Layers of zero probability for conditional states of
many-valued events
Tommaso Flaminio
Lluís Godo
IIIA - CSIC
Universitat Autonoma de Barcelona (Spain)
{tommaso, godo}@iiia.csic.es
ManyVal 2012
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
1 / 26
Outline
1
Introduction
Betting on conditional events
Lexicographic and non-standard probability
2
The case of Łukasiewicz Events
3
Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
4
Future work
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
2 / 26
Introduction
Conditional probability on Boolean algebras
Let B = hB, ∧, ∨, ¬, >, ⊥i be a Boolean algebra, and consider
C = B × B \ {⊥}.
A conditional probability on B is a map P(· | ·) : C → [0, 1] satisfying the
following axioms:
1
P(H | H) = 1 for every H ∈ B \ {⊥};
2
P(· | H) is, for every H ∈ B \ {⊥}, a (finitely additive) probability measure
on B;
3
P((E ∧ A) | H) = P(E | H) · P(A | E ∧ H), for every A ∈ B, and each
E, H ∈ B \ {⊥} such that E ∧ H 6= ⊥.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
3 / 26
Introduction
Betting on conditional events
Outline
1
Introduction
Betting on conditional events
Lexicographic and non-standard probability
2
The case of Łukasiewicz Events
3
Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
4
Future work
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
4 / 26
Introduction
Betting on conditional events
Conditional probability can also be characterized in terms of coherence
(not-sure loss principle).
Betting λ ∈ R on E | H = α means that we accept to pay αλ to the
bookmaker in order to receive, in the possible world V :
λ if both V (E) = 1 and V (H) = 1,
0 if V (E) = 0 and V (H) = 1,
αλ, if V (H) = 0.
Then an assignment
χ : Ei | Hi → αi .
is coherent (it does not ensure a sure loss), iff there is no way of betting on χ
ensuring a sure loss for the bookmaker, i.e. a sure win for the gambler.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
5 / 26
Introduction
Betting on conditional events
Conditional probability can also be characterized in terms of coherence
(not-sure loss principle).
Betting λ ∈ R on E | H = α means that we accept to pay αλ to the
bookmaker in order to receive, in the possible world V :
λ if both V (E) = 1 and V (H) = 1,
0 if V (E) = 0 and V (H) = 1,
αλ, if V (H) = 0.
Then an assignment
χ : Ei | Hi → αi .
is coherent (it does not ensure a sure loss), iff there is no way of betting on χ
ensuring a sure loss for the bookmaker, i.e. a sure win for the gambler.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
5 / 26
Introduction
Betting on conditional events
Conditional probability can also be characterized in terms of coherence
(not-sure loss principle).
Betting λ ∈ R on E | H = α means that we accept to pay αλ to the
bookmaker in order to receive, in the possible world V :
λ if both V (E) = 1 and V (H) = 1,
0 if V (E) = 0 and V (H) = 1,
αλ, if V (H) = 0.
Then an assignment
χ : Ei | Hi → αi .
is coherent (it does not ensure a sure loss), iff there is no way of betting on χ
ensuring a sure loss for the bookmaker, i.e. a sure win for the gambler.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
5 / 26
Introduction
Betting on conditional events
Coherence criterion for conditional assignments, actually characterizes
conditional probability by the following:
Theorem ([1])
Let E1 | H1 , . . . , En | Hn be conditional events, let χ : Ei | Hi 7→ αi an
assignment, and let B be the Boolean algebra generated by the unconditional
events Ei , Hi .
Then the following are equivalent:
1
χ is coherent;
2
There exists a conditional probability P(· | ·) on B such that for each
1 ≤ i ≤ n,
P(Ei | Hi ) = χ(Ei | Hi ) = αi .
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
6 / 26
Introduction
Betting on conditional events
Coherence criterion for conditional assignments, actually characterizes
conditional probability by the following:
Theorem ([1])
Let E1 | H1 , . . . , En | Hn be conditional events, let χ : Ei | Hi 7→ αi an
assignment, and let B be the Boolean algebra generated by the unconditional
events Ei , Hi .
Then the following are equivalent:
1
χ is coherent;
2
There exists a conditional probability P(· | ·) on B such that for each
1 ≤ i ≤ n,
P(Ei | Hi ) = χ(Ei | Hi ) = αi .
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
6 / 26
Introduction
Betting on conditional events
Coherence criterion for conditional assignments, actually characterizes
conditional probability by the following:
Theorem ([1])
Let E1 | H1 , . . . , En | Hn be conditional events, let χ : Ei | Hi 7→ αi an
assignment, and let B be the Boolean algebra generated by the unconditional
events Ei , Hi .
Then the following are equivalent:
1
χ is coherent;
2
There exists a conditional probability P(· | ·) on B such that for each
1 ≤ i ≤ n,
P(Ei | Hi ) = χ(Ei | Hi ) = αi .
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
6 / 26
Introduction
Lexicographic and non-standard probability
Outline
1
Introduction
Betting on conditional events
Lexicographic and non-standard probability
2
The case of Łukasiewicz Events
3
Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
4
Future work
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
7 / 26
Introduction
Lexicographic and non-standard probability
Non-standard and lexicographic probability (cf. [1, 2])
Coherent conditional probability assignments can be characterized in terms
of non-standard probability, and lexicographic probability
Theorem
Let E1 | H1 , . . . , En | Hn be conditional events, let χ : Ei | Hi 7→ αi an
assignment, and let B be the Boolean algebra generated by the unconditional
events Ei , Hi . Then the following are equivalent:
1
χ is coherent (i.e. it does not allow surely winning strategies);
2
There exists a non-standard probability P ∗ : B → ∗ [0, 1] such that
For every i, P ∗ (Hi ) >
0;
For every i, αi = St
3
P ∗ (Ei ∧Hi )
P ∗ (Hi )
.
