Jaśkowski's discussive logic D2 was formulated with the help of the modal logic S5 as follows (see [7,8]): A ∈ D2 iff ♦A • ∈ S5, where (−) • is a translation of discussive formulae from For d into the modal language. We say that a modal logic L defines D2 iff D2 = {A ∈ For d : ♦A • ∈ L}. In [14] and [10] were respectively presented the weakest normal and the weakest regular logic which ( †): have the same theses beginning with '♦' as S5. Of course, all logics fulfilling the above condition, define D2. In [10] it was prowed that in the cases of logics closed under congruence the following holds: defining D2 is equivalent to having the property ( †). In this paper we show that this equivalence holds also for all modal logics which are closed under replacement of tautological equivalents (rte-logics).We give a general method which, for any class of modal logics determined by a set of joint axioms and rules, generates in the given class the weakest logic having the property ( †). Thus, for the class of all modal logics we obtain the weakest modal logic which owns this property. On the other hand, applying the method to various classes of modal logics: rte-logics, congruential, monotonic, regular and normal, we obtain the weakest in a given class logic defining D2.
Abstract. In Béziau (Log Log Philos 15:99-111, 2006) a logic Z was defined with the help of the modal logic S5. In it, the negation operator is understood as meaning 'it is not necessary that'. The strong soundnesscompleteness result for Z with respect to a version of Kripke semantics was also given there. Following the formulation of Z we can talk about Z-like logics or Beziau-style logics if we consider other modal logics instead of S5-such a possibility has been mentioned in [1]. The correspondence result between modal logics and respective Beziau-style logics has been generalised for the case of normal logics naturally leading to soundness-completeness results [see Marcos :279-300, 2005) and Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 34(4): [229][230][231][232][233][234][235][236][237][238][239][240][241][242][243][244][245][246][247][248] 2005)]. In Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 37(3-4): [185][186][187][188][189][190][191][192][193][194][195][196] 2008), (Bull Sect Log 38(3-4):189-203, 2009) some partial results for non-normal cases are given. In the present paper we try to give similar but more general correspondence results for the non-normal-worlds case. To achieve this aim we have to enrich original Beziau's language with an additional negation operator understood as 'it is necessary that not'.
In Jaśkowski’s model of discussion, discussive connectives represent certain interactions that can hold between debaters. However, it is not possible within the model for participants to use explicit modal operators. In the paper we present a modal extension of the discussive logic $\textbf{D}_{\textbf{2}}$ that formally corresponds to an extended version of Jaśkowski’s model of discussion that permits such a use. This logic is denoted by $\textbf{m}\textbf{D}_{\textbf{2}}$. We present philosophical motivations for the formulation of this logic. We also give syntactic characterizations of the logic and propose a comparison with certain other modal systems. In particular, we prove that $\textbf{m}\textbf{D}_{\textbf{2}}$ is neither normal nor regular. On the basis of the axiomatization of $\textbf{D}_{\textbf{2}}$, we give an axiomatization of $\textbf{m}\textbf{D}_{\textbf{2}}$. We also give another axiomatization which is not based on the axiomatization of $\textbf{D}_{\textbf{2}}$. Furthermore, we give a natural Kripke-style semantics for $\textbf{m}\textbf{D}_{\textbf{2}}$ and prove the respective adequacy theorems.
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