Logics of imprecise comparative probability

Ding, Yifeng; Holliday, Wesley H.; Icard, Thomas

doi:10.1016/j.ijar.2021.02.004

Cited by 5 publications

(7 citation statements)

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“…In response to a preprint version of this paper, Koudijs[54] defined notions of dependence bisimulation for our dependence models as well as their modal relational versions, and proved a Characterization Theorem for LFD as a fragment of FOL that is invariant under dependence bisimulations. A similar characterization of LFD was found independently in Pútzstück[68] 31. Even so, some natural epistemic notions lack an obvious match in LFD.…”

supporting

confidence: 74%

“…LFD-style dependence goes by universal quantification over all assignments. But as we observed, one can soften this, as in non-monotonic default logics, by going to models where the semantic dependence holds only in the most plausible cases, or only with high probability in some qualitative sense, [31]. In that case, the agenda for LFD becomes wide open again.…”

Section: Discussionmentioning

confidence: 91%

“…Dependence atoms D x y express knowledge subsumption: 'agent x knows at least as much as agent y', [29], while atoms D X Y stand for the analogue notion of group subsumption. 31 Next, since an equivalence relation is also a partition as used in the traditional semantics of questions, [42], dependence models also have an interrogative interpretation. Variables x represent basic questions, and sets of variables X are joint questions asking for the answers to all the given questions).…”

Section: Other Interpretations: Information Knowledge Questionsmentioning

confidence: 99%

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A Simple Logic of Functional Dependence

Baltag

Benthem

2021

J Philos Logic

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This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are discussed: continuous dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games.

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supporting

confidence: 74%

Section: Discussionmentioning

confidence: 91%

Section: Other Interpretations: Information Knowledge Questionsmentioning

confidence: 99%

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A Simple Logic of Functional Dependence

Baltag

Benthem

2021

J Philos Logic

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“…For a brief example of how probability measures on ortholattices can be used to interpret probabilistic language, we consider comparative probability operators, 'at least as likely as' ( ) and 'more likely than' (≻), as studied in, e.g., Kratzer 1991, Yalcin 2010, and Ding et al 2021 Let PL be the language generated by the grammar…”

Section: Probabilitymentioning

confidence: 99%

“…We now define weak and strict comparative probability operators from our sets of measures, as in Ding et al 2021. As before, we use the same symbols for algebraic operations on propositions as for symbols of our formal language, trusting that no confusion will arise.…”

Section: Probabilitymentioning

confidence: 99%

The Orthologic of Epistemic Modals

Holliday¹,

Mandelkern²

2022

Preprint

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Epistemic modals have peculiar logical features that are challenging to account for in a broadly classical framework. For instance, while a sentence of the form p ∧ ♦¬p ('p, but it might be that not p') appears to be a contradiction, ♦¬p does not entail ¬p, which would follow in classical logic.Likewise, the classical laws of distributivity and disjunctive syllogism fail for epistemic modals. Existing attempts to account for these facts generally either under-or over-correct. Some theories predict that p ∧ ♦¬p, a so-called epistemic contradiction, is a contradiction only in an etiolated sense, under a notion of entailment that does not always allow us to replace p ∧ ♦¬p with a contradiction; these theories underpredict the infelicity of embedded epistemic contradictions. Other theories savage classical logic, eliminating not just rules that intuitively fail, like distributivity and disjunctive syllogism, but also rules like non-contradiction, excluded middle, De Morgan's laws, and disjunction introduction, which intuitively remain valid for epistemic modals. In this paper, we aim for a middle ground, developing a semantics and logic for epistemic modals that makes epistemic contradictions genuine contradictions and that invalidates distributivity and disjunctive syllogism but that otherwise preserves classical laws that intuitively remain valid. We start with an algebraic semantics, based on ortholattices instead of Boolean algebras, and then propose a more concrete possibility semantics, based on partial possibilities related by compatibility. Both semantics yield the same consequence relation, which we axiomatize.Then we show how to extend our semantics to explain parallel phenomena involving probabilities and conditionals. The goal throughout is to retain what is desirable about classical logic while accounting for the non-classicality of epistemic vocabulary.

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