Document Type : Research
Author
Department of Philosophy, Faculty of Literature and Humanities, University of Isfahan, Isfahan, Iran
Abstract
In this note, we study the effect of adding fixed points to justification logics. By making use of the fixed point operators (or diagonal operators) introduced by Smorynski in his Diagonalization Operator Logic, we introduce fixed point extensions of Fitting's quantified logic of proofs QLP. We then formalize the Knower Paradox and various self-reference versions of the Surprise Test Paradox in these fixed point extensions of QLP. By interpreting a surprise statement as a statement for which there is no justification or evidence, we propose a solution to the self-reference version of the Surprise Test paradox. We show that one of the axioms of QLP (the Uniform Barcan Formula) could be the reason for producing contradiction in these paradoxes, and thus by rejecting this axiom we can avoid contradiction in the aforementioned paradoxes. By introducing Mkrtychev models for the fixed point extensions of QLP, we further show that these fixed point extensions (without the Uniform Barcan Formula) are consistent.
Keywords
- Artemov, S.N.: Operational modal logic. (1995).
- Artemov, S.N.: Explicit Provability and Constructive Semantics. Bull. Symb. Log. 7, 1–36 (2001).
- Fitting, M.: A Quantified Logic of Evidence. Electron. Notes Theor. Comput. Sci. 143, 59–71 (2006). https://doi.org/10.1016/j.entcs.2005.04.038.
- Fitting, M.: A quantified logic of evidence. Ann. Pure Appl. Log. 152, 67–83 (2008). https://doi.org/10.1016/j.apal.2007.11.003.
- Smoryński, C.: Self-Reference and Modal Logic. Springer New York, New York, NY (1985). https://doi.org/10.1007/978-1-4613-8601-8.
- O’Connor, D.J.: Pragmatic paradoxes. Mind. 57, 358–359 (1948). https://doi.org/10.1093/mind/lvii.227.358.
- Weiss, P.: The Prediction Paradox. Mind. 61, 265–269 (1952). https://doi.org/10.1093/mind/lxi.242.265.
- Sorensen, R.: Epistemic Paradoxes, https://plato.stanford.edu/entries/epistemic-paradoxes/, last accessed 2021/04/18.
- Quine, W. V.: ON A So-Called Paradox. Mind. 62, 65–67 (1953). https://doi.org/10.1093/mind/lxii.245.65.
- Kripke, S.A.: On Two Paradoxes of Knowledge. In: Philosophical Troubles: Collected Papers. Oxford University Press (2011). https://doi.org/10.1093/acprof:oso/9780199730155.003.0002.
- Chow, T.Y.: The Surprise Examination or Unexpected Hanging Paradox. Am. Math. Mon. 105, 41–51 (1998). https://doi.org/10.1080/00029890.1998.12004847.
- Cheung, L.K.C.: On Two Versions of “the Surprise Examination Paradox.” Philos. (United States). 41, 159–170 (2013). https://doi.org/10.1007/s11406-013-9416-7.
- Gerbrandy, J.: The surprise examination in dynamic epistemic logic. Synthese. 155, 21–33 (2007). https://doi.org/10.1007/s11229-005-2211-7.
- Shaw, R.: The paradox of the unexpected examination. Mind. 67, 382–384 (1958). https://doi.org/10.1093/mind/lxvii.267.382.
- Fitch, F.B.: A Goedelized Formulation of the Prediction Paradox. Am. Philos. Q. 1, (1964).
- Kritchman, S., Raz, R.: The surprise examination paradox and the second incompleteness theorem. Not. AMS. 57, 1454–1458 (2010).
- Ardeshir, M., Ramezanian, R.: A solution to the surprise exam paradox in constructive mathematics. Rev. Symb. Log. 5, 679–686 (2012). https://doi.org/10.1017/S1755020312000160.
- Boolos, G.: The logic of provability. Cambridge university press (1995).
- Montague, R.: Syntactical Treatment of Modality with Corrollaries on Reflexion Principles and Finite Axiomatizability. Acta Filosophica Fenn. 16, 153–165 (1963).
- Égré, P.: The Knower Paradox in the Light of Provability Interpretations of Modal Logic. J. Logic, Lang. Inf. 14, 13–48 (2005).
- Kaplan, D., Montague, R.: A paradox regained. Notre Dame J. Form. Log. 1, 79–90 (1960).
- Ghari, M.: A Note on Fixed Points in Justification Logics and the Surprise Test Paradox. ArXiv e-prints. (2014).
- Mkrtychev, A.: Models for the Logic of Proofs. In: Adian, S. and Nerode, A. (eds.) Logical Foundations of Computer Science, 4th International Symposium, LFCS’97, Yaroslavl, Russia, July 6--12, 1997, Proceedings. pp. 266–275. Springer (1997). https://doi.org/10.1007/3-540-63045-7_27.
- Arló-Costa, H., Kishida, K.: Three Proofs and the Knower in the Quantified Logic of Proofs. In: Online Proceedings of Sixth Annual Formal Epistemology Workshop (FEW 2009). , Carnegie Mellon University, Pittsburg, PA, USA (2009).
- Dean, W., Kurokawa, H.: The Paradox of the Knower revisited. Ann. Pure Appl. Log. 165, 199–224 (2014). https://doi.org/10.1016/j.apal.2013.07.010.
- Dean, W.: Montague’s paradox, informal provability, and explicit modal logic. Notre Dame J. Form. Log. 55, 157–196 (2014). https://doi.org/10.1215/00294527-2420636.
- Fagin, R., Y. Halpern, J., Moses, Y., Y. Vardi, M.: Reasoning about Knowledge. MIT Press (1995).
- Binkley, R.: The Surprise Examination in Modal Logic. J. Philos. 65, 127–136 (1968). https://doi.org/10.2307/2024556.
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