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INTUITIONISTIC EPISTEMIC LOGIC

  • SERGEI ARTEMOV (a1) and TUDOR PROTOPOPESCU (a1)

Abstract

We outline an intuitionistic view of knowledge which maintains the original Brouwer–Heyting–Kolmogorov semantics for intuitionism and is consistent with the well-known approach that intuitionistic knowledge be regarded as the result of verification. We argue that on this view coreflection AKA is valid and the factivity of knowledge holds in the form KA → ¬¬A ‘known propositions cannot be false’.

We show that the traditional form of factivity KAA is a distinctly classical principle which, like tertium non datur A ∨ ¬A, does not hold intuitionistically, but, along with the whole of classical epistemic logic, is intuitionistically valid in its double negation form ¬¬(KA ¬ A).

Within the intuitionistic epistemic framework the knowability paradox is resolved in a constructive manner. We argue that this paradox is the result of an unwarranted classical reading of constructive principles and as such does not have the consequences for constructive foundations traditionally attributed it.

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Corresponding author

*PROGRAMS IN COMPUTER SCIENCE, MATHEMATICS AND PHILOSOPHY THE GRADUATE CENTER, THE CITY UNIVERSITY OF NEW YORK 365 FIFTH AVENUE, RM. 4329 NEW YORK CITY, NY, 10016, NY, USA E-mail: sartemov@gc.cuny.edu
PROGRAM IN PHILOSOPHY THE GRADUATE CENTER, THE CITY UNIVERSITY OF NEW YORK 365 FIFTH AVENUE NEW YORK CITY, NY, 10016, NY, USA E-mail: tprotopopescu@gradcenter.cuny.edu

