CONTRADICTION, PRINCIPLE OF (principium contradictionis), in logic, the term applied to the second of the three primary “laws of thought.” The oldest statement of the law is that contradictory statements cannot both at the same time be true, e.g. the two propositions “A is B” and “A is not B” are mutually exclusive. A may be B at one time, and not at another; A may be partly B and partly not B at the same time; but it is impossible to predicate of the same thing, at the same time, and in the same sense, the absence and the presence of the same quality. This is the statement of the law given by Aristotle (τὸ γὰρ αὐτὸ ὑπάρχειν τε καὶ μὴ ὑπάρχειν ἀδύνατον τῷ αὐτῷ καὶ κατὰ τὸ αὐτό, Metaph. Γ 3, 1005 b 19). It takes no account of the truth of either proposition; if one is true, the other is not; one of the two must be true.
Modern logicians, following Leibnitz and Kant, have generally adopted a different statement, by which the law assumes an essentially different meaning. Their formula is “A is not not-A”; in other words it is impossible to predicate of a thing a quality which is its contradictory. Unlike Aristotle’s law this law deals with the necessary relation between subject and predicate in a single judgment. Whereas Aristotle states that one or other of two contradictory propositions must be false, the Kantian law states that a particular kind of proposition is in itself necessarily false. On the other hand there is a real connexion between the two laws. The denial of the statement “A is not-A” presupposes some knowledge of what A is, i.e. the statement A is A. In other words a judgment about A is implied. Kant’s analytical propositions depend on presupposed concepts which are the same for all people. His statement, regarded as a logical principle purely and apart from material facts, does not therefore amount to more than that of Aristotle, which deals simply with the significance of negation.
See text-books of Logic, e.g. C. Sigwart’s Logic (trans. Helen Dendy, London, 1895), vol. i. pp. 142 foll.; for the various expressions of the law see Ueberweg’s Logik, § 77; also J. S. Mill, Examination of Hamilton, 471; Venn, Empirical Logic.
CONTRAFAGOTTO, Double Bassoon or Contrabassoon (Fr. contrebasson; Ger. Kontrafagott), a wood-wind instrument of the double reed family, which it completes as grand bass, the other members being the oboe, cor anglais, and bassoon. The contrafagotto corresponds to the double bass in strings, to the contrabass tuba in the brass wind, and to the pedal clarinet in the single reed wood wind.
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Fig. 1.—Contrafagotto, German model (Wilhelm Heckel). | From Capt. C. R. Day’s Cat. of Mus. Inst. by permission of Fyre & Spottiswoode. Fig. 2.—Contrafagotto, Haseneier-Morton model. |
There are at the present day three distinct makes of contrafagotto. (1) The modern German (fig. 1) is founded on the older models, resembling the bassoon, the best-known being Heckel’s of Biebrich-am-Rhein, used at Bayreuth and in many German orchestras. In this model the characteristics of the bassoon are preserved, and the tone is of true fagotto quality extended in its lower register. The Heckel contrafagotto consists of a wooden tube 16 ft. 4 in. long with a conical bore, and doubled back four times upon itself to make it less unwieldy. It is thus about the same length as the bassoon and terminates in a bell 4 in. in diameter pointing downwards. The crook consists of a small brass tube about 2 ft. long, having a very narrow bore, to which is bound the double-reed mouthpiece. (2) The modern English double bassoon is one designed by Dr W. H. Stone, and made under his superintendence by Haseneier of Coblenz. It is stated that instruments of this pattern are less fatiguing to blow than those resembling the bassoon. The bore is truly conical, starting with a diameter of ¼ in. at the reed and ending in a diameter of 4 in. at the open end of the tube which points upwards and has no defined bell, being merely finished with a rim. Alfred Morton, in England, has constructed double bassoons on Dr Stone’s design (fig. 2). (3) The third model is of brass and consists of a conical tube of wide calibre some 15 or 16 ft. long, curved round four times upon itself and having a brass tuba or euphonium bell which points upwards. This brass model, usually known as the Belgian or French (fig. 3), was really of Austrian origin, having been first introduced by Schöllnast of Presburg about 1839. B. F. Czerveny of Königgrätz and Victor Mahillon of Brussels both appear to have followed up this idea independently; the former producing a metal contrafagotto in E♭ in 1856 and one in B♭ which he called sub-contrafagotto in 1867, while Mahillon’s was ready in 1868. In the brass contrafagotto the lateral holes are pierced at theoretically correct intervals along the bore, and have a diameter almost equal to the section of the bore at the point where the hole is pierced. The octave harmonic only is obtainable on this instrument owing to the great length of the bore and its large calibre. There are therefore two octave keys which give a chromatic compass
The modern wooden contrafagotto has a pitch one octave below that of the bassoon and three below that of the oboe; its compass extending from 16 ft. C. to middle C. The harmonics of the octave in the middle register and of the 12th in the upper register are obtained by skilful manipulation of the reed with the lips and increased pressure of the breath. The notes of both extremes are difficult to produce.
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| Fig. 3.—The French or Belgian Contrafagotto. | |