ASPEKTI INTENZIONALNOSTI
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мар 17, 2021
Ključne reči:
intenzionalna logika, intenzionalna definicija, dokazno-teorijska semantika, teorija dokaza, implicitna definicija
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Apstrakt
Cilj rada je da čitaoca uvede u razliku između onoga intenzionalnog i onoga ekstenzionalnog kao distinkciju između različitih pristupa značenju. Tvrdićemo da, uprkos opštem uverenju, intenzionalni aspekat matematičkih pojmova može biti, pa i da zapravo jeste uspešno opisan u matematici. Jedan od onih koji su za nas naročito interesantni jeste pojam dedukcije onako kako je prikazan u opštoj teoriji dokaza. Naša razmatranja za rezultat imaju odbranu a) važnosti semantičkog pristupa zasnovanog na pravilima i b) pozicija prema kojima neredukcionistička i donekle cirkularna objašnjenja igraju suštinsku ulogu u opisivanju intenzionalnosti u matematici.
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Kako citirati
MAKSIMOVIĆ, K. (2021). ASPEKTI INTENZIONALNOSTI. Arhe, 17(34), 61–83. https://doi.org/10.19090/arhe.2020.34.61-83
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Reference
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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.
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.