Abstract
Pavel Florensky solves Lewis Carroll’s ‘Barbershop’ paradox to support his reasoning in a previous chapter. Our discussion includes a) the problem (which we also refer to as the p paradox), b) Carroll’s solution, c) Bertrand Russell’s solution, d) Florensky’s solution and then e) a material example proffered by Florensky. Both Russell and Florensky disagree with Carroll’s solution, yet, (ostensibly) unbeknownst to themselves they offer the same solution, which is ‘p implies not-q’. Given Florensky’s material example, the solution seems to tell us something about the logic of belief. We ask whether Florensky’s example has reverse implications for Russell’s solution.
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Notes
Translated by Boris Jakim, with an Introduction by Richard F. Gustafson, Princeton University Press (1997). First published in Russian under the same title, Cтoлп и Утвepждeниe Иcтины: Oпыт Пpaвocлaвнoй Teoдицeи в Двeнaдцaти Пиcьмax, in Moscow in 1914.
For readers unfamiliar with Florensky, here is a very brief introduction. Florensky was born January 9, 1882 in Yevlakh, Azerbaijan and, after roughly 4 years of Soviet imprisonment, was executed 8 December 1937. Most of his youth was spent in Tblisi, Georgia. In 1904, he took his first degree in mathematics at Moscow University, where he studied under N. V. Bugaev (1837–1903). Florensky also studied philosophy at Moscow University, under the influential philosophers S. N. Trubetskoy (1862–1905) and L. M. Lopatin (1855–1920) (PGT, x). He married a Russian peasant, Anna Mikhailovna Giatsintov, in the late summer of 1910, a union which ultimately produced five children. In 1911, having studied at Moscow Theological Academy, he was ordained a Russian Orthodox Priest. (He was raised in a non-religious household and baptized in 1903). A leading figure in 20th century Russian Orthodox philosophy, Florensky is also regarded as a polymath, known to have published in at least ten fields—ranging from mathematics and philosophy to botany, theology and art history. He held teaching appointments in philosophy, physics and engineering, and various other notable posts at research centers and universities. Few of Florensky’s works have been translated into English. The three most notable are The Pillar and Ground of Truth (Princeton: 1997), Beyond Vision: Essays on the Perception of Art, Nicoletta Misler, ed. and Wendy Salmond, trans. (Reaktion Books: 2002) and Iconostasis, Donald Sheehan and Olga Andrejev, trans. (St Vladimir’s Seminary Press: 1996). There are relatively few works on Florensky in English. Two books that give further insight into his life and thought, however, are worthy of mention 1) Avril Pyman’s Pavel Florensky: A Quiet Genius: The Tragic and Extraordinary Life of Russia’s Unknown da Vinci (Continuum 2010); and 2) R. Slesinski Pavel Florensky: A Metaphysics of Love (St Vladimir’s Seminary Press).
In Mind, N. S., (1894) 3(11), 436–38. (Incidentally, Florensky gives 1906; and Couturat, on whom he seems to rely a fair bit, gives 1905; I’ve been unable to determine reason for this.) See also the discussions of Alfred Sidgwick ibid. 3(12), 582 and W. E. Johnson ibid. 3(12), 583; Sidgwick ibid. (1895) 4(13), 143; Johnson ibid., 143–44; E. E. C. Jones ibid. (1905) 14(1), 146–48; W. ibid. (1905) 14(2), 292–3; Jones ibid. (1905) 14(4), 576–78; Arthur W. Burks and Irving M. Copi ibid. (1950) 59(234), 219–22.
Florensky is quoting the French logician Louis Couturat (1868–1914) Les principes des Mathématiques (Paris: 1905), 16. See note 852 on 575. His formulation of Carroll’s problem reads ‘q implique r; mais p implique que q implique non-r; que faut-il en conclure?’ In reply, he says ‘On raisonnera comme il suit: p implique non-q or non-r; or non-r implique non-q; donc p implique non-q. c’est a dire la verite de p implique la faussete de q. Mais Lewis Carroll raisonne autrement’, and he rehearses what we present in section III. He argues though that Carroll’s solution, not-p, is false ‘parce qu’il est possible que q implique a la fois r et non-r; seulement alors q est faux (comme impliquant deux propositions contradictoires)’, [because it is possible that q implies both r and not-r; only then q is false (as implying two contradictory propositions)]. This is false though, according to the parameters of the paradox as put forth by Carroll. For his (1)—namely, if p, then q—is, as he puts it, ‘always true’. Thus, Florensky disagrees with this (cf PGT, 356), arguing that q implies r; yet under the condition p, it is not the case that q implies not-r; but rather that p implies ‘the negation of q’ (ibid.), which entails not-r. Thus, if Allen is out, Brown is out; but in the case that Carr is out, Allen is in and Brown is in. Or, if A, then B; but if C, then if ~A, then ~B (cf LP, 438). And using p, q and r: if q, then r; but if p, then if ~q, then ~r. Hence, under the condition p, q is false: if p, then ~q.
Quoting Couturat Les principes, 16. See note 853 on 575.
See note 2 for Lewis reference.
Cf. B. Russell Principles of Mathematics [PM] (George Allen & Unwin: 1937 [1903]), 16–18. See the comparable solution given by Johnson in Mind (1894) 3(12), 583.
The Florensky text seems to have a typo here. The first statement of this conclusion (at (IV) on page 356) is either p or not-p. The editor seems not to understand the argument in this manner however because this is remedied in the following text when the conclusion is presented as it is presented here.
The p of Russell’s definition is not to be confused with the p of the p paradox. The definition just happens to use the variable p; it is applicable to any instance of negation, however, and not just in this particular instance.
I have changed the script to match the so called ‘new’ orthography.
See The Problem of Consciousness (Oxford: Basil Blackwell, 1991) and The Mysterious Flame: Conscious Minds in a Material World (New York: Basic Books, 1999).
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Rhodes, M. Note on Florensky’s Solution to Carroll’s ‘Barbershop’ Paradox: Reverse Implication for Russell?. Philosophia 40, 607–616 (2012). https://doi.org/10.1007/s11406-011-9333-6
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DOI: https://doi.org/10.1007/s11406-011-9333-6