Epistemic Updates on Algebras

Kurz, Alexander; Palmigiano, Alessandra

doi:10.2168/lmcs-9(4:17)2013

Cited by 26 publications

(64 citation statements)

References 16 publications

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“…In the present section, we collect the needed preliminaries: in 2.1, we review the logic of epistemic actions and knowledge. Our presentation slightly departs from [2], and closely follows [20,18]. 1 In 2.2, we briefly review the intuitionistic version of EAK, the axiomatization of which is directly captured in the rules of the calculus introduced in Section 4.…”

Section: Preliminariesmentioning

confidence: 99%

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Multi-type display calculus for dynamic epistemic logic

Frittella

Greco

Kurz

et al. 2014

J Logic Computation

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In the present paper, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The displayapproach is suitable to modularly chart the space of dynamic epistemic logics on weaker-than-classical propositional base. The presence of types endows the language of the Dynamic Calculus with additional expressivity, allows for a smooth proof-theoretic treatment, and paves the way towards a general methodology for the design of proof systems for the generality of dynamic logics, and certainly beyond dynamic epistemic logic. We prove that the Dynamic Calculus adequately captures Baltag-Moss-Solecki's dynamic epistemic logic, and enjoys Belnap-style cut elimination.

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Section: Preliminariesmentioning

confidence: 99%

“…The (single-agent version of the) intuitionistic logic of epistemic actions and knowledge (IEAK) has been introduced in [18]. In the present subsection we report on its multi-agent version.…”

Section: Intuitionistic Eakmentioning

confidence: 99%

Multi-type display calculus for dynamic epistemic logic

Frittella

Greco

Kurz

et al. 2014

J Logic Computation

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“…In [39,35], an analysis of PAL and EAK has been given from the point of view of algebraic semantics, resulting in the definition of the intuitionistic counterparts of PAL and EAK. In the present subsection, we briefly review the definition of the latter one, as it reveals a more subtle interaction between the various modalities, thus preparing the ground for the even richer picture that will arise from the proof-theoretic analysis.…”

Section: The Intuitionistic Version Of Eakmentioning

confidence: 99%

“…semantically (cf. [35,Definition 4.2. ]), that the occurrences of P re(α) can be removed both in the premises and in the conclusion without affecting either the soundness of the rule or the proof power of the system; this entirely remedies the problem.…”

Section: Operational Rules With Side Conditionsmentioning

confidence: 99%

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A proof-theoretic semantic analysis of dynamic epistemic logic

Frittella

Greco²,

Kurz³

et al. 2014

J Logic Computation

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The present paper provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics. Dynamic epistemic logic is one of the best known members of a family of logical systems which have been successfully applied to diverse scientific disciplines, but the proof theoretic treatment of which presents many difficulties. After an illustration of the proof-theoretic semantic principles most relevant to the treatment of logical connectives, we turn to illustrating the main features of display calculi, a proof-theoretic paradigm which has been successfully employed to give a proof-theoretic semantic account of modal and substructural logics. Then, we review some of the most significant proposals of proof systems for dynamic epistemic logics, and we critically reflect on them in the light of the previously introduced proof-theoretic semantic principles. The contributions of the present paper include a generalisation of Belnap's cut elimination metatheorem for display calculi, and a revised version of the display-style calculus D.EAK [30]. We verify that the revised version satisfies the previously mentioned proof-theoretic semantic principles, and show that it enjoys cut elimination as a consequence of the generalised metatheorem.

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Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

Khudairi

2019

Philosophical Studies Series

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This essay examines the philosophical significance of Ω-logic in Zermelo-Fraenkel set theory with choice (ZFC). The dual isomorphism between algebra and coalgebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of Ω-logical validity can then be countenanced within a coalgebraic logic, and Ω-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of Ω-logical validity correspond to those of second-order logical consequence, Ω-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets. * Forthcoming in d'Alfonso and Berkich (eds),'On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence -Themes from IACAP 2016' (Springer: 2018). further be defined via automata. In Section 3, I examine how models of epistemic modal algebras to which modal coalgebraic automata are dually isomorphic are availed of in the computational theory of mind. Finally, in Section 4, the philosophical significance of the characterization of the modal profile of Ωlogical validity for the philosophy of mathematics is examined. I argue (i) that it vindicates a type of neo-logicism with regard to mathematical truth in the set-theoretic multiverse, and (ii) that it provides a modal and computational account of formal grasp of the concept of 'set', adducing in favor of a realist conception of the cumulative hierarchy of sets. Section 5 provides concluding remarks.

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Epistemic Updates on Algebras

Cited by 26 publications

References 16 publications

Multi-type display calculus for dynamic epistemic logic

Multi-type display calculus for dynamic epistemic logic

A proof-theoretic semantic analysis of dynamic epistemic logic

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

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