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Semantical Proof of Subformula Property for the Modal Logics K4.3, KD4.3, and S4.3

100%

Yazaki D.

Bulletin of the Section of Logic

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2019

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tom 48

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nr 4

EN

The main purpose of this paper is to give alternative proofs of syntactical and semantical properties, i.e. the subformula property and the nite model property, of the sequent calculi for the modal logics K4.3, KD4.3, and S4.3. The application of the inference rules is said to be acceptable, if all the formulas in the upper sequents are subformula of the formulas in lower sequent. For some modal logics, Takano analyzed the relationships between the acceptable inference rules and semantical properties by constructing models. By using these relationships, he showed Kripke completeness and subformula property. However, his method is difficult to apply to inference rules for the sequent calculi for K4.3, KD4.3, and S4.3. Lookinglosely at Takano's proof, we nd that his method can be modied to construct nite models based on the sequent calculus for K4.3, if the calculus has (cut) and all the applications of the inference rules are acceptable. Similarly, we can apply our results to the calculi for KD4.3 and S4.3. This leads not only to Kripke completeness and subformula property, but also to finite model property of these logics simultaneously.

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New Modification of the Subformula Property for a Modal Logic

100%

Takano M.

Bulletin of the Section of Logic

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2020

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tom 49

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nr 3

255-268

EN

A modified subformula property for the modal logic KD with the additionalaxiom □ ◊(A ∨ B) ⊃ □ ◊ A ∨ □ ◊B is shown. A new modification of the notion of subformula is proposed for this purpose. This modification forms a natural extension of our former one on which modified subformula property for the modal logics K5, K5D and S4.2 has been shown ([2] and [4]). The finite model property as well as decidability for the logic follows from this.

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Polyadic algebras over nonclassical logics

88%

Pigozzi D., Salibra A.

Banach Center Publications

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1993

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tom 28

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nr 1

51-66

EN

The polyadic algebras that arise from the algebraization of the first-order extensions of a SIC are characterized and a representation theorem is proved. Standard implicational calculi (SIC)'s were considered by H. Rasiowa [19] and include classical and intuitionistic logic and their various weakenings and fragments, the many-valued logics of Post and Łukasiewicz, modal logics that admit the rule of necessitation, BCK logic, etc.

4

A Sound Interpretation of Leśniewski's Epsilon in Modal Logic KTB

63%

Inoue T.

Bulletin of the Section of Logic

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2021

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tom 50

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nr 4

455-463

EN

In this paper, we shall show that the following translation \(I^M\) from the propositional fragment \(\bf L_1\) of Leśniewski's ontology to modal logic \(\bf KTB\) is sound: for any formula \(\phi\) and \(\psi\) of \(\bf L_1\), it is defined as (M1) \(I^M(\phi \vee \psi) = I^M(\phi) \vee I^M(\psi)\), (M2) \(I^M(\neg \phi) = \neg I^M(\phi)\), (M3) \(I^M(\epsilon ab) = \Diamond p_a \supset p_a . \wedge . \Box p_a \supset \Box p_b .\wedge . \Diamond p_b \supset p_a\), where \(p_a\) and \(p_b\) are propositional variables corresponding to the name variables \(a\) and \(b\), respectively. In the last, we shall give some comments including some open problems and my conjectures.

Ograniczanie wyników

3 Bulletin of the Section of Logic

1 Banach Center Publications

1 Inoue T.

1 Pigozzi D.

1 Salibra A.

1 Takano M.

1 Yazaki D.

1 2021

1 2020

1 2019

1 1993

Semantical Proof of Subformula Property for the Modal Logics K4.3, KD4.3, and S4.3

New Modification of the Subformula Property for a Modal Logic

Polyadic algebras over nonclassical logics

A Sound Interpretation of Leśniewski's Epsilon in Modal Logic KTB