Probability Measures in Gödel $$_\varDelta $$ Logic

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

“…the free n-generated product algebra, correspond, one-one, to regular Borel probability measures on [0, 1] n . 1 Moreover, and quite surprisingly since in the axiomatization of states the product t-norm operation is only indirectly involved via a condition concerning double negation, we prove that every state belongs to the convex closure of product logic valuations. Finally, these results will allow us to introduce a suitable class of probabilistic-like models with respect to which the modal logic we will introduce in Section 6 turns out to be sound and complete.…”

“…For instance, for n = 2, we have p (1,1) = ¬x 1 ∧ ¬x 2 , p (1,2) = ¬x 1 ∧ ¬¬x 2 , p (2,1) = ¬¬x 1 ∧ ¬x 2 , p (2,2) = ¬¬x 1 ∧ ¬¬x 2 , while the following Figure 1 shows how [0, 1] 2 is partitioned by G (1,1) , G (1,2) , G (2,1) and G (2,2) . Figure 1.…”

“…The aims of this contribution are the following: (1) we will introduce and study states for product logic -the remaining fundamental many-valued logic for which such a notion is still lacking-(2) we will prove that our axiomatization results in characterizing Lebesgue integrals of truth-functions of product logic formulas with respect to regular Borel probability measures, and (3) following similar lines to those of [14,20], we will axiomatize a modal expansion of Lukasiewicz logic for probabilistic reasoning on events described by formulas of product logic. In more detail, we show that states of the Lindenbaum algebra of product logic over n variables, i.e.…”

Computing Duals of Finite Gödel Algebras

“…Such type of representations can be found in the literature for locally finite subvarieties of MTL algebras [20], such as nilpotent minimum algebras [21], and revised drastic product algebras [22] and their subvarieties: drastic product algebras and EMTL algebras [23]. Another interesting duality for finite Gödel ∆ algebras is introduced in [24], and in [25] is shown that the dual category of this variety is also dual to the variety of drastic product algebras (studied in [26]). All these varieties are subvarieties of an interesting algebraic variety related to weak negation functions over [0, 1], the variety of WNM algebras.…”

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].…”