Rules

It is hardly possible to find any clear and universal definition of consequence. In fact, the medieval authors seem to presuppose the notion of‘‘following’’ as fundamental, as it well may be. But all of them seem to assume that there is some modal element: if the antecedent of a valid consequence is true, the consequent cannot be false. Among other things, this means that consequence is quite different from our material implication. It must not be simply identified with the modern strict implication either. (Notice, moreover, that only correct inferences were usually called consequences, though it was admitted that the term could also have a wider sense.)

During the heyday of consequence theory, a special problem arose concerning the existence of the components of a consequence. Since the medieval practice was to regard propositions as concrete linguistic acts, truth and falsehood could only belong to actual propositions, and a consequence could hold only if its two components were simultaneously present. This trouble was avoided with a counterfactual assumption: if the two components were formed simultaneously, then the required relationships would occur. Yet, there are problems: in Buridan’s example ‘‘No proposition is negative, therefore no ass is running,’’ the antecedent would obviously be false when asserted, and the consequence would become correct, though it is surely incorrect. The strategy recommended by Buridan is to say that, in a correct consequence, things cannot be as the antecedent signifies, without being as the consequent signifies: a semantical interpretation by means of a step from propositions to states of affairs. But especially conjoined with self-reference, the condition of the existence of propositions was a source of numerous problems.

It is remarkable that consequence theory can consider relations between unanalyzed propositions, without entering the structure of subjects and predicates. In this respect it resembles propositional logic and is simpler than the Aristotelian syllogistics. Burley states, indeed, that propositional consequence rules must be utilized in all logic, and he places consequence theory before syllogistics, because syllogisms are just a species of consequences. Most writers do not say this equally explicitly, even if they often follow the same order. And even in Burley’s case it would be quite mistaken to claim that he builds predicate logic upon propositional logic in the modern sense. (Note, however, that from the case of syllogisms it is easy to see that the antecedent can include several propositions.)

The theory of consequences aimed at finding some universal rules about consequence relations. What were, then, these laws? Let us designate consequence with!. Interestingly, a number of rules were about relations between consequences, about proper steps from one consequence to another. (This is one context where it is best to regard the consequences as inferences and to think that the rules are about the universal validity of some inference forms.) These rules have therefore been occasionally called ‘‘proof-theoretical.’’ In other words, such rules concerned the conduct of inference. One such universal rule was, if P! Q and Q! R, then P! R. (This can be generalized to what is now known as the ‘‘cut rule.’’) Another central rule was that of contraposition: if P! Q, then —Q! —P. These two recur constantly in the texts. Burley, Ockham, and Buridan did not mention the fundamental modus ponens and modus tollens among consequence rules, but later for instance Strode began his exposition by stating these two with perfect elegance.

A different type of rules just declared that a certain single inference form was universally valid. For instance, the consequence from ‘‘All S are P’’ to the particular ‘‘Some S is P’’ was valid, because the universal proposition had existential import, according to the medieval conception. Likewise, many consequences were considered valid because of the connection between ‘‘superior’’ and ‘‘inferior.’’ For example, the consequence from ‘‘Every animal is running’’ to ‘‘Every man is running’’ is valid, since the subject term in the antecedent is superior to the subject term of the consequent. (The study of such relations is connected to the semantical doctrine of suppositions of terms.) No authors attempted to list a large array of present-day propositional tautologies as consequential rules, but a few basic principles occurred in this function. Thus, Burley already proved De Morgan’s laws as consequence rules. He presents them as immediate corollaries of his rule: ‘‘The formal element that is affirmed in one contradictory must be denied in the other.’’ Therefore, if — (P & Q) and P, then —Q, that is, if — (P & Q) then —P V —Q. Such absolutely elementary tautologies of propositional logic asP&Q! P, P! P V Qdo not appear among consequence rules; instead, there are rules about inference from conjunctive to simple terms and from simple to disjunctive terms.

Authors who wrote comprehensively about consequences had obviously systematic purposes, which they expressed by laying down primary rules and then proving derived rules, which were occasionally even called theorems. The proofs could be long and detailed. However, the process did not amount to a calculus in any modern sense, since the principles of derivation were not spelled out beforehand. They were chosen case by case. Consequence treatises also listed examples of erroneous consequence rules: rules that resembled correct rules but produced fallacies.

During the fourteenth century, consequence theory was enriched with modal considerations. An increasing space was given to consequences with necessary or possible antecedents. To take the most elementary example, when the antecedent is necessary the consequent is necessary too. At the same time the correct understanding of modal syllogisms was debated, and modal consequences reflect similar issues. The final addition was that of episte-mic operators. Strode already gave a set of rules for consequences concerning knowledge, doubt, and understanding, and fifteenth century authors then added more attitude operators, like belief. The various modal qualifications multiplied the number of given rules.

The first theorists said that the consequence holds because the negation (or ‘‘opposite’’) of the consequent is repugnant to the antecedent. The ‘‘repugnance’’ was then explained with a modal condition about the necessary relation. The nature of this necessity disturbed logicians somewhat, and they referred to current modal theories. After the birth of possible-worlds semantics, Buridan offered one particularly interesting principle: if one proposition can have more causes of truth, that is, verifying states of affairs, than another, but not conversely, the former follows from the latter.