Algebra and probability in Lukasiewicz logic

Transcription

1 Algebra and probability in Lukasiewicz logic Ioana Leuştean Faculty of Mathematics and Computer Science University of Bucharest Probability, Uncertainty and Rationality Certosa di Pontignano (Siena), 1-3 November, 2009 Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 1 / 32

2 LAP Logic Algebra Probability Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 2 / 32

3 LAP Logic Algebra Probability lap = movement once arround a course Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 2 / 32

4 LAP Logic Algebra Probability lap = movement once arround a course Classical logic Boolean algebras Classical probability theory: the set of events is a Boolean algebra Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 2 / 32

5 LAP Logic Algebra Probability lap = movement once arround a course Classical logic Boolean algebras Classical probability theory: the set of events is a Boolean algebra LAP interaction in Lukasiewicz logic Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 2 / 32

6 Lukasiewicz logic 3-valued (2-valued) Lukasiewicz logic classical logic {0, 1 2,1} {0,1} L 3 L 2 J. Lukasiewicz, Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 3 / 32

7 Lukasiewicz logic n-valued (2-valued) Lukasiewicz logic classical logic 1 {0, n 1, 2 n 2 n 1,..., n 1,1} {0,1} L n L 2 J. Lukasiewicz, J. Lukasiewicz, Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 3 / 32

8 Lukasiewicz logic -valued n-valued (2-valued) Lukasiewicz logic Lukasiewicz logic classical logic [0, 1] {0, 1 n 1, 2 n 1,1} {0,1} L n L 2,..., n 2 n 1 J. Lukasiewicz, J. Lukasiewicz, J. Lukasiewicz, A. Tarski, Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 3 / 32

9 ( -valued) Lukasiewicz logic: Connectives and Lukasiewicz considered also Mp for p is possible Tarski defined Mp = p p Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 4 / 32

10 ( -valued) Lukasiewicz logic: Connectives and Lukasiewicz considered also Mp for p is possible Tarski defined Mp = p p truth-tables p := 1 p, p q := min(1 p + q, 1) (p, q [0, 1] ) Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 4 / 32

11 ( -valued) Lukasiewicz logic: Connectives and Lukasiewicz considered also Mp for p is possible Tarski defined Mp = p p truth-tables p := 1 p, p q := min(1 p + q, 1) (p, q [0, 1] ) Lukasiewicz logic is truth-functional Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 4 / 32

12 ( -valued) Lukasiewicz logic L: Axioms (L1) ϕ (ψ ϕ); (L2) (ϕ ψ) ((ψ χ) (ϕ χ)); (L3) ((ϕ ψ) ψ) ((ψ ϕ) ϕ); (L4) ( ψ ϕ) (ϕ ψ). the deduction rule is modus ponens:{ϕ, ϕ ψ} ψ Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 5 / 32

13 ( -valued) Lukasiewicz logic L: Axioms (L1) ϕ (ψ ϕ); (L2) (ϕ ψ) ((ψ χ) (ϕ χ)); (L3) ((ϕ ψ) ψ) ((ψ ϕ) ϕ); (L4) ( ψ ϕ) (ϕ ψ). the deduction rule is modus ponens:{ϕ, ϕ ψ} ψ L + ((ϕ ϕ) ϕ) classical logic L + A n + {A k k {2,...,(n 2)}, k (n 1)} n-valued logic Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 5 / 32

14 ( -valued) Lukasiewicz logic L: McNaughton Theorem, 1951 (formulas as functions) If f : [0, 1] n [0, 1], TFAE: (a) f = ϕ [0,1] for some formula ϕ of Lukasiewicz logic, (b) f is continuous such that q 1,..., q k : R n R (a 1,...,a n ) [0, 1] n i 1, k f (a 1,...,a n ) = q i (a 1,...,a n ), where q i (a 1,...,a n ) = m 0i + m 1i a m ni a n, m ji Z i, j. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 6 / 32

