Substructural Logics Substructural Logics p.1/46
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1 Substructural Logics Substructural Logics p.1/46
2 (ordered algebraic structures, universal algebra, algebraic logic ) provability deducibility cut Dedekind-MacNeille Substructural Logics p.2/46
3 (abstract algebraic logic. Substructural Logics p.3/46
4 N. Galatos, P. Jipsen, T. Kowalski, HO, Residuated Lattices: an algebraic glimpse at substructural logics, Studies in Logic and the Foundations of Mathematics, vol.151, Elsevier, April, 2007 Ordered Structures in Many-valued Logic: Sorrento (2006) tutorial. C. Holland, D. Mundici, C. Tsinakis, HO Order, Algebra, and Logics: Nashville (2007) tutorial : (2007) Substructural Logics p.4/46
5 Gentzen sequent LJ LJ sequent. m 0. α 1,...,α m β "β follows from assumptions α 1,...,α m ". sequent sequent LJ provably equivalent. (α 1... α m ) β Substructural Logics p.5/46
6 LJ α α sequent Cut (structural rules) Substructural Logics p.6/46
7 Cut rules for implication Cut Γ α Σ,α,Ξ ϕ Σ, Γ, Ξ ϕ (cut) Rules for implication Γ α β, ϕ α β,γ, ϕ ( ) α, Γ β Γ α β ( ) Substructural Logics p.7/46
8 Rules for lattice operations Γ,α, ϕ Γ,β, ϕ Γ,α β, ϕ Γ α Γ α β ( 1) ( ) Γ β Γ α β ( 2) Γ,α, ϕ Γ,α β, ϕ ( 1 ) Γ,β, Γ α Γ β Γ α β ϕ Γ,α β, ϕ ( ) ( 2 ) Substructural Logics p.8/46
9 Structural rules Structural rules sequent. ((i) (o) (w) (weakening rules).) (e) exchange rule (commutativity): (c) contraction rule (square-increasing): (i) left weakening rule (integrality): (o) right weakening rule (minimality of 0): Γ,α,β, ϕ Γ,β,α, ϕ Γ,α,α, ϕ Γ,α, ϕ Γ, ϕ Γ,α, ϕ Γ Γ α Substructural Logics p.9/46
10 Sequent FL sequent FL (Full Lambek Calculus) LJ structural rules. 0,1. Initial sequents: 1- and 0-weakening: 1 0 Γ, ϕ Γ, 1, ϕ (1w) Γ Γ 0 (0w) Substructural Logics p.10/46
11 FL Cut rule Rules for lattice operations LJ conjunction. contraction weakening. conjunction. Substructural Logics p.11/46
12 fusion fusion. fusion. Γ α β Γ, α β ( ) Σ,α,β,Γ ϕ Σ,α β,γ ϕ ( ) FL. α β ϕ is provable iff α, β ϕ is provable. Substructural Logics p.12/46
13 (residuals) exchange rule implication ( ). α\β β/α α β. Γ α Ξ,β, ϕ Ξ, Γ,α\β, ϕ (\ ) α, Γ β Γ α\β ( \) Γ α Ξ,β, ϕ Ξ,β/α,Γ, ϕ (/ ) Γ,α β Γ β/α FL. α β ϕ is provable iff β α\ϕ is provable iff α ϕ/β is provable ( /) Substructural Logics p.13/46
14 0. α = α\0 α =0/α. FL β β γ γ α α β/γ,γ β α, α\(β/γ),γ β α\(β/γ),γ α\β α\(β/γ) (α\β)/γ Substructural Logics p.14/46
15 Sequent FL.. FL : LJ FL e : FL+ exchange FL ew : FL+ exchange + weakening Substructural Logics p.15/46
16 Lambek categorial grammer (J. Lambek, 1958) Relevant logics : weakening rules contraction rule Łukasiewicz, Linear logic : weakening contraction Johansson minimal logic : weakening Substructural Logics p.16/46
17 1 Logics without contraction rules (with Komori), 1985: Kripke Heyting 88 conference, 1988: FL-series Tübingen conference, 1990: MacNeille completions Substructural Logics p.17/46
18 Cut elimination : FL c : weakening contraction interpolation property: L Craig interpolation property (CIP) φ, ψ φ\ψ L δ φ\δ δ\ψ L, Var(δ) Var(φ) Var(ψ). Var(γ) γ. Substructural Logics p.18/46
19 disjunction property: contraction variable sharing property: weakening Maksimova s variable separation property positive fragments Substructural Logics p.19/46
20 Non-classical logics non-classical logics natural deduction Hilbert Fusion Girard resource-sensitive? resource Substructural Logics p.20/46
