Logic

From the middle of the twelfth century, logicians developed various branches of their subject, known as the logica modernorum ("contemporary logic"), that had not been treated specially, or at all, in antiquity. Peter of Spain's widely read Tractatus (often called Summulae logicales) illustrates how parts of the logica modernorum had developed up until about the 1230s. There was a lull in interest and innovation in logic for nearly a century, but in the first half of the fourteenth century writers at Oxford, such as Walter Burley (d. 1344/45), William of Ockham (d. ?1349), Thomas Bradwardine (d. 1349), and William of Heytesbury (d. 1372/23), and, in Paris, John Buridan (d. after 1358) revived the branches of the logica modernorum and brought them to new levels of sophistication. The Logica magna (Great logic) of Paul of Venice (d. 1429) is a vast record of these achievements.

Some branches of the logica modernorum grew directly from the mid-twelfth-century interest in fallacies. For example, sophismata were a sort of disputation, involving a master and his pupils, built around sentences that either are apparently false but can be interpreted so as to make them true (e.g., "The whole Socrates is less than Socrates") or at least are open to different interpretations (e.g., "Every man is"). The ambiguities usually centered around the use of what were called "syncategorematic" words—words other than ordinary nouns, adjectives, and verbs with their own referential content: for instance, "only," "except," "all," "begins," "ceases"—and specialized treatises were devoted to studying these syncategoremata. Another type of disputation, "obligations," involved trying to force an opponent who has agreed to defend a particular statement into a self-contradiction, while following a very strict set of rules for what statements may be accepted or must be rejected. Liar paradoxes ("What I am now saying is false")—"insolubles"—formed another branch of study. In the theory of the "properties of terms" a highly elaborate theory was developed about the reference of words depending on their function within a sentence. Propositional logic was elaborated in treatises on what were called "consequences" (consequentiae), although it remains questionable how far the approach was purely propositional.

Aristotle's texts and methods were not, however, forgotten in the later Middle Ages. Aristotelian syllogistic, studied earlier through Boethius's textbooks, could now be learned directly from the Prior Analytics. It was a basic tool for almost every medieval philosopher or theologian, and a set of mnemonics ("bArbArA," "cElArEnt," etc.) were devised to enable students to remember the valid patterns. On Interpretation continued to be central to discussions about possibility, necessity, and divine prescience, and the Posterior Analytics provided the criteria for organizing branches of knowledge as diverse as grammar and theology.