On the Forcing Semantics for Monoidal t-norm Based Logic

MTL-algebras are algebraic structures for the Esteva-Godo monoidal tnorm 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 X , we define the weak forcing value |φ|X and the forcing value [φ]X , for any formula φ of MTL in X . We establish some arithmetical properties of |.|X and [.]X , and prove the equality [φ]X =‖φ‖X , where ‖φ‖X is the truth value of φ in X .