We considered an 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 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 standard FOL, we obtain a commutative homomorphic diagram, which is valid in each given possible world of an intensional FOL: from a free algebra of the FOL syntax, into its intensional algebra of concepts, and, successively, into an extensional relational algebra (different from Cylindric algebras). Then we show that this composition corresponds to the Tarski's interpretation of the standard extensional FOL in this possible world. 1. Introduction In “über Sinn und edeutung,” Frege concentrated mostly on the senses of names, holding that all names have a sense (meaning). It is natural to hold that the same considerations apply to any expression that has an extension. But two general terms can have the same extension and different cognitive significance; two predicates can have the same extension and different cognitive significance; two sentences can have the same extension and different cognitive significance. So, general terms, predicates, and sentences all have senses as well as extensions. The same goes for any expression that has an extension or is a candidate for extension. The significant aspect of an expression’s meaning is its extension. We can stipulate that the extension of a sentence is its truth-value, and that the extension of a singular term is its referent. The extension of other expressions can be seen as associated entities that contribute to the truth-value of a sentence in a manner broadly analogous to the way in which the referent of a singular term contributes to the truth-value of a sentence. In many cases, the extension of an expression will be what we intuitively think of as its referent, although this need not hold in all cases. While Frege himself is often interpreted as holding that a sentence’s referent is its truth-value, this claim is counterintuitive and widely disputed. We can avoid that issue in the present framework by using the technical term “extension.” In this context, the claim that the
S. Bond and M. Denecker, “I-logic: an intensional logic of informations,” in Proceedings of the 19th Belgian-Dutch Conference on Artificial Intelligence, pp. 49–56, Utrecht, The Netherlands, November 2007.
R. Montague, “The proper treatment of quantification in ordinary English,” in Approaches to Natural Language, J. Hintikka, P. Suppes, J. M. E. Moravcsik, et al., Eds., pp. 221–242, Reidel, Dordrecht, The Netherlands, 1973.
Z. Majki？, “Intensional semantics for RDF data structures,” in Proceedings of the 12th International Symposium on Database Engineering & Applications Systems (IDEAS '08), pp. 69–77, Coimbra, Portugal, September, 2008.
Z. Majki？, “Weakly-coupled ontology integration of P2P database systems,” in Proceedings of the 1st International Workshop on Peer-to-Peer Knowledge Management (P2PKM '04), Boston, Mass, USA, August 2004.
Z. Majki？, “Intensional P2P mapping between RDF ontologies,” in Proceedings of the 6th International Conference on Web Information Systems (WISE '05), M. Kitsuregawa, Ed., vol. 3806 of Lecture Notes in Computer Science, pp. 592–594, New York, NY, USA, November 2005.
Z. Majki？, “Non omniscient intensional contextual reasoning for query-agents in P2P systems,” in Proceedings of the 3rd Indian International Conference on Artificial Intelligence (IICAI '07), Pune, India, December 2007.
Z. Majki？, “Coalgebraic specification of query computation in intensional P2P database systems,” in Proceedings of the International Conference on Theoretical and Mathematical Foundations of Computer Science (TMFCS '08), pp. 14–23, Orlando, Fla, USA, July 2008.
Z. Majki？, “RDF view-based interoperability in intensional FOL for Peer-to-Peer database systems,” in Proceedings of the International Conference on Enterprise Information Systems and Web Technologies (EISWT '08), pp. 88–96, Orlando, Fla, USA, July 2008.