# Logics of Power-Algebras: the Boolean case

## Orvar Lorimer Olsson, FLoV

Using semantics based on algebras commonplace in the study of logics, however this usage can take forms: Many-valued logics are commonly described in terms of valuations into a single algebra of truth values, whereas other treatments concern full classes of algebraic structures such as varieties defined by equation theories. This has given rise to similar constructions within different fields of study, but where the direct connections may not to be fully spelled out.

Working from the many-valued perspective, in [1], by lifting the consideration of valuations into subsets of an algebra, Priest defines what he calls the plurivalent version of a logic. This process essentially constitutes taking the power-algebra of truth values from the original logic, as described by Brink [2], together with a conservative choice of truth condition inherited from the original algerba essentially through member-hood statements of specific elements. In the simplest example of this construction Priest can identify some well known multivalued logics as plurivalent logics on classical truth values of the two valued Boolean algebra [1]. However this identification fails if the construction was conducted on any other than this specific Boolean algebra.

Continuing work towards general methods for logics with team semantics [3] I will in this presentation consider the power-algebras of arbitrary Boolean algebras and establish a sound and complete labelled natural deduction system for entailments of member-hood statements. This gives rise to the definition and presentation of logics that can be viewed as sub-structural versions of the referred many-valued logics with proof systems for which additional rules can be added to obtain the original logic.

We will finish by reflecting on the achieved construction, and consider how it can be generalised from the class of Boolean Algebras to constructing logics based on other equational classes of algebras.

# References

- Graham Priest, Plurivalent logics, The Australasian Journal of Logic, vol. 11 (2014), no. 1.
- Chris Brink, Power structures, Algebra Universalis, vol. 30 (1993), pp. 177–216.
- Fredrik Engström, Orvar Lorimer Olsson, The propositional logic of teams, arXiv preprint, (2023), arXiv.2303.14022