algebra and probability in lukasiewicz...
TRANSCRIPT
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Algebra and probability in Lukasiewicz logic
Ioana Leustean
Faculty of Mathematics and Computer ScienceUniversity of Bucharest
Probability, Uncertainty and RationalityCertosa di Pontignano (Siena), 1-3 November, 2009
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LAP
Logic
Algebra
Probability
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LAP
Logic
Algebra
Probability
lap = movement once arround a course
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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 Leustean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 2 / 32
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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 Leustean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 2 / 32
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Lukasiewicz logic
3-valued (2-valued) Lukasiewicz logic classical logic
l l0, 1
2 , 1 0, 1
‖ ‖
L3 L2
J. Lukasiewicz, 1920.
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Lukasiewicz logic
n-valued (2-valued) Lukasiewicz logic classical logic
l l0, 1
n−1 ,2
n−1 , . . . ,n−2n−1 , 1 0, 1
‖ ‖
Ln L2
J. Lukasiewicz, 1920.
J. Lukasiewicz, 1929.
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Lukasiewicz logic
∞-valued n-valued (2-valued) Lukasiewicz logic Lukasiewicz logic classical logic
l l l[0, 1] 0, 1
n−1 ,2
n−1 , . . . ,n−2n−1 , 1 0, 1
‖ ‖
Ln L2
J. Lukasiewicz, 1920.
J. Lukasiewicz, 1929.
J. Lukasiewicz, A. Tarski, 1930.
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(∞-valued) Lukasiewicz logic:
Connectives
¬ and −→
• Lukasiewicz considered also Mp for ”p is possible”• Tarski defined Mp = ¬p −→ p
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(∞-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] )
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(∞-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
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(∞-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
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(∞-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 + An + Ak | k ∈ 2, . . . , (n − 2), k 6| (n − 1) ⇒ n-valued logic
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(∞-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
∃q1, . . ., qk : Rn → R ∀(a1, . . . , an) ∈ [0, 1]n ∃i ∈ 1, kf (a1, . . . , an) = qi (a1, . . . , an),
where qi (a1, . . . , an) = m0i + m1ia1 + · · ·+ mnian, mji ∈ Z ∀i , j .
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(∞-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
∃q1, . . ., qk : Rn → R ∀(a1, . . . , an) ∈ [0, 1]n ∃i ∈ 1, kf (a1, . . . , an) = qi (a1, . . . , an),
where qi (a1, . . . , an) = m0i + m1ia1 + · · ·+ mnian, mji ∈ Z ∀i , j .
Normal form representation theorem
A. Di Nola, A.Lettieri, 2004
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(∞-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
∃q1, . . ., qk : Rn → R ∀(a1, . . . , an) ∈ [0, 1]n ∃i ∈ 1, kf (a1, . . . , an) = qi (a1, . . . , an),
where qi (a1, . . . , an) = m0i + m1ia1 + · · ·+ mnian, mji ∈ 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 axiomsusing modus ponens and substitution.
