  In this paper we considered the extension of the First-order Logic (FOL) by
Bealer's intensional abstraction operator. Contemporary use of the term
'intension' derives from the traditional logical Frege-Russell's doctrine that
an idea (logic formula) has both an extension and an intension. Although there
is divergence in formulation, it is accepted that the extension of an idea
consists of the subjects to which the idea applies, and the intension consists
of the attributes implied by the idea. From the Montague's point of view, the
meaning of an idea can be considered as particular extensions in different
possible worlds. In the case of the pure FOL we obtain commutative homomorphic
diagram that holds in each given possible world of the intensional FOL, from
the free algebra of the FOL syntax, toward its intensional algebra of concepts,
and, successively, to the new extensional relational algebra (different from
Cylindric algebras). Then we show that it corresponds to the Tarski's
interpretation of the standard extensional FOL in this possible world.
