22C:096
Computation, Information, and Description

Survey of Mathematical Techniques

The Self-Incrimination Principle

Definition of "paradox"

From Webster's New World Dictionary.

paradox

1. a statement that seems contradictory, absurd, etc. but may be true in fact. 2. a statement that is self-contradictory in fact and, hence, false. 3. a person or thing seeming inconsistent and full of contradictions.

From the Oxford English Dictionary.

paradox, sb. (a.). Also 6-7 -oxe. [ad. (perh. through Fr. paradoxe, 14th c. in Hatz.-Darm.) L. paradoxum, -on, sb., properly neuter of paradox-us, Gr. nap oo3-os adj. contrary to received opinion or expectation, f. nap past, beyond, contrary to + o 3a opinion; in Gr. and L. also used subst., esp. in pl. nap oo3a Stoical paradoxes: cf. Cicero Paradoxa, prooe m. 4. In Fr. and Eng. the sb. is the earlier and more important.]

A sb. 1 a A statement or tenet contrary to received opinion or belief; often with the implication that it is marvellous or incredible; sometimes with unfavourable connotation, as being discordant with what is held to be established truth, and hence absurd or fantastic; sometimes with favourable connotation, as a correction of vulgar error. (In actual use rare since 17th c., though often insisted upon by writers as the proper sense.) 1540 PALSGRAVE tr. Acolastus Prol. B ij b, We shall not wytsafe any Paradoxes in noo place, i. we shall not wytsafe (to speake or make mention of..) any thynges, that be aboue or beyonde the common oppynyon of men. 1546 BP. GARDINER Declar. Art. Joye 54 b, Your fonde paradox of only fayth iustifieth. 1581 MARBECK Bk. Notes 791 Paradox is a straunge sentence, contrarie to the opinion of the most part. Or thus: It is a straunge sentence, not easely to be conceiued of the common sort. 1602 SHAKS. Ham. III. i. 115 This was sometime a Paradox, but now the time giues it proofe. 1616 BULLOKAR Eng. Expos., Paradox, an opinion maintained contrary to the common allowed opinion, as if one affirme that the earth doth mooue round, and the heauens stand still. 1653 H. MORE Antid. Ath. II. xii. f17 (1712) 84 That pleasant and true Paradox of the Annual Motion of the Earth. 1656 HOBBES Liberty, Necess., & Chance (1841) 304 The Bishop speaks often of paradoxes with such scorn or detestation, that a simple reader would take a paradox either for felony or some other heinous crime,..whereas perhaps a judicious reader knows..that a paradox, is an opinion not yet generally received. 1697 tr. Burgersdicius' Logic II. xv. 65 A Paradox is said to be a Probleme true against the common Opinion..such as that, viz., the Earth moves; which, tho' it be true, yet may it be so against the common Opinion, and therefore a Paradox. 1854 DE QUINCEY Templars' Dial. Wks. IV. 183 A paradox, you know, is simply that which contradicts the popular opinion-which in too many cases is the false opinion. 1890 Illustr. Lond. News 26 Apr. 535/3 A paradox is a proposition really or apparently contradictory to a commonly received idea... It is, as its name indicates, a conceit contrary to opinion, but not..contrary to reason. A position contrary to reason is a paralogism.
b Rhet. repr. L. paradoxum. A conclusion or apodosis contrary to what the audience has been led up to expect. Obs. 1678 PHILLIPS (ed. 4), Paradox..In Rhetorick, it is something which is cast in by the by, contrary to the opinion or expectation of the Auditor, and is otherwise called Hypomone.
2 a A statement or proposition which on the face of it seems self-contradictory, absurd, or at variance with common sense, though, on investigation or when explained, it may prove to be well-founded (or, according to some, though it is essentially true). spec. in Literary Criticism. 1569 CROWLEY Soph. Dr. Watson i. 187 Your straunge Paradox of Christes eating of his owne fleshe. 1607 J. NORDEN Surv. Dial. IV. 195, I can tell you a pretie paradoxe..Boggy and spungy ground,..though in it owne nature it be too moist, yet if it be overflowed with water often, it will settle and become firme. 1624 HAYWARD Suprem. Relig. 5 Three or foure at the table; who esteemed that which I had said, not for a Para- doxe, but for an Adoxe, or flat Absurditie. 