Constructive Fuzzy Logics

We generalise Kripke’s semantics for Intuitionistic logic to Hajek’s BL and consider the constructive subsystems of GBLewf and Intuitionistic Affine logic or ALi. The genesis of our semantics is the Poset Product construction for GBL-algebras elucidated in a series of papers by Peter Jipsen, Simone Bova, and Franco Montagna. We present natural deduction systems for all of these systems and corresponding deduction theorems for these same. We present the algebraic semantics for each of the logics under consideration, demonstrate their soundness and completeness with respect to these algebraic semantics. We also show how the classical Kripke semantics for Intuitionistic logic can be recast in terms of Poset Products. We then proceed to the main results, showing how a very natural generalisation of the Kripke semantics holds for each of GBLewf , ALi and Hajek’s BL based on the embedding results of Jipsen and Montagna and the decidability results of Bova and Montagna. We demonstrate soundness and completeness of the logics under our semantics in each case, with the exception of ALi, whose robust completeness with respect to the intended models (relational models with frames valued in involutive pocrims) we leave as an open problem for the ambitious reader