There exists a r ∈ N, and a lexicographic probability space hP0 , . . . , Pr i
such that
For every i there exists a 1 ≤ `(i) ≤ r such that P`(i) (Hi ) > 0, and
For every i, αi =
Flaminio, Godo (IIIA - CSIC)
P`(i) (Ei ∧Hi )
P`(i) (Hi ) .
ManyVal - Salerno, July 2012
ManyVal 2012
8 / 26
Introduction
Lexicographic and non-standard probability
Non-standard and lexicographic probability (cf. [1, 2])
Coherent conditional probability assignments can be characterized in terms
of non-standard probability, and lexicographic probability
Theorem
Let E1 | H1 , . . . , En | Hn be conditional events, let χ : Ei | Hi 7→ αi an
assignment, and let B be the Boolean algebra generated by the unconditional
events Ei , Hi . Then the following are equivalent:
1
χ is coherent (i.e. it does not allow surely winning strategies);
2
There exists a non-standard probability P ∗ : B → ∗ [0, 1] such that
For every i, P ∗ (Hi ) >
0;
For every i, αi = St
3
P ∗ (Ei ∧Hi )
P ∗ (Hi )
.
There exists a r ∈ N, and a lexicographic probability space hP0 , . . . , Pr i
such that
For every i there exists a 1 ≤ `(i) ≤ r such that P`(i) (Hi ) > 0, and
For every i, αi =
Flaminio, Godo (IIIA - CSIC)
P`(i) (Ei ∧Hi )
P`(i) (Hi ) .
ManyVal - Salerno, July 2012
ManyVal 2012
8 / 26
Introduction
Lexicographic and non-standard probability
Non-standard and lexicographic probability (cf. [1, 2])
Coherent conditional probability assignments can be characterized in terms
of non-standard probability, and lexicographic probability
Theorem
Let E1 | H1 , . . . , En | Hn be conditional events, let χ : Ei | Hi 7→ αi an
assignment, and let B be the Boolean algebra generated by the unconditional
events Ei , Hi . Then the following are equivalent:
1
χ is coherent (i.e. it does not allow surely winning strategies);
2
There exists a non-standard probability P ∗ : B → ∗ [0, 1] such that
For every i, P ∗ (Hi ) >
0;
For every i, αi = St
3
P ∗ (Ei ∧Hi )
P ∗ (Hi )
.
There exists a r ∈ N, and a lexicographic probability space hP0 , . . . , Pr i
such that
For every i there exists a 1 ≤ `(i) ≤ r such that P`(i) (Hi ) > 0, and
For every i, αi =
Flaminio, Godo (IIIA - CSIC)
P`(i) (Ei ∧Hi )
P`(i) (Hi ) .
ManyVal - Salerno, July 2012
ManyVal 2012
8 / 26
Introduction
Lexicographic and non-standard probability
Non-standard and lexicographic probability (cf. [1, 2])
Coherent conditional probability assignments can be characterized in terms
of non-standard probability, and lexicographic probability
Theorem
Let E1 | H1 , . . . , En | Hn be conditional events, let χ : Ei | Hi 7→ αi an
assignment, and let B be the Boolean algebra generated by the unconditional
events Ei , Hi . Then the following are equivalent:
1
χ is coherent (i.e. it does not allow surely winning strategies);
2
There exists a non-standard probability P ∗ : B → ∗ [0, 1] such that
For every i, P ∗ (Hi ) >
0;
For every i, αi = St
3
P ∗ (Ei ∧Hi )
P ∗ (Hi )
.
There exists a r ∈ N, and a lexicographic probability space hP0 , . . . , Pr i
such that
For every i there exists a 1 ≤ `(i) ≤ r such that P`(i) (Hi ) > 0, and
For every i, αi =
Flaminio, Godo (IIIA - CSIC)
P`(i) (Ei ∧Hi )
P`(i) (Hi ) .
ManyVal - Salerno, July 2012
ManyVal 2012
8 / 26
Introduction
Lexicographic and non-standard probability
Idea of the proof
The class A = {a1 , . . . , am } of atoms generated by the events Ei , Hi can be
stratified in a hierarchy defined by r + 1 probability distributions p0 , . . . , pr , as
follows:
for each j, pj : A → [0, 1],
if Aj = {at ∈ A : pj (at ) = 0}, then pj+1 is 0 on A \ Aj .
for each Hi , there exists a minimum j such that Pj (Hi ) =
This minimum level is `(i): the zero-layer of Hi .
P
a∈Hi
pj (a) > 0.
The nonstandard probability P ∗ respects zero-layers. In fact P ∗ is
defined such that, for each Hi , Hz ,
if `(i) < `(z), then P ∗ (Hz ) << P ∗ (Hi ),
i.e. P ∗ (Hz )/P ∗ (Hi ) is a positive infinitesimal.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
9 / 26
Introduction
Lexicographic and non-standard probability
Idea of the proof
The class A = {a1 , . . . , am } of atoms generated by the events Ei , Hi can be
stratified in a hierarchy defined by r + 1 probability distributions p0 , . . . , pr , as
follows:
for each j, pj : A → [0, 1],
if Aj = {at ∈ A : pj (at ) = 0}, then pj+1 is 0 on A \ Aj .
for each Hi , there exists a minimum j such that Pj (Hi ) =
This minimum level is `(i): the zero-layer of Hi .
