Defusing a Paradox to a Hypodox
DOI:
https://doi.org/10.12775/LLP.2024.020Keywords
Hypodox, Paradox, Bertrand’s Chord Paradox, Liar Paradox, Liar Hypodox, Curry’s Paradox, Yablo’s Paradox, Epimenides Paradox, Truth-teller, Parking Voucher Paradox, Paradoxical dilemma, Dilemma over a hypodox, Ship of Theseus, Water-and-wine ProblemAbstract
One way of resolving a paradox is to defuse it to a hypodox. This way is relatively unknown though. The goal of this paper is to explain this way with varied examples. The hypodoxes are themselves a broad class: both the Truth-teller and the 21st birthday of someone born on 29th February can be construed as hypodoxes. The most familiar kind of relation between paradoxes and hypodoxes is exemplified by the relation between the Liar and the Truth-teller. This article concerns a second kind where a paradox is defused to a hypodox by restricting or rejecting some granted principles. The Liar paradox has this second kind of relation to a Liar hypodox, which will be introduced. In some cases, defusing a paradox to a hypodox is only a partial resolution, as the hypodox itself may then need resolving. Even so, such a partial resolution decomposes a complex problem into more easily understood problems. Moreover, I compare the result of defusing a paradox to a hypodox with the results of resolving paradoxes in other ways. I give four examples. The first is mainly pedagogic, concerning a birthday. The second is a lightweight legal case, presenting a parking voucher paradox. The third is a formal system in which a Liar and Liar-like sentences are hypodoxical. The fourth is a philosophical critique of ways of solving Bertrand’s chord paradox.
References
Bertrand, J., 1889, Calcul des probabilities, Paris: Gauthier-Villars et Fils.
Church, A., 1946, “Alexandre Koyré, ‘The liar. Philosophy and phenomenological research’, vol. 6, no. 3 (1946), pp. 344–362”, The Journal of Symbolic Logic, 11(4): 131–131.
Clark, M., 2012, Paradoxes from A to Z, Routledge.
Cook, R. T., 2009, Dictionary of Philosophical Logic, Edinburgh University Press.
Cook, R. T., 2013, Paradoxes, John Wiley & Sons.
Cook, R. T., 2020, “An intensional theory of truth: an informal report”, The Philosophical Forum, 51(2): 115–126.
Cook, R. T., 2022, “Outline of an intensional theory of truth”, Notre Dame Journal of Formal Logic, 63(1): 81–108.
Da Ré, B., F. Pailos, and D. Szmuc, 2020, “Theories of truth based on four-valued infectious logics”, Logic Journal of the IGPL, 28(5): 712–746. DOI: http://dx.doi.org/10.1093/jigpal/jzy057
Eldridge-Smith, P., 2007, “Paradoxes and hypodoxes of time travel”, in J. Lloyd Jones, P. Campbell, and P. Wylie (eds.), Art and Time, Australian Scholarly Publishing, Melbourne. https://philarchive.org/archive/ELDPAH
Eldridge-Smith, P., 2008, “The liar paradox and its relatives”, PhD Thesis: The Australian National University. https://digitalcollections.anu.edu.au/handle/1885/49284. DOI: http://dx.doi.org/10.25911/5d7a2c4535ea8
Eldridge-Smith, P., 2012, “A hypodox! a hypodox! a disingeneous hypodox!”, The Reasoner, 6(7): 118–19. https://research.kent.ac.uk/reasoning/the-reasoner/the-reasoner-volume-6/
Eldridge-Smith, P., 2019, “The liar hypodox: A truth-teller’s guide to defusing proofs of the liar paradox”, Open Journal of Philosophy, 9(2): 152–171. https://file.scirp.org/pdf/OJPP_2019050716145501.pdf. DOI: http://dx.doi.org/10.4236/ojpp.2019.92011
Eldridge-Smith, P., 2020, “Two fallacies in proofs of the liar paradox”, Philosophia, 48(3): 947–966. DOI: http://dx.doi.org/10.1007/s11406-019-00158-5
Eldridge-Smith, P., 2022, “In search of modal hypodoxes using paradox hypodox duality”, Philosophia, 50:2457–2476. DOI: http://dx.doi.org/10.1007/s11406-022-00546-4
Eldridge-Smith, P., 2023, “The import of hypodoxes for the liar and russell’s paradoxes”, Synthese, 202(4): 105. DOI: http://dx.doi.org/10.1007/s11229-023-04326-9
Fine, K., 1975, “Vagueness, truth and logic”, Synthese: 265–300.
