F. Wolter and M. Zakharyaschev. On the relation between intuitionistic and classical modal logics. Algebra and Logic, 36:121--155, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....= hW;R; R3 ; R ; P i, we denote by F its underlying full MIPC frame hW;R; R3 ; R ; UpW i. An MIPC frame F is a frame for an MIPC logic L when F j= L. An MIPC logic L is D persistent if F j= L holds for any descriptive MIPC frame F for L. It is straightforward to show that MIPC is D persistent (see [14]) 3 Constructing a finite counter MIPC model First, we review some basic notions on MIPC models, which will be used in this section. A valuation on an MIPC frame F = hW;R; R3 ; R ; P i is a mapping from VarL 3 to P . An MIPC model is a pair M = hF; V i where F is an MIPC frame and V is a ....

....Thus, M is a finite counter MIPC model of L for OE. 4 MIPC logics with the finite model property In this section, we give some corollaries of Theorem 3.16 in accessible forms and present some examples of MIPC logics with the finite model property. For this, we use the following result given in [14]. 12 Proposition 4.1 If a 3 IM logic L is characterized by a class of full 3 frames which is closed under elementary equivalence (in the first order language with the predicates = R, R3 and R ) then L is D persistent. From this and Theorem 3.16, we obtain Corollary 4.2 Let L be an ....

F. Wolter and M. Zakharyaschev. On the relation between intuitionistic and classical modal logics. Algebra and Logic, 36:73--92, 1997.

....a Kripke model 1 . We define modal algebras corresponding to CL and constructive S4 below and show how to construct representations for them. Our techniques are standard; for more background on intuitionistic and modal duality theory the reader is referred to (Johnstone 1982) Goldblatt 1976) Wolter and Zakharyaschev (1995, 1996, 1997) The proofs are not difficult. We state the results here for two reasons: first, they provide easy completeness proofs for cS4 and CL (especially compared to the complicated proof given by Fairtlough and 1 More precisely, a general frame; see the discussion below. 9 Mendler) and, ....

Wolter, F., and M. Zakharyaschev. 1996. On the Relation between Intuitionistic and Classical Modal Logics. Algebra and Logic.

....of the form hW;R;R3 ; R ; UpW i. For an MIPC frame F = hW;R; R3 ; R ; P i, we denote by F its underlying full MIPC frame hW;R; R3 ; R ; UpW i. An MIPC logic L is D persistent if F j= L implies F j= L for any descriptive MIPC frame F. It is straightforward to prove that MIPC is D persistent (see [15]) We say an MIPC logic is Kripke complete if L is characterized by a class of full MIPC frames, and universal if there exists a universal first order sentence 8 with the predicates = R, R3 and R such that, for any full MIPC frame F, F validates L iff F satisfies 8. 3 Constructing a finite ....

....M OE is a finite MIPC model of L that refutes OE. 4 MIPC logics with the finite model property In this section, we give some corollaries of Theorem 3.17 in accessible forms and present some examples of MIPC logics with the finite model property. For this, we use the following result given in [15]. Proposition 4.1 If a 3 IM logic L is characterized by a class of full 3 frames which is closed under elementary equivalence (in the first order language with the predicates = R, R3 and R ) then L is D persistent. From this and Theorem 3.17, we obtain Corollary 4.2 Let L be an MIPC logic. ....

F. Wolter and M. Zakharyaschev. On the relation between intuitionistic and classical modal logics. Algebra and Logic, 36:73--92, 1997.

.... 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 [15] 19] 1] In Section 1 we define relational and algebraic semantics for intuitionistic modal logics and combine elements of duality theory for modal and intermediate logics ....

....omit almost all proofs. In Section 2 we show that every normal extension of IntK can be embedded by the (extended) Godel translation into bimodal logics forming an interval [ 0 L; oeL] and that the embedding reflects decidability, Kripke completeness, the finite model property and tabularity. In [21] we prove an unexpected analog of Blok Esakia s theorem (on an isomorphism between the lattices of intermediate logics and extensions of the Grzegorczyk system Grz; see [2] 6] establishing an isomorphism between the lattices of extensions of IntK and oeIntK) Then, in Section 3, using results ....

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F. Wolter and M. Zakharyaschev. On the relation between intuitionistic and classical modal logics. Manuscript, 1995.

