¹ See Nye, AndreaWords of Power: a feminist reading of the history of logic (Routledge: London 1990)Google Scholar. Cf. Plumwood, Val ‘The Politics of Reason: Towards a Feminist Logic’: Australasian Journal of Philosophy 71 (1993), pp. 436–462CrossRefGoogle Scholar.

² See most recently Priest, GrahamDoubt Truth to be a Liar (Clarendon Press: Oxford 2006)Google Scholar. Priest does not doubt all the received laws: ‘though nothing is completely uncontentious, there is hardly any disagreement about the correctness of [the Law of Identity]’: ibid., p. 13.

³ Maddy, Penelope ‘A Naturalistic Look at Logic’: Proceedings and Addresses of the American Philosophical Association 76 (2) 2002, pp. 61–90CrossRefGoogle Scholar.

⁴ Strictly, it might be better to express these immediately as ontological and not logical claims: everything is identical with itself; nothing is both itself and something other than itself; everything is one thing or another. But this would be to assume their metaphysical status from the beginning (and make their contentiousness too obvious). I prefer to begin by examining them as if they were merely logical, rules about how to speak and think.

⁵ Which is at least intimated in Parmenides: see Sorabji, RichardTime, Creation and the Continuum (Duckworth: London 1983)Google Scholar for discussion of atomistic and continuous theories of time and change.

⁶ A readable account of the reasonings that have led to these conclusions can be found in Greene, BrianThe Elegant Universe (Cape: London 1999)Google Scholar.

⁷ See also Greene, BrianThe Fabric of the Cosmos (Penguin: London 2004), p. 351Google Scholar.

⁸ For further logical exegesis of the so-called sorites paradox see Beall, J.C., ed., Liars and Heaps (Clarendon Press: Oxford 2003)Google Scholar. Unfortunately few if any of the papers in this volume are comprehensible to any but dedicated logicians, and none make any attempt to persuade non-logicians that this matters.

⁹ See Ganeri, JonardonPhilosophy in Classical India (Routledge: London 2001), pp. 132ffGoogle Scholar.

¹⁰ A point that explains why Aristotelian logic does not permit obversion, the inference from a's not being X to its being not-X: the traditional Square of Opposition is made coherent by the assumption that propositions of the form A & I have existential import, while those of the form E and O do not. And ‘it is false that x is A’ implies that ‘x is not A’ only if that in turn does not imply that ‘x is not-A’ (which is actually reasonable enough: it is false (or at least, not true) that the King of France is bald, but he doesn't for that reason have a head of hair.)

¹¹ Aristotle De Partibus Animalium 643b10ff; see Physics 5.227a10f.

¹² See Aristotle On Interpretation ch.9. There is a large literature on the interpretation of this one chapter. My reading is that Aristotle accepts that it will be true tomorrow that there is either a sea-battle or there isn't, but that it is not now true either that there will be or there won't. Symbolically, F(p v ¬p) does not imply Fp v F¬p, any more than x's not being A implies that x is not-A. At the moment, nothing determines either that there will be or that won't be a sea-battle: neither claim is yet true.

¹³ As Pelletier, F.J. & Stainton, R.J. point out (against Timothy Williamson Vagueness (Routledge: London 1994))Google Scholar in ‘On “the Denial of Bivalence is Absurd”’, , in Australasian Journal of Philosophy 81 (2003), pp. 369–382Google Scholar.

¹⁴ For an introduction to the very large recent literature on vagueness see Keefe, R. & Smith, P., eds., Vagueness: A Reader (MIT Press: Cambridge, MA 1996)Google Scholar. See also the bibliography maintained by Justin Needle at http://www.btinternet.com/~justin.needle/articles.htm (accessed 1.5.2006).

¹⁵ See Pylyshyn, Zenon W., ed., The Robots Dilemma: The Frame Problem in Artificial Intelligence (Ablex: Norwood, New Jersey 1987)Google Scholar.

