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Łukasiewicz–Moisil algebras (LM''n'' algebras) were introduced in the 1940s by Grigore Moisil (initially under the name of Łukasiewicz algebras〔Andrei Popescu, ''(Łukasiewicz-Moisil Relation Algebras )'', Studia Logica, Vol. 81, No. 2 (Nov., 2005), pp. 167-189〕) in the hope of giving algebraic semantics for the ''n''-valued Łukasiewicz logic. However, in 1956 Alan Rose discovered that for ''n'' ≥ 5, the Łukasiewicz–Moisil algebra does not model the Łukasiewicz logic. A faithful model for the ℵ0-valued (infinitely-many-valued) Łukasiewicz–Tarski logic was provided by C. C. Chang's MV-algebra, introduced in 1958. For the axiomatically more complicated (finite) ''n''-valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MV''n''-algebras. MV''n''-algebras are a subclass of LM''n''-algebras, and the inclusion is strict for ''n'' ≥ 5.〔Iorgulescu, A.: Connections between MV''n''-algebras and ''n''-valued Łukasiewicz-Moisil algebras—I. Discrete Math. 181, 155–177 (1998) 〕 In 1982 Roberto Cignoli published some additional constraints that added to LM''n''-algebras produce proper models for ''n''-valued Łukasiewicz logic; Cignoli called his discovery proper Łukasiewicz algebras.〔R. Cignoli, Proper n-Valued Łukasiewicz Algebras as S-Algebras of Łukasiewicz ''n''-Valued Propositional Calculi, Studia Logica, 41, 1982, 3–16, 〕 Moisil however published in 1964 a logic to match his algebra (in the general ''n'' ≥ 5 case), now called Moisil logic.〔 After coming in contact with Zadeh's fuzzy logic, in 1968 Moisil also introduced an infinitely-many-valued logic variant and its corresponding LM''θ'' algebras.〔Georgescu, G., Iourgulescu, A., Rudeanu, S.: "(Grigore C. Moisil (1906–1973) and his School in Algebraic Logic )." International Journal of Computers, Communications & Control 1, 81–99 (2006)〕 Although the Łukasiewicz implication cannot be defined in a LM''n'' algebra for ''n'' ≥ 5, the Heyting implication can be, i.e. LM''n'' algebras are Heyting algebras; as a result, Moisil logics can also be developed (from a purely logical standpoint) in the framework of Brower’s intuitionistic logic.〔, Theorem 3.6〕 == Definition == A LM''n'' algebra is a De Morgan algebra (a notion also introduced by Moisil) with ''n''-1 additional unary, "modal" operations: , i.e. an algebra of signature where ''J'' = . (Some sources denote the additional operators as to emphasize that they depend on the order ''n'' of the algebra.〔Cignoli, R., “The algebras of Lukasiewicz many-valued logic - A historical overview,” in S. Aguzzoli et al.(Eds.), Algebraic and Proof-theoretic Aspects of Non-classical Logics, LNAI 4460, Springer, 2007, 69-83. 〕) The additional unary operators ∇''j'' must satisfy the following axioms for all ''x'', ''y'' ∈ ''A'' and ''j'', ''k'' ∈ ''J'':〔 # # # # # # if for all ''j'' ∈ ''J'', then ''x'' = ''y''. (The adjective "modal" is related to the (failed ) program of Tarksi and Łukasiewicz to axiomatize modal logic using many-valued logic.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Łukasiewicz–Moisil algebra」の詳細全文を読む スポンサード リンク
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