A very simple many-valued predicate calculus is presented; a completeness theorem is proved and the arithmetical complexity of some notions concerning provability is determined.

The connection between similarity logic and the theory of closure operators is examined. Indeed one proves that the consequence relation defined in [14] can be obtained by composing two closure operators and that the resulting operator is still a closure operator. Also, we extend any similarity into a similarity which is compatible with the logical equivalence, and we prove that this gives the same consequence relation.