P
a∈Hi
pj (a) > 0.
The nonstandard probability P ∗ respects zero-layers. In fact P ∗ is
defined such that, for each Hi , Hz ,
if `(i) < `(z), then P ∗ (Hz ) << P ∗ (Hi ),
i.e. P ∗ (Hz )/P ∗ (Hi ) is a positive infinitesimal.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
9 / 26
Introduction
Lexicographic and non-standard probability
Idea of the proof
The class A = {a1 , . . . , am } of atoms generated by the events Ei , Hi can be
stratified in a hierarchy defined by r + 1 probability distributions p0 , . . . , pr , as
follows:
for each j, pj : A → [0, 1],
if Aj = {at ∈ A : pj (at ) = 0}, then pj+1 is 0 on A \ Aj .
for each Hi , there exists a minimum j such that Pj (Hi ) =
This minimum level is `(i): the zero-layer of Hi .
P
a∈Hi
pj (a) > 0.
The nonstandard probability P ∗ respects zero-layers. In fact P ∗ is
defined such that, for each Hi , Hz ,
if `(i) < `(z), then P ∗ (Hz ) << P ∗ (Hi ),
i.e. P ∗ (Hz )/P ∗ (Hi ) is a positive infinitesimal.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
9 / 26
Introduction
Lexicographic and non-standard probability
Idea of the proof
The class A = {a1 , . . . , am } of atoms generated by the events Ei , Hi can be
stratified in a hierarchy defined by r + 1 probability distributions p0 , . . . , pr , as
follows:
for each j, pj : A → [0, 1],
if Aj = {at ∈ A : pj (at ) = 0}, then pj+1 is 0 on A \ Aj .
for each Hi , there exists a minimum j such that Pj (Hi ) =
This minimum level is `(i): the zero-layer of Hi .
P
a∈Hi
pj (a) > 0.
The nonstandard probability P ∗ respects zero-layers. In fact P ∗ is
defined such that, for each Hi , Hz ,
if `(i) < `(z), then P ∗ (Hz ) << P ∗ (Hi ),
i.e. P ∗ (Hz )/P ∗ (Hi ) is a positive infinitesimal.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
9 / 26
Introduction
Lexicographic and non-standard probability
Idea of the proof
The class A = {a1 , . . . , am } of atoms generated by the events Ei , Hi can be
stratified in a hierarchy defined by r + 1 probability distributions p0 , . . . , pr , as
follows:
for each j, pj : A → [0, 1],
if Aj = {at ∈ A : pj (at ) = 0}, then pj+1 is 0 on A \ Aj .
for each Hi , there exists a minimum j such that Pj (Hi ) =
This minimum level is `(i): the zero-layer of Hi .
P
a∈Hi
pj (a) > 0.
The nonstandard probability P ∗ respects zero-layers. In fact P ∗ is
defined such that, for each Hi , Hz ,
if `(i) < `(z), then P ∗ (Hz ) << P ∗ (Hi ),
i.e. P ∗ (Hz )/P ∗ (Hi ) is a positive infinitesimal.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
9 / 26
Introduction
Lexicographic and non-standard probability
G. Coletti and R. Scozzafava, Probabilistic Logic in a Coherent Setting.
Trends in Logic, vol. 15, Kluwer, 2002.
J. H. Halpern, Lexicographic probability, conditional probability, and
nonstandard probability. In Proceedings of the Eighth Conference on
Theoretical Aspects of Rationality and Knowledge, 2001, pp. 17–30.
[arXiv:cs/0306106v2]
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
10 / 26
The case of Łukasiewicz Events
Łukasiewicz events and states
An MV-algebra is a structure A = (A, ⊕, ¬, ⊥) of type (2, 1, 0) such that
(A, ⊕, ⊥) is a commutative monoid with neutral element ⊥, and the following
equations hold:
¬(¬x) = x
x ⊕ > = >, where > = ¬⊥
x ⊕ ¬(x ⊕ ¬y ) = y ⊕ ¬(y ⊕ ¬x)
(?) The real unit interval [0, 1] with operations ⊕ and ¬ defined by
x ⊕ y = min{1, x + y }, and ¬x = 1 − x
is an MV-algebra denoted by [0, 1]MV and called the standard MV-algebra.
(?) Fix a k ∈ N. Then the set of all functions from [0, 1]k into [0, 1] that are
continuous, piecewise linear, and such that each piece has integer coefficient,
with the operations ⊕ and ¬ defined as a pointwise application of those of
[0, 1]MV is an MV-algebra (actually the free MV-algebra over k generators).
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
11 / 26
The case of Łukasiewicz Events
Łukasiewicz events and states
An MV-algebra is a structure A = (A, ⊕, ¬, ⊥) of type (2, 1, 0) such that
(A, ⊕, ⊥) is a commutative monoid with neutral element ⊥, and the following
equations hold:
¬(¬x) = x
x ⊕ > = >, where > = ¬⊥
x ⊕ ¬(x ⊕ ¬y ) = y ⊕ ¬(y ⊕ ¬x)
(?) The real unit interval [0, 1] with operations ⊕ and ¬ defined by
x ⊕ y = min{1, x + y }, and ¬x = 1 − x
is an MV-algebra denoted by [0, 1]MV and called the standard MV-algebra.
