Conditional probability on MV-algebras

Kroupa, Tomáš

doi:10.1016/j.fss.2004.04.010

Cited by 28 publications

(20 citation statements)

References 8 publications

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“…Indeed, in the last years, different answers have been provided to the question "what is a conditional state?" (see [5,11,15]). Roughly speaking there have been two main approaches: in the first one, exploited in [5,11], and similarly to what happens e.g.…”

Section: Towards a Logic Of Conditional Probability For Fuzzy Eventsmentioning

confidence: 99%

“…• A weak probabilistic model for Ł∀ evaluates a modal formula P ( ) by means of a finitely additive measure (or state) defined over the MV-algebra of provably equivalent Łukasiewicz formulas (see [16][17][18][19][20][21] for a detailed definition of state over an MV-algebra).…”

Section: (Fp1) P (¬ ∨ ) → (P ( ) → P ( )) (Fp2) P (¬ ) ≡ ¬P ( ) (Fpmentioning

confidence: 99%

“…The other approach has been essentially developed by Kroupa in [15] where the notion of conditional state has been introduced as definable from the notion of state on a MV-algebra enriched with a product operation. A structure A = (A, ⊕, , ¬, 0) is an MV-algebra with product (those algebras are also called PMV-algebras in [19]) if A = (A, ⊕, ¬, 0) is an MV-algebra and is a commutative and associative binary operation on A such that for all a, b, c ∈ A:…”

Section: Towards a Logic Of Conditional Probability For Fuzzy Eventsmentioning

confidence: 99%

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A logic for reasoning about the probability of fuzzy events

Flaminio

Godo

2007

Fuzzy Sets and Systems

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In this paper we present the logic F P (Ł n , Ł) which allows to reason about the probability of fuzzy events formalized by means of the notion of state in a MV-algebra. This logic is defined starting from a basic idea exposed by Hájek [Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998]. Two kinds of semantics have been introduced, namely the class of weak and strong probabilistic models. The main result of this paper is a completeness theorem for the logic F P (Ł n , Ł) w.r.t. both weak and strong models. We also present two extensions of F P (Ł n , Ł): the first one is the logic F P (Ł n , RP L), obtained by expanding the F P (Ł n , Ł)-language with truth-constants for the rationals in [0, 1], while the second extension is the logic F CP (Ł n , Ł 1 2 ) allowing to reason about conditional states.

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Section: Towards a Logic Of Conditional Probability For Fuzzy Eventsmentioning

confidence: 99%

Section: (Fp1) P (¬ ∨ ) → (P ( ) → P ( )) (Fp2) P (¬ ) ≡ ¬P ( ) (Fpmentioning

confidence: 99%

Section: Towards a Logic Of Conditional Probability For Fuzzy Eventsmentioning

confidence: 99%

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A logic for reasoning about the probability of fuzzy events

Flaminio

Godo

2007

Fuzzy Sets and Systems

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“…Our next step will involve relaxing the Boolean constraint, a relaxation which implies a substantial generalization of the Rejection constraint as well and that may have a significant impact on our understanding of conditional many-valued probability, a topic to which considerable research effort has been devoted in the past decade, see e.g. [9,12,17,18]). Another interesting prospective (pointed out by one of the referees) is to look at the conditional as a partial operation on a Boolean algebra and apply techniques of theory of partial algebras [3].…”

Section: Conclusion and Further Workmentioning

confidence: 99%

On the Algebraic Structure of Conditional Events

Flaminio

Godo

Hosni

2015

Lecture Notes in Computer Science

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Abstract. This paper initiates an investigation of conditional measures as simple measures on conditional events. As a first step towards this end we investigate the construction of conditional algebras which allow us to distinguish between the logical properties of conditional events and those of the conditional measures which we can be attached to them. This distinction, we argue, helps us clarifying both concepts.

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“…Indeed, in the last years, different answers have been provided to the question "what is a conditional state?" (see Di Nola et al 1999;Gerla 2001;Kroupa 2005). Roughly speaking there have been two main approaches: in the first one, exploited in Di Nola et al (1999), Gerla (2001), and similarly to what happens e.g.…”

Section: Conditional Probabilitymentioning

confidence: 99%

Strong non-standard completeness for fuzzy logics

Flaminio¹

2007

Soft Comput

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In this paper we are going to introduce the notion of strong non-standard completeness (SNSC) for fuzzy logics. This notion naturally arises from the well known construction by ultraproduct. Roughly speaking, to say that a logic C is strong non-standard complete means that, for any countable theory over C and any formula ϕ such that C ϕ, there exists an evaluation e of C-formulas into a C-algebra A such that the universe of A is a non-Archimedean extension [0, 1] of the real unit interval [0, 1], e is a model for , but e(ϕ) < 1. Then we will apply SNSC to prove that various modal fuzzy logics allowing to deal with simple and conditional probability of infinite-valued events are complete with respect to classes of models defined starting from non-standard measures, that is measures taking value in [0, 1] .

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Conditional probability on MV-algebras

Cited by 28 publications

References 8 publications

A logic for reasoning about the probability of fuzzy events

A logic for reasoning about the probability of fuzzy events

On the Algebraic Structure of Conditional Events

Strong non-standard completeness for fuzzy logics

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