Deductibility Logic: the modal axiom B and adjunction

Abstract

Logic TK formalizes in the propositional environment the notion of the Tarski
consequence operator. Because that it is called the logic of deductibility. In this process
of formalizing the notion of deductibility, the logic obtained has a modal character. It is
obtained from the classical propositional logic plus an unary operator relative to the Tarski
operator. Logic TK has as its algebraic model the TK-algebras and as topological / set
theoretic model the quasi-topological spaces, or Tarski spaces. The modal operators of
logic TK are associated, in their topological counterpart, with the concepts of closure and
interior. Galois connections, in turn, are originated from Galois theory. They are obtained
from the Galois pairs, which act over order structures with changes in the in the orders that
define these structures. In a first approach, we find that the interior and close operators of
almost topological spaces do not determine a Galois pair. When analysed in the logical
context, if we include the well-known modal axiom B on the logic TK the
operators determine an adjunction that is a Galois pair. The counterpart of such procedure,
in almost topological context, led us to reach an adjunction from the closing and interior
operators.