Hirokazu Nishimura

Descriptively complete process logic

SummaryThe main purpose of this paper is to contribute to the development of Pratts [12, 13] process logic by presenting a modest language for this logic in which some reasonably powerful class of connectives is definable. In particular, Parikhs [8] formidable language SOAPL is shown to be interpretable in our new language. Semantically, Pneulis [9, 10] temporal semantics is incorporated into that of process logic.

Sequential Method in Quantum Logic

Since Birkhoff and von Neumann [2] a new area of logical investigation has grown up under the name of quantum logic. However it seems to me that many authors have been inclined to discuss algebraic semantics as such (mainly lattices of a certain kind) almost directly without presenting any axiomatic system, far from developing any proof theory of quantum logic. See, e.g., Gunson [9], Jauch [10], Varadarajan [15], Zeirler [16], etc. In this sense many works presented under the name of quantum logic are algebraic in essence rather than genuinely logical, though it is absurd to doubt the close relationship between algebraic and logical study on quantum mechanics. The main purpose of this paper is to alter this situation by presenting an axiomatization of quantum logic as natural and as elegant as possible, which further proof-theoretical study is to be based on. It is true that several axiomatizations of quantum logic are present now. Several authors have investigated the so-called material implication α → β ( = ¬α∨(α ∧ β)) very closely with due regard to its importance. See, e.g., Finch [5], Piziak [11], etc. Indeed material implication plays a predominant role in any axiomatization of a logic in Hilbert-style. Clark [4] has presented an axiomatization of quantum logic with negation ¬ and material implication → as primitive connectives. In this paper we do not follow this approach. First of all, this approach is greatly complicated because orthomodular lattices are only locally distributive.

Sequential method in propositional dynamic logic

SummaryRecently prepositional modal logic of programs, called ‘prepositional dynamic logic’, has been developed by many authors, following the ideas of Fisher and Ladner [1] and Pratt [12]. The main purpose of this paper is to present a Gentzen-type sequential formulation of this logic and to establish its semantical completeness with due regard to sequential formulation as such. In a sense our sequential formulation might be regarded as a powerful tool to establish the completeness theorem of already familiar axiomatizations of prepositional dynamic logic such as seen in Harel [4], Parikh [11] or Segerberg [15]. Indeed our method is powerful enough in completeness proof to yield a desired structure directly without making a detour through such intermediate constructs as a ‘pseudomodel’ or a ‘nonstandard structure’, which can be seen in Parikh [11]. We also show that our sequential system of prepositional dynamic logic does not enjoy the so-called cut-elimination theorem.

Model theory for tense logic: Saturated and special models with applications to the tense hierarchy

The aims of this paper are: (1) to present tense-logical versions of such classical notions as saturated and special models; (2) to establish several fundamental existence theorems about these notions; (3) to apply these powerful techniques to tense complexity.In this paper we are concerned exclusively with quantifiedK1 (for linear time) with constant domain. Our present research owes much to Bowen [2], Fine [5] and Gabbay [6].