21 (provability) consequence relation (deducibility). (. Σ α α Σ FL (Σ FL α) FL sequents γ ( γ Σ) sequent α. Substructural Logics p.21/46
22 L Σ L FL α Σ L α consequence relation L. provability deducibility Substructural Logics p.22/46
23 (deduction theorem). ( : union) Σ,α β Σ α β. α α 2 FL provable α FL α 2. α α α α ( ) Substructural Logics p.23/46
24 (PLDT) FL (GO 2006). Σ,α FL β iff there exist iterated conjugates δ i of α (i m for some m) such that Σ FL ( δ i )\β. α (iterated conjugate) α λ ϕ (x) =(ϕ\xϕ) 1 ρ ψ (x) =(ψx/ψ) 1.. PLDT Czelakowski-Dziobiak. Substructural Logics p.24/46
25 LJ. FL weakening exchange. if Γ, θ is provable then Γ,ψ 1, θ is provable, if Γ,α,β, θ is provable then both Γ,β,λ β (α), θ and Γ,ρ α (β),α, θ are provable. Substructural Logics p.25/46
26 exchange. λ ϕ (x) =λ ϕ (x) =x 1 PLDT (LDT). Σ,α FLe β iff Σ FLe (α 1) m β for some m. m (local). Substructural Logics p.26/46
27 cut. (cf. FL ec. FL c. ) FL e. ( Lincoln, Mitchell, Scedrov & Shankar (1992). cf. HO 1997, Buszkowski 2007) Substructural Logics p.27/46
28 Hilbert deduciblity. PLDT. L. FL L, ϕ ϕ\ψ L ψ L, ϕ L ϕ 1 L, ϕ L γ γ\ϕγ γϕ/γ L, L. Substructural Logics p.28/46
29 .. Substructural Logics p.29/46
30 2 Residuated lattices Hájek HO, "Logics without contraction rule and residuated lattices I" AsubL workshop universal algebra, ordered algebraic structures, abstract algebraic logic BJO algebraic cut elimination GO 2006a in Studia Logica, 2006b in JSL Kihara-O algebraic characterization of logical properties Substructural Logics p.30/46
31 Residuated lattices A = A,,,, \,/,1 residuated lattice (RL) A. A,,, A,, 1, for all x, y, z A, x y z iff y x\z iff x z/y. A = A,,,, \,/,1, 0 FL- A,,,, \,/,1 RL x = x\0 x =0/x. Substructural Logics p.31/46
32 RL FL-algebra Dilworth, Ward (1930 ). lattice ordered group RL. a\b = a 1 b, b/a = ba 1. Mulvey, Rosenthal quantale RL. Łukasiewicz MV FL-. Substructural Logics p.32/46
33 FL- FL- A α 1,α 2,...,α m β valid A α 1 α 2... α m β. exchange x y y x (commutativity), contraction x x 2, left weakening x y x and x y y (integrality), right weakening 0 x. Substructural Logics p.33/46
34 α 1,α 2,...,α m β FL- A {α 1,α 2,...,α m } (deductive) filter β. deductive filter. ( lattice ordered group G H a H c G cac 1 H H. Substructural Logics p.34/46
35 Varieties equational classes K variety H, S, P. K variety HSP (Tarski). Σ Mod(Σ) Σ s t A = s t. K equational class K =Mod(Σ) Σ. varieties = equational classes (Birkhoff) RL RL FL- FL variety. Substructural Logics p.35/46
36 terms: s,t,u,... formulas: α,β,... s = t s t, i.e. (s\t) (t\s) 1 α, i.e. α 1=1 α the variety of Boolean algebras classical logic the variety of Heyting algebras intuitionistic logic subvarieties of RL (FL)? Substructural Logics p.36/46
37 Algebraization a la Lindenbaum 1. FL subvariety V, L(V) ={α; V =1 α}. 2. L, {s t;(s t) L} FL subvariety V(L) 3. L, V.. substructural logics are logics of residuated lattices. RL sequent (1974) Substructural Logics p.37/46
38 implication. monoid explicit. residual implication.. HO, Trends in Logic: 50 Years of Studia Logica, 2003 Substructural Logics p.38/46
39 Equational consequence FL subvariety V equational consequence = V. {u i v i ; i I} {s t} {u i v i ; i I} = V s t the first-order formula " {u i v i ; i I} implies s t" holds always in every algebra in V, or equivalently, s t follows from equations {ui v i ; i I} with "axioms" of V in equational calculus. Substructural Logics p.39/46