(?) Fix a k ∈ N. Then the set of all functions from [0, 1]k into [0, 1] that are
continuous, piecewise linear, and such that each piece has integer coefficient,
with the operations ⊕ and ¬ defined as a pointwise application of those of
[0, 1]MV is an MV-algebra (actually the free MV-algebra over k generators).
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
11 / 26
The case of Łukasiewicz Events
Łukasiewicz events and states
An MV-algebra is a structure A = (A, ⊕, ¬, ⊥) of type (2, 1, 0) such that
(A, ⊕, ⊥) is a commutative monoid with neutral element ⊥, and the following
equations hold:
¬(¬x) = x
x ⊕ > = >, where > = ¬⊥
x ⊕ ¬(x ⊕ ¬y ) = y ⊕ ¬(y ⊕ ¬x)
(?) The real unit interval [0, 1] with operations ⊕ and ¬ defined by
x ⊕ y = min{1, x + y }, and ¬x = 1 − x
is an MV-algebra denoted by [0, 1]MV and called the standard MV-algebra.
(?) Fix a k ∈ N. Then the set of all functions from [0, 1]k into [0, 1] that are
continuous, piecewise linear, and such that each piece has integer coefficient,
with the operations ⊕ and ¬ defined as a pointwise application of those of
[0, 1]MV is an MV-algebra (actually the free MV-algebra over k generators).
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
11 / 26
The case of Łukasiewicz Events
The free MV-algebra Free(k ) over k generators, is the Lindenbaum algebra of
Łukasiewicz propositional logic.
A Łukasiewicz event will be, for us, any equivalence class of formulas [θ], that
is, any McNaughton function f ∈ Free(k ).
A state [6] on an MV-algebra A is any map s : A → [0, 1] such that
s(>) = 1,
whenever x y = ⊥, then s(x ⊕ y ) = s(x) + s(y ).
A state s : A → [0, 1] is said to be faithful if s(x) = 0, implies x = ⊥.
A state s : A → ∗ [0, 1] is said to be a hyperstate.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
12 / 26
The case of Łukasiewicz Events
The free MV-algebra Free(k ) over k generators, is the Lindenbaum algebra of
Łukasiewicz propositional logic.
A Łukasiewicz event will be, for us, any equivalence class of formulas [θ], that
is, any McNaughton function f ∈ Free(k ).
A state [6] on an MV-algebra A is any map s : A → [0, 1] such that
s(>) = 1,
whenever x y = ⊥, then s(x ⊕ y ) = s(x) + s(y ).
A state s : A → [0, 1] is said to be faithful if s(x) = 0, implies x = ⊥.
A state s : A → ∗ [0, 1] is said to be a hyperstate.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
12 / 26
The case of Łukasiewicz Events
The free MV-algebra Free(k ) over k generators, is the Lindenbaum algebra of
Łukasiewicz propositional logic.
A Łukasiewicz event will be, for us, any equivalence class of formulas [θ], that
is, any McNaughton function f ∈ Free(k ).
A state [6] on an MV-algebra A is any map s : A → [0, 1] such that
s(>) = 1,
whenever x y = ⊥, then s(x ⊕ y ) = s(x) + s(y ).
A state s : A → [0, 1] is said to be faithful if s(x) = 0, implies x = ⊥.
A state s : A → ∗ [0, 1] is said to be a hyperstate.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
12 / 26
The case of Łukasiewicz Events
The free MV-algebra Free(k ) over k generators, is the Lindenbaum algebra of
Łukasiewicz propositional logic.
A Łukasiewicz event will be, for us, any equivalence class of formulas [θ], that
is, any McNaughton function f ∈ Free(k ).
A state [6] on an MV-algebra A is any map s : A → [0, 1] such that
s(>) = 1,
whenever x y = ⊥, then s(x ⊕ y ) = s(x) + s(y ).
A state s : A → [0, 1] is said to be faithful if s(x) = 0, implies x = ⊥.
A state s : A → ∗ [0, 1] is said to be a hyperstate.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
12 / 26
The case of Łukasiewicz Events
The free MV-algebra Free(k ) over k generators, is the Lindenbaum algebra of
Łukasiewicz propositional logic.
A Łukasiewicz event will be, for us, any equivalence class of formulas [θ], that
is, any McNaughton function f ∈ Free(k ).
A state [6] on an MV-algebra A is any map s : A → [0, 1] such that
s(>) = 1,
whenever x y = ⊥, then s(x ⊕ y ) = s(x) + s(y ).
A state s : A → [0, 1] is said to be faithful if s(x) = 0, implies x = ⊥.
A state s : A → ∗ [0, 1] is said to be a hyperstate.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
12 / 26
Conditional probability (states) on MV-algebras
There are several attempts to generalize conditional probability (conditional
states) on MV-algebras
Gerla [1]: Axiomatic approach to conditional probability on MV-algebras.
A conditional probability s(· | ·) is a primitive notion.
Kroupa [2]: Conditional probability is definable from a simple probability:
s(f | g) =
s(f · g)
,
s(g)
whenever s(g) > 0. (Montagna [3] provided a characterization of
Kroupa’s approach in terms of a not-sure loss principle)
Mundici [7, 8]: Conditional probability as a state on a quotient algebra
(the quotient is obtained by forcing an antecedent), and Rényi
conditional probability on MV-algebras.
Montagna et al. [5]: Stable coherence, and characterization of stable
coherent (complete) assignments through unconditional hyperstates (i.e.
nonstandard-valued states).
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
13 / 26
Conditional probability (states) on MV-algebras
There are several attempts to generalize conditional probability (conditional
states) on MV-algebras
Gerla [1]: Axiomatic approach to conditional probability on MV-algebras.
