We considered an extension of the first-order logic (FOL) by Bealerââ?¬â?¢s intensional abstraction operator. Contemporary use of the\r\nterm ââ?¬Å?intensionââ?¬Â derives from the traditional logical Frege-Russell doctrine that an idea (logic formula) has both an extension and\r\nan intension. Although there is divergence in formulation, it is accepted that the ââ?¬Å?extensionââ?¬Â of an idea consists of the subjects to\r\nwhich the idea applies, and the ââ?¬Å?intensionââ?¬Â consists of the attributes implied by the idea. From the Montagueââ?¬â?¢s point of view, the\r\nmeaning of an idea can be considered as particular extensions in different possible worlds. In the case of standard FOL, we obtain\r\na commutative homomorphic diagram, which is valid in each given possible world of an intensional FOL: from a free algebra of\r\nthe FOL syntax, into its intensional algebra of concepts, and, successively, into an extensional relational algebra (different from\r\nCylindric algebras). Then we show that this composition corresponds to the Tarskiââ?¬â?¢s interpretation of the standard extensional FOL\r\nin this possible world.
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