Inquisitive Heyting Algebras

0546145 - FLÚ 2022 RIV NL eng J - Journal Article
Punčochář, Vít
Inquisitive Heyting Algebras.
Studia Logica. Roč. 109, č. 5 (2021), s. 995-1017. ISSN 0039-3215. E-ISSN 1572-8730
R&D Projects: GA ČR(CZ) GA20-18675S
Institutional support: RVO:67985955
Keywords : Logic of questions * Inquisitive logic * Intuitionistic logic * Superintuitionistic logics * Heyting algebras * Algebraic semantics * Antichains
OECD category: Philosophy, History and Philosophy of science and technology
Impact factor: 0.833, year: 2021
Method of publishing: Limited access
https://doi.org/10.1007/s11225-020-09936-9

In this paper we introduce a class of inquisitive Heyting algebras as algebraic structures that are isomorphic to algebras of finite antichains of bounded implicative meet semilattices. It is argued that these structures are suitable for algebraic semantics of inquisitive superintuitionistic logics, i.e. logics of questions based on intuitionistic logic and its extensions. We explain how questions are represented in these structures (prime elements represent declarative propositions, non-prime elements represent questions, join is a question-forming operation) and provide several alternative characterizations of these algebras. For instance, it is shown that a Heyting algebra is inquisitive if and only if its prime filters and filters generated by sets of prime elements coincide and prime elements are closed under relative pseudocomplement. We prove that the weakest inquisitive superintuitionistic logic is sound with respect to a Heyting algebra iff the algebra is what we call a homomorphic p-image of some inquisitive Heyting algebra. It is also shown that a logic is inquisitive iff its Lindenbaum–Tarski algebra is an inquisitive Heyting algebra.
Permanent Link: http://hdl.handle.net/11104/0322696