References

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Artemov, S. (2001). Explicit Provability and Constructive Semantics. Bulletin of Symbolic Logic, 7(1), 136.
Artemov, S. (2008). The logic of justification. Review of Symbolic Logic, 1(4), 477513.
Artemov, S., & Beklemishev, L. D. (2005). Provability logic. In Gabbay, D., and Guenthner, F., editors. Handbook of Philosophical Logic (second edition), Vol. 13. Springer, Dordrecht, pp. 189360.
Artemov, S., & Fitting, M. (2012). Justification logic. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy (Fall 2012 ed.), The Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University.
Artemov, S., & Protopopescu, T. (2010). Knowability from a Logical Point of View. Technical Report TR 2010008, CUNY Ph.D. Program in Computer Science, 3349–3376.
Artemov, S., & Protopopescu, T. (2012). Discovering knowability: A semantical analysis. Synthese, 190(16).
Artemov, S., & Protopopescu, T. (2014, June). Intuitionistic epistemic logic. ArXiv, math.LO 1406.1582v1.
Božić, M., & Došen, K. (1984). Models for normal intuitionistic modal logics. Studia Logica, 43(3), 217245.
Božić, M., & Došen, K. (1985). Models for stronger normal intuitionistic modal logics. Studia Logica, An International Journal for Symbolic Logic, 44(1), 3970.
Brogaard, B., & Salerno, J. (2009). Fitch’s paradox of knowability. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy (Fall 2009 ed.), The Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University.
Brouwer, L. (1981). Brouwer’s Cambridge Lectures on Intuitionism. Cambridge University Press, Cambridge.
Buss, S. (1998). Introduction to proof theory. In Buss, S., editor, Handbook of Proof Theory, Chapter 1. Elsevier, Amsterdam, pp. 178.
Chagrov, A., & Zakharyaschev, M. (1997). Modal Logic. Oxford, UK: Clarendon Press.
Church, A. (2009). Referee Reports on Fitch’s “A Definition of Value”. See Salerno (2009), pp. 1320.
Clarke, E. M., & Kurshan, R. P. (1996). Computer-aided verification. IEEE Spectrum, 33(6), 6167.
Constable, R. (1998). Types in logic, mathematics and programming. In Buss, S., editor, Handbook of Proof Theory. Elsevier, Amsterdam, pp. 683786.
Contu, P. (2006). The justification of the logical laws revisited. Synthese, 148(3), 573588.
De Vidi, D., & Solomon, G. (2001). Knowability and intuitionistic logic. Philosophia, 28(1), 319334.
Dean, W., & Kurakawa, H. (2009). From the knowability paradox to the existence of proofs. Synthese, 176(2), 177225.
Descartes, R. (1642). Meditations on first philosophy. In The Philosophical Writings of Descartes, Vol. II. Cambridge, MA: Cambridge University Press, pp. 362.
Došen, K. (1984). Intuitionistic double negation as a necessity operator. Publications de L’Institute Mathématique (Beograd)(NS), 35(49), 1520.
Dummett, M. (1963). Realism. In Truth and Other Enigmas. Cambridge, MA: Harvard University Press, pp. 145165.
Dummett, M. (1970). Enforcing the encyclical. New Blackfriars, 51(600), 229234.
Dummett, M. (1973). The philosophical basis of intuitionistic logic. In Truth and Other Enigmas. Cambridge, MA: Harvard University Press, pp. 215247.
Dummett, M. (1976). What is a theory of meaning II. In The Seas of Language. Oxford University Press, New York, pp. 3493.
Dummett, M. (1977). Elements of Intuitionism. Clarendon Press Oxford.
Dummett, M. (1979). What does the appeal to use do for the theory of meaning? In The Seas of Language, Chapter 4. Oxford Univeristy Press, New York, pp. 106116.
Dummett, M. (1991). The Logical Basis of Metaphysics. Cambridge, MA: Harvard University Press. The William James Lectures, 1976.
Dummett, M. (1998). Truth from a constructive point of view. Theoria, 64, 122138.
Dummett, M. (2001). Victor’s error. Analysis, 61, 12.
Dummett, M. (2009). Fitch’s paradox of knowability. See Salerno (2009), Chapter 4, pp. 5152.
Fitch, F. (1963). A logical analysis of some value concepts. Journal of Symbolic Logic, 28(2), 135142.
Floridi, L. (1998). Mathematical skepticism: A sketch with historian in foreground. In Zande, J., and Popkin, R., editors, The Skeptical Tradition Around 1800. Kluwer, Dordrecht, pp. 4160.
Floridi, L. (2000). Mathematical skepticism. The Proceedings of the Twentieth World Congress of Philosophy, 2000, 217265.
Floridi, L. (2004). Mathematical skepticism: The debate between Hobbes and Wallis. In Skepticism in Renaissance and Post-Renaissance Thought: New Interpretations. Humanity Books.
Frege, G. (1884). The Foundations of Arithmetic: A Logical-Mathematical Investigation into the Concept of Number. New York, NY: Pearson Education.