15 ( -valued) Lukasiewicz logic L: McNaughton Theorem, 1951 (formulas as functions) If f : [0, 1] n [0, 1], TFAE: (a) f = ϕ [0,1] for some formula ϕ of Lukasiewicz logic, (b) f is continuous such that q 1,..., q k : R n R (a 1,...,a n ) [0, 1] n i 1, k f (a 1,...,a n ) = q i (a 1,...,a n ), where q i (a 1,...,a n ) = m 0i + m 1i a m ni a n, m ji Z i, j. Normal form representation theorem A. Di Nola, A.Lettieri, 2004 Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 6 / 32

16 ( -valued) Lukasiewicz logic L: McNaughton Theorem, 1951 (formulas as functions) If f : [0, 1] n [0, 1], TFAE: (a) f = ϕ [0,1] for some formula ϕ of Lukasiewicz logic, (b) f is continuous such that q 1,..., q k : R n R (a 1,...,a n ) [0, 1] n i 1, k f (a 1,...,a n ) = q i (a 1,...,a n ), where q i (a 1,...,a n ) = m 0i + m 1i a m ni a n, m ji Z i, j. Normal form representation theorem A. Di Nola, A.Lettieri, 2004 Rose and Rosser (1958) A formula ϕ is a [0, 1]-tautology of L iff it can be derived from the axioms using modus ponens and substitution. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 6 / 32

17 Algebra: historical overview Gr.C. Moisil: 3-valued and 4-valued Lukasiewicz algebras (1940) n-valued Lukasiewicz algebras (1941) Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 7 / 32

18 Algebra: historical overview Gr.C. Moisil: 3-valued and 4-valued Lukasiewicz algebras (1940) n-valued Lukasiewicz algebras (1941) A. Rose (1965): for n 5 the n-valued Lukasiewicz algebra 1 K n = {0,, 1} is not closed to Lukasiewicz implication n 1, n 2 n 1 Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 7 / 32

19 Algebra: historical overview Gr.C. Moisil: 3-valued and 4-valued Lukasiewicz algebras (1940) n-valued Lukasiewicz algebras (1941) A. Rose (1965): for n 5 the n-valued Lukasiewicz algebra 1 K n = {0, n 1, n 2 n 1, 1} is not closed to Lukasiewicz implication Gr.C. Moisil: ϑ-valued Lukasiewicz algebras (1968) Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 7 / 32

20 Algebra: historical overview Gr.C. Moisil: 3-valued and 4-valued Lukasiewicz algebras (1940) n-valued Lukasiewicz algebras (1941) A. Rose (1965): for n 5 the n-valued Lukasiewicz algebra 1 K n = {0, n 1, n 2 n 1, 1} is not closed to Lukasiewicz implication Gr.C. Moisil: ϑ-valued Lukasiewicz algebras (1968) Argentininan school: A. Monteiro, R.Cignoli, L. Monteiro, M. Abad, etc.... C.C. Chang: MV-algebras (1958) Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 7 / 32

21 Algebra: historical overview Gr.C. Moisil: 3-valued and 4-valued Lukasiewicz algebras (1940) n-valued Lukasiewicz algebras (1941) A. Rose (1965): for n 5 the n-valued Lukasiewicz algebra 1 K n = {0, n 1, n 2 n 1, 1} is not closed to Lukasiewicz implication Gr.C. Moisil: ϑ-valued Lukasiewicz algebras (1968) Argentininan school: A. Monteiro, R.Cignoli, L. Monteiro, M. Abad, etc.... C.C. Chang: MV-algebras (1958) R. Grigolia: MV n -algebras (1977) Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 7 / 32

22 Algebra: historical overview Gr.C. Moisil: 3-valued and 4-valued Lukasiewicz algebras (1940) n-valued Lukasiewicz algebras (1941) A. Rose (1965): for n 5 the n-valued Lukasiewicz algebra 1 K n = {0, n 1, n 2 n 1, 1} is not closed to Lukasiewicz implication Gr.C. Moisil: ϑ-valued Lukasiewicz algebras (1968) Argentininan school: A. Monteiro, R.Cignoli, L. Monteiro, M. Abad, etc.... C.C. Chang: MV-algebras (1958) R. Grigolia: MV n -algebras (1977) R. Cignoli: proper n-valued Lukasiewicz algebras (1982) Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 7 / 32

