Abstract
Intuitionistic Public Announcement Logic (IntPAL) proposed by Ma et al. (2014) aims at formalizing changes of an agent’s knowledge in a constructive manner. IntPAL can be regarded as an intuitionistic generalization of Public Announcement Logic (PAL) whose modal basis is the intuitionistic modal logic IK by Fischer Servi (1984) and Simpson (1994). We also refer to IK for the basis of this paper. Meanwhile, Nomura et al. (2015) provided a cut-free labelled sequent calculus based on the study of Maffezioli et al. (2010). In this paper, we introduce a labelled sequent calculus for IntPAL (we call it \(\mathbf {GIntPAL}\)) as both an intuitionistic variant of \(\mathbf {GPAL}\) and a public announcement extension of Simpson’s labelled calculus, and show that all theorems of the Hilbert axiomatization of IntPAL are also derivable in \(\mathbf {GIntPAL}\) with the cut rule. Then we prove the admissibility of the cut rule in \(\mathbf {GIntPAL}\) and also the soundness result for birelational Kripke semantics. Finally, we derive the semantic completeness of \(\mathbf {GIntPAL}\) as a corollary of these theorems.
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- 1.
Labelled sequent calculus (cf. [17]) is one of the most uniform approaches for sequent calculus for modal logic, where each formula has a label corresponding to an element (sometimes called a possible world) of a domain in Kripke semantics for modal logic.
- 2.
Epistemic logics are basically based on the modal system S5, but the most primitive modal system K is usually the starting point in the case of constructing a proof system of a modal logic; and we also follow the custom and employ IntK for its semantics.
- 3.
Two conditions, (F1) and (F2), are required to show hereditary (and validity of axioms) in IntK on which \(\mathbf {GIntPAL}\) is based. In fact, one more condition is added to the two in [14] for some specific purpose in their paper. That is \(R_{a}=(\leqslant \circ R_{a})\cap (R_{a}\circ \geqslant )\).
- 4.
Note that the above IntK frame satisfies the conditions since \((R_{a}\circ \leqslant )=(\leqslant \circ R_{a})=(\geqslant \circ R_{a})=(R_{a}\circ \leqslant )=\{w_{1},w_{1}\}^2\).
- 5.
Note that \(\lnot p\) is an abbreviation of \(p\rightarrow \bot \).
- 6.
We would like to thank the anonymous reviewers for their constructive comments to our manuscript. This work of the first author was supported by Grant-in-Aid for JSPS Fellows, and that of the second author was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) Grant Numbers 15K21025. This work was conducted by JSPS Core-to-Core Program (A. Advanced Research Networks).
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Nomura, S., Sano, K., Tojo, S. (2015). A Labelled Sequent Calculus for Intuitionistic Public Announcement Logic. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2015. Lecture Notes in Computer Science(), vol 9450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48899-7_14
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