Gabbay, D. M., Kurucz, A., Wolter, F., & Zakharyaschev, M. (2003). Many-Dimensional Modal Logics: Theory and Applications. Studies in Logic and the Foundations of Mathematics. Elsevier, Dordrecht.
Gettier, E. (1963). Is knowledge justified true belief? In Pojman, L., editor, The Theory of Knowledge. Belmont, California: Wadsworth Thompson, pp. 125127.
Glivenko, V. (1929). On Some Points of the Logic of Mr. Brouwer. See Mancosu (1998), Chapter 22, pp. 301305.
Gödel, K. (1933). An interpretation of the intuitionistic propositional calculus. In Feferman, S., Dawson, J. W., Goldfarb, W., Parsons, C., and Solovay, R. M., editors, Collected Works, Vol. 1. Oxford Univeristy Press, Dordrecht, pp. 301303.
Hart, W. D. (1979). Access and inference. Proceedings of the Aristotelian Society, LIII, 153166.
Hazlett, A. (2010). The myth of factive verbs. Philosophy and Phenomenological Research, 80(3), 497522.
Hazlett, A. (2012). Factive presupposition and the truth condition on knowledge. Acta Analytica, 27(4), 461478.
Heyting, A. (1930). On intuitionistic logic. See Mancosu (1998), Chapter 23, pp. 306310.
Heyting, A. (1964). The intuitionistic foundations of mathematics. In Putnam, H., and Benacerraf, P., editors, Philosophy of Mathematics, Selected Readings (first edition). Prentice-Hall, Upper Saddle River, pp. 4249.
Heyting, A. (1966). Intuitionism: An Introduction (second revised edition). Studies in Logic and the Foundations of Mathematics, Vol. 41. North-Holland, Amsterdam.
Hilbert, D., & Bernays, P. (1932). Grundlagen der Mathematik II (Grundlehren der mathematischen Wissenschaften). Springer-Verlag, Dordrecht.
Hirai, Y. (2010a). An intuitionistic epistemic logic for sequential consistency on shared memory. In Logic for Programming, Artificial Intelligence, and Reasoning. Springer, Dordrecht, pp. 272289.
Hirai, Y. (2010b). Disjunction Property and Finite Model Property for an Intuitionistic Epistemic Logic unpublished manuscript, https://yoichihirai.com/nasslli2010hirai.pdf.
Horsten, L. (2014). Philosophy of mathematics. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy (Spring 2014 ed.).
Husserl, E. (1901). Logical Investigations (2nd Ed edition). Routledge & Kegan Paul, London.
Khlentzos, D. (2004). Naturalistic Realism and the Anti-Realist Challenge. MIT Press, Cambridge.
Kolmogorov, A. N. (1925). On the principle of excluded middle. In van Heijenoort, J., editor, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge, pp. 415437.
Kopylov, A., & Nogin, A. (2001). Markov’s principle for propositional type theory. In Computer Science Logic. Springer, Dordrecht, pp. 570584.
Kreisel, G. (1962). Foundations of intuitionistic logic. In Nagel, E., Suppes, P., & Tarski, A., editors, Logic, Methodology and Philosohy of Science: Proceedings of the 1960 International Congress. Stanford University Press, Redwood City, pp. 198210.
Krupski, V. N., & Yatmanov, A. (2016). Sequent calculus for intuitionistic epistemic logic IEL. In Artemov, S. & Nerode, A., editors, Logical Foundations of Computer Science. Lecture Notes in Computer Science, Vol. 9537. Springer, Dordrecht, pp. 187201.
Löb, M. H. (1955). Solution of a problem of Leon Henkin. Journal of Symbolic Logic, 20(2), 115118.
Mancosu, P., editor (1998). From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920’s. Oxford Univeristy Press, Oxford.
Martin-Löf, P. (1987, December). Truth of Proposition, Evidence of Judgement, Validity of a Proof. Synthese, 73(3), 407420.
Martin-Löf, P. (1990). A path from logic to metaphyiscs. In Atti del Congresso Nuovi problemi della logica e della filosofia della scienza. Società Italiana di Logica e Filosofia delle Scienze, Viareggio, pp. 141149.
Martin-Löf, P. (1998). Truth and knowability on the principles C and K of Michael Dummett. In Dales, H., and Olivieri, G., editors, Truth in Mathematics, Chapter 5. Oxford Univeristy Press, New York, pp. 105114.
Martino, E., & Usberti, G. (1994). Temporal and Atemporal truth in Intuitionistic Mathematics. Topoi, 13(2), 8392.
Marton, P. (2006). Verificationists versus realists: The battle over knowability. Synthese, 151(1), 8198.
Murzi, J. (2010). Knowability and bivalence: Intuitionistic solutions ot the paradox of knowability. Philosophical Studies, 149, 269281.
Percival, P. (1990). Fitch and Intuitionistic Knowability. Analysis, 50(3), 182187.
Prawitz, D. (1980). Intuitionistic logic: A philosophical challenge. In von Wright, G. H., editor, Logic and Philosophy. Martinus Nijhoff, The Hague, pp. 110.