23 Algebra: historical overview Gr.C. Moisil: 3-valued and 4-valued Lukasiewicz algebras (1940) n-valued Lukasiewicz algebras (1941) A. Rose (1965): for n 5 the n-valued Lukasiewicz algebra 1 K n = {0, n 1, n 2 n 1, 1} is not closed to Lukasiewicz implication Gr.C. Moisil: ϑ-valued Lukasiewicz algebras (1968) Argentininan school: A. Monteiro, R.Cignoli, L. Monteiro, M. Abad, etc.... C.C. Chang: MV-algebras (1958) R. Grigolia: MV n -algebras (1977) R. Cignoli: proper n-valued Lukasiewicz algebras (1982) J.M. Font, A.J. Rodriguez, A. Torrens: Wajsberg algebras (1984) Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 7 / 32

24 Algebra: historical overview Gr.C. Moisil: 3-valued and 4-valued Lukasiewicz algebras (1940) n-valued Lukasiewicz algebras (1941) A. Rose (1965): for n 5 the n-valued Lukasiewicz algebra 1 K n = {0, n 1, n 2 n 1, 1} is not closed to Lukasiewicz implication Gr.C. Moisil: ϑ-valued Lukasiewicz algebras (1968) Argentininan school: A. Monteiro, R.Cignoli, L. Monteiro, M. Abad, etc.... C.C. Chang: MV-algebras (1958) R. Grigolia: MV n -algebras (1977) R. Cignoli: proper n-valued Lukasiewicz algebras (1982) J.M. Font, A.J. Rodriguez, A. Torrens: Wajsberg algebras (1984) MV-algebras and Wajsberg algebras MV n -algebras and proper n-valued Lukasiewicz algebras Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 7 / 32

25 Algebra: historical overview Gr.C. Moisil: 3-valued and 4-valued Lukasiewicz algebras (1940) n-valued Lukasiewicz algebras (1941) A. Rose (1965): for n 5 the n-valued Lukasiewicz algebra 1 K n = {0, n 1, n 2 n 1, 1} is not closed to Lukasiewicz implication Gr.C. Moisil: ϑ-valued Lukasiewicz algebras (1968) Argentininan school: A. Monteiro, R.Cignoli, L. Monteiro, M. Abad, etc.... C.C. Chang: MV-algebras (1958) R. Grigolia: MV n -algebras (1977) R. Cignoli: proper n-valued Lukasiewicz algebras (1982) J.M. Font, A.J. Rodriguez, A. Torrens: Wajsberg algebras (1984) MV-algebras and Wajsberg algebras MV n -algebras and proper n-valued Lukasiewicz algebras Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 7 / 32

26 the n-valued case Boolean logic ւ ց Lukasiewicz n-valued logic Post n-valued logic ւ ց L-proper LM n -algebras MV n -algebras Post n -algebras (Cignoli,1982) (Grigolia,1977) (Rosenbloom,1942) LM n -algebras (Moisil,1941) Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 8 / 32

27 n-valued Lukasiewicz-Moisil algebras LM n -algebra (L,,,, ϕ 1,...,ϕ n 1, 1) (LM0)(L,,, ) De Morgan algebra (LM1) ϕ i (ϕ j (x)) = ϕ j (x) (LM2) ϕ i (x y) = ϕ i (x) ϕ i (y) (LM3) ϕ i (x) (ϕ i (x)) = 1 (LM4) ϕ i (x ) = (ϕ n i (x)) (LM5) ϕ 1 (x) ϕ n 1 (x) (LM6) ϕ i (x) = ϕ i (y) for any i 1, n 1 x = y Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 9 / 32

