FACETS OF INTENSIONALITY
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Published:
Mar 17, 2021
Keywords:
intensional logic, intensional definition, proof-theoretic semantics, proof theory, implicit definitions
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Abstract
The goal of this paper is to introduce the reader to the distinction between intensional and extensional as a distinction between different approaches to meaning. We will argue that despite the common belief, intensional aspects of mathematical notions can be, and in fact have been successfully described in mathematics. One that is for us particularly interesting is the notion of deduction as depicted in general proof theory. Our considerations result in defending a) the importance of a rule-based semantical approach and b) the position according to which non-reductive and somewhat circular explanations play an essential role in describing intensionality in mathematics.
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MAKSIMOVIĆ, K. (2021). FACETS OF INTENSIONALITY. Arhe, 17(34), 61–83. https://doi.org/10.19090/arhe.2020.34.61-83
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References
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Crocco, G., Van Atten, M., Cantu, P., Engelen, E. M. (eds.) (2017) Kurt Gödel Maxims and Philosophical Remarks Volume X.
Davidson, D. (1967) “Truth and Meaning”, Synthese Vol. 17, pp. 304-323.
Devlin, K. (2003) “The forgotten revolution”, available online: (https://www.maa.org/external_archive/devlin/devlin_03_03.html) (accessed on September 1st, 2020).
Došen, K. (1989) “Logical constants as punctuation marks”, Notre Dame Journal of Formal Logic, Vol. 30, pp. 362-381.
Došen, K. (1997) “Logical consequence: A turn in style”, Logic and Scientific Methods, Chiara, M. et al. (eds.), pp. 289-311.
Došen, K. and Petrić, Z. (2004) “Identity of proofs based on Normalization and Generality“, available online: (https://arxiv.org/pdf/math/0208094.pdf) (accessed on September 1st, 2020).
Došen, K. (2001) “Abstraction and application in adjunction” available online: (arXiv:math/0111061) (accessed on September 1st, 2020).
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Došen, K. (2017) “Should the explicans be simpler than the explicandum?”, Proof Theory as Mathesis Universalis, Loveno di Menaggio (Como), abstract available online: http://www.mi.sanu.ac.rs/~kosta/K% 20Dosen_Como.pdf (accessed on September 1st, 2020).
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Dummett, M. (1991) The Logical Basis of Metaphysics, London: Duckworth.
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Frege, G. (1884) Die Grundlagen der Arithmetik: eine logisch mathematische Untersuchung über den Begriff der Zahl (English translation: The basic laws of arithmetic: The exposition of the system, Furth, M. (ed.), Berkeley, CA: University of California Press, 1982).
Frege, G. (1891) “Funktion und Begriff”, in: Funktion, Begriff, Bedeutung: Fünf logische Studien, Patzig, G. (ed.), Göttingen: Vandenhoeck & Ruprecht, 1962.
Frege, G. (1948) “Sense and reference”, The Philosophical Review Vol. 57, No. 3, pp. 209-230.
Gentzen, G. (1935) “Untersuchungen über das logische Schliessen”, Matematische Zeitschrift Vol. 39, pp. 176-210, (English translation: “Investigations into logical deduction”, The Collected Papers of Gerhard Gentzen, Scabo, M. (ed.), Amsterdam: North-Holland Publishing Company, pp. 68-132, 1969).
Gödel, K. (1944), “Russell’s Mathematical Logic”, in Collected Works, II: Publications 1938–1974, S. Feferman, J. W. Dawson, S. C. Kleene, G. H. Moore, R. M. Solovay, Jean van Heijenoort (eds.), Oxford:Oxford University Press, pp. 119-141
Grice, H. P. (1989) Studies in the way of words, Cambridge, MA: Harvard University Press.
Hamkins, J. D. & Lewis, A. (1998) “Infinite time Turing machines”, available online: (arXiv:math/9808093v1) (accessed on September 1st, 2020).
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Kostić, J. “Gödel on the theory of concepts”, (manuscript in preparation).
Kostić, J. & Maksimović, K. (2020) “Growing into deduction”, Theoria, Vol. 63, pp. 87-106.
Kostić, J., Maksimović, K., Milošević, S. “Is natural deduction natural”, (manuscript in preparation).
Lambek, J. (1972) “Deductive systems and categories III: Cartesian closed categories, intuitionist propositional calculus, and combinatory logic”, in: Toposes, Algebraic Geometry and Logic (Lecture Notes in Mathematic 274), Lawvere F.W. (ed.), Berlin: Springer, pp. 57-82.