Prawitz, D. (1998a). Comments on Goran Sundholm’s paper: “Proofs as acts and proofs as objects”. Theoria, 64, 318329.
Prawitz, D. (1998b). Comments on Lars Bergström’s paper: “Prawitz’s version of verificationism”. Theoria, 64, 293303.
Prawitz, D. (1998c). Comments on Michael Dummett’s Paper “Truth from a constructive point of view”. Theoria, 64, 283292.
Prawitz, D. (1998d). Truth and objectivity from a verificationist point of view. In Dales, H., and Olivieri, G., editors, Truth in Mathematics, Chapter 2. Oxford Univeristy Press, New York, pp. 4151.
Prawitz, D. (1998e). Truth from a constructive perspective. In Martinez, C., Rivas, U., and Villegas-Forero, L., editors, Truth in Perspective, Recent Issues in Logic, Representation and Ontology, Chapter 2. Ashgate, Hoboken, pp. 2335.
Prawitz, D. (2005). Logical consequence from a constructive point of view. In Shapiro, S., editor, The Oxford Handbook of Philosophy of Mathematics and Logic, Chapter 22. Oxford Univeristy Press, New York, pp. 671695.
Prawitz, D. (2006). Meaning approached via proofs. Synthese, 148(3), 507524.
Proietti, C. (2012). Intuitionistic epistemic logic, Kripke models and Fitch’s Paradox. Journal of Philosophical Logic, 41(5), 877900.
Proof (2014). Proof. In Encyclopedia Brittanica.
Protopopescu, T. (2015). Intuitionistic epistemology and modal logics of verification. In van der Hoek, W., and Holliday, W., editors, Logics, Rationality and Interaction (LORI 2015). Lecture Notes in Computer Science, Vol.9394. Springer, Dordrecht, pp. 295307.
Protopopescu, T. (2016). An arithmetical interpretation of verification and intuitionistic knowledge. ArXiv, math.LO 1601.03059. Post-print correcting various publisher’s errors. Published in Artemov, S. and Nerode, A., editors, Logical Foundations of Computer Science. Lecture Notes in Computer Science, Vol. 9537. Springer, pp. 317330.
Putnam, H. (1977). Realism and reason. Proceedings and Addresses of the American Philosophical Association, 50(6), 483498.
Rasmussen, S. (2009). The paradox of knowability and the mapping objection. See Salerno (2009), Chapter 5, pp. 5375.
Salerno, J., editor (2009). New Essays on the Knowability Paradox. Oxford University Press, New York.
Schroeder-Heister, P. (2006). Validity concepts in proof-theoretic semantics. Synthese, 148(3), 525571.
Sundholm, G. (2002). Proof theory and meaning. In Gabbay, D., and Guenther, F., editors, Handbook of Philosophical Logic (second edition), Vol. 9. Kluwer, Dordrecht, pp. 165198.
Tarski, A. (1969). Truth and proof. Scientific American, 220(6), 6377.
Tennant, N. (1997). The Taming of the True. Oxford Univeristy Press, Oxford.
Tennant, N. (2009). Revamping the restriction strategy. In Salerno, J., editor, New Essays on the Knowability Paradox. New York, NY: Oxford University Press, pp. 223238.
Usberti, G. (2006). Towards a semantics based on the notion of justification. Synthese, 148(3), 675699.
van Dalen, D. (2002). Intuitionistic logic. In Handbook of Philosophical Logic (2nd edition), Vol. 5. Springer, pp. 1114.
van Dalen, D. (2004). Logic and Structure. Springer, Dordrecht.
van Dalen, D., & Troelstra, A. (1988a). Constructivism in Mathematics An Introduction, Vol. I. Studies in Logic and the Foundations of Mathematics, Vol. 121. Elsevier, Amsterdam.
van Dalen, D., & Troelstra, A. (1988b). Constructivism in Mathematics An Introduction, Vol. II. Studies in Logic and the Foundations of Mathematics, Vol. 121. Elsevier, Amsterdam.
Voevodsky, V., et al. (2013). Homotopy Type Theory. Univalent Foundations Program.
Wansing, H. (2010). Proofs, disproofs and their duals. In Goranko, V., Beklemishev, L., and Shehtman, V., editors, Advances in Modal Logic, Vol. 8. College Publications, London, pp. 483505.
Williamson, T. (1982). Intuitionism disproved? Analysis, 42(4), 203207.
Williamson, T. (1988). Knowability and constructivism. Philosophical Quarterly, 38(153), 422432.
Williamson, T. (1992). On intuitionistic modal epistemic logic. Journal of Philosophical Logic, 21(1), 6389.
Williamson, T. (1994). Never say never. Topoi, 13, 135145.
Wright, C. (1982). Strict finitism. See Wright (1993b), Chapter 4, pp. 107175.
Wright, C. (1993a). Can a Davidsonian meaning-theory be construed in terms of assertibility? See Wright (1993b), Chapter 14, pp. 403432.
Wright, C. (1993b). Realism, Meaning, and Truth (2nd edition). Blackwell, Oxford.
Wright, C. (1994). Truth and Objectivity. Harvard University Press, Cambridge.
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