28 n-valued Lukasiewicz-Moisil algebras LM n -algebra (L,,,, ϕ 1,...,ϕ n 1, 1) (LM0)(L,,, ) De Morgan algebra (LM1) ϕ i (ϕ j (x)) = ϕ j (x) (LM2) ϕ i (x y) = ϕ i (x) ϕ i (y) (LM3) ϕ i (x) (ϕ i (x)) = 1 (LM4) ϕ i (x ) = (ϕ n i (x)) (LM5) ϕ 1 (x) ϕ n 1 (x) (LM6) ϕ i (x) = ϕ i (y) for any i 1, n 1 x = y L-proper LM n -algebra = LM n -algebra + {F ik } i,k + axioms Post n -algebra = LM n -algebra +{c i } i=1,n 2 + axioms LM n -algebras, MV n -algebras, Post n -algebras are equational classes. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 9 / 32

29 Examples: Example 1: 0 1 n 1 2 n 2 n 1... n 1 1 ϕ ϕ ϕ n ϕ n Example 2: L = L X n = {f { f : X L n } 1, f (x) n i ϕ i (f )(x) = n 1 0, otherwise Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

30 Moisil s determination principle L is LM n -algebra, MV n -algebra or P n -algebra L B(L) = {x x x = 1} (the Boolean reduct) L x ϕ 1 (x),...,ϕ n 1 (x) B(L) x = y iff ϕ i (x) = ϕ i (y) for any i 1, n 1 Any element is characterized by (n 1) Boolean nuances. The functor B : LukMoisil m Bool has a right adjoint. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

31 Moisil s determination principle L is LM n -algebra, MV n -algebra or P n -algebra L B(L) = {x x x = 1} (the Boolean reduct) L x ϕ 1 (x),...,ϕ n 1 (x) B(L) x = y iff ϕ i (x) = ϕ i (y) for any i 1, n 1 Any element is characterized by (n 1) Boolean nuances. The functor B : LukMoisil m Bool has a right adjoint. Any element of L can be recovered from its Boolean nuances iff L is a Post n -algebra Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

32 Determination principle for subalgebras Is it true that for S, T A ϕ i (S) = ϕ i (T) for any i 1, n 1 S = T? Yes, if S and T are proper ideals. Not in general, if S and T are subalgebras. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

33 Determination principle for subalgebras I.L.,2008. L - LM n -algebra S L {x L : J i (x) I i i} I S J 1 (L),..., J n 2 (L), J n 1 (L) = B(S) I 1,..., I n 2, I n 1 B(L) I injective, S surjective I(S) = I(T) S = T S is L-proper (MV n -algebra) I(S) satisfies (MV) (n 5) (MV): I i I k I n i+k 1, 3 i n 2, 1 k n 4, k < i S is P n -algebra I 1 = = I n 1 = B(S) The Boolean nuances of a subalgebra are (n 1) Boolean ideals Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

34 Nuances of truth vs truth degrees G. Georgescu, A. Popescu, 2006 Moisil logic is derived from the classical logic by the idea of nuancing, mathematically expressed by a categorical adjunction Starting from a logical system and using the idea of nuance, it is possible to construct an n-nuanced logical system on the top of the given one. Nuancing the Lukasiewicz logic, they defined the n-nuanced MV-algebras and they proved that there is a pair of adjoint functors between the two categories. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

35 MV-algebras Lukasiewicz -valued logic C.C.Chang, 1958 An MV-algebra is a structure (A,,, 0) such that: 1 (A,, 0) abelian monoid, 2 (x ) = x, 3 (x y) y = (y x) x, 4 0 x = 0. ([0, 1],,, 0) MV-algebra, x y = min(x + y, 0), x = 1 x MV-algebras are bounded distributive lattices with 1 = 0, x y = (y x) x, x y = (x y) MV-algebras are reziduated lattices with x y = (x y), x y = (x y ). Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

36 MV-algebras Lukasiewicz -valued logic C.C.Chang, 1958 An MV-algebra is a structure (A,,, 0) such that: 1 (A,, 0) abelian monoid, 2 (x ) = x, 3 (x y) y = (y x) x, 4 0 x = 0. R. Cignoli, I.M.L. D Ottaviano, D. Mundici, Algebraic foundations of many-valued reasoning, Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

