Probability Measures in the Logic of Nilpotent Minimum

Abstract

We axiomatize the notion of state over finitely generated free NM-algebras, the Lindenbaum algebras of pure Nilpotent Minimum logic. We show that states over the free n-generated NM-algebra NM n exactly correspond to integrals of elements of NM n with respect to Borel probability measures.

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

“…Moreover, such states are shown to correspond to convex combinations of finitely many truth-value assignments. Similar results have been obtained for the case Gödel logic expanded with Baaz-Monteiro operator ∆ [1], and for the case of Nilpotent Minimum logic [4].…”

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

“…The theory of Płonka sums is intrinsically connected with the notion of partition function 3 which we recall in the following. Definition 3.…”

“…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].…”

On some questions concerning the axiomatisation of WNM-algebras and their subvarieties

“…The papers [4,6,8,9,17] present several results concerning combinatorial representation and combinatorially defined categories dually equivalent to subvarieties of WNM.…”