Mac Lane, S. (1998) Categories for the Working Mathematician, second edition, Berlin: Springer.
Maksimović, К. (2016) “Uses of the Language of Mathematics”, Theoria, Vol. 59, No. 1, pp. 26-41.
Poincare, H. (1902) La Science et L’Hypoth`ese, Paris: Flammarion, English translation: Science and Hypothesis, The Value of Science, Science and Method., Halstead, G. B. (ed.), Lanham, MD: University Press of America, 1982.
Prawitz, D. (1965) Natural Deduction: A Proof-Theoretical Study, Stockholm: Almqvist & Wiksell.
Prawitz, D. (1971). “Ideas and Results in Proof Theory”, Proceedings of the Second Scandinavian Logic Symposium (Oslo 1970), Jens E. Fenstad (ed.), Amsterdam: North-Holland, pp. 235–308.
Schroeder-Heister, P. (2012) “Proof-theoretic semantics”, Zalta, E.N. (ed.), The Stanford Encyclopedia of Philosophy, available online: https://plato.stanford.edu/entries/proof-theoretic-semantics/ (accessed on September 1st, 2020).
Schroeder-Heister, P. (2013) “Definitional Reflection and Basic Logic”, Annals of Pure and Applied Logic, Vol. 164, No. 4, pp. 491–501.
Schroeder-Heister, P. (2015) “Proof-theoretic validity based on elimination rules”. In: Haeusler, E. H., de Campos Sanz, W. and Lopes B., (eds.), Why is this a Proof? Festschrift for Luiz Carlos Pereira. London: College Publications, pp. 159–176.
Wang, H. (1996) A Logical Journey: from Gödel to Philosophy, Cambridge: MIT Press.
Wittgenstein, L. (1958) Philosophical Investigations, Oxford: Basil Blackwell.
Berarducci, A. & Dezani-Ciancaglini, M. (1999) “Infinite A-calculus and types”, Theoretical Computer Science, Vol. 212, pp. 29-75.
Church, A. (1951) “A Formulation of the Logic of Sense and Denotation”, in Structure, Method, and Meaning; Essays in honor of Henry M. Sheffer, Henle, P., Kallen, H. (eds.), Liberal Arts Press, NY., USA, pp. 3-24.
Church, A. (1973) “Outline of a Revised Formulation of the Logic of Sense and Denotation (Part I)”, Noûs, Vol. 7, No. 1, pp. 24-33.
Church, A. (1974) “Outline of a Revised Formulation of the Logic of Sense and Denotation (Part II)”, Noûs, Vol. 8, No. 2, pp. 135-156.
Church, A. (1993) “A Revised Formulation of the Logic of Sense and Denotation. Alternative (1)”, Noûs, Vol. 27, No. 2, pp. 141-157.
Crocco, G., Van Atten, M., Cantu, P., Engelen, E. M. (eds.) (2017) Kurt Gödel Maxims and Philosophical Remarks Volume X.
Davidson, D. (1967) “Truth and Meaning”, Synthese Vol. 17, pp. 304-323.
Devlin, K. (2003) “The forgotten revolution”, available online: (https://www.maa.org/external_archive/devlin/devlin_03_03.html) (accessed on September 1st, 2020).
Došen, K. (1989) “Logical constants as punctuation marks”, Notre Dame Journal of Formal Logic, Vol. 30, pp. 362-381.
Došen, K. (1997) “Logical consequence: A turn in style”, Logic and Scientific Methods, Chiara, M. et al. (eds.), pp. 289-311.
Došen, K. and Petrić, Z. (2004) “Identity of proofs based on Normalization and Generality“, available online: (https://arxiv.org/pdf/math/0208094.pdf) (accessed on September 1st, 2020).
Došen, K. (2001) “Abstraction and application in adjunction” available online: (arXiv:math/0111061) (accessed on September 1st, 2020).
Došen, K. (2013) Основна логика, available online: (http://www.mi.sanu.ac.rs/~kosta/Osnovna%20logika.pdf) (accessed on September 1st, 2020).
Došen, K. (2017) “Should the explicans be simpler than the explicandum?”, Proof Theory as Mathesis Universalis, Loveno di Menaggio (Como), abstract available online: http://www.mi.sanu.ac.rs/~kosta/K% 20Dosen_Como.pdf (accessed on September 1st, 2020).
Dummett, M. (1978) Truth and Other Enigmas, Cambridge, MA: Harvard University Press.
Dummett, M. (1991) The Logical Basis of Metaphysics, London: Duckworth.
