David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Studia Logica 53 (3):397 - 416 (1994)
In this note, we will study four implicational logicsB, BI, BB and BBI. In , Martin and Meyer proved that a formula is provable inBB if and only if is provable inBBI and is not of the form of » . Though it gave a positive solution to theP - W problem, their method was semantical and not easy to grasp. We shall give a syntactical proof of the syntactical relation betweenBB andBBI logics. It also includes a syntactical proof of Powers and Dwyer's theorem that is proved semantically in . Moreover, we shall establish the same relation betweenB andBI logics asBB andBBI logics. This relation seems to say thatB logic is meaningful, and so we think thatB logic is the weakest among meaningful logics. Therefore, by Theorem 1.1, our Gentzentype system forBI logic may be regarded as the most basic among all meaningful logics. It should be mentioned here that the first syntactical proof ofP - W problem is given by Misao Nagayama .
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References found in this work BETA
Hiroakira Ono & Yuichi Komori (1985). Logics Without the Contraction Rule. Journal of Symbolic Logic 50 (1):169-201.
M. E. Szabo (1971). The Collected Papers of Gerhard Gentzen. Journal of Philosophy 68 (8):238-265.
E. P. Martin & R. K. Meyer (1982). Solution to the P-W Problem. Journal of Symbolic Logic 47 (4):869-887.
Yuichi Komori (1986). Predicate Logics Without the Structure Rules. Studia Logica 45 (4):393 - 404.
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