A conditional probability s(· | ·) is a primitive notion.
Kroupa [2]: Conditional probability is definable from a simple probability:
s(f | g) =
s(f · g)
,
s(g)
whenever s(g) > 0. (Montagna [3] provided a characterization of
Kroupa’s approach in terms of a not-sure loss principle)
Mundici [7, 8]: Conditional probability as a state on a quotient algebra
(the quotient is obtained by forcing an antecedent), and Rényi
conditional probability on MV-algebras.
Montagna et al. [5]: Stable coherence, and characterization of stable
coherent (complete) assignments through unconditional hyperstates (i.e.
nonstandard-valued states).
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
13 / 26
Conditional probability (states) on MV-algebras
There are several attempts to generalize conditional probability (conditional
states) on MV-algebras
Gerla [1]: Axiomatic approach to conditional probability on MV-algebras.
A conditional probability s(· | ·) is a primitive notion.
Kroupa [2]: Conditional probability is definable from a simple probability:
s(f | g) =
s(f · g)
,
s(g)
whenever s(g) > 0. (Montagna [3] provided a characterization of
Kroupa’s approach in terms of a not-sure loss principle)
Mundici [7, 8]: Conditional probability as a state on a quotient algebra
(the quotient is obtained by forcing an antecedent), and Rényi
conditional probability on MV-algebras.
Montagna et al. [5]: Stable coherence, and characterization of stable
coherent (complete) assignments through unconditional hyperstates (i.e.
nonstandard-valued states).
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
13 / 26
Conditional probability (states) on MV-algebras
There are several attempts to generalize conditional probability (conditional
states) on MV-algebras
Gerla [1]: Axiomatic approach to conditional probability on MV-algebras.
A conditional probability s(· | ·) is a primitive notion.
Kroupa [2]: Conditional probability is definable from a simple probability:
s(f | g) =
s(f · g)
,
s(g)
whenever s(g) > 0. (Montagna [3] provided a characterization of
Kroupa’s approach in terms of a not-sure loss principle)
Mundici [7, 8]: Conditional probability as a state on a quotient algebra
(the quotient is obtained by forcing an antecedent), and Rényi
conditional probability on MV-algebras.
Montagna et al. [5]: Stable coherence, and characterization of stable
coherent (complete) assignments through unconditional hyperstates (i.e.
nonstandard-valued states).
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
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13 / 26
Conditional probability (states) on MV-algebras
There are several attempts to generalize conditional probability (conditional
states) on MV-algebras
Gerla [1]: Axiomatic approach to conditional probability on MV-algebras.
A conditional probability s(· | ·) is a primitive notion.
Kroupa [2]: Conditional probability is definable from a simple probability:
s(f | g) =
s(f · g)
,
s(g)
whenever s(g) > 0. (Montagna [3] provided a characterization of
Kroupa’s approach in terms of a not-sure loss principle)
Mundici [7, 8]: Conditional probability as a state on a quotient algebra
(the quotient is obtained by forcing an antecedent), and Rényi
conditional probability on MV-algebras.
Montagna et al. [5]: Stable coherence, and characterization of stable
coherent (complete) assignments through unconditional hyperstates (i.e.
nonstandard-valued states).
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
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Conditional probability (states) on MV-algebras
Montagna’s result
A complete real-valued assignment
Λ : f1 | g1 7→ α1 , . . . , fn | gn 7→ αn , g1 7→ β1 , . . . , gn 7→ βn
is stably coherent if there is another assessment
Λ0 : f1 | g1 7→ α10 , . . . , fn | gn 7→ αn0 , g1 7→ β10 , . . . , gn 7→ βn0
such that α10 , . . . , αn0 , β10 , . . . , βn0 belong to a nonstandard extension ∗ [0, 1] of
[0, 1], and in addition:
(i) Λ and Λ0 differ by an infinitesimal, that is, for i = 1, . . . , n, |αi0 − αi | and
|βi0 − βi | are infinitesimal;
(ii) for i = 1, . . . n, βi0 > 0
(iii) Λ0 avoids sure loss.
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Conditional probability (states) on MV-algebras
Montagna’s result
Theorem
Let f1 . . . fn , g1 , . . . gn be Łukasiewicz events, and let
Λ : fi | gi 7→ αi , gi 7→ βi (i = 1, . . . , n)
be a complete assignment. Then the following are equivalent:
Λ is stably coherent;
There exists a faithful hyperstate s∗ such that
For every i, St(s∗ (gi )) = βi ;
For every i, St(s∗ (fi · gi )) = αi · βi .
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Conditional probability (states) on MV-algebras
Montagna’s result
Theorem
Let f1 . . . fn , g1 , . . . gn be Łukasiewicz events, and let
Λ : fi | gi 7→ αi , gi 7→ βi (i = 1, . . . , n)
be a complete assignment. Then the following are equivalent:
Λ is stably coherent;
There exists a faithful hyperstate s∗ such that
For every i, St(s∗ (gi )) = βi ;
For every i, St(s∗ (fi · gi )) = αi · βi .
Flaminio, Godo (IIIA - CSIC)
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Conditional probability (states) on MV-algebras
Montagna’s result
Theorem
Let f1 . . . fn , g1 , . . . gn be Łukasiewicz events, and let
Λ : fi | gi 7→ αi , gi 7→ βi (i = 1, . . . , n)
be a complete assignment. Then the following are equivalent:
Λ is stably coherent;
There exists a faithful hyperstate s∗ such that
For every i, St(s∗ (gi )) = βi ;
For every i, St(s∗ (fi · gi )) = αi · βi .
