States in Łukasiewicz logic correspond to probabilities of rational polyhedra

It will be shown that probabilities of infinite-valued events represented by formulas in Łukasiewicz propositional logic are in one-to-one correspondence with tight probability measures over rational polyhedra in the unit hypercube. This result generalizes a recent work on rational measures of polyhedra and provides an elementary geometric approach to reasoning under uncertainty with states in Łukasiewicz logic.

► We study a relation of states in Lukasiewicz logic to probabilities of rational polyhedra in the unit n-dimensional cube. ► We provide a purely geometrical proof of integral representation of states. ► We show that there is a 1–1 correspondence between the states and the tight probabilities on rational polyhedra.