# zbMATH — the first resource for mathematics

The paradox of the knower revisited. (English) Zbl 1348.03057
Summary: The Paradox of the Knower was originally presented by D. Kaplan and R. Montague [Notre Dame J. Formal Logic 1, 79–90 (1960; Zbl 0112.00409)] as a puzzle about the everyday notion of knowledge in the face of self-reference. The paradox shows that any theory extending Robinson arithmetic with a predicate $$K(x)$$ satisfying the factivity axiom $$K(\overline A)\to A$$ as well as a few other epistemically plausible principles is inconsistent. After surveying the background of the paradox, we will focus on a recent debate about the role of epistemic closure principles in the Knower. We will suggest this debate sheds new light on the concept of knowledge which is at issue in the paradox – i.e. is it a “thin” notion divorced from concepts such as evidence or justification, or is it a “thick” notion more closely resembling mathematical provability? We will argue that a number of features of the paradox suggest that the latter option is more plausible. Along the way, we will provide a reconstruction of the paradox using a quantified extension of S. N. Artemov’s Logic of Proofs [Bull. Symb. Log. 7, No. 1, 1–36 (2001; Zbl 0980.03059)], as well as a series of results linking the original formulation of the paradox to reflection principles for formal arithmetic. On this basis, we will argue that while the Knower can be understood to motivate a distinction between levels of knowledge, it does not provide a rationale for recognizing a uniform hierarchy of knowledge predicates in the manner suggested by C. A. Anderson [“The paradox of the knower ”, J. Philos. 80, No. 6, 338–355 (1984; doi:10.2307/2026335)].

##### MSC:
 03F30 First-order arithmetic and fragments 03B42 Logics of knowledge and belief (including belief change) 03F40 Gödel numberings and issues of incompleteness 03F45 Provability logics and related algebras (e.g., diagonalizable algebras)
Full Text:
##### References:
 [1] Anderson, C. A., The paradox of the knower, J. Philos., 80, 6, 338-355, (1984) [2] Artemov, S. N., Explicit provability and constructive semantics, Bull. Symbolic Logic, 7, 1, 1-36, (2001) · Zbl 0980.03059 [3] Beklemishev, L., Reflection principles and provability algebras in formal arithmetic, Russian Math. Surveys, 60, 2, 197-268, (2005) · Zbl 1097.03054 [4] Bezboruah, A.; Shepherdson, J. C., Gödelʼs second incompleteness theorem for $$\mathsf{Q}$$, J. Symbolic Logic, 41, 2, 503-512, (1976) · Zbl 0328.02017 [5] Cross, C., The paradox of the knower without epistemic closure, Mind, 110, 438, 319-333, (2001) [6] Cross, C., More on the paradox of the knower without epistemic closure, Mind, 113, 449, 109-114, (2004) [7] Cross, C., The paradox of the knower without epistemic closure—corrected, Mind, 121, 457-566, (2012) [8] Dean, W., Montagueʼs paradox, informal provability, and explicit modal logic, Notre Dame J. Form. Log., 55, 2, (2014) · Zbl 1352.03062 [9] Detlefsen, M., Hilbertʼs program, Synth. Libr., vol. 182, (1986), Kluwer Dordrecht [10] Feferman, S., Transfinite recursive progressions of axiomatic theories, J. Symbolic Logic, 27, 3, 259-316, (1962) · Zbl 0117.25402 [11] Fitting, M., A quantified logic of evidence, Ann. Pure Appl. Logic, 152, 1-3, 67-83, (2008) · Zbl 1133.03008 [12] Franks, C., The autonomy of mathematical knowledge: hilbertʼs program revisited, (2009), Cambridge University Press · Zbl 1192.00012 [13] Franzén, T., Inexhaustibility: A non-exhaustive treatment, (2004), A.K. Peters, Ltd. · Zbl 1094.03001 [14] Friedman, H.; Sheard, M., An axiomatic approach to self-referential truth, Ann. Pure Appl. Logic, 33, 1-21, (1987) · Zbl 0634.03058 [15] Gentzen, G., The consistency proof for elementary number theory, (The Collected Papers of Gerhard Gentzen, (1969)), 132-213 [16] Gettier, E., Is justified true belief knowledge?, Analysis, 23, 6, 121-123, (1963) [17] Gödel, K., Lecture at zilselʼs, (Feferman, S., Collected Works. Vol. III. Unpublished Lectures and