A Method of Generating Modal Logics Defining Jaśkowski’s Discussive Logic D2

Abstract

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.

Simplified Kripke-Style Semantics for Some Normal Modal Logics

“…We obtain that a modal logic is a normal extension of K45 if and only if it is determined by a subclass of the class of finite semiuniversal frames. 3 Furthermore, we obtain that a modal logic is a normal extension of KD45 (resp. KB4; S5) if and only if it is determined by a set consisting of finite semi-universal frames with A = ∅ (resp.…”

“…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 .…”

“…Ad 2. If a logic is determined by a subclass of F ∅ ∪ U fin , then it is normal and contains (4), (5) and (B), since these formulas are valid in this subclass, by Lemma 2.1 (1,3).…”

A Kotas-Style Characterisation of Minimal Discussive Logic

“…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]). )…”

“…(We use standard results from modal logic, for details see for example, References [12,13].) Of course, if for a given normal modal logic S, we have (D) ∈ S, then D ⊆ S. It is known (see Reference [6]) that one can consider various accessibility relations but the resulting discussive logic would be still the same. By definition, D S5 = D 2 .…”