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Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
Outline
1
Introduction
Betting on conditional events
Lexicographic and non-standard probability
2
The case of Łukasiewicz Events
3
Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
4
Future work
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Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
Let f1 , g1 , . . . , fn , gn be Łukasiewicz events in Free(k ).
Let ∆ be a minimal unimodular triangulation of the hypercube [0, 1]k that
linearizes each fi and gi . Also let
Ver (∆) = {x1 , . . . , xm }
be the set of vertices of ∆.
Let h1 , . . . , hm be the normalized Shauder hats corresponding to the
vertices in Ver (∆). Each hi is a McNaughton function, and hence
hi ∈ Free(k ).
Lm
- For distinct hi , hj ∈ H, hi hj = 0, and t=1 ht = 1;
- For each i = 1, . . . , n,
m
m
M
M
fi =
ht · fi (xt ), and gi =
ht · gi (xt ).
t=1
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t=1
ManyVal 2012
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Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
Let f1 , g1 , . . . , fn , gn be Łukasiewicz events in Free(k ).
Let ∆ be a minimal unimodular triangulation of the hypercube [0, 1]k that
linearizes each fi and gi . Also let
Ver (∆) = {x1 , . . . , xm }
be the set of vertices of ∆.
Let h1 , . . . , hm be the normalized Shauder hats corresponding to the
vertices in Ver (∆). Each hi is a McNaughton function, and hence
hi ∈ Free(k ).
Lm
- For distinct hi , hj ∈ H, hi hj = 0, and t=1 ht = 1;
- For each i = 1, . . . , n,
m
m
M
M
fi =
ht · fi (xt ), and gi =
ht · gi (xt ).
t=1
Flaminio, Godo (IIIA - CSIC)
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t=1
ManyVal 2012
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Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
Let f1 , g1 , . . . , fn , gn be Łukasiewicz events in Free(k ).
Let ∆ be a minimal unimodular triangulation of the hypercube [0, 1]k that
linearizes each fi and gi . Also let
Ver (∆) = {x1 , . . . , xm }
be the set of vertices of ∆.
Let h1 , . . . , hm be the normalized Shauder hats corresponding to the
vertices in Ver (∆). Each hi is a McNaughton function, and hence
hi ∈ Free(k ).
Lm
- For distinct hi , hj ∈ H, hi hj = 0, and t=1 ht = 1;
- For each i = 1, . . . , n,
m
m
M
M
fi =
ht · fi (xt ), and gi =
ht · gi (xt ).
t=1
Flaminio, Godo (IIIA - CSIC)
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t=1
ManyVal 2012
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Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
Let Free(k )+ be the MV-algebra generated by
Free(k ) ∪ {fi · gi : i = 1, . . . , n}.
Every state s on Free(k ) can be extended to a state (s)+ on Free(k )+ by
stipulating: for every p ∈ Free(k )+
(s)+ (p) =
m
X
t=1
s(ht ) · p(xt ).
In a similar way, every hyperstate s∗ on Free(k ) extends to a hyperstate
(s∗ )+ on Free(k )+ .
Given a class {d0 , . . . P
, dr } of mappings from {h1 , . . . , hm } into [0, 1]
m
satisfying, for each j, t=1 dj (ht ) = 1 (we will henceforth call them
distributions), we define the zero-layer of a function p in Free(k )+ as
`(p) = min{j : ∃t ≤ m, dj (ht ) > 0, p(xt ) > 0}
if such a j exists, and `(p) = ∞ otherwise.
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Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
Let Free(k )+ be the MV-algebra generated by
Free(k ) ∪ {fi · gi : i = 1, . . . , n}.
Every state s on Free(k ) can be extended to a state (s)+ on Free(k )+ by
stipulating: for every p ∈ Free(k )+
(s)+ (p) =
m
X
t=1
s(ht ) · p(xt ).
In a similar way, every hyperstate s∗ on Free(k ) extends to a hyperstate
(s∗ )+ on Free(k )+ .
Given a class {d0 , . . . P
, dr } of mappings from {h1 , . . . , hm } into [0, 1]
m
satisfying, for each j, t=1 dj (ht ) = 1 (we will henceforth call them
distributions), we define the zero-layer of a function p in Free(k )+ as
`(p) = min{j : ∃t ≤ m, dj (ht ) > 0, p(xt ) > 0}
if such a j exists, and `(p) = ∞ otherwise.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
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Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
Let Free(k )+ be the MV-algebra generated by
Free(k ) ∪ {fi · gi : i = 1, . . . , n}.
Every state s on Free(k ) can be extended to a state (s)+ on Free(k )+ by
stipulating: for every p ∈ Free(k )+
(s)+ (p) =
m
X
t=1
s(ht ) · p(xt ).
In a similar way, every hyperstate s∗ on Free(k ) extends to a hyperstate
(s∗ )+ on Free(k )+ .