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Algebra: historical overview
Gr.C. Moisil: 3-valued and 4-valued Lukasiewicz algebras (1940)n-valued Lukasiewicz algebras (1941)
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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 algebraKn = 0, 1
n−1 ,n−2n−1 , 1 is not closed to Lukasiewicz implication
Ioana Leustean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 7 / 32
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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 algebraKn = 0, 1
n−1 ,n−2n−1 , 1 is not closed to Lukasiewicz implication
Gr.C. Moisil: ϑ-valued Lukasiewicz algebras (1968)
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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 algebraKn = 0, 1
n−1 ,n−2n−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)
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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 algebraKn = 0, 1
n−1 ,n−2n−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: MVn-algebras (1977)
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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 algebraKn = 0, 1
n−1 ,n−2n−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: MVn-algebras (1977)
R. Cignoli: proper n-valued Lukasiewicz algebras (1982)
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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 algebraKn = 0, 1
n−1 ,n−2n−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: MVn-algebras (1977)
R. Cignoli: proper n-valued Lukasiewicz algebras (1982)
J.M. Font, A.J. Rodriguez, A. Torrens: Wajsberg algebras (1984)
Ioana Leustean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 7 / 32
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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 algebraKn = 0, 1
n−1 ,n−2n−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: MVn-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 algebrasMVn-algebras and proper n-valued Lukasiewicz algebras
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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 algebraKn = 0, 1
n−1 ,n−2n−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: MVn-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 algebrasMVn-algebras and proper n-valued Lukasiewicz algebras
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the n-valued case
Boolean logicւ ց
Lukasiewiczn-valued logic
Postn-valued logic
ւ ց ↓ L-proper LMn-algebras ≃ MVn-algebras ⊃ Postn-algebras(Cignoli,1982) (Grigolia,1977) (Rosenbloom,1942)
∩LMn-algebras
(Moisil,1941)
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n-valued Lukasiewicz-Moisil algebras
LMn-algebra
(L,∨,∧,∗ , ϕ1, . . . , ϕn−1, 1)(LM0)(L,∨,∧,∗ ) De Morgan algebra (LM1) ϕi (ϕj(x)) = ϕj(x)(LM2) ϕi (x ∨ y) = ϕi (x) ∨ ϕi (y) (LM3) ϕi (x) ∨ (ϕi (x))∗ = 1(LM4) ϕi (x∗) = (ϕn−i (x))∗ (LM5) ϕ1(x) ≤ · · · ≤ ϕn−1(x)
(LM6) ϕi (x) = ϕi (y) for any i ∈ 1, n − 1⇒ x = y
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n-valued Lukasiewicz-Moisil algebras
LMn-algebra
(L,∨,∧,∗ , ϕ1, . . . , ϕn−1, 1)(LM0)(L,∨,∧,∗ ) De Morgan algebra (LM1) ϕi (ϕj(x)) = ϕj(x)(LM2) ϕi (x ∨ y) = ϕi (x) ∨ ϕi (y) (LM3) ϕi (x) ∨ (ϕi (x))∗ = 1(LM4) ϕi (x∗) = (ϕn−i (x))∗ (LM5) ϕ1(x) ≤ · · · ≤ ϕn−1(x)
(LM6) ϕi (x) = ϕi (y) for any i ∈ 1, n − 1⇒ x = y
L-proper LMn-algebra = LMn-algebra + Fiki ,k + axioms
Postn-algebra = LMn-algebra +cii=1,n−2+ axioms
LMn-algebras, MVn-algebras, Postn-algebras are equational classes.
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Examples:
Example 1:
0 1n−1
2n−1 . . . n−2
n−1 1
ϕ1 0 0 0 . . . 0 1ϕ2 0 0 0 . . . 1 1...
......
......
......
ϕn−2 0 0 1 . . . 1 1ϕn−1 0 1 1 . . . 1 1
Example 2:
L = LXn = f | f : X −→ Ln
ϕi (f )(x) =
1, f (x) ≥ n−in−1
0, otherwise
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Moisil’s determination principle
L is LMn-algebra, MVn-algebra or Pn-algebraL ⊃ B(L) = x | x ∨ x∗ = 1 (the Boolean reduct)
L ∋ x 7→ ϕ1(x),. . .,ϕn−1(x) ∈ B(L)x = y iff ϕi (x) = ϕi (y) for any i ∈ 1, n − 1
Any element is characterized by (n − 1) Boolean nuances.
The functor B : LukMoisilm → Bool has a right adjoint.
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Moisil’s determination principle
L is LMn-algebra, MVn-algebra or Pn-algebraL ⊃ B(L) = x | x ∨ x∗ = 1 (the Boolean reduct)
L ∋ x 7→ ϕ1(x),. . .,ϕn−1(x) ∈ B(L)x = y iff ϕi (x) = ϕi (y) for any i ∈ 1, n − 1
Any element is characterized by (n − 1) Boolean nuances.
The functor B : LukMoisilm → Bool has a right adjoint.
Any element of L can be ”recovered” from its Boolean nuancesiff L is a Postn-algebra
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Determination principle for subalgebras
Is it true that for S , T ⊆ A
ϕi (S) = ϕi (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.
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Determination principle for subalgebras
I.L.,2008.
