Nonclassical Logics

Fuzzy TOPSIS Pub Date : 1900-01-01 DOI:10.1201/9781003168416-2-2

M. El Alaoui

引用次数: 6

Abstract

Solution. A5 = ♦A ⊃ ♦A = ¬ ¬A ⊃ ¬ ¬A. We show that F 6|= ¬Ki¬F ⊃ Ki¬Ki¬F for some formula F if and only if F 6|= ¬KiG ⊃ Ki¬KiG for some formula G. Let F 6|= ¬Ki¬F ⊃ Ki¬Ki¬F for some formula F . That is, (M, w) 6|= ¬Ki¬F ⊃ Ki¬Ki¬F for some world w in some interpretationM based on F . Choosing G = ¬F we obtain (M, w) 6|= ¬KiG ⊃ Ki¬KiG and therefore also F 6|= ¬KiG ⊃ Ki¬KiG. Conversely, let F 6|= ¬KiG ⊃ Ki¬KiG for some formula G. That is, (M, w) 6|= ¬KiG ⊃ Ki¬KiG for some world w in some interpretation M based on F . Furthermore we have (M, w) 6|= ¬Ki¬¬G ⊃ Ki¬Ki¬¬G. Choosing F = ¬G we obtain (M, w) 6|= ¬Ki¬F ⊃ Ki¬Ki¬F and therefore also F 6|= ¬Ki¬F ⊃ Ki¬Ki¬F .

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模逻辑

解决方案。A5 =♦a、♦a =♦a、我们证明了对于某个公式F，当且仅当对于某个公式g, F 6 bbb = ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω。即(M, w) 6|= _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _选择G = F，我们得到(M, w) 6|=¬KiG、kikig，因此也得到F 6|=¬KiG、kikig。反过来，对于某个公式g，设F 6 bbb =¬KiG、Ki¬KiG，即(M, w) 6 bbb =¬KiG、Ki¬KiG，对于基于F的某个解释M中的某个世界w。更进一步，我们有(M, w) 6|= ω ω ω， ω ω ω。选择F = G，我们得到(M, w) 6 bbb =¬¬¬F、¬¬¬F，因此也是F 6 bbb =¬¬¬F、¬¬¬F。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Fuzzy TOPSIS

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