Frege, G. (1879) Begriffsschrift, a Formula Language, Modeled upon that of Arithmetic, for Pure Thought. In From Frege to Gödel, edited by Jean van Heijenoort. Cambridge, MA: Harvard University Press, 1967. Originally published as Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (Halle: L. Nebert).
Frege, G. (1884) Die Grundlagen der Arithmetik: eine logisch mathematische Untersuchung über den Begriff der Zahl (English translation: The basic laws of arithmetic: The exposition of the system, Furth, M. (ed.), Berkeley, CA: University of California Press, 1982).
Frege, G. (1891) “Funktion und Begriff”, in: Funktion, Begriff, Bedeutung: Fünf logische Studien, Patzig, G. (ed.), Göttingen: Vandenhoeck & Ruprecht, 1962.
Frege, G. (1948) “Sense and reference”, The Philosophical Review Vol. 57, No. 3, pp. 209-230.
Gentzen, G. (1935) “Untersuchungen über das logische Schliessen”, Matematische Zeitschrift Vol. 39, pp. 176-210, (English translation: “Investigations into logical deduction”, The Collected Papers of Gerhard Gentzen, Scabo, M. (ed.), Amsterdam: North-Holland Publishing Company, pp. 68-132, 1969).
Gödel, K. (1944), “Russell’s Mathematical Logic”, in Collected Works, II: Publications 1938–1974, S. Feferman, J. W. Dawson, S. C. Kleene, G. H. Moore, R. M. Solovay, Jean van Heijenoort (eds.), Oxford:Oxford University Press, pp. 119-141
Grice, H. P. (1989) Studies in the way of words, Cambridge, MA: Harvard University Press.
Hamkins, J. D. & Lewis, A. (1998) “Infinite time Turing machines”, available online: (arXiv:math/9808093v1) (accessed on September 1st, 2020).
Hilbert, D. (1899) Grundlagen der Geometrie. Tuebner, (English translation:The Foundations of Geometry, Chicago, IL: The Open Court Publishing Company, La Salle, 1950).
Kostić, J. “Gödel on the theory of concepts”, (manuscript in preparation).
Kostić, J. & Maksimović, K. (2020) “Growing into deduction”, Theoria, Vol. 63, pp. 87-106.
Kostić, J., Maksimović, K., Milošević, S. “Is natural deduction natural”, (manuscript in preparation).
Lambek, J. (1972) “Deductive systems and categories III: Cartesian closed categories, intuitionist propositional calculus, and combinatory logic”, in: Toposes, Algebraic Geometry and Logic (Lecture Notes in Mathematic 274), Lawvere F.W. (ed.), Berlin: Springer, pp. 57-82.
Mac Lane, S. (1998) Categories for the Working Mathematician, second edition, Berlin: Springer.
Maksimović, К. (2016) “Uses of the Language of Mathematics”, Theoria, Vol. 59, No. 1, pp. 26-41.
Poincare, H. (1902) La Science et L’Hypoth`ese, Paris: Flammarion, English translation: Science and Hypothesis, The Value of Science, Science and Method., Halstead, G. B. (ed.), Lanham, MD: University Press of America, 1982.
Prawitz, D. (1965) Natural Deduction: A Proof-Theoretical Study, Stockholm: Almqvist & Wiksell.
Prawitz, D. (1971). “Ideas and Results in Proof Theory”, Proceedings of the Second Scandinavian Logic Symposium (Oslo 1970), Jens E. Fenstad (ed.), Amsterdam: North-Holland, pp. 235–308.
Schroeder-Heister, P. (2012) “Proof-theoretic semantics”, Zalta, E.N. (ed.), The Stanford Encyclopedia of Philosophy, available online: https://plato.stanford.edu/entries/proof-theoretic-semantics/ (accessed on September 1st, 2020).
Schroeder-Heister, P. (2013) “Definitional Reflection and Basic Logic”, Annals of Pure and Applied Logic, Vol. 164, No. 4, pp. 491–501.
Schroeder-Heister, P. (2015) “Proof-theoretic validity based on elimination rules”. In: Haeusler, E. H., de Campos Sanz, W. and Lopes B., (eds.), Why is this a Proof? Festschrift for Luiz Carlos Pereira. London: College Publications, pp. 159–176.
Wang, H. (1996) A Logical Journey: from Gödel to Philosophy, Cambridge: MIT Press.
Wittgenstein, L. (1958) Philosophical Investigations, Oxford: Basil Blackwell.