A Relevant Logic of Questions
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A Relevant Logic of Questions ´ r1 V´ıt Punˇcochaˇ Received: 28 December 2018 / Accepted: 3 December 2019 / © Springer Nature B.V. 2020
Abstract This paper introduces the inquisitive extension of R, denoted as InqR, which is a relevant logic of questions based on the logic R as the background logic of declaratives. A semantics for InqR is developed, and it is shown that this semantics is, in a precisely defined sense, dual to Routley-Meyer semantics for R. Moreover, InqR is axiomatized and completeness of the axiomatic system is established. The philosophical interpretation of the duality between Routley-Meyer semantics and the semantics for InqR is also discussed. Keywords Inquisitive logic · Relevant logic · The logic R · Routley-Meyer semantics · Relevance · Questions
1 Introduction Basic propositional inquisitive logic (InqB) is a logical system for a standard propositional language enriched with one additional binary connective, called “inquisitive disjunction”, by means of which questions are formed (see, e.g., [14]). InqB can be described as a conservative extension of classical propositional logic in the sense that in the declarative fragment of the language, i.e. if inquisitive disjunction is omitted, the logic validates exactly those formulas that are classically valid. However, InqB is not “schematically conservative”, by which I mean that there are some schemata (e.g., ¬¬ϕ → ϕ) that are valid for the declarative part of the language, but their validity does not extend to the whole language. This is related to a striking feature of InqB, namely that it is not closed under uniform substitution. The failure of uniform substitution is desirable and well-motivated in inquistive logic. The reason is that atomic formulas always represent declarative sentences and questions are present in the language only in the form of complex formulas – generated by inquisitive disjunction. Moreover, the logical behaviour of V´ıt Punˇcoch´aˇr
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Institute of Philosophy, Czech Academy of Sciences, Jilsk´a 1, 110 00 Prague, Czech Republic
V. Punˇcoch´arˇ
questions is supposed to differ from the logical behaviour of declarative sentences and so one cannot always substitute a question for an atomic declarative in a valid formula without loosing its validity (see [14], § 2.5.5). A consequence of this feature of InqB is that it is not possible to formulate a semantics for InqB that could be seen as an extension of truth-table semantics for classical logic. Inquisitive logic contrasts in this respect with other, more familiar conservative extensions of classical logic like, for example, normal modal logics. In Kripke semantics for normal modal logics, formulas are evaluated with respect to possible worlds and in every possible world (of any Kripke model) the non-modal logical operators, whether applied to non-modal or modal sentences, behave exactly like they do in the standard truth-table semantics for classical logic. But something like that is not possible in the case of InqB since it would imply sch
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