37 MV-algebras Lukasiewicz -valued logic C.C.Chang, 1958 An MV-algebra is a structure (A,,, 0) such that: 1 (A,, 0) abelian monoid, 2 (x ) = x, 3 (x y) y = (y x) x, 4 0 x = 0. Chang s completeness theorem For a formula ϕ TFAE: (a) ϕ provable, (d) ϕ holds in L n for any n 2, (b) ϕ holds in any MV-algebra, (e) ϕ holds in [0, 1] Q, (c) ϕ holds in any MV-chain, (f) ϕ holds in [0, 1]. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

38 Mundici s categorical equivalence Mundici s categorical equivalence (1986) The category of MV-algebras is equivalent with the category of abelian lattice-ordered groups with strong unit. For any MV-algebra A there exists an abelian lattice-ordered group with strong unit (G, u) such that A [0, u] G. u strong unit: u 0, for any x G there is n 1 s.t. x nu Γ(G, u) = ([0, u] G,,, 0): x y = (x + y) 1, x = 1 x 0 u G Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

39 Mundici s categorical equivalence Mundici s categorical equivalence (1986) The category of MV-algebras is equivalent with the category of abelian lattice-ordered groups with strong unit. For any MV-algebra A there exists an abelian lattice-ordered group with strong unit (G, u) such that A [0, u] G. u strong unit: u 0, for any x G there is n 1 s.t. x nu Γ(G, u) = ([0, u] G,,, 0): x y = (x + y) 1, x = 1 x strong unit logical interpretation existence maximal ideals Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

40 Functional representation A MV-algebra, I A is an ideal if: (x I, y x y I) and (x, y I x y I) for any MV-algebra A, the maximal ideal space MaxA with the spectral topology is a compact Hausdorff space (open sets: r(i) = {M MaxA I M} for some ideal I). A is semisimple if {M M Max(A)} = C(MaxA) = {f : MaxA [0, 1] f continuous} Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

41 Functional representation A MV-algebra, I A is an ideal if: (x I, y x y I) and (x, y I x y I) for any MV-algebra A, the maximal ideal space MaxA with the spectral topology is a compact Hausdorff space (open sets: r(i) = {M MaxA I M} for some ideal I). A is semisimple if {M M Max(A)} = C(MaxA) = {f : MaxA [0, 1] f continuous} L.P.Belluce, 1986 Any semisimple MV-algebra A is isomorphic with a separating MV-subalgebra of C(MaxA) (with pointwise operations). ι: A C(MaxA) embedding M 1 M 2 f ι(a) (f (M 1 ) = 0 and f (M 2 ) 0) Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

42 Functional representation ι: A C(MaxA) embedding A. Di Nola, S.Sessa, 1995 A σ-complete MaxA basically disconnected A complete MaxA extremally disconnected Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

43 Functional representation ι: A C(MaxA) embedding A. Di Nola, S.Sessa, 1995 A σ-complete MaxA basically disconnected A complete MaxA extremally disconnected (V. Marra, I.L.) We characterized those MV-algebras A with the property that A C(X) for some compact Hausdorff space X. We proved that the category of compact Hausdorff spaces and continuous maps is equivalent with a full subcategory of MV-algebras. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

44 Semantical and sintactical consequences in L For a set Θ of formulas, define Θ = sintactic consequences of Θ Θ = = semantic consequences of Θ Theorem TFAE: Θ = Θ = L(Θ) (the Lindenbaum-Tarski algebra of Θ) is semisimple. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

45 Semantical and sintactical consequences in L For a set Θ of formulas, define Θ = sintactic consequences of Θ Θ = = semantic consequences of Θ Theorem TFAE: Θ = Θ = L(Θ) (the Lindenbaum-Tarski algebra of Θ) is semisimple. R. Cignoli, I.M.L. D Ottaviano, D. Mundici, Algebraic foundations of many-valued reasoning, P. Hájek, Metamathematics of fuzzy logic, Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

46 Di Nola s representation theorem, 1991 Theorem Up to isomorphism, every MV-algebra A is an algebra of [0, 1] -valued functions over some set, where [0, 1] is an ultrapower of [0, 1], only depending on th cardinatlity of A. [0, 1] is the unit interval of R (non-standard reals) Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