1694 BENTLEY Boyle Lect. 66 'Tis no less a truth than a paradox, that there are no greater fools than atheistical wits; and none so credulous as infidels. A. 1716 SOUTH Serm. (1744) XI. 127 If you will admit the paradox, it makes a man do more than he can do. A. 1806 HORSLEY Serm. (1811) 369 Of the two parts..of a paradox, both are often true, and yet, when proved to be true, may continue paradoxical. 1809-10 COLERIDGE Friend (1865) 54 The legal paradox, that a libel may be the more a libel for being true. 1885 SEELEY Introd. Polit. Sc. i. (1896) 3 In my opinion, to lecture on political science is to lecture on history. Here is the Paradox-I use the word in its original sense of a proposition which is really true, though it sounds false. 1902 Daily Chron. 30 Oct. 3/1 Perhaps the only immortal paradoxes are the divine paradoxes called Beatitudes; for each generation sees their truth, but as no one ever acts upon them, their paradox comes with perpetual freshness to every age.1939 BROOKS & WARREN Understanding Poetry VI. 637 Paradox, a statement which seems on the surface contradictory, but which involves an element of truth. Because of the element of contrast between the form of the statement and its true implications, paradox is closely related to irony. 1942 C. BROOKS in A. Tate Language of Poetry 37 Few of us are prepared to accept the statement that the language of poetry is the language of paradox... Yet there is a sense in which paradox is the language appropriate and inevitable to poetry. 1947 C. BROOKS Well-Wrought Urn 230 Paradox, as a device for contrasting the conventional views of a situation, or the limited and special view of it such as those taken in practical and scientific discourse, with a more inclusive view. 1960 Commentary Nov. 369 He had been instructed in paradox, tension, and ambiguity in a course called `Introduction to Literature'.
b Often applied to a proposition or statement that is actually self-contradictory, or contradictory to reason or ascertained truth, and so, essentially absurd and false. Hence some (cf. quot. 1639) have denied statements to be paradoxes when they can be proved after all to be true, or have called them `apparent paradoxes' (quot. 1876), when they are real paradoxes in sense 2. 1570 FOXE A. & M. (ed. 2) 1299 This monstrous paradoxe of transubstantiation was neuer induced or receaued publickly in the Churche, before the tyme of y[e] Lateran Councell. 1588 SHAKS. L.L.L. IV. iii. 253 Berow..No face is faire that is not full so blacke. Kin. O paradoxe, Blacke is the badge of hell. 1628 WITHER Brit. Rememb. III. 39 Vulgar men, doe such expressions hold To be but idle Paradoxes. 1639 FULLER Holy War III. iv. (1840) 121 It is therefore no paradox to say, that in some case the strength of a kingdom doth consist in the weakness of it. 1645 MILTON Tetrach. (Matt. xix. 7-8) Wks. (1851) 215 The most grosse and massy paradox that ever did violence to reason and religion. 1777 PRIESTLEY Disc. Philos. Necess. ix. 110 This will be no paradox, but a most important and necessary truth. 1822 LD. JEFFREY in Life (1852) II. 211 The dulness is increased in proportion to the density, and the book becomes ten times more tedious by its compression. This is not a paradox now, but a simple truth. 1851 GLADSTONE Glean. VI. xxvi. 17 To my mind there could be no more monstrous paradox, than such a proposition would involve. 1876 L. STEPHEN Eng. Th. 18th Cent. II. 375 The apparent paradox that while no man sets a higher value upon truthfulness..than Johnson, no man could care less for the foundations of speculative truth.
c Logic. A statement or proposition which, from an acceptable premise and despite sound reasoning, leads to a conclusion that is against sense, logically unacceptable, or self-contradictory; freq. distinguished by name, esp. of its propounder or of the type of problem it raises. Cf. LIAR 1 d, Russell's paradox s.v. RUSSELL. 1903 B. RUSSELL Princ. Math. xliii. 358 This paradox, which, as I shall show, is strictly correlative to the Achilles, may be called for convenience the Tristram Shandy. 1921 W. E. JOHNSON Logic I. iii. 45 The paradox of implication assumes many forms, some of which are not easily recognised as involving mere varieties of the same fundamental principle. 