CVS Symposium, October 2005 p.1/22
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1 CVS Symposium, October 2005 p.1/22
2 Characteristica universalis CVS Symposium, October 2005 p.2/22
3 G. Boole, The Mathematical Analysis of Logic being an essay towards a calculus of deductive reasoning, 1847 CVS Symposium, October 2005 p.3/22
4 Polish school Łukasiewicz, Tarski, Lindenbaum Mostowski, Sikorski, Rasiowa H. Rasiowa-R. Sikorski, The Mathematics of Metamathematics, 1963 A. and S. Feferman, Alfred Tarski, life and logic, 2004 Universal algebra Tarski, Jónsson, McKenzie, Blok, Pigozzi A.I. Mal cev CVS Symposium, October 2005 p.4/22
5 Rasiowa, Sikorski 60 Kripke. Kripke frame = Stone, Jónsson-Tarski dual space Fine, Thomason, HO, Shehtman CVS Symposium, October 2005 p.5/22
6 Fine canonical elementarily determined Goldblatt, Hodkinson, Venema Erdös graph Residuated lattice universal algebra algebraic logic Algebraic and topological methods in non-classical logics II, Barcelona, 2005 P. Jipsen, T. Kowalski, HO, Residuated Lattices: an algebraic glimpse at substructural logics CVS Symposium, October 2005 p.6/22
7 CVS Symposium, October 2005 p.7/22
8 .. CVS Symposium, October 2005 p.8/22
9 LK LJ FL LJ FL e FL+ exchange FL ew FL+ exchange + weakening Logics without contraction rules (with Komori), 1985 Kripke Heyting 88 conference, 1988 van Benthem, Girard FL-series Tübingen conference, 1990 MacNeille completions CVS Symposium, October 2005 p.9/22
10 Weakening rule (left): Weakening rule (right): Contraction rule: Exchange rule: Γ ϕ α, Γ ϕ Γ Γ α α, α, ϕ α, ϕ Γ,α,β, ϕ Γ,β,α, ϕ CVS Symposium, October 2005 p.10/22
11 Lambek categorial grammer (J. Lambek, 1958) Relevant logics weakening rules contraction rule Łukasiewicz, Linear logic Resource-sensitive? CVS Symposium, October 2005 p.11/22
12 fusion LJ conjunction. fusion. fusion. Γ α β Γ, α β ( ) Σ,α,β,Γ ϕ Σ,α β,γ ϕ ( ). ϕ α β is provable iff ϕ α β is provable CVS Symposium, October 2005 p.12/22
13 Residuated Lattices 1 commutative. L;, 1; p.o. monoid. L;, L;, 1 (commutative) monoid. x y x z y z p.o. monoid residuated. x y z x y z CVS Symposium, October 2005 p.13/22
14 Residuated Lattices 2 residuated p.o. monoid L; L;,,, 1, residuated lattice (. lattice ordered groups RL. RL monoid. y z =max{x : x y z} CVS Symposium, October 2005 p.14/22
15 RL RL L s 1,s 2,...,s m t valid L s 1 s 2... s m t. exchange x y y x, contraction x x 2, weakening (left) x y x and x y y, weakening (right) 0 x. CVS Symposium, October 2005 p.15/22
16 Algebraization validity. RL (equational class).. equational class H, S, P (variety). deducibility variety equational consequence Blok-Pigozzi algebraizability CVS Symposium, October 2005 p.16/22
17 residuated lattices CVS Symposium, October 2005 p.17/22
18 residuated lattices implication. monoid explicit. residual implication.. Trends in Logic: 50 Years of Studia Logica, 2003 CVS Symposium, October 2005 p.18/22
19 cut elimination, Craig s interpolation property amalgamation property, Robinson s consistency property cut elimination. CVS Symposium, October 2005 p.19/22
20 equational interpolation = amalgamation + congruence extension property (universal algebra) Maksimova, Wroński, HO algebraization (substructural logics) Galatos-HO interpolation and amalgamation (abstract algebraic logic) Czelakowski-Pigozzi residuated structure MacNeille completion Rasiowa-Sikorski lemma CVS Symposium, October 2005 p.20/22
21 Algebraic cut elimination Belardinelli-Jipsen-HO, cf. Rasiowa, Maehara, Okada cut elimination variety MacNeille completion. MacNeille completion Heyting algebra variety trivial Boolean algebra variety Bezhanishvili-Harding CVS Symposium, October 2005 p.21/22
22 Rasiowa, Jónsson, Blok Trends in Logic III Conference in memoriam A. Mostowski, H. Rasiowa, C. Rauszer, September, 2005, Warszawa & Ruciane-Nida CVS Symposium, October 2005 p.22/22
Substructural Logics Substructural Logics p.1/46
Substructural Logics Substructural Logics p.1/46 (ordered algebraic structures, universal algebra, algebraic logic ) provability deducibility cut Dedekind-MacNeille Substructural Logics p.2/46 (abstract
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