¹⁶ In a fully determinist world this would be absolutely true: nothing could be any different than it is without altering everything else (see Alexander On Fate 191, 30–192, 28: Long, A.A. & Sedley, D.N. eds., The Hellenistic Philosophers (Cambridge University Press: Cambridge 1987) 55 N: vol.1, p. 337)Google Scholar, and a fully competent intelligence could therefore deduce every from any truth. Our own world, probably, has a little more flexibility, but it is not utterly discontinuous. Scientific theory aspires to the absolutely unified condition (on which more below).

¹⁷ See Keith Lehrer ‘Why Not Scepticism?’ in The Philosophical Forum 5.1971, pp. 289–298; reprinted in The Theory of Knowledge: Classical and Contemporary Readings (third edition), ed. Louis Pojman (Belmont: Wadsworth, 2003), pp. 56–63.

¹⁸ The occasional abrupt response that we have no positive evidence for the existence of such ‘Googols’ is clearly irrelevant. How, after all, could we expect such evidence if they do indeed exist? And their mere possibility is enough to show that we don't know they don't, and that ‘therefore’ we know nothing of what we formerly believed.

¹⁹ Cf. Augustine De Doctrina Christiana 1.1: ‘these thoughts which the Lord has already vouchsafed to me with a view to undertaking this work will, as soon as I begin to impart them to others, be multiplied by His grace.’

²⁰ Babbage, Charles, The Ninth Bridgwater Treatise: a Fragment (London, 1838; re-issued by Frank Cass: London 1967), pp. 36ffGoogle Scholar. The treatise is so called not because Babbage had written eight earlier ones, but because it was an uncanonical addition to the eight Bridgwater Treatises composed by other leading 19^th century thinkers. The machine which Babbage describes is the fragment of his much larger, projected machine, and was built in 1833. Babbage himself suggests that the computer (‘the Calculating Engine’) is operating a ‘new rule’, while the actual mechanical process that engenders it is just the same.

²¹ Robert Chambers followed Babbage in proposing this model for evolutionary change: what were once dinosaurs begin to beget birds when the proper moment comes: see Chambers, RobertVestiges of the Natural History of Creation (1844: reissued by Leicester University Press: Leicester 1969)Google Scholar. I have discussed this at greater length in Biology and Christian Ethics (Cambridge University Press: Cambridge 2000), pp. 22ff.

²² Wells, H.G.First and Last Things (Constable: London 1908)Google Scholar, chapter 1.5 ‘The Classificatory Assumption’, cited by Ray Sorenson in the Stanford Encyclopedia essay on vagueness http://plato.stanford.edu/entries/vagueness/, accessed 22 August 2006. Wells' text can be found at Project Gutenberg http://www.gutenberg.org/dirs/etext03/frsls10.txt, accessed 22 August 2006.

²³ See Ganeri op.cit., pp. 106ff on the Yogacara Buddhist, Dinnaga, who identified concepts primarily by what they excluded.

²⁴ See also Priest, GrahamBeyond the Limits of Thought (Clarendon Press: Oxford 2002; 2nd edition), pp. 271ffCrossRefGoogle Scholar.

²⁵ Ong, Walter J.Orality and Literacy: the technologizing of the word (Methuen: London 1982) pp. 52–3CrossRefGoogle Scholar, after Luria, A.R.Cognitive Development: its Cultural and Social Foundations, ed. Cole, M., tr. Lopez-Morillas, M. & Solotaroff, L. (Harvard University Press: Cambridge, Mass 1976), pp. 108fGoogle Scholar.

²⁶ ibid, p. 53, after Fernandez, James, in Karp, I. & Bird, C.S., eds., Explorations in African Systems of Thought (Indiana University Press: Bloomington, Indiana 1980), pp. 44–59Google Scholar.

²⁷ Shestov, LevPotestas Clavium, tr. Martin, B. (Ohio University Press: Athens, Ohio 1968; 1st ed. 1923), p. 341Google Scholar. See also Horowitz, Brian ‘The Tension between Athens and Jerusalem in the Philosophy of Lev Shestov’: Slavic and East European Journal 43 (1999), pp. 156–73CrossRefGoogle Scholar.