L - LMn-algebra
S ≤ LI→ J1(L), . . ., Jn−2(L), Jn−1(L) = B(S)
x ∈ L : Ji (x) ∈ Ii∀iS← I1, . . ., In−2, In−1 ≤ B(L)
I injective, S surjective
I(S) = I(T )⇒ S = T
S is L-proper (MVn-algebra) ⇔ I(S) satisfies (MV) (n ≥ 5)(MV): Ii ∩ Ik ⊆ In−i+k−1, 3 ≤ i ≤ n − 2, 1 ≤ k ≤ n − 4, k < i
S is Pn-algebra ⇔ I1 = · · · = In−1 = B(S)
The Boolean nuances of a subalgebra are (n − 1) Boolean ideals
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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 ispossible to construct an n-nuanced logical system on the top of thegiven one.
Nuancing the Lukasiewicz logic, they defined the n-nuanced
MV-algebras and they proved that there is a pair of adjoint functorsbetween the two categories.
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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 with1 = 0∗, x ∨ y = (y∗ ⊕ x)∗ ⊕ x , x ∧ y = (x∗ ∨ y)∗
MV-algebras are reziduated lattices withx −→ y = (x∗ ⊕ y), x ⊙ y = (x∗ ⊕ y∗)∗.
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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, 2000.
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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 Ln 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].
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Mundici’s categorical equivalence
Mundici’s categorical equivalence (1986)
The category of MV-algebras is equivalent with the category of abelianlattice-ordered groups with strong unit. For any MV-algebra A there existsan 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
- G0 u
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Mundici’s categorical equivalence
Mundici’s categorical equivalence (1986)
The category of MV-algebras is equivalent with the category of abelianlattice-ordered groups with strong unit. For any MV-algebra A there existsan 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
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Functional representation
A MV-algebra, ∅ 6= 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 thespectral topology is a compact Hausdorff space(open sets: r(I ) = M ∈ MaxA | I 6⊆ M for some ideal I ).
A is semisimple if⋂
M | M ∈ Max(A) = ∅
C (MaxA) = f : MaxA −→ [0, 1] | f continuous
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Functional representation
A MV-algebra, ∅ 6= 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 thespectral topology is a compact Hausdorff space(open sets: r(I ) = M ∈ MaxA | I 6⊆ 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 separatingMV-subalgebra of C (MaxA) (with pointwise operations).
ι : A→ C (MaxA) embedding∀ M1 6= M2 ∃ f ∈ ι(A) (f (M1) = 0 and f (M2) 6= 0)
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Functional representation
ι : A→ C (MaxA) embedding
A. Di Nola, S.Sessa, 1995
A σ-complete ⇒ MaxA basically disconnected
A complete ⇒ MaxA extremally disconnected
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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 ofcompact Hausdorff spaces and continuous maps is equivalent with a fullsubcategory of MV-algebras.
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Semantical and sintactical consequences in L
For a set Θ of formulas, defineΘ⊢ = sintactic consequences of ΘΘ|= = semantic consequences of Θ
Theorem
TFAE:
Θ⊢ = Θ|=
L(Θ) (the Lindenbaum-Tarski algebra of Θ) is semisimple.
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Semantical and sintactical consequences in L
For a set Θ of formulas, defineΘ⊢ = sintactic consequences of ΘΘ|= = 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 ofmany-valued reasoning, 2000.
P. Hajek, Metamathematics of fuzzy logic, 1998.
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Di Nola’s representation theorem, 1991
Theorem
Up to isomorphism, every MV-algebra A is an algebra of [0, 1]∗-valuedfunctions over some set, where [0, 1]⋆ is an ultrapower of [0, 1], onlydepending on th cardinatlity of A.
[0, 1]⋆ is the unit interval of R⋆ (non-standard reals)
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MV-algebras are twofold structures
generalization of Boolean algebras
intervals in abelian lattice ordered groups with strong unit
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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 andlattice-ordered groups as special cases.
