De Finetti theorem and Borel states in [0,1]-valued algebraic logic

Kühr, Jan; Mundici, Daniele

doi:10.1016/j.ijar.2007.02.005

Cited by 68 publications

(46 citation statements)

References 9 publications

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“…We recall the MV-algebraic approach to de Finetti's notion of coherence [19] and we prove a similar coherence criterion for Riesz MV-algebras. We also provide a logical expression of the coherence criterion when the events are represented by formulas RL.…”

Section: Linear Combinations Of Formulas and De Finetti's Coherence Cmentioning

confidence: 99%

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Łukasiewicz logic and Riesz spaces

Nola

Leustean

2014

Soft Comput

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We initiate a deep study of Riesz MV-algebras which are MV-algebras endowed with a scalar multiplication with scalars from [0, 1]. Extending Mundici's equivalence between MV-algebras and ℓ-groups, we prove that Riesz MV-algebras are categorically equivalent with unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent with the class of commutative unital C * -algebras. The propositional calculus RL that has Riesz MV-algebras as models is a conservative extension of Lukasiewicz ∞-valued propositional calculus and it is complete with respect to evaluations in the standard model [0, 1]. We prove a normal form theorem for this logic, extending McNaughton theorem for Lukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in RL and we relate them with the analogue of de Finetti's coherence criterion for RL.

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Section: Linear Combinations Of Formulas and De Finetti's Coherence Cmentioning

confidence: 99%

“…The following definition generalizes de Finetti's notion of coherence and provides an algebraic approach within Lukasiewicz logic [19]. By Remark 3.12, any morphsim of Riesz MV-algebras is just a morphism between the MV-algebra reducts of its domain and codomain.…”

Section: Linear Combinations Of Formulas and De Finetti's Coherence Cmentioning

confidence: 99%

Łukasiewicz logic and Riesz spaces

Nola

Leustean

2014

Soft Comput

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“…In this section, we will generalize to pseudo-BCK algebras the identity between de Finetti maps and Bosbach states, following the results proved by Kühr and Mundici in [28] who showed that de Finetti's coherence principle that has an origin in Dutch book making, has a strong relationship with MV-states on MV-algebras. Then we generalize this also for state-measures on unital pseudo-BCK algebras that are downwardsdirected.…”

Section: Coherence De Finetti Maps and Borel Statesmentioning

confidence: 99%

“…We recall the following definition and notations used in [28]. Let A be a nonempty set, [0,1] A the set of all functions V : A → [0, 1] endowed with the product topology.…”

Section: Coherence De Finetti Maps and Borel Statesmentioning

confidence: 99%

Measures, states and de Finetti maps on pseudo-BCK algebras

Ciungu

Dvurečenskij

2010

Fuzzy Sets and Systems

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Abstract. In this paper, we extend the notions of states and measures presented in [12] to the case of pseudo-BCK algebras and study similar properties. We prove that, under some conditions, the notion of a state in the sense of [12] coincides with the Bosbach state, and we extend to the case of pseudo-BCK algebras some results proved by J. Kühr only for pseudo-BCK semilattices. We characterize extremal states, and show that the quotient pseudo-BCK algebra over the kernel of a measure can be embedded into the negative cone of an archimedean ℓ-group. Additionally, we introduce a Borel state and using results by J. Kühr and D. Mundici from [28], we prove a relationship between de Finetti maps, Bosbach states and Borel states.

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“…In the last few years, the notion of a state has been studied by many experts in MV-algebras, e.g, [13,12].…”

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confidence: 99%

Effect algebras with state operator

Pulmannová¹

EPiC Series in Computing

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This paper is a joint work with Anna Jenčová.Effect algebras have been introduced by Foulis and Bennett [2] (see also [3,4] for equivalent definitions) for modeling unsharp measurements in quantum mechanical systems [5]. They are a generalization of many structures which arise in the axiomatization of quantum mechanics (Hilbert space effects [7]), noncommutative measure theory and probability (orthomodular lattices and posets, [6]), fuzzy measure theory and many-valued logic 9]).A state, as an analogue of a probability measure, is a basic notion in algebraic structures used in the quantum theories (see e.g., [8]), and properties of states have been deeply studied by many authors.In MV-algebras, states as averaging the truth value were first studied in [11]. In the last few years, the notion of a state has been studied by many experts in MV-algebras, e.g, [13,12].Another approach to the state theory on MV-algebras has been presented recently in [15]. Namely, a new unary operator was added to the MV-algebras structure as an internal state (or so-called state operator). MV-algebras with the added state operator are called state MValgebras. The idea is that an internal state has some properties reminiscent of states, but, while a state is a map from an MV-algebra into [0, 1], an internal state is an operator of the algebra. State MV-algebras generalize, for example, Hájek's approach [14] to fuzzy logic with modality P r (interpreted as probably with the following semantic interpretation: The probability of an event a is presented as the truth value of P r(a). For a more detailed motivation of state MV-algebras and their relation to logic, see [15].In [1], the notion of a state operator was extended from MV-algebras to the more general frame of effect algebras. A state operator is there defined as an additive, unital and idempotent operator on E. A state operator on E is called strong, if it satisfies the additional condition(1)Since MV-algebras form a special subclass of effect algebras, so-called MV-effect algebras, it was shown that the definition of a state operator on effect algebras coincides with the original definition on MV-algebras if and only if the state operator is strong. Moreover, if τ is faithful, i.e., τ (a) = 0 implies a = 0, then property (1) is automatically satisfied.In the present paper, we show that state operators on an effect algebra E are related with states on E in the following way: (1) every state on E induces a state operator on the tensor product [0, 1] ⊗ E. (2) If E admits an ordering set of states, then every state operator on E induces a state on E. We study state operators mainly on convex effect algebras. Convex effect algebras as effect algebras with an additional convexity structure were introduced and studied in [17,16]. It was proved in [17], that every convex effect algebra is isomorphic with the interval [0, u] in an ordered real linear space (V, V + ), where u is an order unit. Moreover, (V, u) is an order unit space, i.e. u is an archimedean order unit, if and only if E ad...

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De Finetti theorem and Borel states in [0,1]-valued algebraic logic

Cited by 68 publications

References 9 publications

Łukasiewicz logic and Riesz spaces

Łukasiewicz logic and Riesz spaces

Measures, states and de Finetti maps on pseudo-BCK algebras

Effect algebras with state operator

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