Abstract
The paper expands upon the work by Wang [4], who proposes a new framework based on quantifier-free predicate language extended by a new modality \(\exists x\Box \) and axiomatizes the logic over S5 frames. This paper gives the logics over K, D, T, 4, S4 frames with increasing and constant domains. And we provide a general strategy for proving completeness theorems for logics w.r.t. the increasing domain and logics w.r.t. the constant domain respectively.
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Notes
- 1.
Know-wh stands for verb know followed by a wh-question word.
- 2.
The rule name might seem a little odd, since the rule goes in the direction “\(\lnot \Box \)to\(\lnot \exists \Box \)”. But if we rewrite the rule by the converse-negative propositions, we will find that in a sense the direction is exactly “\(\Box \)to\(\exists \Box \)”.
- 3.
The condition for Re\(\exists \Box \) can be relaxed to “\(\varphi [y/x]\) is admissible and y is not free in \(\varphi \)”. To ease the presentation, here we strengthen the condition.
- 4.
- 5.
Note that S5 models must be constant domain models. In this section we do not talk about S5.
- 6.
In [4], the worlds of the canonical model is all the maximal consistent sets with \(\exists \)-property. The reason \(\exists \)-property works is that \(\vdash _{{\text {S5}}^{\mathrm {c}}_{\exists \Box }}(\exists x\Box \varphi \rightarrow \Box \varphi [y/x])\rightarrow \Box (\exists x\Box \varphi \rightarrow \Box \varphi [y/x])\). Based on this, we could find a unified witness for each \(\exists x\Box \varphi \), which ensures that the existence lemma works.
- 7.
S5 models must be constant domain models, thus \({\text {S5}}^{\mathrm {i}}_{\exists \Box }={\text {S5}}^{\mathrm {c}}_{\exists \Box }\).
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Acknowledgments
This work was supported by the NSSF grant 19BZX135 for the project Decidable Fragments of First-order Modal Logic. The author thanks Yanjing Wang for giving the author the ideas of the axiomations of these systems. The author thanks the anonymous reviewers for their insightful comments. In particular, one of the reviewers indicated that the results might be extended to a richer fragment.
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Wang, X. (2021). Completeness Theorems for \(\exists \Box \)-Fragment of First-Order Modal Logic. In: Ghosh, S., Icard, T. (eds) Logic, Rationality, and Interaction. LORI 2021. Lecture Notes in Computer Science(), vol 13039. Springer, Cham. https://doi.org/10.1007/978-3-030-88708-7_20
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