W.B.Ewald, Intuitionistic Tense and Modal Logic, Journal of Symbolic Logic, 51 (1986), pp.166-179.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are derived for two distinct classes of model structures of QLL. 1 Introduction The logic under consideration in this paper is an intuitionistic logic which possesses a single modality. Other intuitionistic modal logics exist (see for example [BdP96] Fit48] PS86] Sim94] FS80] [Ewa86]) though it is not always clear what their semantics ought to be, or what they are useful for. Lax Logic is novel in that the modal operator has a flavour of both necessity and possibility. In fact, the general form of this logic has been independently studied by looking at an extension of Moggi s ....

W. B. Ewald. Intuitionistic tense and modal logic. Journal of Symbolic Logic, 51, 1986.

....of a set of worlds W and two accessibility relations, one intuitionistic and the other modal R, it is not so clear how these relations should interact (frame conditions) and just how they should be used to interpret specifically the # modality. The mainstream approach as exemplified by Ewald [Ewa86], Fischer Servi [FS80] Plotkin and Stirling [PS86] Simpson [Sim94] is based on the analogy of # with and of # with # quantification over the modal accessibility R. Reading these quantifiers intuitionistically, relative to one arrives at the semantic interpretation w #v. ##u. v R ....

....#B)is automatically validated. It is not only the fallible worlds but also the extension by sets #, capturing hereditary refutation information, that distinguishes the representation of constructive modal logic, such as CS4, from that for standard intuitionistic modal logics, such as those of [PS86,FS80,Ewa86]. Indeed, if the axioms and #(# ## are adopted the sets # and fallible worlds become redundant. Without these axioms, however, we also need the negative information in # to characterise truth at a world fully. It is also worthwhile to note that the model representation of Thm. 3 for ....

W. B. Ewald. Intuitionistic tense and modal logic. Journal of Symbolic Logic, 51, 1986.

....of a set of worlds W and two accessibility relations, one intuitionistic and the other modal R, it is not so clear how these relations should interact (frame conditions) and just how they should be used to interpret specifically the # modality. The mainstream approach as exemplified by Ewald [Ewa86], Fischer Servi [FS80] Plotkin and Stirling [PS86] Simpson [Sim94] is based on the analogy of # with 8 and of # with 9 quantification over the modal accessibility R. Reading these quantifiers intuitionistically, relative to , one arrives at the semantic interpretation w j= #A iff 8v: w v ) 8u: v ....

.... #B) is automatically validated. It is not only the fallible worlds but also the extension by sets , capturing hereditary refutation information, that distinguishes the representation of constructive modal logic, such as CS4, from that for standard intuitionistic modal logics, such as those of [PS86,FS80,Ewa86]. Indeed, if the axioms :# and #( # # are adopted the sets and fallible worlds become redundant. Without these axioms, however, we also need the negative information in to characterise truth at a world fully. It is also worthwhile to note that the model representation of Thm. 3 for CS4 ....

W. B. Ewald. Intuitionistic tense and modal logic. Journal of Symbolic Logic, 51, 1986.

....of a set of worlds W and two accessibility relations, one intuitionistic and the other modal R, it is not so clear how these relations should interact (frame conditions) and just how they should be used to interpret specifically the 3 modality. The main stream approach as exemplified by Ewald [Ewa86], Fischer Servi [FS80] Plotkin and Stirling [PS86] Simpson [Sim94] is based on the analogy of 2 with 8 and of 3 with 9 quantification over the modal accessibility R. Reading these quantifiers intuitionistically, relative to , one arrives at the semantic interpretation w j= 2A iff 8v: w v ) ....

....metatheory. It is not only the possibility of fallible worlds but also the extension by sets , capturing hereditary refutation information, that distinguishes the representation of constructive modal logic, such as CS4, from that for standard intuitionistic modal logics, such as those of [PS86,FS80,Ewa86]. Indeed, if the axioms :3 and 3( 3 3 are adopted the sets and fallible worlds become redundant. Without these axioms, however, we also need this negative information to characterise truth at a world fully. It is also worthwhile to note that the model representation of Thm. 3 for ....

W. B. Ewald. Intuitionistic tense and modal logic. Journal of Symbolic Logic, 51, 1986.

.... 9 3 Constraint Models for PLL Kripke style analyzes have been given for other intuitionistic modal logics, for instance by Simpson [Simpson, 1994] and Plotkin and Stirling [Plotkin and Stirling, 1986] for system IK, by Fischer Servi [Fischer Servi, 1980] for the class of ( IC systems, and by Ewald [Ewald, 1986] for an intuitionistic tense logic. The approach taken here most closely follows [Plotkin and Stirling, 1986] in using one set of worlds but two separate frame relations to interpret fl and oe. This satisfies our requirement that fl be given a Kripke style interpretation. As a result of our ....

