Łukasiewicz logic and Riesz spaces

Abstract

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.

Infinitary logic and basically disconnected compact Hausdorff spaces

“…A Riesz MV-algebra [11] is a structure (R, ⊕, * , 0, {r | r ∈ [0, 1]}), where (R, ⊕, * , 0) is an MV-algebra and {r | r ∈ [0, 1]} is a family of unary operations such that the following properties hold for any x, y ∈ A and r, q ∈ [0, 1]: r(x ⊙ y * ) = (rx) ⊙ (ry) * , (r ⊙ q * ) · x = (rx) ⊙ (qx) * , r(qx) = (rq)x, 1x = x.…”

“…For what follows, a particularly important full subcategory of Riesz MValgebras is the one of norm-complete Riesz MV-algebras [11]. In any Riesz MV-algebra R it is possible to define the unit seminorm · u : R → [0, 1] by x u = min{r ∈ [0, 1] | x ≤ r1 R } for any x ∈ R. Such a seminorm induces a pseudometric ρ · u ; when R is semisimple, · u is a norm and ρ · u is a metric.…”

“…Finally, our system enjoys standard completeness with respect to the real interval [0, 1].[25] led to a categorical equivalence between Riesz MV-algebras and Riesz spaces with strong unit. Consequently, one can develop a logical system that extends Lukasiewicz logic and has Riesz MV-algebras as models [11].The strong connection between functional analysis and Riesz spaces should be reflected by their underlying logical systems and in the present paper we test the expressive power of the "logic of Riesz Spaces" in order to provide a logical system that is closer to what we may think of as the "logic of (some) Hausdorff spaces". Remarkably, by adding a countable disjunction to the logic of Riesz MV-algebras we were able to obtain the desired bridge between logic, topology and functional analysis.Our construction rests on Kakutani's duality between abstract M-spaces and compact Hausdorff spaces [18].…”

“…Not very known is the interplay between the theory of Riesz spaces and Lukasiewicz logic: given any positive element u of a Riesz space V , the interval [0, u] = {x ∈ V | 0 ≤ x ≤ u} can be endowed with a stucture of Riesz MValgebra [11]. These structures are expansions of MV-algebras, the standard semantics of the ∞-valued Lukasiewicz logic [8,27].…”

“…Remarkably, by adding a countable disjunction to the logic of Riesz MV-algebras we were able to obtain the desired bridge between logic, topology and functional analysis.Our construction rests on Kakutani's duality between abstract M-spaces and compact Hausdorff spaces [18]. Indeed, from a categorical point of view, the norm-complete Riesz MV-algebras 1 are equivalent to M-spaces, and henceforth dual to compact Hausdorff spaces [11]. Our aim is to express the property of being "norm-complete" in a logical setting.…”