Substitution of Variables in Aristotle

Substitution of Variables in Aristotle DEMETRIUS J. HADGOPOULOS HISTORIANS OF LOGICand logicians have praised Aristotle's introduction of variables into logic as one of the greatest discoveries or inventions in the field.1They have pointed out that Aristotle not oniy uses term-variables but that he also uses propositional variables ,z However, some of them believe, as we shall see, that the only kind of substitution for variables that Aristotle knew was either the substitution of a universal term for a term-variable or the substitution of the same universal term for two different termvariables or the substitution of a proposition for a propositional variable. They categorically state that Aristotle was not aware of the identification of two variables, i.e., that he never substituted variables for variables. For example, Lukasiewicz writes: "[The] identification of variables [was] not known to Aristotle.''8 "Aristotle never substitutes for a variable A another variable B.''~ "There is no passage in the Prior Analytics where two variables are identified.''6 His belief concerning the substitution of variables in Aristotle is also shared by Bochefiski, who writes: Where we have been speaking not of propositions but of forms, seeing that Aristotelian syllogisms always contain letters ('A', 'B', '1" etc.) in place of words, which are evidently to be interpreted as variables, Aristotle himself gives examples of how substitutions can be made in them. That is indeed the only kind of substitution for variables known to him: he has, for example, no thought of substituting variables for variables. Nevertheless this is an immense discovery: the use of letters instead of constant words gave birth to formal logic,e The variables that Lukasiewicz and Bochefiski have in mind in the above quotations are term-variables, but since Bochefisld knows that Aristotle also uses propositional variables , what he says about the substitution of variables in Aristotle may be taken to cover both sorts of variables. In discussing a passage from the Prior Analytics, where Aristotle substitutes the same universal term for two different term-variables (without identifying them) in order to show that from two opposite premises a conclusion can be drawn in the second and third x Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint o/Modern Formal Logic, 2nd ed. (Oxford, 1957),p. 7. I. M. Bochefi.~ki,A History o/ Formal Logic (New York: Chelsea Publishing Co., 1970), p. 69. W. and M. Kneale, The Development o~ Logic (Oxford: Oxford University Press, 1966), p. 61. 2 Bochefiski, p. 77; Kneale, p. 91. e Lvk~qiewicz, p. 221. 4Ibid., p. 8. e Ibid., p. 9. e Boch~ki, p. 69. [133] 134 HISTORY OF PHILOSOPHY figure, Lukaciewicz writes: "The heavy roundabout way by means of concrete terms... is quite unnecessary. It seems that the straight way in this problem, i.e. the way by identifying variables, was not seen by Aristotle.''7 Later came Alexander of Aphrodisias (ft. c. 205 A.D.)who saw the way, and he identified variables in his proof of the conversion of the universal negative proposition. Alexander's proof is as follows: "If one says that the universal negative does not convert, let A belong to no B; if it does not convert, let B belong to some A. There results by means of the first figure that A does not belong to some A, which is absurd.''s In this proof, Lukasiewicz says9 that Alexander identifies the variables A and C, substituting A for C in the mood of the first figure known as Ferio: "If A belongs to no B and B belongs to some C, necessarily A does not belong to some C.''1~ We notice that in the above proof both Ferio and the identification of variables are not written down explicitly. Nevertheless, Lukasiewicz writes: "This [Alexander's proof] is perhaps the neatest example of an argument by substitution derived from an ancient sol.lrce. ''11 Now I shall try to show that Lukasiewicz and Bochefiski are mistaken in their belief that the only kind of substitution of variables found in Aristotle is the one they mention. As we shall later see, Aristotle identifies two variables and substitutes one for the other, though the passage in which...