On the Forcing Semantics for Monoidal t-norm Based Logic

MTL-algebras are algebraic structures for the Esteva-Godo monoidal t-norm based logic (MTL), a many-valued propositional calculus that formalizes the structure of the real interval [0, 1], induced by a left-continuous t-norm. Given a complete MTL-algebra Χ, we define the weak forcing value |φ|χ and the forcing value [φ]χ, for any formula φ of MTL in Χ. We establish some arithmetical properties of|.|χ and [.]χ, and prove the equality [φ]χ=||φ||χ, where ||φ||χ is the truth value of φ in Χ.