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The paper is concerned with propositional calculi having arbitrary modus inference operations that are analogous to the modus ponens operation. For these calculi the question on the existence and cardinality of sets of generators is examined. A criterion for propositional calculi with arbitrary modus inference operations to be finitely generated is put forward. For some calculi, the existence of a finite complete system of tautologies is shown to imply the existence of a basis of arbitrarily large finite cardinality. We show the existence of propositional calculi with countable basis, without any basis, and calculi having a complete subsystem without any basis.

Criterion for propositional calculi to be finitely generated