²⁸ Some years ago, my young son, engaged in putting marbles into a jar, asked me if the jar was full. When I said it was, he said ‘Aha!’, and poured a glass of water into the jar.

²⁹ O'Flaherty, WendyHindu Myths: a sourcebook translated from the Sanskrit (Penguin: Harmondsworth 1975) p. 83fGoogle Scholar (from The Mahabharata). See also Douglas, MaryPurity and Danger: an analysis of concepts of pollution and taboo (Routledge & Kegan Paul: London 1966)CrossRefGoogle Scholar for further anthropological reflections on boundary-crossing.

³⁰ E. Husserl Logische Untersuchungen I, p. 117, cited by Shestov, LevPotestas Clavium, tr. Martin, B. (Henry Regnery: Chicago 1970; 1^st published 1923), p. 305Google Scholar

³¹ Aristotle Metaphysics 1.1012b, cited by Shestov op.cit., p. 296. See Degnan, Michael ‘Does Aristotle Beg the Question in His Defense of the Principle of Non-Contradiction?’: Proceedings of the American Catholic Philosophical Association, 1990, pp. 146–159Google Scholar. Cf. Priest, Graham ‘“To Be and Not to Be – That Is the Answer”: On Aristotle on the Law of Non-Contradiction’ in Logical Analysis and History of Philosophy, Volume 1: Philosophiegeschichte im Uberblick, eds. Newen, Albert & Meixner, Uwe, (Schoningh: Paderborn, 1998), pp. 91–130Google Scholar, reprinted with revisions in Priest Doubt Truth to be a Liar op.cit., pp. 7–42.

³² See H.W. Johnstone ‘The Law of Non-Contradiction’: Logique et Analyse 3: pp. 3–10 for a critical discussion of this ‘regulative’ conception of the law.

³³ See Priest Doubt Truth to be a Liar op.cit., pp. 31–3 on negation as cancellation.

³⁴ See Priest Beyond the Limits of Thought op.cit., ch.10.

³⁵ ‘In this respect physicists are like ordinary people. If they can't resolve a contradiction, and the contradiction is not pressing, they just disregard it and give their attention to those aspects of the theory (or theories) that are pleasantly consistent’: Malin, ShimonNature Loves to Hide (Oxford University Press: New York 2001), p. 90Google Scholar.

³⁶ ‘Complete self-confidence is not merely a sin; complete self-confidence is a weakness. Believing utterly in one's self is a hysterical and superstitious belief like believing in Joanna Southcott’: Chesterton, G.K.Orthodoxy (Fontana: London 1961; 1^st published 1908), p. 14Google Scholar

³⁷ Chesterton ibid., p. 32. See also Blake, William, speaking of ‘doubt which is self-contradiction’: The Everlasting Gospel (1910): Complete Works, ed. Keynes, Geoffrey (Oxford University Press: London 1966), p. 753Google Scholar

³⁸ Is it enough to reply that I believe that I could be mistaken but that in actual fact, fortunately, I'm not, and so that I believe all my beliefs are true? But believing in such good fortune seems even more irrational than believing that not all my beliefs are true.

³⁹ Priest Doubt Truth to be a Liar op.cit., pp. 56–71 rejects ‘trivialism’ (that every proposition is true, including, presumably, the proposition that not all propositions are true), because this amounts to endorsing any number of utterly insane and unreasonable propositions. This is not necessarily a good argument.

⁴⁰ See Lewis, David K.On the Plurality of Worlds (Blackwell: Oxford 1986)Google Scholar; Deutsch, DavidThe Fabric of Reality (Penguin: Harmondsworth 1998)Google Scholar.

