4-valued Reasoning with Stratified Bilattices

4-VALUED REASONING WITH STRATIFIED BILATTICES Hisashi Komatsu School of Information Sciences, Hiroshima City University, 3-4-1 Ozuka-higashi, Asa-minami-ku, Hiroshima 731-3194, Japan e-mail: komatsu©cs.hiroshima-cu.ac.jp ABSTRACT Since [5], 4-valued logic is known to be a useful tool to capture the human reasoning: it is paraconsistent, can treat incompleteness and inconsistency of information etc. In this paper, I propose a 4-valued reasoning system with stratified bilattices of [12]. It inherits desirable formal properties of 4-valued logic, and further realizes a certain kind of default reasoning and truth maintenance system with a simple, lucid LK-style calculus without esoteric, exotic 4-valued operations in [6, 7, 2, 3] etc. 1 Preliminaries The logical method provides a basic tool to capture the human knowledge and human reasoning. But the traditional 2-valued logic has certain defects in formulating them. I.e., it cannot treat the incomplete or contradictory information which often appears in the human epistemic state. Furthermore, the 2-valued logic is not paraconsistent, i.e., from the contradiction, all statements are derived. In order to avoid such faults, [5, 1] propose the 4-valued logic. It is based on a very simple idea that the traditional two truth values 'true' and 'false' are treated independently so that the following four truth values are assumed in the 4-valued logic': (1) i) T (true): A is true. ii) F (false): A is false. iii) B (both): A is both true and false. iv) N (none): A is neither true nor false. The new truth values B and N represent the contradictory and the incomplete information respectively, i.e. the knowledge states that the traditional 2-valued logic cannot treat because it assumes the following relations between 'true' and 'false' (2) A is true .#> A is not false, A is false 4 A is not true so that B and N are impossible at all. Although the introduction of B and N enables the 4-valued logic to treat the contradictory and the incom- plete knowledge state, it is also true that such states tend to change to the contradiciton- and incompleteness- free states, i.e. to T or F. In this sense, T and F in the traditional 2-valued logic are the final ideal states which the human knowledge state with contradiction and incompleteness aims at. In order to treat these tendencies, several logical systems have been developed which are classified to the following two types: (3) i) N� ii) B�T or F: default logic, circumscription, negation as failure etc. T or F: truth maintenance system etc. In this paper, I present a 4-valued logical system called the situational 4-valued logic S4Val. S4Val can describe the knowledge state and the reasoning with contradictory and incomplete information. Further it can formulate — not all but — some interesting types of the default logic and the truth maintenance system. In the following, I present the language, the semantics, and the proof theory of S4Val, and then the above-mentioned non-monotonic reasoning. 'The idea stems from [11]. 2 Language of S4Val The language of S4Val is the same as the …

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