Towards a probability theory for product logic: States, integral representation and reasoning

Abstract

The aim of this paper is to extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel probability measure. Furthermore, the relation between states and measures is shown to be one-one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal logic for probabilistic reasoning on product logic events and prove soundness and completeness with respect to probabilistic spaces, where the algebra is a free product algebra and the measure is a state in the above sense.

Projectivity and unification in substructural logics of generalized rotations

“…The variety of Gödel algebras is known to indeed have strong unitary unification type (it follows for instance from the results in [29]), and the same result follows for nilpotent minimum algebras without negation fixpoint, since they are categorically equivalent to Gödel algebras ( [35]). Our approach allows to obtain these results uniformly, together with a new result about the whole variety of nilpotent minimum algebras.…”

“…Product logic has been introduced by Hájek, Godo, Esteva in [45], and has been deeply studied in recent years. In particular, relevant results have been obtained about: the functional representation of its free finitely generated algebras [26]; structural completeness [27]; categorical representation [52] and duality [36]; SMT-solvers [58]; modal extensions [59]; probability theory [35]. However, to the best of our knowledge, the unification type of product logic remained an open problem so far.…”

Corrigendum to “Towards a probability theory for product logic: States, integral representation and reasoning” [Int. J. Approx. Reason. 93 (2018) 199–218]

“…In our recent paper [2] we introduced states of free, finitely generated, product-algebras. For the sake of completeness, let us recall that, letting F P (n) be the free n-generated product algebra, a state of F P (n) is a map For a better reading of this note, let us recall our main result of [2] which shows an integral representation theorem for states of free, finitely generated, product algebras. Theorem 1.1 ([2, Corollary 4.10]).…”

Probability over Plonka sums of Boolean algebras: states, metrics and topology

“…Thanks to the introduction of states, it is possible to render the probability of certain fuzzy events. Subsequently, the theory of states have attracted successful attention and have been studied for the algebraic semantics of other fuzzy logics, such as G ödel-Dummet [4], G ödel ∆ [1], the logic of nilpotent minimum [3] and product logic [17]. Within the same strand of research, probability measures have been defined and studied also for other algebraic structures (connected to logic), such as Heyting algebras [41], De Morgan algebras [33], orthomodular lattices [5] and effect algebras [19].…”