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INTUITIONISTIC EPISTEMIC LOGIC

Published online by Cambridge University Press:  21 March 2016

SERGEI ARTEMOV  and
TUDOR PROTOPOPESCU
SERGEI ARTEMOV*
Affiliation:
The Graduate Center, The City University of New York
TUDOR PROTOPOPESCU*
Affiliation:
The Graduate Center, The City University of New York
*
*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
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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 A → KA 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 KA → A 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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Type
Research Article
Information
The Review of Symbolic Logic , Volume 9 , Issue 2 , June 2016 , pp. 266 - 298
Copyright
Copyright © Association for Symbolic Logic 2016 

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References

BIBLIOGRAPHY

Artemov, S. (2001). Explicit Provability and Constructive Semantics. Bulletin of Symbolic Logic, 7(1), 1–36.CrossRefGoogle Scholar
Artemov, S. (2008). The logic of justification. Review of Symbolic Logic, 1(4), 477–513.CrossRefGoogle Scholar
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. 189–360.Google Scholar
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.Google Scholar
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.Google Scholar
Artemov, S., & Protopopescu, T. (2012). Discovering knowability: A semantical analysis. Synthese, 190(16).Google Scholar
Artemov, S., & Protopopescu, T. (2014, June). Intuitionistic epistemic logic. ArXiv, math.LO 1406.1582v1.Google Scholar
Božić, M., & Došen, K. (1984). Models for normal intuitionistic modal logics. Studia Logica, 43(3), 217–245.CrossRefGoogle Scholar
Božić, M., & Došen, K. (1985). Models for stronger normal intuitionistic modal logics. Studia Logica, An International Journal for Symbolic Logic, 44(1), 39–70.Google Scholar
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.Google Scholar
Brouwer, L. (1981). Brouwer’s Cambridge Lectures on Intuitionism. Cambridge University Press, Cambridge.Google Scholar
Buss, S. (1998). Introduction to proof theory. In Buss, S., editor, Handbook of Proof Theory, Chapter 1. Elsevier, Amsterdam, pp. 1–78.Google Scholar
Chagrov, A., & Zakharyaschev, M. (1997). Modal Logic. Oxford, UK: Clarendon Press.CrossRefGoogle Scholar
Church, A. (2009). Referee Reports on Fitch’s “A Definition of Value”. See Salerno (2009), pp. 13–20.CrossRefGoogle Scholar
Clarke, E. M., & Kurshan, R. P. (1996). Computer-aided verification. IEEE Spectrum, 33(6), 61–67.CrossRefGoogle Scholar
Constable, R. (1998). Types in logic, mathematics and programming. In Buss, S., editor, Handbook of Proof Theory. Elsevier, Amsterdam, pp. 683–786.CrossRefGoogle Scholar
Contu, P. (2006). The justification of the logical laws revisited. Synthese, 148(3), 573–588.CrossRefGoogle Scholar
De Vidi, D., & Solomon, G. (2001). Knowability and intuitionistic logic. Philosophia, 28(1), 319–334.CrossRefGoogle Scholar
Dean, W., & Kurakawa, H. (2009). From the knowability paradox to the existence of proofs. Synthese, 176(2), 177–225.CrossRefGoogle Scholar
Descartes, R. (1642). Meditations on first philosophy. In The Philosophical Writings of Descartes, Vol. II. Cambridge, MA: Cambridge University Press, pp. 3–62.Google Scholar
Došen, K. (1984). Intuitionistic double negation as a necessity operator. Publications de L’Institute Mathématique (Beograd)(NS), 35(49), 15–20.Google Scholar
Dummett, M. (1963). Realism. In Truth and Other Enigmas. Cambridge, MA: Harvard University Press, pp. 145–165.Google Scholar
Dummett, M. (1970). Enforcing the encyclical. New Blackfriars, 51(600), 229–234.CrossRefGoogle Scholar
Dummett, M. (1973). The philosophical basis of intuitionistic logic. In Truth and Other Enigmas. Cambridge, MA: Harvard University Press, pp. 215–247.Google Scholar
Dummett, M. (1976). What is a theory of meaning II. In The Seas of Language. Oxford University Press, New York, pp. 34–93.Google Scholar
Dummett, M. (1977). Elements of Intuitionism. Clarendon Press Oxford.Google Scholar
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. 106–116.Google Scholar
