Abstract
In this paper, we define a multi-type calculus for inquisitive logic, which is sound, complete and enjoys Belnap-style cut-elimination and subformula property. Inquisitive logic is the logic of inquisitive semantics, a semantic framework developed by Groenendijk, Roelofsen and Ciardelli which captures both assertions and questions in natural language. Inquisitive logic adopts the so-called support semantics (also known as team semantics). The Hilbert-style presentation of inquisitive logic is not closed under uniform substitution, and some axioms are sound only for a certain subclass of formulas, called flat formulas. This and other features make the quest for analytic calculi for this logic not straightforward. We develop a certain algebraic and order-theoretic analysis of the team semantics, which provides the guidelines for the design of a multi-type environment accounting for two domains of interpretation, for flat and for general formulas, as well as for their interaction. This multi-type environment in its turn provides the semantic environment for the multi-type calculus for inquisitive logic we introduce in this paper.
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Notes
- 1.
Recall that \(\mathsf {L}\) is an intermediate logic if \(\mathbf {IPL} \subseteq \mathsf {L}\subseteq \mathbf {CPL} \).
- 2.
A Heyting algebra is perfect if it is complete, completely distributive and completely join-generated by its completely join-prime elements. Equivalently, any perfect algebra can be characterized up to isomorphism as the complex algebra of some partially ordered set.
- 3.
We follow the notational conventions introduced in [10], according to which each structural connective in the upper row of the synoptic tables is interpreted as the logical connective(s) in the two slots below it in the lower row. Specifically, each of its occurrences in antecedent (resp. succedent) position is interpreted as the logical connective in the left-hand (resp. right-hand) slot. Hence, for instance, the structural symbol \(\sqsupset \) is interpreted as classical implication
when occurring in succedent position and as classical disimplication \(\mapsto \) (i.e. \(\alpha \mapsto \beta : = {\sim } \alpha \sqcap \beta \)) when occurring in antecedent position. - 4.
A sequent \(x \vdash y\) is type-uniform if x and y are of the same type.
when occurring in succedent position and as classical disimplication References
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Acknowledgements
This research has been made possible by the NWO Vidi grant 016.138.314, by the NWO Aspasia grant 015.008.054, and by a Delft Technology Fellowship awarded in 2013.
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Appendices
Appendix I
The derivation of (A3) \(({\downarrow } \alpha \rightarrow (A \vee B)) \rightarrow ({\downarrow } \alpha \rightarrow A) \vee ({\downarrow } \alpha \rightarrow B)\):
Appendix II
The derivation of (A4) \(\lnot \lnot {\downarrow } \alpha \rightarrow {\downarrow } \alpha \):
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Frittella, S., Greco, G., Palmigiano, A., Yang, F. (2016). A Multi-type Calculus for Inquisitive Logic. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science(), vol 9803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52921-8_14
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