axiomatization, determine which arithmetical notions can be defined on finite and on infinite models, and conversely, we discuss which logical notions can be defined out of purely arithmetical ones, and what sort of (non-)classical logics can be induced.Next, we investigate a series of strengthenings of MFO(#), again using normal form methods. The monadic second-order version is close, in a precise sense, to additive Presburger Arithmetic, while versions with the natural device of tuple counting take us to Diophantine equations, making the logic undecidable. We also define a system ML(#) that combines basic modal logic over binary accessibility relations with counting, needed to formulate ubiquitous reasoning patterns such as the Pigeonhole Principle. We prove decidability of ML(#), and provide a new kind of bisimulation matching the expressive power of the language.As a complement to the fragment approach pursued here, we also discuss two other ways of lowering the complexity of FO(#) by changing the semantics of counting in natural ways. A first approach replaces cardinalities by abstract but well-motivated values of 'mass' or other mereological aggregating notions. A second approach keeps the cardinalities but generalizes the meaning of counting to work in models that allow dependencies between variables.Finally, we return to our starting point in natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary and syntax, as well as with natural reasoning modules such as the monotonicity calculus. In addition to these encounters with formal semantics, we discuss the role of counting in semantic evaluation procedures for quantifier expressions and determine, for instance, which binary quantifiers are computable by finite 'semantic automata'. We conclude with some general thoughts on yet further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on connecting our analysis to empirical findings in cognitive science.

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Section: Fact 92 Lmentioning

confidence: 99%

Interleaving Logic and Counting

VAN BENTHEM,

ICARD

2023

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“… See Holliday & Icard (2013b) andHolliday & Icard (2018).18 SeeLassiter (2010Lassiter ( , 2011Lassiter ( , 2015Lassiter ( , 2017,Moss (2013Moss ( , 2015Moss ( , 2018,Rothschild (2012),Swanson (2006), and Yalcin (2007 for discussion of alternative motivations for probabilistic semantics. SeeLassiter (2015) for criticisms of Holliday & Icard's semantics distinct from those I raise below.…”

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confidence: 99%

Probability Modals and Infinite Domains

Marushak

2020

J Philos Logic

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Recent years have witnessed a proliferation of attempts to apply the mathematical theory of probability to the semantics of natural language probability talk. These sorts of "probabilistic" semantics are often motivated by their ability to explain intuitions about inferences involving likely and probable-intuitions that Angelika Kratzer's canonical semantics fails to accommodate through a semantics based solely on an ordering of worlds and a qualitative ranking of propositions. However, recent work by Wesley Holliday and Thomas Icard has been widely thought to undercut this motivation: they present a world-ordering semantics that yields essentially the same logic as probabilistic semantics. In this paper, I argue that the challenge remains: defenders of world-ordering semantics have yet to offer a plausible semantics that captures the logic of comparative likelihood. Holliday & Icard's semantics yields an adequate logic only if models are restricted to Noetherian pre-orders. But I argue that the Noetherian restriction faces problems in cases involving infinitely large domains of epistemic possibilities. As a result, probabilistic semantics remains the better explanation of the data.

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A Simple Logic of Concepts

Icard

Moss

2022

J Philos Logic

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Axiomatization in the Meaning Sciences

Cited by 4 publications

References 37 publications

Interleaving Logic and Counting

Interleaving Logic and Counting

Probability Modals and Infinite Domains

A Simple Logic of Concepts

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