....Moreover, it turns out that in general the relationship between the lattices NExtU and NExtK4 is similar to that between ExtInt and NExtS4 discussed in Section 1. To show this, we take advantage of the results on embeddings of intuitionistic modal logics into classical polymodal logics obtained in [27, 28]. Let ML 2 be the language with two necessity operators 2 I and 2 (and the implication ) and let T 00 be the translation from ML , into ML 2 prefixing 2 I to all subformulas and replacing , with . Given logics L 1 and L 2 in the unimodal languages ML 2 Gamma 2 and ML 2 Gamma 2 I , ....

....the smallest bimodal logic in ML 2 to contain L 1 [L 2 . By IntK we mean the minimal normal intuitionistic modal logic in the language ML , i.e. the smallest set of formulas containing Int, the modal axiom of K and closed under modus ponens, substitution and necessitation) As is shown in [27], i) the map aeM = f 2 ML , T 00 ( 2 Mg; is a lattice homomorphism from NExt(S4 Omega K) onto NExtIntK (preserving the finite model property and decidability) 17 (ii) each logic IntK Phi Gamma is embedded by T 00 into any logic M in the interval (S4 Omega K) Phi T 00 ....

F. Wolter and M. Zakharyaschev. On the relation between intuitionistic and classical modal logics. Algebra and Logic, 1996. To appear.

.... belongs to the former iff ST ( is a theorem of the latter. According to Simpson [1994] this result was proved by C. Stirling; see also Grefe [1997] Various extensions of FS were studied by Bull [1966] Ono [1977] Fischer Servi [1977, 1980, 1984] Amati and Pirri [1994] Ewald [1986] Wolter and Zakharyaschev [1996], Wolter [1997b] The best known one is probably the logic MIPC = FS Phi 2p p Phi 2p 22p Phi 3p 23p Phi p 3p Phi 33p 3p Phi 32p 2p introduced by Prior [1957] Bull [1966] noticed that the translation defined by (p i ) P i (x) fi ) fi , for fi 2 f; ....

....by that translation. In this paper we restrict attention only to the classes of normal intuitionistic modal logics introduced above. An interesting example of a non normal system was constructed by Wijesekera [1990] A general model theory for such logics is developed by Sotirov [1984] and Wolter and Zakharyaschev [1996]. 2 Now let us consider the algebraic and relational semantics for the logics introduced in the preceding section. All the semantical concepts to be defined below turn out to be natural combinations of the corresponding notions developed for classical modal and superintuitionistic logics. For ....

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F. Wolter and M. Zakharyaschev. On the relation between intuitionistic and classical modal logics. Algebra and Logic, 1996. To appear. 13

....3 R3 follow from (3) and (4) repectively 9 . With each ML frame G we associate the ML algebra G defined by G = hA; 2; 3; gi: We leave the straightforward proof that G is a ML algebra to the reader. One may also combine proofs of similar results from [29] and [42]. A filtered ML frame is a structure F = hG; F i such that G is a ML frame and F is a filter in G . A pointed ML frame is a structure hG; xi such that G is a ML frame and x 2 g. We identify hG; xi with the filtered ML frame hG; fa 2 G : x 2 agi and we identify a ML frame G with the ....

....and (4) will be quite useful. 4 KRIPKE SEMANTICS 9 fi[A] ffi(a) a 2 Ag; where fi(a) fX : a 2 Xg: We leave it to the reader to prove that A is a ML frame and that (A ) A. The proof is standard by using Lemma 2. The reader may also combine proofs of similar results from [19] 7] [42], 29] Clearly the converse, i.e. G ) G, only holds for a special class of frames: We call following [42] a ML frame G descriptive iff (8x; y) x Delta y , 8a 2 A) x 2 a ) y 2 a) 8x; y) xR2 y , 8a 2 A) x 2 2a ) y 2 a) 8x; y) xR3 y , 8a 2 A) y 2 a ) x 2 3a) and for ....

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F. Wolter & M. Zakharyaschev. On the relation between intuitionistic and classical modal logics, to appear in Algebra and Logic, 1996

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F. Wolter and M. Zakharyaschev. On the relation between intuitionistic and classical modal logics. Algebra and Logic, 36:121--155, 1997.

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F. Wolter and M. Zakharyaschev. `The relation between intuitionistic and classical modal logics'. Algebra and logic 36, 73--92 (1997). 13

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F. Wolter and M. Zakharyaschev, `The relation between intuitionistic and classical modal logics, Algebra and logic 36, 73-92, 1997. 20

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F. Wolter and M. Zakharyaschev. On the relation between intuitionistic and classical modal logics. Algebra and Logic, 1996. To appear.

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