M. Bozic and K.Dosen, Models for normal intuitionistic logics, Studia Logica 43 (1984), pp. 217-245.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of presenting an intuitionistic concept of modality was faced in Fitch [Fit49] and Prior [Pri57] Prior proposed a modal extension of IPL which turns out to be S5 once the axiom of excluded middle is added. They were also considered as counterparts of classical modal logics in Bozic and Dosen [BD84] and Fischer Servi [Fis77] In [BD84] the intuitionistic modal logic IntK was introduced as the smallest set of formulas including the standard axioms of IPL and the axiom: and closed under modus ponens, substitution and necessitation. An intuitionistic modal logic is a set of formulas ....

....concept of modality was faced in Fitch [Fit49] and Prior [Pri57] Prior proposed a modal extension of IPL which turns out to be S5 once the axiom of excluded middle is added. They were also considered as counterparts of classical modal logics in Bozic and Dosen [BD84] and Fischer Servi [Fis77] In [BD84], the intuitionistic modal logic IntK was introduced as the smallest set of formulas including the standard axioms of IPL and the axiom: and closed under modus ponens, substitution and necessitation. An intuitionistic modal logic is a set of formulas including IntK and closed under modus ....

M. Bozic and K. Dosen, Models for normal intuitionistic logics, Studia Logica, 43, 1984, pp. 217-245. 133

....[3] It is worth noticing that in [21] a semantics with a Kripke type frame is also adopted. The modal relation is introduced as an extension of the intuitionistic accessibility relation. General criteria to find the intuitionistic counterpart of many classical modal systems are both proposed in [1, 6] and in [8, 9, 10, 11, 12] In [1] two systems HK2 and HK3 are introduced, as intuitionistic counterparts of the classical system K. The first one deals only with 2 and the latter only with 3. HK3 turns out to be sound and complete with respect to the Kripke type frames satisfying one of the two ....

....a semantics with a Kripke type frame is also adopted. The modal relation is introduced as an extension of the intuitionistic accessibility relation. General criteria to find the intuitionistic counterpart of many classical modal systems are both proposed in [1, 6] and in [8, 9, 10, 11, 12] In [1] two systems HK2 and HK3 are introduced, as intuitionistic counterparts of the classical system K. The first one deals only with 2 and the latter only with 3. HK3 turns out to be sound and complete with respect to the Kripke type frames satisfying one of the two connecting properties we are ....

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M. Bozic and K.Dosen, Models for normal intuitionistic logics, Studia Logica 43 (1984), pp. 217-245.

....operators we call them intuitionistic modal logics and classical polymodal logics. That the Godel translation can be extended to an embedding of at least a few particular intuitionistic modal systems into some classical polymodal logics was observed by several authors (cf. 13] 25] 26] [5]) Fischer Servi [13] 15] used a variant of the translation to define true intuitionistic analogues of a number of classical modal systems. In [27] we exploited the translation proposed by Shehtman [25] to embed intuitionistic modal logics with the single necessity operator 2 of K into bimodal ....

....called 3 normal if fl(p q) flp flq and : fl belong to it. In this case we write 3 instead of fl and call it the possibility operator. Every 3 normal logic is closed under : 3 . The smallest 3 normal logic is denoted by IntK3 . Some particular 2 normal IM systems have been investigated in [5], 21] and [26] general results on the finite model property of such logics can be found in [27] 3 normal systems have been considered in [5] and [26] As in classical modal logic, given a 2 normal IM logic L, we can define the dual operator 3 by taking 3 = 2: Likewise, in a 3 normal logic ....

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M. Bozi'c and K. Dosen. Models for normal intuitionistic logics. Studia Logica, 43:217--245, 1984.

....results on the finite model property (and so decidability) from classical bimodal logics to their intuitionistic fragments. Here we consider extensions of IntK, intuitionistic propositional logic with the necessity operator 2 of the minimal classical modal system K, introduced and investigated in [3], 19] The possibility operator 3 is defined via 2 and : as in classical modal logic (although this does not mean that 2 and 3 are dual as in K; see [11] In the next paper [21] we extend a part of the theory developed here to systems with weaker connections between 2 and 3, e.g. to those in ....

....under any valuation x j= for every point x in F and every 2 IntK. And using the standard canonical model technique one can prove that every IM logic L is characterized by a suitable class C of IM frames in the sense that 2 L iff F j= for all F 2 C (all essential details can be found in [3]) An IM frame F = hW; R; P i is called a Kripke IM frame if P consists of all cones in W . The underlying Kripke IM frame of an IM frame F is denoted by F. An IM logic is Kripke complete if it is characterized by a class of Kripke IM frames. An IM algebra is a structure A = hA; 2i such ....

M. Bozi'c and K. Dosen. Models for normal intuitionistic logics. Studia Logica, 43:217--245, 1984.

....I 4 (Kuznetsov used 4 instead of 2) the intuitionistic analog of the Godel Lob classical provability logic GL. It can be obtained by adding to IntK2 (and even to Int) the axioms p 2p; 2p p) p; p q) p) 2q p) A model theory for logics in ExtIntK2 was developed by Ono [1977] Bozi c and Dosen [1984], Dosen [1985] Sotirov [1984] and Wolter and Zakharyaschev [1997] we discuss it in the next section. Font [1984, 1986] considered these logics from the algebraic point of view, and Luppi [1996] investigates their interpolation property by proving, in particular, that the superamalgamability of ....

....logic where 9 and 8 are not dual. Another family of normal intuitionistic modal logics can be defined in the language L3 by taking as the basic system the smallest logic in L3 to contain the axioms 3(p q) 3p 3q and :3 ; it will be denoted by IntK3 . Logics in ExtIntK3 were studied by Bozi c and Dosen [1984], Dosen [1985] Sotirov [1984] and Wolter [1997b] Finally, we can define intuitionistic modal logics with independent 2 and 3. These are extensions of IntK23 , the smallest logic in the language L23 containing both IntK2 and IntK3 . Fischer Servi [1980, 1984] constructed an interesting logic in ....

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M. Bozi'c and K. Dosen. Models for normal intuitionistic logics. Studia Logica, 43:217--245, 1984.

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M. Bozic and K.Dosen, Models for normal intuitionistic logics, Studia Logica 43 (1984), pp. 217-245.

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