Ewald, W. B. (1986). Intuitionistic tense and modal logic. Journal of Symbolic Logic, 51.

....MIPC (see [2, 3, 21] The forcing relations for both modal operators are analogous to those of the intuitionistic quantifiers on Kripke 2 frames ( 5] see Definition 3. 6) Two connecting properties for the Kripke type frames are given to characterize the intuitionistic modal logics (see also [7]) We show that IK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IK45 and IS5 have the Disjunction Property. A similar path in tense logic has been followed by Ewald in [7] In his paper, besides a set of states of knowledge (worlds) and the intuitionistic partial order relation, a set of ....

....Two connecting properties for the Kripke type frames are given to characterize the intuitionistic modal logics (see also [7] We show that IK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IK45 and IS5 have the Disjunction Property. A similar path in tense logic has been followed by Ewald in [7]. In his paper, besides a set of states of knowledge (worlds) and the intuitionistic partial order relation, a set of times and a temporal ordering within each world are introduced. Two dual derived relations on the set of worlds can be defined: one of them turns out to satisfy both the connecting ....

W.B.Ewald, Intuitionistic Tense and Modal Logic, Journal of Symbolic Logic, 51 (1986), pp.166-179.

.... Phi 3p p (respectively, IntK23 Phi 2p p) A well motivated 23 IM logic containing IntK23 was introduced by Fischer Servi [14] 15] FS = IntK23 Phi 3(p q) 2p 3q) Phi (3p 2q) 2(p q) Extensions of FS will be called FS logics. Some of them were studied in [15] 1] and [10]. By adding to a consistent IM logic in the language LMn the Law of the Excluded Middle p:p, we obtain a classical logic with n modal operators. We denote by C, R and K the monomodal logics IntC Phi p:p, IntR Phi p:p and IntK2 Phi p :p, respectively. 2 Semantics and Duality The logics ....

W.B. Ewald. Intuitionistic tense and modal logic. The Journal of Symbolic Logic, 51:166--179, 1986.

....are derived for two distinct classes of model structures of QLL. 1 Introduction The logic under consideration in this paper is an intuitionistic logic which possesses a single modality. Other intuitionistic modal logics exist (see for example [BdP96] Fit48] PS86] Sim94] FS80] [Ewa86]) though it is not always clear what their semantics ought to be, or what they are useful for. Lax Logic is novel in that the modal operator has a flavour of both necessity and possibility. In fact, the general form of this logic has been independently studied by looking at an extension of Moggi s ....

W. B. Ewald. Intuitionistic tense and modal logic. Journal of Symbolic Logic, 51, 1986.

....models. 4 Kripke Models for PLL Kripke style analyzes have been given for other intuitionistic modal logics, for instance by Simpson [Simpson, 1994] and Plotkin and Stirling [Plotkin and Stirling, 1986] for system IK, by Fischer Servi [Fischer Servi, 1980] for the class of ( IC systems, and by Ewald [Ewald, 1986] for an intuitionistic tense logic. The approach taken here most closely follows [Plotkin and Stirling, 1986] in using one set of worlds but two separate frame relations to interpret fl and oe. Here we identify a natural class of Kripke models for which PLL is sound and complete. The models ....

Ewald, W. B. (1986). Intuitionistic tense and modal logic. Journal of Symbolic Logic, 51.

....Zakharyaschev School of Information Science, JAIST, Tatsunokuchi, Ishikawa 923 12, Japan (e mails: wolter jaist.ac.jp and mikhail jaist.ac.jp) 0 Intuitionistic modal logics originate from different sources and have different areas of application. They include philosophy (see e.g. Prior [1957] Ewald [1986], Williamson [1992] the foundations of mathematics (Kuznetsov [1985] Kuznetsov and Muravitskij [1986] and computer science (Plotkin and Stirling [1986] Stirling [1987] Wijesekera [1990] Modalities are added to intuitionistic logic in the framework of studying new intuitionistic ....

....first order logic: 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) ....

W.B. Ewald. Intuitionistic tense and modal logic. Journal of Symbolic Logic, 51:166--179, 1986.

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W.B.Ewald, Intuitionistic Tense and Modal Logic, Journal of Symbolic Logic, 51 (1986), pp.166-179.

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