The Basic Constructive Logic for Absolute Consistency defined with a Propositional Falsity Constant

The logic BKc6 is the basic constructive logic in the ternary relational semantics (without a set of designated points) adequate to consistency understood as absolute consistency, i.e., non-triviality. Negation is introduced in BKc6 with a negation connective. The aim of this paper is to define the logic BKc6F. In this logic negation is introduced via a propositional falsity constant. We prove that BKc6 and BKc6F are definitionally equivalent. Then, we show how to extend BKc6F within the spectrum of logics delimited by contractionless intuitionistic logic. All logics defined in the paper are paraconsistent logics.