Dummett, M. (1991). The Logical Basis of Metaphysics. Cambridge, MA: Harvard University Press. The William James Lectures, 1976.Google Scholar
Dummett, M. (1998). Truth from a constructive point of view. Theoria, 64, 122–138.CrossRefGoogle Scholar
Dummett, M. (2001). Victor’s error. Analysis, 61, 1–2.CrossRefGoogle Scholar
Dummett, M. (2009). Fitch’s paradox of knowability. See Salerno (2009), Chapter 4, pp. 51–52.CrossRefGoogle Scholar
Fitch, F. (1963). A logical analysis of some value concepts. Journal of Symbolic Logic, 28(2), 135–142.CrossRefGoogle Scholar
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. 41–60.CrossRefGoogle Scholar
Floridi, L. (2000). Mathematical skepticism. The Proceedings of the Twentieth World Congress of Philosophy, 2000, 217–265.CrossRefGoogle Scholar
Floridi, L. (2004). Mathematical skepticism: The debate between Hobbes and Wallis. In Skepticism in Renaissance and Post-Renaissance Thought: New Interpretations. Humanity Books.Google Scholar
Frege, G. (1884). The Foundations of Arithmetic: A Logical-Mathematical Investigation into the Concept of Number. New York, NY: Pearson Education.Google Scholar
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.Google Scholar
Gettier, E. (1963). Is knowledge justified true belief? In Pojman, L., editor, The Theory of Knowledge. Belmont, California: Wadsworth Thompson, pp. 125–127.Google Scholar
Glivenko, V. (1929). On Some Points of the Logic of Mr. Brouwer. See Mancosu (1998), Chapter 22, pp. 301–305.Google Scholar
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. 301–303.Google Scholar
Hart, W. D. (1979). Access and inference. Proceedings of the Aristotelian Society, LIII, 153–166.Google Scholar
Hazlett, A. (2010). The myth of factive verbs. Philosophy and Phenomenological Research, 80(3), 497–522.CrossRefGoogle Scholar
Hazlett, A. (2012). Factive presupposition and the truth condition on knowledge. Acta Analytica, 27(4), 461–478.CrossRefGoogle Scholar
Heyting, A. (1930). On intuitionistic logic. See Mancosu (1998), Chapter 23, pp. 306–310.Google Scholar
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. 42–49.Google Scholar
Heyting, A. (1966). Intuitionism: An Introduction (second revised edition). Studies in Logic and the Foundations of Mathematics, Vol. 41. North-Holland, Amsterdam.Google Scholar
Hilbert, D., & Bernays, P. (1932). Grundlagen der Mathematik II (Grundlehren der mathematischen Wissenschaften). Springer-Verlag, Dordrecht.Google Scholar
Hirai, Y. (2010a). An intuitionistic epistemic logic for sequential consistency on shared memory. In Logic for Programming, Artificial Intelligence, and Reasoning. Springer, Dordrecht, pp. 272–289.CrossRefGoogle Scholar
Hirai, Y. (2010b). Disjunction Property and Finite Model Property for an Intuitionistic Epistemic Logic unpublished manuscript, https://yoichihirai.com/nasslli2010hirai.pdf.Google Scholar
Horsten, L. (2014). Philosophy of mathematics. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy (Spring 2014 ed.).Google Scholar
Husserl, E. (1901). Logical Investigations (2nd Ed edition). Routledge & Kegan Paul, London.Google Scholar
Khlentzos, D. (2004). Naturalistic Realism and the Anti-Realist Challenge. MIT Press, Cambridge.CrossRefGoogle Scholar
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. 415–437.Google Scholar
Kopylov, A., & Nogin, A. (2001). Markov’s principle for propositional type theory. In Computer Science Logic. Springer, Dordrecht, pp. 570–584.CrossRefGoogle Scholar
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. 198–210.Google Scholar
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. 187–201.CrossRefGoogle Scholar
Löb, M. H. (1955). Solution of a problem of Leon Henkin. Journal of Symbolic Logic, 20(2), 115–118.CrossRefGoogle Scholar
Mancosu, P., editor (1998). From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920’s. Oxford Univeristy Press, Oxford.Google Scholar
Martin-Löf, P. (1987, December). Truth of Proposition, Evidence of Judgement, Validity of a Proof. Synthese, 73(3), 407–420.CrossRefGoogle Scholar
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. 141–149.Google Scholar
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. 105–114.CrossRefGoogle Scholar
