21.10.2015 10:00 @ Applied Mathematical Logic
We investigate the lattice of extensions of the four-valued Belnap-Dunn logic, called super-Belnap logics by analogy to super-intuitionistic logics. Although the Belnap-Dunn logic itself has been studied extensively, very little is known about the landscape of super-Belnap logics. In an attempt to fill this gap, we prove some new completeness theorems, identify some splittings of this lattice, and show that it is non-modular and has the cardinality of the continuum. Moreover, we describe the protoalgebraic and self-extensional super-Belnap logics. Surprisingly, it turns out that there is a close connection between super-Belnap logics and finite graphs: for example, the lattice of classes of finite graphs closed under homomorphisms dually embeds into the lattice of finitary (explosive) extensions of the Belnap-Dunn logic.
