Abstract
Term-modal logics, developed by Fitting et al., enable us to index a modal operator by a term of the first-order logic and even to quantify variables in the index of the modal operator. In this paper, we expand term-modal logics by allowing a modal operator to be indexed by a finite sequence of terms as well as a single term. The expanded logics are generalizations of both term-modal logics and quantified modal logics. We provide sound Hilbert-style axiomatizations (without Barcan-like axioms) for the logics and establish the strong completeness results for some of the logics. We also propose sequent calculi for the logics and show cut elimination theorems and Craig interpolation theorems for some of the calculi.
We would like to thank three reviewers for their constructive comments to our manuscript. The work of all authors was partially supported by the research supported by JSPS Grant-in-Aid for Scientific Research (B) (KAKENHI 17H02258). The work of the second author was partially supported also by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) Grant Number 19K12113 and JSPS Core-to-Core Program (A. Advanced Research Networks).
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Notes
- 1.
We can find the very idea of term-modal logic even in Hintikka’s Knowledge and Belief, where for a sentence like “a knows that P” he says “a is a name of a person or [...] a definite description referring to a human being.” [6, p. 3] He also considers substitution of such names by equality axioms. [6, ch. 6]
- 2.
- 3.
A frame over \(\mathcal {D}\) for \(\mathcal {L}\) must satisfy that \(\bigcap _{w \in W} D_w \ne \varnothing \) if \(\mathsf {Con} \ne \varnothing \) in \(\mathcal {L}\).
- 4.
The strong completeness results of \(\mathsf {H}(\mathbf {tK}\varSigma )\) for all \(\varSigma \subseteq \bigcup _{n\in \mathbb {N}} \{\,{\mathrm {D}_n,\mathrm {T}_n,\mathrm {4}_n,\mathrm {B}_n}\,\}\) such that \(\mathrm {B}_k \in \varSigma \) for some \(k\in \mathbb {N}\) are not presented in this paper. For example, the Hilbert system \(\mathsf {H}(\mathbf {tKB}_1)\) is not yet proved to be strongly complete, as the ordinal canonical model construction is not so straightforward for \(\mathbf {tKB_1}\). The step-by-step method introduced in [2, p. 223] might be applicable for the strong completeness results of the Hilbert systems for such logics, but we have not done yet.
- 5.
- 6.
This treatment is also found in the definition of the semantics of modal predicate logic given by Gamut in [5, pp. 59–60].
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Sawasaki, T., Sano, K., Yamada, T. (2019). Term-Sequence-Modal Logics. In: Blackburn, P., Lorini, E., Guo, M. (eds) Logic, Rationality, and Interaction. LORI 2019. Lecture Notes in Computer Science(), vol 11813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-60292-8_18
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