47 MV-algebras are twofold structures generalization of Boolean algebras intervals in abelian lattice ordered groups with strong unit Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

48 MV-algebras are twofold structures generalization of Boolean algebras intervals in abelian lattice ordered groups with strong unit The theory of MV-algebras is a possible answer to Birkhoff s problem: develop a common abstraction which includes Boolean algebras and lattice-ordered groups as special cases. G. Birkhoff, Lattice Theory, Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

49 Probability MV-algebras ւ ց Boolean algebras Lattice-ordered groups with strong unit Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

50 Probability MV-algebras ւ ց Boolean algebras Lattice-ordered groups with strong unit measures states σ-continuity finite additivity m : B [0, ] s : G R m(0) = 0, s(1) = 1, m( n a n) = Σ 1 m(a n) s(x + y) = s(x) + s(y), a k a n = 0, k n s(g + ) = R + Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

51 Probability vs truth degree P. Hájek, Metamathematics of fuzzy logic, Q1: The patient is young Q2: The patient will survive next week The sentence Q1 is true to some degree - the lower the age of the patient, the more the sentence is true. The sentence Q2 is a crisp sentence (T F), but we do not know which is the case; we may have some probability (degree of belief) that the sentence is true. Most many-valued logics are truth-functional, while the probability calculus is not, since P(a b) cannot be determined using P(a) and P(b). Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

52 Probability vs truth degree P. Hájek, Metamathematics of fuzzy logic, Q1: The patient is young Q2: The patient will survive next week The sentence Q1 is true to some degree - the lower the age of the patient, the more the sentence is true. The sentence Q2 is a crisp sentence (T F), but we do not know which is the case; we may have some probability (degree of belief) that the sentence is true. Most many-valued logics are truth-functional, while the probability calculus is not, since P(a b) cannot be determined using P(a) and P(b). Truth of a many-valued sentence is a matter of degree. Probability is not a degree of truth. Probability is a degree of belief. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

53 States (finitely additive case) D. Mundici, 1995 Definition A state on (A,,, 0) is a map s : A [0, 1] s.t s(1) = 1 and s(x y) = s(x) + s(y) whenever x y (x y = 0). The state s is faithfull if s(x) = 0 implies x = 0. s(p) is the average degree of truth of p Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

54 States (finitely additive case) D. Mundici, 1995 Definition A state on (A,,, 0) is a map s : A [0, 1] s.t s(1) = 1 and s(x y) = s(x) + s(y) whenever x y (x y = 0). The state s is faithfull if s(x) = 0 implies x = 0. s(p) is the average degree of truth of p extension results, characterization of the state space,... Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

55 de Finetti s coherence criterion for MV-algebras D. Mundici, 2006, ϕ 1,..., ϕ n are formulas of Lukasiewicz, α 1,..., α n [0, 1]. TFAE: for all λ 1,..., λ n R there exists a valuation V s.t. n i=1 λ i(α i V(ϕ i )) 0 (the probabilistic assessment (P(ϕ i ) = α i ) i is coherent), there exists a state s defined on the Lindenbaum-Tarski algebra of L s.t. s([ϕ i ]) = α i for all i {1,...,n}. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

56 de Finetti s coherence criterion for MV-algebras D. Mundici, 2006, ϕ 1,..., ϕ n are formulas of Lukasiewicz, α 1,..., α n [0, 1]. TFAE: for all λ 1,..., λ n R there exists a valuation V s.t. n i=1 λ i(α i V(ϕ i )) 0 there exists a state s defined on the Lindenbaum-Tarski algebra of L s.t. s([ϕ i ]) = α i for all i {1,...,n}. MV-algebra with internal state(t. Flaminio, F. Montagna,?) An SMV-algebra is a structure (A, σ) such that A is an MV-algebra and σ : A A s.t.: σ(0) = 0, σ(x y) = σ(x) σ(y (x y) ), σ(x ) = σ(x), σ(σ(x) σ(y)) = σ(x) σ(y). Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

