Abstract
The paper introduces the notion of state for involutive bisemilattices, a variety which plays the role of algebraic counterpart of weak Kleene logics and whose elements are represented as Płonka sums of Boolean algebras. We investigate the relations between states over an involutive bisemilattice and probability measures over the (Boolean) algebras in the Płonka sum representation and, the direct limit of these algebras. Moreover, we study the metric completion of involutive bisemilattices, as pseudometric spaces, and the topology induced by the pseudometric.
Highlights
Probability theory is grounded on the notion of event
We have shown how to define a notion of state on Płonka sums of Boolean algebras, with the aim of expressing the probability for elements of an involutive bisemilattice, a variety associated to the logic PWK
We have shown that the elements of the class N GIB always carry a state
Summary
Probability theory is grounded on the notion of event. Events are traditionally interpreted as elements of a (σ-complete) Boolean algebra. The idea motivating the present work is to further extend the theory of states to nonclassical events; in particular, to one of the three-valued logics in the weak Kleene family, whose algebraic semantics is played by the variety of involutive bisemilattices (see [9]) The peculiarity of such variety is that each of its members has a representation in terms of Płonka sums of Boolean algebras. The axiomatisation of states we propose, which is motivated by the logic PWK (Paraconsistent Weak Kleene), allows to “break” a state into a family of (finitely additive) probability measures over the Boolean algebras in the Płonka sum representation of an involutive bisemilattice.
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