Intensional logic concerns the content of concepts, (i.e., intensions), and the relations between them, whereas an extensional logic concerns the extension of concepts, (i.e., sets), and the relations between them. Intensional logic is thus a special part of logic having its own restricted subject matter. There are at least two specific groups of problems to which extensional logic has failed to give satisfactory answers, and which intensional logic aims to study:

1. Two non-identical concepts can be co-extensional in the given universe of discourse. Then, especially, in the universe of discourse of all possible things, all contradictions would be co-extensional, i.e. the empty set. The empty set being a subset of every set, it follows that all contradictions would be included in the extension of every concept. So, evidently, concepts and their extensions cannot be identified in all cases.

2. The logical inference formalized by using a material implication satisfies the requirement that from the true propositions only true propositions follow. However, since it is always true when the antecedent is false or the consequent is true, its use may lead to some paradoxical situations, that is, “the paradox of material implication”.

Thus, the reciprocity between extensional and intensional logic is not one-one.

1. Intensional vs. extensional logic

In this first lesson, we will present several reasons to separate intensional logic from extensional logic. We will give a historical development of logic starting from Aristotle’s square of opposites and his idea of demonstrative science and their connection to Leibniz, who was trying to develop both intensional and extensional logic within same formalism. In 1960’s inspired by Leibniz’s idea of possible worlds, Stig Kanger, Jaakko Hintikka, and Saul Kripke independently developed the possible world semantic for modal logic. We will show why in these approaches intensional notions are either reduced to extensional set-theoretic constructs in diversity of worlds or as being non-logical notions left unexplained. So, when developing an adequate presentation of an intensional logic it must consider both formal (logic) and contentual (epistemic) aspects of concepts and their relationships.

2. An introduction to an axiomatic intensional concept theory

In this second lesson, we will introduce the most essential relation between concepts, which is an intensional containment relation based on the intensions of concepts. Then, based on this intensional containment relation between concepts an axiomatic intensional concept theory inspired by Leibniz’s logic, will be presented.

3. Some developments and problems of intensional concept theory

In this third lesson, we will present some further developments of intensional concept theory, for example intensional relational concepts. Then we will present the problem of intensional negation, since dealing with definition of negation, difference between intensional and extensional logic is the most striking.

References

Hintikka, J., 1969: Models for Modalities. Dordrecht: D. Reidel.

Kauppi, R., 1960: Über die Leibnizsche Logik mit besonderer Berücksichtigung des Problems der Intension und der Extension. Acta Philosophica Fennica, Fasc. XII. Helsinki: Societas Philosophica Fennica.

Kauppi, R., 1967: Einführung in die Theorie der Begriffssysteme. Acta Universitatis Tamperensis. Ser. A. Vol. 15. Tampere: Tampereen yliopisto.

Kripke, S., 1959: “A completeness theorem in modal logic”, The Journal of Symbolic Logic, vol. 24, pp. 1–14.

Leibniz, G. W., 1966: Logical Papers: A Selection Translated and Edited with Introduction. Ed. Parkinson, G. H. R., Clarendon Press.

Leibniz, G. W., 1997: Philosophical Writings. Ed. G. H. R. Parkinson. Trans. M. Morris and G. H. R. Parkinson. London: The Everyman Library.

Montague, R., 1974: Formal Philosophy. Ed. R. Thomason. New Haven and London: Yale University Press.

Palomäki, J., 1994: From Concepts to Concept Theory: Discoveries, Connections, and Results. Acta Universitatis Tamperensis, ser. A, vol. 416. Tampere: Tampereen yliopisto.

Palomäki, J., 2012: “God, Intensional Negation - and the Devil(s)”. Philosophy Study 5, Vol.2., 323-336.

Palomäki, J., 2012: “An Axiomatic Approach to Relational Concepts”. Information Modelling and Knowledge Bases, XXVI. Eds. B. Thalheim, H. Jaakkola, Y. Kiyoki, and N. Yoshida. Amsterdam, Berlin, Oxford, Tokyo, Washington, DC.: IOS Press, 355-360.