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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Branching quantifier ： ウィキペディア英語版 Branching quantifier In logic a branching quantifier, also called a Henkin quantifier, finite partially ordered quantifier or even nonlinear quantifier, is a partial ordering :$\backslash langle\; Qx\_1\backslash dots\; Qx\_n\backslash rangle$ of quantifiers for Q∈. It is a special case of generalized quantifier. In classical logic, quantifier prefixes are linearly ordered such that the value of a variable ''ym'' bound by a quantifier ''Qm'' depends on the value of the variables :y1,...,ym-1 bound by quantifiers :Qy1,...,Qym-1 preceding ''Qm''. In a logic with (finite) partially ordered quantification this is not in general the case. Branching quantification first appeared in a 1959 conference paper of Leon Henkin.〔Henkin, L. "Some Remarks on Infinitely Long Formulas". ''Infinitistic Methods: Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 2–9 September 1959'', Panstwowe Wydawnictwo Naukowe and Pergamon Press, Warsaw, 1961, pp. 167-183. 〕 Systems of partially ordered quantification are intermediate in strength between first-order logic and second-order logic. They are being used as a basis for Hintikka's and Gabriel Sandu's independence-friendly logic. ==Definition and properties== The simplest Henkin quantifier $Q\_H$ is :$(Q\_Hx\_1,x\_2,y\_1,y\_2)\backslash phi(x\_1,x\_2,y\_1,y\_2)\backslash equiv\backslash begin\backslash forall\; x\_1\; \backslash exists\; y\_1\backslash \backslash \; \backslash forall\; x\_2\; \backslash exists\; y\_2\backslash end\backslash phi(x\_1,x\_2,y\_1,y\_2)$. It (in fact every formula with a Henkin prefix, not just the simplest one) is equivalent to its second-order Skolemization, i.e. :$\backslash exists\; f\; \backslash exists\; g\; \backslash forall\; x\_1\; \backslash forall\; x\_2\backslash phi\; (x\_1,x\_2,f(x\_1),g(x\_2))$. It is also powerful enough to define the quantifier $Q\_\}x)\backslash phi\; (x)\backslash equiv\backslash exists\; a(Q\_Hx\_1,x\_2,y\_1,y\_2)(a\backslash land\; (x\_1=x\_2\; \backslash leftrightarrow\; y\_1=y\_2)\; \backslash land\; (\backslash phi\; (x\_1)\backslash rightarrow\; (\backslash phi\; (y\_1)\backslash land\; y\_1\backslash neq\; a)))$. Several things follow from this, including the nonaxiomatizability of first-order logic with $Q\_H$ (first observed by Ehrenfeucht), and its equivalence to the $\backslash Sigma\_1^1$-fragment of second-order logic (existential second-order logic)—the latter result published independently in 1970 by Herbert Enderton〔Jaakko Hintikka and Gabriel Sandu, "Game-theoretical semantics", in ''Handbook of logic and language'', ed. J. van Benthem and A. ter Meulen, Elsevier 2011 (2nd ed.) citing Enderton, H.B., 1970. Finite ordered quantifiers. Z. Math. Logik Grundlag. Math. 16, 393–397 .〕 and W. Walkoe.〔 citing W. Walkoe, Finite ordered quantification, J. Symbolic Logic 35 (1970) 535-555. 〕 The following quantifiers are also definable by $Q\_H$.〔 * Rescher: "The number of φs is less than or equal to the number of ψs" :$(Q\_Lx)(\backslash phi\; x,\backslash psi\; x)\backslash equiv\; Card(\backslash \; )\backslash leq\; Card(\backslash \; )\; \backslash equiv\; (Q\_Hx\_1x\_2y\_1y\_2)(\backslash leftrightarrow\; y\_1=y\_2)\; \backslash land\; (\backslash phi\; x\_1\; \backslash rightarrow\; \backslash psi\; y\_1))$ * Härtig: "The φs are equinumerous with the ψs" :$(Q\_Ix)(\backslash phi\; x,\backslash psi\; x)\backslash equiv\; (Q\_Lx)(\backslash phi\; x,\backslash psi\; x)\; \backslash land\; (Q\_Lx)(\backslash psi\; x,\backslash phi\; x)$ * Chang: "The number of φs is equinumerous with the domain of the model" :$(Q\_Cx)(\backslash phi\; x)\backslash equiv\; (Q\_Lx)(x=x,\backslash phi\; x)$ The Henkin quantifier $Q\_H$ can itself be expressed as a type (4) Lindström quantifier.〔 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「Branching quantifier」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.