Franciscan Studies

THE LOGIC OF NEGATIVE TERMS IN BOETHIUS H istorians of logic have recently been turning their attention to the De Syllogismo Hypothetico of Boethius, and have found in it a quite highly developed propositional calculus.1 So far as we are aware, however, his De Syllogismo Categórica and his Introductio ad Syllogismos Categóricos have not yet been subjected to similar scrutiny; and in the latter work at least there are features of considerable interest. The Introductio ad Syllogismos Categóricos resembles the De Syl­ logismo Hypothetico in exhibiting a special interest in the results of attaching a negative particle to an element or to the elements of a proposition. Just as he gives in the latter work an exhaustive account of such varieties of the conditional proposition as ‘If p then not q’, ‘If not p then q’, ‘If not p then not q’, ‘If p then if q then not r’, and so on, so in the Introductio he considers the relations of opposition, entailment , and so on which hold between categorical propositions with and without negative (or as he calls them ‘infinite’) terms. In doing this he does not use variables such as ‘a ’ and ‘b ’ , but the concrete terms which he uses are selected on a definite principle, which we shall now illustrate. He compares, to begin with, propositions in which neither term is negative, and the corresponding propositions in which both terms are. He sets out2 a list of twenty propositions of each kind, the first four pairs being the following: Omnis homo rationalis est — Omnis non homo non rationalis est. Nullus homo rationalis est — Nullus non homo non rationalis est. Quidam homo rationalis est — Quidam non homo non rationalis est. Quidam homo rationalis non est — Quidam non homo non rationalis non est. 1 See, in particular, K . Diirr, The Propositional Logic of Boethius (NorthHolland Publishing Co., 1951); R- van den Driessche, “ Sur le ‘ de syllogismo hypothetico’ de Boece,” Methodos Vol. I, No. 3; I. M. Bochenski, Ancient Formal Logic (North-Holland Publishing Co., 1951), pp. 106-109. 2 On p. 708 B in the edition of Boethius’s philosophical works which forms Vol. 64 of Migne’s Patrología (Series Latina). I 2 A . N. Prior All the remaining four groups of four pairs have the same forms in the same order (A, E, I, O), and all have the same subject, homo; but the predicate in the four after the above is not rationalis but grammaticus ; in the next four, lapis; in the next four, justus; and in the last four, risibilis. The discussion which follows this tabulation makes it clear that the point of all this repetition is that these various pre­ dicates, with homo as subject, exemplify five different possible relations between classes. (I) The class of men is wholly included in that of rational beings, but not coextensive with it (for there are divinae sub­ stantiae which are rational but not human). (II) The class of men wholly includes that of grammarians, but is again not coincident with it. (Ill) The class of men and that of stones are mutually exclusive. (IV) The class of men neither wholly excludes nor wholly includes nor is wholly included in the class of just beings (the divinae substantiae being again appealed to as just beings which are not men). And (V) the class of men coincides with the class of beings capable of laughter. These five are, of course, precisely the possibilities illustrated by Euler’s five diagrams, and Boethius uses them, exactly as Euler’s diagrams might be used, to determine whether there is any necessary connection between the truth-value of one of his left-hand propositions and that of the corresponding right-hand proposition. Using ‘ Uba’ for ‘ Every b is a’, ‘Yba’ for ‘No b is a ’, ‘Iba’ for ‘ Some b is a ’, and ‘ Oba’ for ‘ Some b is not a’, ‘na’ for the negative term ‘non-a’, his method may be schematized by means of the following table: Man, Man, Man, Man, Man, rational grammarian stone just risible Uba True False False False True Unbna False True False False True Yba False False True False False Ynbna False...