Abstract
In this section, we will give a brief introduction to proof theory for two important branches of nonclassical logics, that is, modal logics and substructural logics. They are important because both of them include vast varieties of logics that have been actively studied.
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Notes
- 1.
The formula \(\Box \alpha \) in the lower sequent of rules \(\mathsf{(K), (4), (T), (S4)}\), and also all formulas in \(\Box \varGamma \) in the lower sequent of rules \(\mathsf{(K)}\) and \(\mathsf{(D)}\) must be regarded as principal formulas (see Sect. 1.2) of the corresponding rules, when we prove cut elimination (see Theorem 4.1).
- 2.
Analytic cut property of \(\mathbf{GS5}\) is proved by Takano (1992). In addition to it, he introduced also sequent systems for modal logics \(\mathbf{KB, KDB, KTB}\) and \(\mathbf{KB4}\), in which every rule for modality \(\Box \) is acceptable, and showed that each of them has analytic cut property.
- 3.
The name ‘full Lambek caliculus’ and the series of sequent systems prefixed by \(\mathbf{FL}\) were firstly introduced in Ono (1990).
- 4.
The result was shown in Kiriyama and Ono (1991) by using the technique developed in Meyer (1966).
- 5.
Precisely speaking, they should be called involutive commutative substructural logics. Here, a commutative logic means a logic having exchange rule(s).
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Ono, H. (2019). Modal and Substructural Logics. In: Proof Theory and Algebra in Logic. Short Textbooks in Logic. Springer, Singapore. https://doi.org/10.1007/978-981-13-7997-0_4
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DOI: https://doi.org/10.1007/978-981-13-7997-0_4
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