The countable versus uncountable branching recurrences in computability logic

Abstract

This paper introduces a new simplified version of the countable branching recurrence ▪ of Computability Logic, proves its equivalence to the old one, and shows that the basic logic induced by ▪ (i.e., the one in the signature {¬,∧,∨,▪,▪}) is a proper superset of the basic logic induced by the uncountable branching recurrence ▪ (i.e., the one in the signature {¬,∧,∨,▪,▪}). A further result of this paper is showing that ▪ is strictly weaker than ▪ in the sense that ▪F logically implies ▪F but not vice versa.