Given a class {d0 , . . . P
, dr } of mappings from {h1 , . . . , hm } into [0, 1]
m
satisfying, for each j, t=1 dj (ht ) = 1 (we will henceforth call them
distributions), we define the zero-layer of a function p in Free(k )+ as
`(p) = min{j : ∃t ≤ m, dj (ht ) > 0, p(xt ) > 0}
if such a j exists, and `(p) = ∞ otherwise.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
ManyVal 2012
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Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
Theorem
Let f1 | g1 , . . . , fn | gn be as above, and let χ : fi | gi 7→ αi , gi 7→ βi (for
i = 1, . . . , n) be a real-valued complete assignment. Then the following are
equivalent:
(i) There exists a faithful hyperstate s∗ : Free(k ) → ∗ [0, 1] such that for
every gi , St(s∗ (gi )) = βi , and for every i = 1, . . . , n,
∗ +
(s ) (fi · gi )
αi = St
.
s∗ (gi )
(ii) There exist states s0 , . . . , sr over Free(k ) such that, for every
i = 1, . . . , n, there exists `(i) ∈ {0, . . . , r } such that s`(i) (gi ) > 0.
Moreover, if βx > 0, `(gx ) = 0, s0 (gx ) = βx ; and for every i,
αi =
Flaminio, Godo (IIIA - CSIC)
(s`(i) )+ (fi · gi )
.
s`(i) (gi )
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Conditional probability (states) on MV-algebras
∗
Defining layers of zero-probability for Łukasiewicz events
∗
(⇒). Let s : Free(k ) → [0, 1] be a faithful hyperstate, and let H the class of
normalized Schauder hats given by fi , gi .
(1) Define:
H0 = H, and d0 : H → [0, 1] as d0 (h) = St(s∗ (h));
Hi+1 = {h ∈ Hi : di (h) = 0}.
If Hi+1 = ∅, then we stop;
L
{h ∈ Hi+1 }, and
∗
s (h)
di+1 (h) = St
.
s∗ (Φi+1 )
If Hi+1 6= ∅, define Φi+1 =
(2) The process stops in finitely many steps, giving a class of distributions
d0 , . . . , dr such that, for each gi , `(gi ) < ∞.
(3) It holds
αi ·
m
X
t=1
Flaminio, Godo (IIIA - CSIC)
d`(gi ) (ht ) · gi (xt ) =
m
X
t=1
d`(gi ) (ht ) · (fi · gi )(xt ).
ManyVal - Salerno, July 2012
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Conditional probability (states) on MV-algebras
∗
Defining layers of zero-probability for Łukasiewicz events
∗
(⇒). Let s : Free(k ) → [0, 1] be a faithful hyperstate, and let H the class of
normalized Schauder hats given by fi , gi .
(1) Define:
H0 = H, and d0 : H → [0, 1] as d0 (h) = St(s∗ (h));
Hi+1 = {h ∈ Hi : di (h) = 0}.
If Hi+1 = ∅, then we stop;
L
{h ∈ Hi+1 }, and
∗
s (h)
di+1 (h) = St
.
s∗ (Φi+1 )
If Hi+1 6= ∅, define Φi+1 =
(2) The process stops in finitely many steps, giving a class of distributions
d0 , . . . , dr such that, for each gi , `(gi ) < ∞.
(3) It holds
αi ·
m
X
t=1
Flaminio, Godo (IIIA - CSIC)
d`(gi ) (ht ) · gi (xt ) =
m
X
t=1
d`(gi ) (ht ) · (fi · gi )(xt ).
ManyVal - Salerno, July 2012
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Conditional probability (states) on MV-algebras
∗
Defining layers of zero-probability for Łukasiewicz events
∗
(⇒). Let s : Free(k ) → [0, 1] be a faithful hyperstate, and let H the class of
normalized Schauder hats given by fi , gi .
(1) Define:
H0 = H, and d0 : H → [0, 1] as d0 (h) = St(s∗ (h));
Hi+1 = {h ∈ Hi : di (h) = 0}.
If Hi+1 = ∅, then we stop;
L
{h ∈ Hi+1 }, and
∗
s (h)
di+1 (h) = St
.
s∗ (Φi+1 )
If Hi+1 6= ∅, define Φi+1 =
(2) The process stops in finitely many steps, giving a class of distributions
d0 , . . . , dr such that, for each gi , `(gi ) < ∞.
(3) It holds
αi ·
m
X
t=1
Flaminio, Godo (IIIA - CSIC)
d`(gi ) (ht ) · gi (xt ) =
m
X
t=1
d`(gi ) (ht ) · (fi · gi )(xt ).
ManyVal - Salerno, July 2012
ManyVal 2012
20 / 26
Conditional probability (states) on MV-algebras
∗
Defining layers of zero-probability for Łukasiewicz events
∗
(⇒). Let s : Free(k ) → [0, 1] be a faithful hyperstate, and let H the class of
normalized Schauder hats given by fi , gi .
(1) Define:
H0 = H, and d0 : H → [0, 1] as d0 (h) = St(s∗ (h));
Hi+1 = {h ∈ Hi : di (h) = 0}.
If Hi+1 = ∅, then we stop;
L
{h ∈ Hi+1 }, and
∗
s (h)
di+1 (h) = St
.
s∗ (Φi+1 )
If Hi+1 6= ∅, define Φi+1 =
(2) The process stops in finitely many steps, giving a class of distributions
d0 , . . . , dr such that, for each gi , `(gi ) < ∞.
(3) It holds
αi ·
m
X
t=1
Flaminio, Godo (IIIA - CSIC)
d`(gi ) (ht ) · gi (xt ) =
m
X
t=1
d`(gi ) (ht ) · (fi · gi )(xt ).
ManyVal - Salerno, July 2012
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20 / 26
Conditional probability (states) on MV-algebras
∗
Defining layers of zero-probability for Łukasiewicz events
∗
(⇒). Let s : Free(k ) → [0, 1] be a faithful hyperstate, and let H the class of
normalized Schauder hats given by fi , gi .
