Truth functionality and measure-based logics

We present a truth-functional semantics for necessity-valued logics, based on the forcing technique. We interpret possibility distributions (which correspond to necessity measures) as informational states, and introduce a suitable language (basically, an extension of classical logic, similar to Pavelka’s language). Then we define the relation of “forcing” between an informational state and a formula, meaning that the state contains enough information to support the validity of the formula. The subsequent step is the definition of a many-valued truth-functional semantics, by simply taking the truth value of a formula to be the set of all informational states that force the truth of the formula. A proof system in sequent calculus form is provided, and validity and completeness theorems are proved.