Does Ockham Know of Material Implication?

DOES OCKHAM KNOW OF MATERIAL IMPLICATION? The present study has two main purposes. First, it will try to determine whether Ockham had any idea of material implication as understood by modern logicians; secondly, through the pertinent discussions, it will serve as a test for ascertaining whether certain recently discovered tracts on logic can be attributed to Ockham. In a previous article1 we showed that at least three systematical works on logic, or Summae of logic, are attributed in certain manuscripts to Ockham. Let us briefly summarize the status quaestionis. There is no doubt that the famous Summa Logicae is a work of Ockham; its authenticity in toto and in its parts is well documented . In the following discussions, this Summa will be called Sl. A manuscript of Munich (Staatsbibliothek 1060) written in 1348 attributes another Summa (Sl) to Ockham. It is called by the author Elementarium, and by the scribe Tractatus médius logicae Ockham. Another manuscript, Assisi 690, written probably in the 14th century, also attributes a logic to Ockham. The scribe calls it Tractatus minor logicae Ockham (S3). All three Summae are distinct works. Sl is the longest, S3 the shortest. There is no question, however, of any one of the Summae being an abbreviation of another, but all three are independent, though related, treatments of the same subject. S2 seems to be closer to S3. All of these logics show characteristic features of Ockham's teaching and arrangement of subject matter. Up to now, our study of these Summae has not showed any serious discrepancy between them, except perhaps one, which will be dealt with in this article, and, as we hope, explained in the course of our discussion. Thus we hope to shed new light on the problem of their authenticity; at the same time, we intend to determine whether an equivalent of the material implication of modern logicians is to be found in Ockham's logic. In all three Summae there are two distinct places where information on our problem is to be found, viz., in the treatment of conditional propositions and of the consequentiae. 1. "Three Sums of Logic attributed to William Ockham," in Franciscan Studies XI (1951), 173-193. [203] [204]OCKHAM AND MATERIAL IMPLICATION I. On Conditional Propositions According to medieval logic, hypothetical propositions in contradistinction to categorical propositions are understood as compound propositions, that is, as propositions which are composed of at least two categorical propositions. Usually five main subclasses of hypothetical propositions are enumerated, though as Ockham does not fail to emphasize, more can be adduced. There are the conditional, the copulative, the disjunctive, the temporal, and the causal propositions. In order to reach an exact understanding of the conditional proposition and the relation prevailing between its component parts, we shall deal first with the relations existing between the parts of conjunctive, disjunctive, and causal propositions. This will be done by comparing the texts, especially of Sl and S2, since S3 has very little to say about the present matter. In order to make a more exact presentation of the following discussions , we shall resort to the following symbols: p, q, r, s are symbols of propositions is the symbol of "and" — in front of, or above, another symbol is the symbol of negation ? is the symbol of "or" / is the symbol of not specified implication (sequitur ) J'is the symbol of material implication /*is the symbol of strict implication = is the symbol of equivalence Impis the symbol for "impossible that" Possis the symbol for "possible that" Parentheses and brackets etc. are used to distinguish parts of complex expressions. A conjunctive or copulative proposition is composed of at least two categorical propositions linked up by the statement—connective "and" (.). Necessary condition for such a proposition is the truth of all its parts. Hence, if (p . q) is true, then ? is true and q is true. Therefore, inference from the whole copulative proposition to each part is valid, but not vice versa. Hence: (p . q) J ? or (p . q) J q PHILOTHEUS BOEHNER, O.F.M.[205] The inverse inference from the part to the whole is incorrect. If one part of a copulative proposition is false the entire proposition is...