The " -calculus" is a formal logical system, which as classical one, treats (true and false) assertions in the grounds, but unlike - in the semantic sense - their truth (falsity) values are differently set. Besides, here are of interest only (always) false formulas or negations etc. We formulate a number of formal theorems of the calculus within its "propositional," "predicate" and "arithmetical" versions and put forward the (meta-theoretical) requirements for completeness and consistency of these systems. It is shown that they largely "share fate" of classical formalizations: when it is about, say, completeness of the propositional calculus, or the (in)completeness of predicate calculus and formal number theory et al. Finally, we bring a version of this formalism as a sequential calculus, as well as a constructive proof of its consistency (after Gentzen).