57 de Finetti s coherence criterion for MV-algebras D. Mundici, 2006,T. Flaminio, F.Montagna, 2009 ϕ 1,..., ϕ n are formulas of Lukasiewicz, α 1,..., α n [0, 1] Q. TFAE: for all λ 1,..., λ n R there exists a valuation V s.t. n i=1 λ i(α i V(ϕ i )) 0 there exists a state s defined on the Lindenbaum-Tarski algebra of L s.t. s([ϕ i ]) = α i for all i {1,...,n}. for any 1 i n, any non-trivial SMV-algebra satisfies the equations where V(ϕ i ) = n i m i for any i. (m i 1)x i = x i and σ(ϕ i ) = n i x i, Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

58 σ-states Definition Let A be a σ-complete MV-algebra (i.e. closed to countable suprema). The state s St(A) is a σ-state if x n ր x implies lim s(x n ) = s(x) for all x, x 1, x 2,... A. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

59 σ-states Definition Let A be a σ-complete MV-algebra (i.e. closed to countable suprema). The state s St(A) is a σ-state if x n ր x implies lim s(x n ) = s(x) for all x, x 1, x 2,... A. study of measures on abstract algebra in Lukasiewicz logic Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

60 σ-states Definition Let A be a σ-complete MV-algebra (i.e. closed to countable suprema). The state s St(A) is a σ-state if x n ր x implies lim s(x n ) = s(x) for all x, x 1, x 2,... A. study of measures on abstract algebra in Lukasiewicz logic B. Riečan and D. Mundici, Probability on MV-algebras, Handbook of Measure Theory, 2002 Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

61 Finite additivity (linearity) vs σ-continuity? Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

62 Finite additivity (linearity) vs σ-continuity? Riesz representation theorem Let X be a compact Hausdorff space, θ : C(X) R a positive linear functional. Then there exists a unique regular measure µ defined on the Borel σ-algebra of X such that θ(f ) = X fdµ for any f C(X). Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

63 Finite additivity (linearity) vs σ-continuity? Riesz representation theorem Let X be a compact Hausdorff space, θ : C(X) R a positive linear functional. Then there exists a unique regular measure µ defined on the Borel σ-algebra of X such that Kroupa-Panti theorem θ(f ) = X fdµ for any f C(X). Let A be a semisimple MV-algebra. For any state s : A [0, 1], there exists a unique regular measure µ defined on the Borel σ-algebra of X = Max(A) such that s(f ) = X fdµ for any f A. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

64 Riesz spaces (vector lattices) A Riesz space are lattice-ordered real vector spaces A s.t.: x y implies x + z y + z for any z A, r 0 and x 0 implies r x 0, where : R A A is the scalar multiplication. Riesz, Kantorovich, Freundenthal, most of the standard real function spaces are Riesz spaces, and in a very natural way. G. Birkhoff, Lattice Theory, Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

65 Riesz MV-algebras L is a Riesz space, : R L L scalar multiplication, u L is a strong unit : [0,1] [0,u] L [0,u] L Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

66 Riesz MV-algebras L is a Riesz space, : R L L scalar multiplication, u L is a strong unit : [0,1] [0,u] L [0,u] L The structure Γ(L, u) = ([0, u] L, ) is called Riesz MV-algebra. The class of Riesz MV-algebras is equational. Riesz MV-algebras are categorically equivalent with Riesz spaces with strong unit. A. Di Nola, P.Flondor, I.L., MV-modules, Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

67 The logic of Riesz MV-algebras L RMV can be developed following the algebraic approach of H. Rasiowa L Luk {δ r : r [0, 1]} Axioms= Axioms of Lukasiewicz logic + Axioms reflecting the equational description of Riesz MV-algebras L RMV is complete w.r.t. [0, 1]-evaluations. For any r [0, 1] the formula r = δ r (ϕ ϕ) has the value r under any [0, 1]-evaluation. Hence the truth-constants are represented in logic, making possible the Pavelka style approach. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