G. Birkhoff, Lattice Theory, 1973.
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Probability
MV-algebrasւ ց
Boolean algebras Lattice-ordered groupswith strong unit
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Probability
MV-algebrasւ ց
Boolean algebras Lattice-ordered groupswith strong unit
↓ ↓measures states↓ ↓
σ-continuity finite additivitym : B → [0,∞] s : G → R
m(0) = 0, s(1) = 1,m(
∨
n an) = Σ∞
1 m(an) s(x + y) = s(x) + s(y),
ak ∧ an = 0, k 6= n s(G+) = R+
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Probability vs truth degree
P. Hajek, Metamathematics of fuzzy logic, 1998.
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 thepatient, the more the sentence is true.
The sentence Q2 is a crisp sentence (T | F ), but we do not knowwhich 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 probabilitycalculus is not, since P(a ∧ b) cannot be determined using P(a) andP(b).
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Probability vs truth degree
P. Hajek, Metamathematics of fuzzy logic, 1998.
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 thepatient, the more the sentence is true.
The sentence Q2 is a crisp sentence (T | F ), but we do not knowwhich 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 probabilitycalculus is not, since P(a ∧ b) cannot be determined using P(a) andP(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.
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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 ands(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
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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 ands(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, ...
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de Finetti’s coherence criterion for MV-algebras
D. Mundici, 2006,
ϕ1, . . ., ϕn are formulas of Lukasiewicz, α1, . . ., αn ∈ [0, 1]. TFAE:
for all λ1, . . ., λn ∈ R there exists a valuation V s.t.∑n
i=1 λi (αi − V (ϕi )) ≥ 0
(the probabilistic assessment (P(ϕi ) = αi )i is coherent),
there exists a state s defined on the Lindenbaum-Tarski algebra of Ls.t. s([ϕi ]) = αi for all i ∈ 1, . . . , n.
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de Finetti’s coherence criterion for MV-algebras
D. Mundici, 2006,
ϕ1, . . ., ϕn are formulas of Lukasiewicz, α1, . . ., αn ∈ [0, 1]. TFAE:
for all λ1, . . ., λn ∈ R there exists a valuation V s.t.∑n
i=1 λi (αi − V (ϕi )) ≥ 0
there exists a state s defined on the Lindenbaum-Tarski algebra of Ls.t. s([ϕi ]) = αi 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 andσ : 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).
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de Finetti’s coherence criterion for MV-algebras
D. Mundici, 2006,T. Flaminio, F.Montagna, 2009
ϕ1, . . ., ϕn are formulas of Lukasiewicz, α1, . . ., αn ∈ [0, 1]∩Q. TFAE:
for all λ1, . . ., λn ∈ R there exists a valuation V s.t.∑n
i=1 λi (αi − V (ϕi )) ≥ 0
there exists a state s defined on the Lindenbaum-Tarski algebra of Ls.t. s([ϕi ]) = αi for all i ∈ 1, . . . , n.
for any 1 ≤ i ≤ n, any non-trivial SMV-algebra satisfies the equations
(mi − 1)xi = x∗i and σ(ϕi ) = nixi ,
where V (ϕi ) = ni
mifor any i .
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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 xn ր x implies lim s(xn) = s(x) for allx , x1, x2, . . . ∈ A.
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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 xn ր x implies lim s(xn) = s(x) for allx , x1, x2, . . . ∈ A.
⇓study of measures on abstract algebra in Lukasiewicz logic
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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 xn ր x implies lim s(xn) = s(x) for allx , x1, x2, . . . ∈ A.
⇓study of measures on abstract algebra in Lukasiewicz logic
B. Riecan and D. Mundici, Probability on MV-algebras,Handbook of Measure Theory, 2002
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Finite additivity (linearity) vs σ-continuity?
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Finite additivity (linearity) vs σ-continuity?
Riesz representation theorem
Let X be a compact Hausdorff space, θ : C (X ) −→ R a positive linearfunctional. Then there exists a unique regular measure µ defined on theBorel σ-algebra of X such that
θ(f ) =∫
Xfdµ for any f ∈ C (X ).
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Finite additivity (linearity) vs σ-continuity?
Riesz representation theorem
Let X be a compact Hausdorff space, θ : C (X ) −→ R a positive linearfunctional. Then there exists a unique regular measure µ defined on theBorel σ-algebra of X such that
θ(f ) =∫
Xfdµ for any f ∈ C (X ).