(1) Define:
H0 = H, and d0 : H → [0, 1] as d0 (h) = St(s∗ (h));
Hi+1 = {h ∈ Hi : di (h) = 0}.
If Hi+1 = ∅, then we stop;
L
{h ∈ Hi+1 }, and
∗
s (h)
di+1 (h) = St
.
s∗ (Φi+1 )
If Hi+1 6= ∅, define Φi+1 =
(2) The process stops in finitely many steps, giving a class of distributions
d0 , . . . , dr such that, for each gi , `(gi ) < ∞.
(3) It holds
αi ·
m
X
t=1
Flaminio, Godo (IIIA - CSIC)
d`(gi ) (ht ) · gi (xt ) =
m
X
t=1
d`(gi ) (ht ) · (fi · gi )(xt ).
ManyVal - Salerno, July 2012
ManyVal 2012
20 / 26
Conditional probability (states) on MV-algebras
∗
Defining layers of zero-probability for Łukasiewicz events
∗
(⇒). Let s : Free(k ) → [0, 1] be a faithful hyperstate, and let H the class of
normalized Schauder hats given by fi , gi .
(1) Define:
H0 = H, and d0 : H → [0, 1] as d0 (h) = St(s∗ (h));
Hi+1 = {h ∈ Hi : di (h) = 0}.
If Hi+1 = ∅, then we stop;
L
{h ∈ Hi+1 }, and
∗
s (h)
di+1 (h) = St
.
s∗ (Φi+1 )
If Hi+1 6= ∅, define Φi+1 =
(2) The process stops in finitely many steps, giving a class of distributions
d0 , . . . , dr such that, for each gi , `(gi ) < ∞.
(3) It holds
αi ·
m
X
t=1
Flaminio, Godo (IIIA - CSIC)
d`(gi ) (ht ) · gi (xt ) =
m
X
t=1
d`(gi ) (ht ) · (fi · gi )(xt ).
ManyVal - Salerno, July 2012
ManyVal 2012
20 / 26
Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
(⇐). Let d0 , . . . , dr distributions on H whose associated states s0 , . . . , sr are
as in (ii).
Let ε > 0 be a positive infinitesimal, and define s∗ : Free(k ) → ∗ [0, 1] as:
s∗ (f ) = K ·
m
X
t=1
ε`(t) · d`(t) (ht ) · f (xt )
where
K =
m
X
t=1
Flaminio, Godo (IIIA - CSIC)
!−1
ε
`(t)
· d`(t) (ht )
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Conditional probability (states) on MV-algebras
Defining layers of zero-probability for Łukasiewicz events
(⇐). Let d0 , . . . , dr distributions on H whose associated states s0 , . . . , sr are
as in (ii).
Let ε > 0 be a positive infinitesimal, and define s∗ : Free(k ) → ∗ [0, 1] as:
s∗ (f ) = K ·
m
X
t=1
ε`(t) · d`(t) (ht ) · f (xt )
where
K =
m
X
t=1
Flaminio, Godo (IIIA - CSIC)
!−1
ε
`(t)
· d`(t) (ht )
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Future work
Future work
- Complexity for the problem of establishing the coherence of a stable
coherent complete assignment using zero-layers
- Find an axiomatization of conditional states on MV-algebras characterizing
stable coherent complete assignments;
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Future work
Future work
- Complexity for the problem of establishing the coherence of a stable
coherent complete assignment using zero-layers
- Find an axiomatization of conditional states on MV-algebras characterizing
stable coherent complete assignments;
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Future work
Future work
- Complexity for the problem of establishing the coherence of a stable
coherent complete assignment using zero-layers
- Find an axiomatization of conditional states on MV-algebras characterizing
stable coherent complete assignments;
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Future work
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9.$5&5#:#4;*
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Future work
0?56@A@?5!9*BC?D!D@9@AE*
==*F!5E=G!9HI6*IGI5A+*==*
P (A · B) = P (A | B) · P (B)
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J*
+,&-./*0$1/2/3(/*4**
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J*
9/"#($:2&;1#(**
;2$-&-#.#,<*
Flaminio, Godo (IIIA - CSIC)
5$3=),&3>&2>**
;2$-&-#.#,<*
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Future work
References:
B. Gerla, Conditioning a State by a Łukasiewicz Event: A Probabilistic
Approach to Ulam Games. Theor. Comput. Sci. 230(1-2): 149-166 (2000)
T. Kroupa, Conditional probability on MV-algebras. Fuzzy Sets and
Systems 149(2): 369-381 (2005).
F. Montagna, A Notion of Coherence for Books on Conditional Events in
Many-valued Logic. J. Log. Comput. 21(5): 829-850 (2011).
F. Montagna, Partially Undetermined Many-Valued Events and Their
Conditional Probability. J. Philosophical Logic 41(3): 563-593 (2012).
Flaminio, Godo (IIIA - CSIC)
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Future work
F. Montagna, M. Fedel, G. Scianna, Non-standard probability, coherence
and conditional probability on many-valued events. Manuscript, 2012.
D. Mundici, Averaging the truth-value in Lukasiewicz logic. Studia Logica
55(1): 113-127 (1995).
D. Mundici, Bookmaking over infinite-valued events. Int. J. Approx.
Reasoning 43(3): 223-240 (2006)
D. Mundici, Advanced Łukasiewicz calculus and MV-algebras, Trends in
Logic, Vol. 35, Springer 2011.
Flaminio, Godo (IIIA - CSIC)
ManyVal - Salerno, July 2012
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