⁴¹ See Egan, GregQuarantine (Gollancz: London 1992)Google Scholar for an almost successful account of what it would be like if this were not so: ‘until one of our ancestors [long before they were human] learnt this trick [of collapsing the wave function and fixing one possibility as the only real], the universe must have been a radically different place from the one we know. Everything happened simultaneously; all possibilities coexisted. The wave function never collapsed, it just kept on growing more and more complex.’ (p. 117)

⁴² This was Richard Feynman's interpretation of the results: see Greene op.cit., pp. 108ff. Stephen Hawking and Thomas Hertog have recently drawn the conclusion that all possible histories have been traversed up to the present moment, and that, correspondingly, we are ourselves responsible for determining what ‘really’ happened: Amanda Gefter ‘Mr Hawking's Universe’, New Scientist 22 April 2006, pp. 28–32. Another speculation, founded in the thought of Eugene Wigner, is that only the last possible observer will determine what has happened: see Baxter, StephenTimelike Infinity (Collins: London 1992)Google Scholar for a fictional version of this imagining, after Wigner, E.P. ‘Remarks on the mind-body question’ in Good, I.J., ed., The Scientist Speculates (Heinemann: London 1962), pp. 284–301Google Scholar.

⁴³ See Moore, WardBring the Jubilee (Heinemann: London 1955)Google Scholar.

⁴⁴ Benford, GregoryTimescape (Pocket Books: New York 1981), pp. 73ffGoogle Scholar: if a ‘past’ receiver and ‘future’ transmitter are so linked that the receiver turns off the transmitter before it sends out a message pastwards, if and only if the receiver receives the message from the future, the resultant state hovers between transmission and non-transmission, though ‘we don't know what the intermediate, hung-up state means’.

⁴⁵ See my ‘Orwell and the Anti-Realists’ in Philosophy 67 (1992), pp. 141–54.

⁴⁶ Rather as Einstein insisted, contra standard interpretations now associated, perhaps wrongly, with Heisenberg, that particles really must have some definite position and velocity even if it will forever be impossible to discover both at once. Most modern physicists maintain that he was wrong (see Greene op.cit., p. 114).

⁴⁷ See Card, Orson ScottPastwatch: the Redemption of Christopher Columbus (Tom Doherty: New York 1996)Google Scholar.

⁴⁸ Shestov, LevAthens and Jerusalem, tr. Martin, B. (Ohio University Press: Athens, Ohio 1966), p. 418Google Scholar.

⁴⁹ The only witness to this remark is Peter Arnett, who attributed it, in the New York Times on 8 February 1968, to one Major Brown after the destruction of the Vietnamese village Ben Tre. The remark has since taken on a life entirely of its own, and is variously assigned to colonel or marine or press officer, with the added, gratuitous, assumption that the speaker was unconscious of its oddity. Arnett's own later account, in The Daily Mirror on 1 April 2003, reads ‘During the Tet Offensive in Vietnam, I entered a US-held town which had been totally destroyed. The Viet Cong had taken over and were threatening the commander's building so he called down an artillery strike which killed many of his own men. The Major with us [presumably ‘Major Brown’] asked: “How could this happen?” A soldier replied: “Sir, we had to destroy the town to save it.”’(from http://www.mirror.co.uk/archive/tm_method=full%26objectid=12795678%26siteid=89520-name_page.html, accessed 16 June 2007). Others have denied that the remark was ever made at all.

⁵⁰ See Plato Phaedo 102.

⁵¹ As Feyerabend, Paul pointed out, in Against Method (NLB: London 1975), p. 49Google Scholar: ‘After Aristotle and Ptolemy, the idea that the earth moves - that strange, ancient and “entirely ridiculous”, Pythagorean view – was thrown on the rubbish heap of history, only to be revived by Copernicus and to be forged by him into a weapon for the defeat of its defeaters.’

⁵² Chesterton, G.K.Fancies versus Fads (Methuen: London 1923), p. 95Google Scholar.

⁵³ See Claude-Pierre, PeggyThe Secret Language of Eating Disorders: The Revolutionary New Approach to Understanding and Curing Anorexia and Bulimia (Doubleday: New York 1998)Google Scholar for an empathetic and helpful account of the condition.

⁵⁴ See Bell, Rudolph M.Holy Anorexia (University of Chicago Press: Chicago 1986)Google Scholar.

