Multivalued logics are commonly described in terms of valuations into an algebra of truth values. In [1], by lifting the consideration of valuations into subsets of the same algebra, Priest defines what he calls the Plurivalent version of a logic. This process essentially constitutes taking the power-algebra [2] of truth values from the original logic and memberhood of the originally designated elements as the new designation. In the simplest example of this construction Priest can identify his own logic of paradox (LP), and a logic known as analytic logic (AL) as plurivalent logics on classical truth values of the two valued Boolean algebra [1]. Furthermore we note that, by changing the designation notion, strong Kleene three valued logic \((\mathrm{K}_3)\) can also be identified by the same algebra through similar memberhood conditions.

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 memberhood statements. This gives rise to the definition and presentation of substructural versions of the aforementioned logics, with proof systems for which additional rules can be added to obtain the original logic.

Bibliography

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