Existence of the least and the greatest elements of a subset of the Lindenbaum algebra (Q1086230)

The purpose of this note is to answer the following problem: Let P be a propositional letter and G(P) a formula. Does the set \(\{\) \(A|\vdash G(A)\}\) always have the least and the greatest elements unless it is empty? To be accurate, of course, the above problem must be understood in the Lindenbaum algebra. The answer is ''no'' if \(\vdash\) denotes provability in the intuitionistic logic, while ''yes'' if the classical logic is concerned with.