The most interesting part of consequence theory for present-day readers is perhaps the discussion about various kinds of consequence. Logicians always gave much attention to such distinctions of types. Usually the first distinction is drawn between consequences ut nunc (‘‘as-of-now’’) and simpliciter. It stems from Boethius, but its precise meaning caused confusion.
A consequence P! Q held simply if it was valid always without temporal qualifications, and as-of-now if P and the opposite of Q were incompatible only now, or alternatively, for a certain period. The consequence is thus valid ut nunc if the antecedent cannot now be true without the consequent, under present conditions. Simple consequences were studied more, and they are more important than the temporally determined ones, but modern scholars have shown great interest in consequences ut
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Nunc. The reason is that, as formulated for instance by Ockham, it reminds of the modern truth-functional material implication. Yet it is probable that even consequence ut nunc is essentially modal. Mere truth of the consequent does not make the consequence valid but something more is required, since some propositions that are now true could, under present conditions, be false. Examples of this depend on modal theories, but at least propositions about future contingents as consequents can lead to such situations - the temporal import of ‘‘now’’ must be taken seriously. Keeping strictly to one single moment, the outcome can amount to material implication. During the fifteenth century, the role of ut nunc declined.
On the other hand, another issue kept its full importance, that is, the question about formality and materiality. This is founded on the conviction that even valid consequences can be valid for different reasons, some for more remarkable reasons than others. Let us consider the basic definition that the antecedent and the opposite ofthe consequent are repugnant or incompatible. What can be the explanation of such repugnance? Obviously, it often results from the contents of the propositions in question: the natures or essences of the mentioned things are such that the antecedent and the opposite of the consequent are in conflict. This was noted by the thirteenth century logicians, like Kilwardby, who said that this consequence was ‘‘natural.’’ But there existed even other cases where the repugnance was undeniable but had nothing to do with the contents of the propositions, with the natures of things, and here they spoke about accidental consequence.
We see the meaning of this distinction when we look at the doctrine of Burley that retains basically the same scheme. For him, a consequence holds naturally when the antecedent ‘‘includes’’ the consequent, whereas the consequence ‘‘a man is a donkey, thus you are sitting’’ is only accidental because it involves no internal connection between the propositions. Attempting to get further, he introduced the notions of formal and material consequence, which were to become generally used. According to him, consequences are formal, when they depend either on logical theorems or on formal relations of inclusion between terms; otherwise they are material. This implies that ‘‘a man is a donkey, thus you are sitting’’ would be only material, but so would the consequence ‘‘Socrates is sitting, thus Socrates is not running’’ be as well. The incompatibility of man and donkey, sitting and running, are no formal conceptual features. Burley is possibly utilizing the Scotistic doctrine of formalities, but his presentation is rather obscure.
Later, authors followed and explicated the distinction between formal and material in different ways. Ockham’s characteristic idea is to claim that consequences are valid because of ‘‘mediums,’’ which are either ‘‘external’’ general logical rules or some ‘‘intrinsic’’ principles that are more like additional premises and true because of the meanings of the terms in the propositions. Both cases can produce formal consequences: ‘‘Socrates is sitting, thus Socrates is not running’’ is formal for him, and so are syllogisms as well. On the other hand, at least ‘‘a man is a donkey, thus there is no God’’ is a material consequence. The reason is that the external rule guaranteeing its validity becomes applicable only because of the terms ‘‘man’’ and ‘‘donkey.’’ Ockham hints towards the notion of logical form, but his text is here very brief.
The idea of logical form was fully developed in Buridan’s logic. According to him, the form of a proposition consists of its structure of syncategorematic elements and the distribution of categorematic elements. And as he defines it, ‘‘a consequence is formal if any proposition with similar form would, when stated, be valid.’’ This means that formally valid consequences hold merely because of the logical forms of the propositions and do not depend on what the categorematic elements are. A consequence is formally valid if all its substitution instances are valid. The same definition is encountered in modern logic. Formal consequences are sheer instances of strict logical theorems, whereas material consequences are valid because they could be completed to formal ones by adding necessary propositions as supplementary premises. (Hence, Buridan called them enthymematic, and his pupils called them imperfect.) ‘‘A man runs, thus an animal runs’’ is not formal, since the substitution ‘‘A horse walks, thus a wood walks’’ is not valid, but it is materially valid since the addition of ‘‘Every man is an animal’’ leads to a formal consequence.
Buridan’s conception is a remarkable achievement in the history of formal logic. Some followers embraced it, such as Pseudo-Scotus in his famous logic commentary, and Albert of Saxony. But it did not win general acceptance; on the contrary, critics argued that it did not catch the point of logical consequence, which was in conceptual inclusion. Actually, it became customary to see the formality of consequences as a matter of inclusion relations (analyticity). It has been argued that such an interpretation of consequence is psychologistic, but this is not necessary. Apparently the important writers were not mainly interested in psychology but in the contents of forms; that is why they called the corresponding consequences formal.
Expressly psychologistic positions developed only toward the end of the fifteenth century.
Gradually, a number of logicians understood that the two criteria concerned different things, two senses of formality. For example, Paul of Venice explains that a consequence is formal if the contradictory of the consequent is formally incompatible with the antecedent (but his account of this formal incompatibility is not quite clear). On the other hand, a consequence is valid de forma if every consequence of the same logical form is valid. This distinction led to somewhat awkward labels like ‘‘formally formal,’’ that is, formal in both senses. The double characterization was elaborated during the fifteenth century by several logicians, such as Paul of Pergula.
One final point must be mentioned. The basic definition of consequence immediately implies that anything follows from an impossible antecedent (ex impossibili quodlibet), and equivalently, the necessary follows from everything. These are the ‘‘paradoxes of implication.’’ Abelard pointed them out and regarded them as so intolerable that he rejected the standard definition of consequence. But in later literature they were normally accepted and sometimes even definitely defended. For most authors, they were fundamental examples of material consequences. This conviction faltered only in the psychologistic wave near 1500.
See also: > Boethius > Garlandus the Computist > John Buridan > Logic > Paul of Venice > Peter Abelard
> Ralph Strode > Syncategoremata > Walter Burley
> William of Ockham
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