Martino, E., & Usberti, G. (1994). Temporal and Atemporal truth in Intuitionistic Mathematics. Topoi, 13(2), 83–92.CrossRefGoogle Scholar
Marton, P. (2006). Verificationists versus realists: The battle over knowability. Synthese, 151(1), 81–98.CrossRefGoogle Scholar
Murzi, J. (2010). Knowability and bivalence: Intuitionistic solutions ot the paradox of knowability. Philosophical Studies, 149, 269–281.CrossRefGoogle Scholar
Percival, P. (1990). Fitch and Intuitionistic Knowability. Analysis, 50(3), 182–187.CrossRefGoogle Scholar
Prawitz, D. (1980). Intuitionistic logic: A philosophical challenge. In von Wright, G. H., editor, Logic and Philosophy. Martinus Nijhoff, The Hague, pp. 1–10.Google Scholar
Prawitz, D. (1998a). Comments on Goran Sundholm’s paper: “Proofs as acts and proofs as objects”. Theoria, 64, 318–329.Google Scholar
Prawitz, D. (1998b). Comments on Lars Bergström’s paper: “Prawitz’s version of verificationism”. Theoria, 64, 293–303.Google Scholar
Prawitz, D. (1998c). Comments on Michael Dummett’s Paper “Truth from a constructive point of view”. Theoria, 64, 283–292.CrossRefGoogle Scholar
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. 41–51.CrossRefGoogle Scholar
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. 23–35.Google Scholar
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. 671–695.CrossRefGoogle Scholar
Prawitz, D. (2006). Meaning approached via proofs. Synthese, 148(3), 507–524.Google Scholar
Proietti, C. (2012). Intuitionistic epistemic logic, Kripke models and Fitch’s Paradox. Journal of Philosophical Logic, 41(5), 877–900.CrossRefGoogle Scholar
Proof (2014). Proof. In Encyclopedia Brittanica.Google Scholar
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. 295–307.CrossRefGoogle Scholar
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. 317–330.CrossRefGoogle Scholar
Putnam, H. (1977). Realism and reason. Proceedings and Addresses of the American Philosophical Association, 50(6), 483–498.CrossRefGoogle Scholar
Rasmussen, S. (2009). The paradox of knowability and the mapping objection. See Salerno (2009), Chapter 5, pp. 53–75.CrossRefGoogle Scholar
Salerno, J., editor (2009). New Essays on the Knowability Paradox. Oxford University Press, New York.CrossRefGoogle Scholar
Schroeder-Heister, P. (2006). Validity concepts in proof-theoretic semantics. Synthese, 148(3), 525–571.Google Scholar
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. 165–198.CrossRefGoogle Scholar
Tarski, A. (1969). Truth and proof. Scientific American, 220(6), 63–77.CrossRefGoogle ScholarPubMed
Tennant, N. (1997). The Taming of the True. Oxford Univeristy Press, Oxford.Google Scholar
Tennant, N. (2009). Revamping the restriction strategy. In Salerno, J., editor, New Essays on the Knowability Paradox. New York, NY: Oxford University Press, pp. 223–238.CrossRefGoogle Scholar
Usberti, G. (2006). Towards a semantics based on the notion of justification. Synthese, 148(3), 675–699.CrossRefGoogle Scholar
van Dalen, D. (2002). Intuitionistic logic. In Handbook of Philosophical Logic (2nd edition), Vol. 5. Springer, pp. 1–114.Google Scholar
van Dalen, D. (2004). Logic and Structure. Springer, Dordrecht.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Voevodsky, V., et al. (2013). Homotopy Type Theory. Univalent Foundations Program.Google Scholar
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. 483–505.Google Scholar
Williamson, T. (1982). Intuitionism disproved? Analysis, 42(4), 203–207.CrossRefGoogle Scholar
Williamson, T. (1988). Knowability and constructivism. Philosophical Quarterly, 38(153), 422–432.CrossRefGoogle Scholar
Williamson, T. (1992). On intuitionistic modal epistemic logic. Journal of Philosophical Logic, 21(1), 63–89.Google Scholar
Williamson, T. (1994). Never say never. Topoi, 13, 135–145.CrossRefGoogle Scholar
Wright, C. (1982). Strict finitism. See Wright (1993b), Chapter 4, pp. 107–175.CrossRefGoogle Scholar
Wright, C. (1993a). Can a Davidsonian meaning-theory be construed in terms of assertibility? See Wright (1993b), Chapter 14, pp. 403–432.Google Scholar
Wright, C. (1993b). Realism, Meaning, and Truth (2nd edition). Blackwell, Oxford.Google Scholar
Wright, C. (1994). Truth and Objectivity. Harvard University Press, Cambridge.Google Scholar