68 The logic of Riesz MV-algebras L RMV can be developed following the algebraic approach of H. Rasiowa L Luk {δ r : r [0, 1]} Axioms= Axioms of Lukasiewicz logic + Axioms reflecting the equational description of Riesz MV-algebras L RMV is complete w.r.t. [0, 1]-evaluations. For any r [0, 1] the formula r = δ r (ϕ ϕ) has the value r under any [0, 1]-evaluation. Hence the truth-constants are represented in logic, making possible the Pavelka style approach. F. Esteva, J. Gispert, L. Godo, C. Noguera, Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

69 The logic of Riesz MV-algebras L RMV can be developed following the algebraic approach of H. Rasiowa L Luk {δ r : r [0, 1]} Axioms= Axioms of Lukasiewicz logic + Axioms reflecting the equational description of Riesz MV-algebras L RMV is complete w.r.t. [0, 1]-evaluations. For any r [0, 1] the formula r = δ r (ϕ ϕ) has the value r under any [0, 1]-evaluation. Hence the truth-constants are represented in logic, making possible the Pavelka style approach. ϕ formula, Θ theory truth degree: ϕ Θ = inf {e(ϕ) : e [0, 1]-evaluation, e(θ) = {1}}, provability degree: ϕ Θ = sup{r [0, 1] : Θ r ϕ}. Pavelka completeness. ϕ Θ = ϕ Θ Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

70 Riesz MV-algebras with faithful states A Riesz MV-algebra, s : A [0, 1] a faithful state (s(x) = 0 x = 0) ρ s (x, y) = s(d(x, y)) = s((x y ) (y x )) (A, ρ s ) metric space Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

71 Riesz MV-algebras with faithful states A Riesz MV-algebra, s : A [0, 1] a faithful state (s(x) = 0 x = 0) ρ s (x, y) = s(d(x, y)) = s((x y ) (y x )) (A, ρ s ) metric space Example If (Y, Ω, µ) a measure space then denote L 1 (µ) the Riesz space of all integrable functions. The constant function 1 is a strong unit of L 1 (µ), so Γ(L 1 (µ),1) is a Ries MV-algebra. Define the state s : Γ(L 1 (µ),1) [0, 1], s(f ) = Y fdµ. Then (Γ(L 1 (µ),1), ρ s ) is a complete metric space. Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

72 A logical approach to metric spaces? Theorem For any pair (A, s) s.t. A is a Riesz MV-algebra, s : A [0, 1] is a faithful state and (A, ρ s ) is a complete metric space, there exists a measure space (Y, Ω, µ) s.t.: (a) Y is an extremally disconnected compact Hausdorff space, (b) Ω is the Borel σ-algebra of Y, (c) A Γ(L 1 (µ),1). Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

73 A logical approach to metric spaces? Theorem For any pair (A, s) s.t. A is a Riesz MV-algebra, s : A [0, 1] is a faithful state and (A, ρ s ) is a complete metric space, there exists a measure space (Y, Ω, µ) s.t.: (a) Y is an extremally disconnected compact Hausdorff space, (b) Ω is the Borel σ-algebra of Y, (c) A Γ(L 1 (µ),1). Y is the maximal ideal space of the complete Boolean algebra of all closed ideals of A MV-algebraic expresssion of Kakutani s representation for abstract L-spaces (1942) Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

74 A logical approach to metric spaces? Theorem For any pair (A, s) s.t. A is a Riesz MV-algebra, s : A [0, 1] is a faithful state and (A, ρ s ) is a complete metric space, there exists a measure space (Y, Ω, µ) s.t.: (a) Y is an extremally disconnected compact Hausdorff space, (b) Ω is the Borel σ-algebra of Y, (c) A Γ(L 1 (µ),1). Y is the maximal ideal space of the complete Boolean algebra of all closed ideals of A MV-algebraic expresssion of Kakutani s representation for abstract L-spaces (1942) Problem Is it possible to extend the above connection to a categorical equivalence? Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32

75 Thank you! Ioana Leuştean (UB) Algebra and Probability in Lukasiewicz logic PURa / 32