On Modal Logics Defining Jaśkowski’s D2-Consequence

Nasieniewski, Marek; Pietruszczak, Andrzej

doi:10.1007/978-94-007-4438-7_9

Cited by 5 publications

(7 citation statements)

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“…It is known that (5 c ) ∈ KD4, (D) ∈ K5 c and (4) ∈ K55 c [see, e.g., 1,3,4]. Hence KD4 = K45 c and KD45 = K55 c .…”

Section: ( R N )mentioning

confidence: 99%

“…We have the following pairs: emptiness to (Q); reflexivity to (T); quasi-reflexivity to (T q ); vacuity to (T c ); seriality to (D); symmetry to (B); transitivity to (4); Euclideanness to (5); the condition (xi) n to (Alt n ); the condition (xii) n to (Talt n ), for any n > 0; the condition (xiii) n to (C n ), for any n 0. Determination theorems for the logic K and its normal extensions by some of the formulas (T), (T c ), (D), (B), (4) and 5…”

Section: Determination Theorems For Some Normal Logicsmentioning

confidence: 99%

“…We prove that any frame of C is either empty or universal. In fact, if W, R ∈ C Λ , then R is symmetric and transitive, because (B) and (4) are valid in W, R . Assume for a contradiction that W, R satisfies condition (c ), i.e., W is not a singleton and there is a unique w ∈ W such that R = W × (W \{w}).…”

Section: Nagle's Factmentioning

confidence: 99%

“…Ad 10. If a logic is determined by a subclass of the class of finite semiuniversal frames which are universal or have 0 < CardA n, then it is normal and contains (D), (4), (5) and (Talt n ), since these formulas are valid in this subclass, by Lemma 2.1 (1,4) and the assumption.…”

Section: If a Logic Is Determined By A Subclass Of The Class Of Finitmentioning

confidence: 99%

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Simplified Kripke-Style Semantics for Some Normal Modal Logics

2019

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proved that the normal logics K45, KB4 (= KB5), KD45 are determined by suitable classes of simplified Kripke frames of the form W, A , where A ⊆ W. In this paper, we extend this result. Firstly, we show that a modal logic is determined by a class composed of simplified frames if and only if it is a normal extension of K45. Furthermore, a modal logic is a normal extension of K45 (resp. KD45; KB4; S5) if and only if it is determined by a set consisting of finite simplified frames (resp. such frames with A = ∅; such frames with A = W or A = ∅; such frames with A = W). Secondly, for all normal extensions of K45, KB4, KD45 and S5, in particular for extensions obtained by adding the so-called "verum" axiom, Segerberg's formulas and/or their T-versions, we prove certain versions of Nagle's Fact (

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“…It is known that (5 c ) ∈ KD4, (D) ∈ K5 c and (4) ∈ K55 c [see, e.g., 1,3,4]. Hence KD4 = K45 c and KD45 = K55 c .…”

Section: ( R N )mentioning

confidence: 99%

Section: Determination Theorems For Some Normal Logicsmentioning

confidence: 99%

Section: Nagle's Factmentioning

confidence: 99%

Section: If a Logic Is Determined By A Subclass Of The Class Of Finitmentioning

confidence: 99%

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Simplified Kripke-Style Semantics for Some Normal Modal Logics

2019

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“…What is only used, is the so-called M-counterpart of the logic S5 (for investigations on this notion see [2]). In various papers it has been shown (see [3][4][5][6][7]) that to be able to formulate D 2 , one can use various modal logics. (Jaśkowski's logic was meant to be a basis for a consequence relation and also in this case there can be given other systems than S5 which also allow to express D 2 -consequence relation (see [8]). )…”

Section: And the Minimal Variant Of Discussive Logicmentioning

confidence: 99%

A Kotas-Style Characterisation of Minimal Discussive Logic

Mruczek-Nasieniewska

Nasieniewski

2019

Axioms

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In this paper, we discuss a version of discussive logic determined by a certain variant of Jaśkowski’s original model of discussion. The obtained system can be treated as the minimal discussive logic. It is determined by frames with serial accessibility relation. As the smallest one, this logic can be treated as a basis which could be extended to richer discussive logics that are obtained by varying accessibility relation and resulting in a lattice of discussive logics. One has to remember that while formulating discussive logics there is no one-to-one determination of discussive logics by modal logics. For example, it is proved that Jaśkowski’s logic D 2 can be expressed by other than S 5 modal logics. In this paper we consider a deductive system for the sketchily described minimal logic. While formulating the deductive system, we apply a method of Kotas that was used to axiomatize D 2 . The obtained system determines a logic D 0 as a set of theses that is contained in D 2 . Moreover, any discussive logic that would be expressed by means of the provided model of discussion would contain D 0 , so it is the smallest discussive logic.

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Sette’s Logics, Revisited

Omori

2017

Logic, Rationality, and Interaction

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On Modal Logics Defining Jaśkowski’s D2-Consequence

Cited by 5 publications

References 10 publications

Simplified Kripke-Style Semantics for Some Normal Modal Logics

Simplified Kripke-Style Semantics for Some Normal Modal Logics

A Kotas-Style Characterisation of Minimal Discussive Logic

Sette’s Logics, Revisited

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