40 Algebraization a la Blok-Pigozzi algebraization theorem (GO 2006) 1. FL subvariety V, {u i v i ; i I} = V s t iff {u i v i ; i I} L(V ) s t, 2. L, {β j ; j J} L α iff {1 β j ; j J} = V (L) 1 α, 3. L, V. Substructural Logics p.40/46
41 deducibility variety equational consequence. Algebraization theorem Commutative RLs quasi-equational theory Glivenko (GO 2006) interpolation amalgamation ( O) Substructural Logics p.41/46
42 Cut elimination inductive argument. inductive.. interpolation theorem. cut elimination. (1991), (1996), (1999), Jipsen-Tsinakis (2002), BJO (2004) Substructural Logics p.42/46
43 F. Belardinelli, P. Jipsen & HO, Algebraic aspects of cut elimination, Studia Logica, (GO, Galatos-Jipsen, Buszkowski) Substructural Logics p.43/46
44 L sequent S L Gentzen matrix partial structures. (cf. Font, Jansana, Pigozzi) S L Gentzen matrix Q V(L) B (quasi-embeddable). B Q (quasi completion). Substructural Logics p.44/46
45 V(L) A S L Gentzen matrix. A MacNeille. L sequent S L variety V(L) MacNeille. variety MacNeille. (Bezhanishvili-Harding) Substructural Logics p.45/46
46 ( ). ( Ciabattoni Galatos ). ). Topological and Algebraic Methods in Logics (TAL), 2009 at Amsterdam Substructural Logics p.46/46
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Title 最適年金の理論 Author(s) 藤井, 隆雄 ; 林, 史明 ; 入谷, 純 ; 小黒, 一正 Citation Issue Date Type Technical Report Text Version publisher URL
Title 最適年金の理論 Author(s) 藤井, 隆雄 ; 林, 史明 ; 入谷, 純 ; 小黒, 一正 Citation Issue 2012-06 Date Type Technical Report Text Version publisher URL http://hdl.handle.net/10086/23085 Right Hitotsubashi University Repository
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Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara 80 1963 Sp(2, R) p L holomorphic discrete series Eichler Brandt Eichler
1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe
![1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe](/thumbs/86/93804893.jpg "1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe")
3 del Pezzo (Hirokazu Nasu) 1 [10]. 3 V C C, V Hilbert scheme Hilb V [C]. C V C S V S. C S S V, C V. Hilbert schemes Hilb V Hilb S [S] [C] ( χ(s, N S/V ) χ(c, N C/S )), Hilb V [C] (generically non-reduced)
; Modus Ponens 1
1 9 00 ; 1402 5 16 20 1502 1 11 1 1 2 1 2 Modus Ponens 1 1 2 Modus Ponens 2 1 12 Gentzen 13 Modus ponens 1 100 2 100 100 P Q P Q Q 2 P Q P Q Modus ponens Y Y 1 Modus Ponens P 2 Y Y Y Y Y Y C D Y F (C D)
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
218.8.21 1 2, (Theorem 1) (Theorem 2) 1 2, (Theorem 1) (Theorem 2) . (c.f. De Donno and Praelli, 24) (Ω, F, (F ) [,T ], P):, usual condiions, T < P : predicable σ-field on Ω [, T ] H 2 = { } X : (index
2 A id A : A A A A id A def = {(a, a) A A a A} 1 { } 1 1 id 1 = α: A B β : B C α β αβ : A C αβ def = {(a, c) A C b B.((a, b) α (b, c) β)} 2.3 α
20 6 18 1 2 2.1 A B α A B α: A B A B Rel(A, B) A B (A B) A B 0 AB A B AB α, β : A B α β α β def (a, b) A B.((a, b) α (a, b) β) 0 AB AB Rel(A, B) 1 2 A id A : A A A A id A def = {(a, a) A A a A} 1 { } 1
Microsoft PowerPoint - logic ppt [互換モード]
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寄せられた質問 : 演習問題について この講義の範囲に含まれる適切な演習問題が載っている参考書がありますか? できれば解答や解説が付いているものがあると良いのですが 第 回の授業の中で 演習問題に取り組む方法を説明しますこの授業は 回だけ行うもので 書籍の1 冊分に比べると少ない分量しかカバーしていません 回の講義の概観 : 1 完全性と不完全性 命題論理 命題論理 ( 真理値 ) ( 公理と推論規則