Kroupa-Panti theorem
Let A be a semisimple MV-algebra. For any state s : A −→ [0, 1], thereexists a unique regular measure µ defined on the Borel σ-algebra ofX = Max(A) such that
s(f ) =∫
Xfdµ for any f ∈ A.
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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, 1936-1940.
... most of the standard real function spaces are Riesz spaces, and in avery natural way.
G. Birkhoff, Lattice Theory, 1973.
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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
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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 withstrong unit.
A. Di Nola, P.Flondor, I.L., MV-modules, 2004.
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The logic of Riesz MV-algebras
LRMV can be developed following the algebraic approach of H. RasiowaLLuk ∪ δr : r ∈ [0, 1]Axioms= Axioms of Lukasiewicz logic + Axioms reflecting the equationaldescription of Riesz MV-algebras
LRMV is complete w.r.t. [0, 1]-evaluations.
For any r ∈ [0, 1] the formula r = δr (ϕ −→ ϕ) has the value r underany [0, 1]-evaluation. Hence the truth-constants are represented inlogic, making possible the Pavelka style approach.
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The logic of Riesz MV-algebras
LRMV can be developed following the algebraic approach of H. RasiowaLLuk ∪ δr : r ∈ [0, 1]Axioms= Axioms of Lukasiewicz logic + Axioms reflecting the equationaldescription of Riesz MV-algebras
LRMV is complete w.r.t. [0, 1]-evaluations.
For any r ∈ [0, 1] the formula r = δr (ϕ −→ ϕ) has the value r underany [0, 1]-evaluation. Hence the truth-constants are represented inlogic, making possible the Pavelka style approach.
F. Esteva, J. Gispert, L. Godo, C. Noguera, 2007.
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The logic of Riesz MV-algebras
LRMV can be developed following the algebraic approach of H. RasiowaLLuk ∪ δr : r ∈ [0, 1]Axioms= Axioms of Lukasiewicz logic + Axioms reflecting the equationaldescription of Riesz MV-algebras
LRMV is complete w.r.t. [0, 1]-evaluations.
For any r ∈ [0, 1] the formula r = δr (ϕ −→ ϕ) has the value r underany [0, 1]-evaluation. Hence the truth-constants are represented inlogic, 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 = supr ∈ [0, 1] : Θ ⊢ r −→ ϕ.
Pavelka completeness. |ϕ|Θ =‖ ϕ ‖Θ
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Riesz MV-algebras with faithful states
A Riesz MV-algebra, s : A −→ [0, 1] a faithful state (s(x) = 0⇒ x = 0)ρs(x , y) = s(d(x , y)) = s((x ⊙ y∗) ∨ (y ⊙ x∗))
(A, ρs) metric space
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Riesz MV-algebras with faithful states
A Riesz MV-algebra, s : A −→ [0, 1] a faithful state (s(x) = 0⇒ x = 0)ρs(x , y) = s(d(x , y)) = s((x ⊙ y∗) ∨ (y ⊙ x∗))
(A, ρs) metric space
Example
If (Y ,Ω, µ) a measure space then denote L1(µ) the Riesz space of allintegrable functions. The constant function 1 is a strong unit of L1(µ), soΓ(L1(µ), 1) is a Ries MV-algebra. Define the state
s : Γ(L1(µ), 1) −→ [0, 1], s(f ) =∫
Yfdµ.
Then (Γ(L1(µ), 1), ρs) is a complete metric space.
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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 afaithful state and (A, ρs) is a complete metric space, there exists ameasure 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(L1(µ), 1).
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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 afaithful state and (A, ρs) is a complete metric space, there exists ameasure 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(L1(µ), 1).
Y is the maximal ideal space of the complete Boolean algebra of allclosed ideals of A
MV-algebraic expresssion of Kakutani’s representation for abstractL-spaces (1942)
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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 afaithful state and (A, ρs) is a complete metric space, there exists ameasure 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(L1(µ), 1).
Y is the maximal ideal space of the complete Boolean algebra of allclosed ideals of A
MV-algebraic expresssion of Kakutani’s representation for abstractL-spaces (1942)
Problem
Is it possible to extend the above connection to a categorical equivalence ?
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Thank you!
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