Gilbert. W. S., and A. S. Sullivan, 1879, The Pirates of Penzance, Chappell 1919.
James, W., 1907, “Lecture ii: What pragmatism means”, in Pragmatism: A New Name for Some Old Ways of Thinking, Project Gutenberg. https://www.gutenberg.org/files/5116/5116-h/5116-h.htm#link2H_4_0004
Jaynes, E. T., 1968, “Prior probabilities”, IEEE Transactions on systems science and cybernetics, 4(3): 227–241.
Jaynes, E. T., 1973, “The well-posed problem”, Foundations of Physics, 3(4): 477–492.
Lycan, W. G., 2010, “What, exactly, is a paradox?”, Analysis, 70(4): 615–622.
Mackie, J. L., 1973, Truth, Probability and Paradox: Studies in Philosophical Logic, Oxford University Press.
Marinoff, L., 1994, “A resolution of bertrand’s paradox”, Philosophy of Science, 61(1): 1–24.
McGee, V., 1990, Truth, Vagueness, and Paradox: An Essay on the Logic of Truth, Hackett Publishing.
Mikkelson, J. M., 2004, “Dissolving the wine/water paradox”, The British Journal for the Philosophy of Science, 55(1): 137–145.
Mortensen, Ch., and G. Priest, 1981, “The truth teller paradox”, Logique et Analyse, 24(95/96): 381–388.
OED, 2020a, “Anniversary, n. and adj.”, in Oxford English Dictionary, Oxford University Press. www.oed.com/view/Entry/7909
OED, 2020b, “Birthday, n.”, in Oxford English Dictionary, Oxford University Press. www.oed.com/view/Entry/19398
Olin, D., 2003, Paradox, vol. 12, McGill-Queen’s Press-MQUP.
Prior, A. N., 1958, “Epimenides the cretans”, Journal of Symbolic Logic, 23(3): 261–266.
Prior, A. N., 1961, “On a family of paradoxes”, Notre Dame Journal of Formal Logic, 2(1): 16–32.
Quine, W. V. O., 1976, “The ways of paradox”, pages 1–18 in The Ways of Paradox and Other Essays, Harvard University Press Cambridge, MA.
Quine, W. V. O., 1995, “Truth, paradox and Gödel’s theorem”, His Selected Logic Papers (Enlarged ed., pp. 236–241), Cambridge, MA: Harvard University Press.
Rescher, N., 2001, “Paradoxes”, Their Roots, Range and Resolution, Chicago and La Salle, Illinois: Open Court.
Rosenblatt, L., and C. Gallovich, 2022, “Paradoxicality in Kripke’s theory of truth”, Synthese, 200(2): 1–23. DOI: http://dx.doi.org/10.1007/s11229-022-03625-x
Rosenblatt, L., and D. E. Szmuc, 2014, “On pathological truths”, The Review of Symbolic Logic, 7(4): 601–617. DOI: http://dx.doi.org/10.1017/S1755020314000239
Rossi, L., 2019, “A unified theory of truth and paradox”, The Review of Symbolic Logic, 12(2): 209–254. DOI: http://dx.doi.org/10.1017/S1755020319000078
Sainsbury, R. M., 2009, Paradoxes, Cambridge University Press.
Schiffer, S. R., 2003, The Things we Mean, Oxford University Press.
Shackel, N., 2007, “Bertrand’s paradox and the principle of indifference”, Philosophy of Science, 74(2): 150–175.
Smullyan, R. M., 1957, “Languages in which self reference is possible”, The Journal of Symbolic Logic, 22(1): 55–67.
Sorensen, R., 2003, A Brief History of the Paradox: Philosophy and the Labyrinths of the Mind, Oxford University Press.
Tarski, A., 1936, “The concept of truth in formalized languages”, pages 152–278 in J. Corcoran (ed.), Logic, Semantics and Metamathematics, Hackett Pub. Co., Indianapolis.
Tarski, A., 1944, “The semantic conception of truth and the foundations of semantics”, Philosophy and Phenomenological Research, 4: 341–376.
Tourville. N., and R. T. Cook, 2020, “Embracing intensionality: Paradoxicality and semi-truth operators in fixed point models”, Logic Journal of the IGPL, 28(5): 747–770. DOI: http://dx.doi.org/10.1093/jigpal/jzy058
Wrangham, R., 2019, The Goodness Paradox: How evolution made us both more and less violent, Profile Books.
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