Measuring Inconsistency in Some Logics with Tense Operators

Abstract

This paper starts the systematic study of inconsistency measures for propositional logics enriched with operators involving time. We use Prior’s operators for tense logic: H, G, P, and F; however, we apply different semantics to them. We define two logics. The first one, $\mathit{ATPL}$, allows formulas with the application of any of the four operators any number of times to propositional logic formulas. The semantics is given in terms of TPL structures. We then show how to measure the inconsistency of a set of $\mathit{ATPL}$ formulas using appropriately modified versions of standard inconsistency measures for propositional logic. Additionally, these new measures are checked for the satisfaction of various properties. We also investigate a new concept of inconsistency that cannot be defined for propositional logic. Then, $\mathit{ATPL}$ is extended to $\mathit{CTPL}$ by applying the propositional connectives to $\mathit{ATPL}$ formulas and measuring inconsistency is extended to sets of $\mathit{CTPL}$ formulas.