1948 H. C. BRODIE in Brodie & Coleman Chwistek's Limits of Sci. p. xxxiii, Typical of such paradoxes is the contradiction of Burali-Forti. 1950 R. CARNAP Logical Found. Probability vii. 469 This is an instance of what Hempel calls the paradox of confirmation. 1955 A. N. PRIOR Formal Logic III. i. 224 As with Lewis's paradoxes, these appear less startling when the definitions of the terms used are considered. 1966 W. V. QUINE Ways of Paradox i. 7 The paradoxes in this class are called antinomies, and it is they that bring on the crises in thought. An antinomy produces a self contradiction by accepted ways of reasoning. 1971 Brit. Jrnl. Philos. Sci. XXII. 337 The Nelson-Grelling paradox requires separate notice, because it can be presented with an explicit distinction between predicative phrases and what they are supposed to express. 1972 T. STOPPARD Jumpers I. 29 Zeno overlooked the fallacy which is exemplified at its most picturesque in his famous paradoxes, which showed in every way but experience that an arrow could never reach its target. 1973 J. L. MACKIE Truth, Probability & Paradox vi. 237 There is a group of paradoxes..which includes the Epimenides and other forms of the liar, heterologicality, Russell's class paradox..and so on.
3 (Without a or plural.) Paradoxical character, condition, or quality; PARADOXY. 1589 PUTTENHAM Eng. Poesie I. xxix. (Arb.) 71 It may be true in manner of Paradoxe. 1788 GIBBON Decl. & F. xliv. (ed. Milman) IV. 186 They imbibed..the love of paradox..and a minute attachment to words and verbal distinctions. A. 1852 WEBSTER Wks. (1877) II. 91 A distinguished lover of liberty of our time, said, with apparent paradox, that the quantity of liberty in any country is exactly equal to the quantity of restraint. 1869 J. MARTINEAU Ess. II. 88 A perpetual source of fallacy and paradox.
4 transf. A phenomenon that exhibits some contradiction or conflict with preconceived notions of what is reasonable or possible; a person of perplexingly inconsistent life or behaviour. hydrostatic paradox: see HYDROSTATIC. A. 1625 FLETCHER Woman's Prize IV. ii, Not let his wife come near him in his sicknes?.. Is she refused? and two old Paradoxes, Peeces of five and fifty without faith, Clapt in upon him? A. 1687 PETTY Pol. Arith. (1690) 92 The wonderful Paradox that Englishmen..pay Customs as Foreigners for all they spend in Ireland. 1706 E. WARD Wooden World Diss. (1708) 45 He's a down-right Paradox. 1846 LYTTON Lucretia II. xviii, One of those strange living paradoxes that can rarely be found out of a commercial community.
5 A shortening of the specific name paradoxus of the Platypus (Ornithorhynchus paradoxus). 1815 in O'Hara Hist. N.S.W. (1817) 452 The water-mole, or paradox, also abounds in all the rivers and ponds.
6 attrib. and Comb., as paradox-monger. 1642 FULLER Holy & Prof. St. II. iv. 62 A Paradox-monger, loving to hold strange yea dangerous Opinions. 1879 Spectator 23 Aug. 1069 Which made the same brilliant paradoxmonger [Prof. Clifford] enjoy saying, `There is one thing in the world more wicked than the desire to command, and that is the will to obey'. B adj. = PARADOXICAL a. Obs. 1624 CAPT. SMITH Virginia VI. 220 Let no man then condemne this paradox opinion. 1654 H. L'ESTRANGE Chas. I (1655) 61 Though paradox it may seem, and out of the rode of common beleef. 1660 BARROW Euclid III. xvi. Cor., Many Paradox and wonderful Consectaries.

'paradox, v. rare. [f. prec. sb.]

1 trans. To affect with a paradox, to cause to show a paradox or contradiction. Obs. rare. 1627-77 FELTHAM Resolves II. xv. 189 The same City that bred him a slave, for his virtues chose him a King; and to his eternal Honour, left his Statue paradox'd with Servitude and Royalty.
2 To bring or drive by paradox. nonce-use. 1692 R. L'ESTRANGE Josephus, Life (1733) 807 Paradoxing soberer Men than himself out of their Senses.
3 intr. To utter paradoxes. Also to paradox it. Hence 'paradoxing vbl. sb. 1647 WARD Simp. Cobler 55 If that Parliament will prescribe what they ought, without such paradoxing. 1694 R. L'ESTRANGE Fables lviii. (1714) 73 There must be no Paradoxing or Playing Tricks with Things Sacred. A. 1811 R. CUMBERLAND in T. Mitchell Aristoph. II. 46 I could..dogmatize..and dispute And paradox it with the best of you.