Essays, (1995), Clarendon), 62-113 [18] Goldman, A. I., Discrimination and perceptual knowledge, J. Philos., 73, 20, 771-791, (1976) [19] Grim, P., Operators in the paradox of the knower, Synthese, 94, 409-428, (1993) · Zbl 0803.03003 [20] Grzegorczyk, A., Undecidability without arithmetization, Studia Logica, 79, 2, 163-230, (2005) · Zbl 1080.03004 [21] Hájek, P.; Pudlák, P., Metamathematics of first-order arithmetic, Perspect. Math. Log., (1998), Springer Berlin · Zbl 0889.03053 [22] Halbach, V., Axiomatic theories of truth, (2011), Cambridge University Press · Zbl 1223.03001 [23] Halbach, V.; Horsten, L., Axiomatizing kripkeʼs theory of truth, J. Symbolic Logic, 71, 2, 677-712, (2006) · Zbl 1101.03005 [24] Hale, B.; Wright, C., The reasonʼs proper study, (2001), Oxford University Press Oxford [25] Hintikka, J., Knowledge and belief: an introduction to the logic of the two notions, (1962), Cornell University Press [26] Kaplan, D.; Montague, R., A paradox regained, Notre Dame J. Form. Log., 1, 3, 79-90, (1960) · Zbl 0112.00409 [27] Ketland, J., Deflationism and tarskiʼs paradise, Mind, 108, 429, 69-94, (1999) [28] Kreisel, G., Foundations of intuitionistic logic, Stud. Logic Found. Math., 44, 198-210, (1962) [29] Kreisel, G.; Lévy, A., Reflection principles and their use for establishing the complexity of axiomatic systems, Math. Log. Q., 14, 7-12, 97-142, (1968) · Zbl 0167.01302 [30] Kripke, S., Outline of a theory of truth, J. Philos., 72, 19, 690-716, (1975) · Zbl 0952.03513 [31] Kurata, R., Paris-harrington principles, reflection principles and transfinite induction up to $$\epsilon_0$$, Ann. Pure Appl. Logic, 31, 237-256, (1986) · Zbl 0603.03015 [32] Leitgeb, H., On formal and informal provability, (New Waves in the Philosophy of Mathematics, (2009), Palgrave Macmillan) · Zbl 1261.03035 [33] Maitzen, S., The knower paradox and epistemic closure, Synthese, 114, 2, 337-354, (1998) · Zbl 0932.03002 [34] McGee, V., How truthlike can a predicate be? A negative result, J. Philos. Logic, 14, 4, 399-410, (1985) · Zbl 0583.03002 [35] Montague, R., Semantic closure and non-finite axiomatizability, (Infinitistic Methods, (1961)), 45-69 · Zbl 0116.00703 [36] Montague, R., Syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability, Acta Philos. Fenn., 16, 153-167, (1963) · Zbl 0117.01302 [37] Myhill, J., Some remarks on the notion of proof, J. Philos., 57, 461-471, (1960) [38] Nozick, R., Philosophical explanations, (1981), Harvard University Press Cambridge [39] Pudlák, P., On the lengths of proofs of consistency, (Collegium Logicum - Annals of the Kurt Gödel Society, (1996)), 65-86 · Zbl 0855.03032 [40] Reinhardt, W. N., Epistemic theories and the interpretation of Gödelʼs incompleteness theorem, J. Philos. Logic, 15, 4, 427-474, (1986) · Zbl 0631.03009 [41] Schmerl, U. R., A fine structure generated by reflection formulas over primitive recursive arithmetic, (Boffa, M.; van Dalen, D.; McAloon, K., Logic Colloquium 78, (1979), Elsevier), 335-350 [42] Shapiro, S., Proof and truth: through thick and thin, J. Philos., 95, 10, 493-521, (1998) [43] Skyrms, B., An immaculate conception of modality or how to confuse use and mention, J. Philos., 75, 7, 368-387, (1978) [44] Smorynski, C., The incompleteness theorems, (Handbook of Mathematical Logic, (1977), North-Holland), 821-865 [45] Smoryński, C., Self-reference and modal logic, (1985), Springer · Zbl 0596.03001 [46] Tait, W., Finitism, J. Philos., 78, 9, 524-546, (1981) [47] Takeuti, G., Proof theory, Stud. Logic Found. Math., vol. 81, (1987), North-Holland/Elsevier Amsterdam/New York [48] Tarski, A., The concept of truth in formalized languages, (Logic, Semantics, Metamathematics, (1956), Clarendon) [49] Tennant, N., Deflationism and the Gödel phenomena: reply to ketland, Mind, 114, 453, 89-96, (2005) [50] Uzquiano, G., The paradox of the knower without epistemic closure?, Mind, 113, 449, 95-107, (2004) [51] Williamson, T., Knowledge and its limits, (2000), Oxford University Press
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.