⁵⁵ ‘The madman is not the man who has lost his reason. The madman is the man who has lost everything except his reason’: Chesterton Orthodoxy op.cit, p. 19.

⁵⁶ See Horton, Robin‘African Traditional Thought and Western Science’, in Africa 37 (1967): 50–71CrossRefGoogle Scholar and 155–187 (also in Horton, RobinPatterns of Thought in Africa and the West: Essays on Magic, Religion and Science (Cambridge University Press: Cambridge, 1993), pp. 197–258)CrossRefGoogle Scholar.

⁵⁷ See Galilei, GalileoDialogues concerning Two New Sciences, tr. Crewe, Henry & de Salvio, Alfonso (Macmillan: New York 1914Google Scholar, reissued by Dover Publications 1952; 1st published 1638), pp. 63f.

⁵⁸ Aristotle Nicomachean Ethics 5.1137b30f.

⁵⁹ Chesterton, G.K.St Thomas Aquinas (Hodder & Stoughton: London 1933), p. 199Google Scholar. See my God, Religion and Reality (SPCK: London 1998).

⁶⁰ ibid., p. 216.

⁶¹ Chesterton, G.K. ‘The Blue Cross’, Father Brown Stories (Cassell: London 1929), p. 27Google Scholar.

⁶² See Sextus Empiricus Against the Logicians II.480f?

⁶³ Plato The Republic VI.567.

⁶⁴ This standard interpretation of Parmenides has been interestingly challenged by Kingsley, PeterReality (Golden Sufi Center: Inverness, California 2003)Google Scholar.

⁶⁵ Boethius Consolation of Philosophy: Medieval Latin Lyrics, tr. Helen Waddell (Penguin: Harmondsworth 1952; 1^st published 1929), p. 59.

⁶⁶ Another game is to concoct plausible explanations for the contradictions within, say, the Sherlock Holmes stories: is Dr J.H. Watson's first name John or James? John is the canonical version, but his wife addresses him as ‘James’. Perhaps his second name was Hamish, and ‘James’ an affectionately mocking Anglicisation: Sayers, Dorothy L.Unpopular Opinions (Gollancz: London 1946), pp. 148–51Google Scholar. Or perhaps Conan Doyle forgot.

⁶⁷ See Blake: ‘Do what you will, this Life's a Fiction/ and is made up of Contradiction’: Complete Works op.cit., p. 751.

⁶⁸ Most commentators assume that this is a satire on bad argument, or at best an incentive to make better arguments. Perhaps so, but Carl Levenson argues, persuasively, that it is the record of a sort of initiation, and the purpose of the ‘arguments’ to loosen one's ties to the familiar: Socrates among the Corybantes (Spring Publications: Connecticut 1999).

⁶⁹ Borges, J.L. ‘Avatars of the Tortoise’ (1939), in Labyrinths, eds., Yates, Donald A. & Irby, James E. (Penguin: Harmondsworth 1970; 1^st published 1964), pp. 237–42: p. 242Google Scholar (also in Other Inquisitions 1937–1952, tr. Ruth L.C. Simms (Washington Square Press: New York), pp. 114–20, reading – very oddly – ‘tenuous, eternal interstices of injustice’ rather than ‘tenuous and eternal crevices of unreason’).

⁷⁰ Plato The Republic VI.509b; see Plotinus Enneads VI.8 [39].16 et al.

⁷¹ James, WilliamThe Principles of Psychology (Macmillan: New York 1890), vol. 1, pp. 288fGoogle Scholar. I have drawn out some radical implications of this notion in ‘Nothing without Mind’, in Consciousness Evolving, ed. James H. Fetzer (John Benjamins: Amsterdam & Philadelphia 2002), pp. 139–60.

⁷² Plotinus Enneads V.1 [10].2, 28, after Homer Iliad 20.65.

⁷³ Plotinus Enneads II.4 [12].5, 18

⁷⁴ Plotinus Enneads I.8 [51].15, 24ff.

⁷⁵ See Plotinus Enneads I.6 [1].9, 34ff.