exceptions to a general statement, which is indeed the natural way of meeting it, we employ the third figure. The statement that "all metals are solids" would at once be disproved by the exception mercury, as follows: Mercury is not solid, Mercury is a metal ; Therefore some metal is not solid. Were any one to assert that what is incomprehensible cannot exist, we meet it at once with the argument that Infinity is incomprehensible, but that infinity certainly exists, because we cannot otherwise explain the nature of a curve line, or of a quantity varying continuously; therefore something that is incomprehensible exists. In this case even one exception is sufficient entirely to negative the proposition, which really means that because a thing is incomprehensible it cannot exist. But if one incomprehensible thing does exist, others may also; and all authority is taken from the statement. According to the Aristotelian system the third figure must also be employed whenever the middle term is a singular term, because in Aristotle's view of the subject a singular term could not stand as the predicate of a proposition. LESSON XVII. REDUCTION OF THE IMPERFECT FIGURES OF THE SYLLOGISM. IN order to facilitate the recollection of the nineteen valid and useful moods of the syllogism, logicians invented, at least six centuries ago, a most curious system of artificial words, combined into mnemonic verses, which may be readily committed to memory. This device, however in genious, is of a barbarous and wholly unscientific character; but a knowledge of its construction and use is still expected from the student of logic, and the verses are therefore given and explained below. 5 'Barbara, Celarent, Darii," Ferioque, prioris; Bokardo, Ferison, habet; Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison. The words printed in ordinary type are real Latin words, signifying that four moods whose artificial names are Barbara, Celarent, Darii and Ferio, belong to the first figure; that four others belong to the second; six more to the third; while the fourth figure moreover contains five moods. Each artificial name contains three vowels, which indicate the propositions forming a valid mood; thus, CElArEnt signifies the mood of the first figure, which has E for a major premise, A for the minor, and E for the conclusion. The artificial words altogether contain exactly the series of combinations of vowels shown in p. 140, excepting those in brackets. These mnemonic lines also contain indications of the mode in which each mood of the second, third and fourth figures can be proved by reduction to a corresponding mood of the first figure. Aristotle looked upon the first figure as a peculiarly evident and cogent form of argument, the Dictum de omni et nullo being directly applicable to it, and he therefore called it the Perfect Figure. The fourth figure was never recognised by him, and it is often called the Galenian figure, because the celebrated Galen is supposed to have discovered it. The second and third figures were known to Aristotle as the Imperfect Figures, which it was necessary to reduce to the first ΙΟ figure by certain conversions and transpositions of the premises, for which directions are to be found in the artificial words. These directions are as follows :— s indicates that the proposition denoted by the preceding vowel is to be converted simply. indicates that the proposition is to be converted per accidens, or by limitation. m indicates that the premises of the syllogism art to be transposed, the major being made the minor of a new syllogism, and the old minor the new major. The mis derived from the Latin mutare, to change. B, C, D, E, the initial consonants of the names, indicate the moods of the first figure, which are produced by reduction; thus Cesare, Camestres and Camenes are reducible to Celarent, Darapti, &c., to Darii, Fresison to Ferio and so on. k denotes that the mood must be reduced or proved by a distinct process called Indirect reduction, or reductio ad impossibile, which will shortly be considered. Let us now take some syllogism, say in Camestres, and follow the directions for reduction. Let the example be All stars are self-luminous ......... All planets are not self-luminous.... Therefore no planets are stars................. (1) (2) (3) The first s in Camestres shows that we are to convert simply the minor premise. The m instructs us to change the order of the premises, and the final s to convert the conclusion simply. When all these changes are made we obtain No self-luminous bodies are planets....... Converse of (2) All stars are self-luminous Therefore no stars are planets......... (1) .Converse of (3) This, it will be found, is a syllogism in Celarent, as might be known from the initial C in Camestres. As another example let us take Fesapo, for instance: All planets are round bodies; Therefore some round bodies are not fixed stars. According to the directions in the name, we are to convert simply the major premise, and by limitation the minor premise. We have then the following syllogism in Ferio: No planets are fixed stars, Some round bodies are planets; Therefore some round bodies are not fixed stars. The reader will easily apply the same process of conversion or transposition to the other moods, according to the directions contained in their names, and the only moods it will be necessary to examine especially are Bramantip, Baroko and Bokardo. As an example of Bramantip we may take: All metals are material substances, All material substances are gravitating bodies; The name contains the letter m, which instructs us to transpose the premises, and the letter p, which denotes conversion by limitation; effecting these changes we have: All material substances are gravitating bodies, Therefore some metals are gravitating bodies. This is not a syllogism in Barbara, as we might have expected, but is the weakened mood AAI of the first figure. It is evident that the premises yield the conclusion "all metals are gravitating bodies,” and we must take the letter to indicate in this mood that the conclusion is weaker than it might be. In truth the fourth figure is so imperfect and unnatural in form, containing nothing but ill-arranged syllogisms, which would have been better stated in the first figure, that Aristotle, the founder of logical science, never allowed the existence of the figure at all. It is to be regretted that so needless an addition was made to the somewhat complicated forms of the syllogism. Indirect reduction. The moods Baroko and Bokardo give a good deal of trouble, because they cannot be reduced directly to the first figure. To show the mode of treating these moods we will take X, Y, Z to represent the major, middle and minor terms of the syllogism, and Baroko may then be stated as follows: Therefore All X's are Y's, Some Z's are not Y's; Some Z's are not X's. Now if we convert the major premise by Contrapɔsition (p. 83) we have "all not- Y's are not-X's," and, making this the major premise of the syllogism, we have All not-y's are not X's, Some Z's are not-y's; Although both the above premises appear to be negative, this is really a valid syllogism in Celarent, because two of the negative particles merely affect the middle term (see p. 134), and we have therefore effected the reduction of the syllogism. Bokardo, when similarly stated, is as follows: |