Handbook of the History of Logic: - Fordham University Faculty

16 John Marenbon
and those using ‘if’ (si/ cum). In practice, he concerns himself almost entirely
with ‘if’. Early in the work [I.3.6-7], he makes a distinction to which he does not
return. There are two sorts of conditionals (hypothetical propositions using si or
cum): accidental (secundum accidens) and those which have a consequence of nature
(ut habeant aliquam naturae consequentiam). An accidental conditional is, for
instance, ‘If (cum) fire is hot, the sky is round’. Those which have a consequence
of nature are of two types. In one type the consequence is necessary, but does not
rest on the positing of the terms (positio terminorum), as for example: ‘If (cum)
it’s a human, it’s an animal’. The other type has a necessary consequence that
does rest on the positing of the terms, as for example, ‘If (si) there is an interposition
of the earth, an eclipse of the moon follows’. From the example he gives,
the distinction that Boethius seems to have in mind is that, in a true accidental
conditional, the antecedent are both necessarily true, but the truth of the one is
entirely unrelated to the truth of the other. In a natural conditional, the truth of
the consequent is related to that of the antecedent, either by dependency where
the conditional rests on the positing of terms — it is because there is an interposition
of the earth that the moon is eclipsed — or by relevance that is not causal
dependency — something’s being a human gives us reason to say it is an animal,
but Boethius would want to say that it is not because it’s a human that it’s an
animal, but rather, because it’s a mortal, rational animal that it’s a human. The
distinction between accidental and natural conditional was one which would become
enormously important in the history of Latin logic from the twelfth century
onwards [Martin, 2005]. But Boethius does not dwell on it; his standard example
conditional, ‘If it is day it is light’, seems to be a natural conditional resting on
the positing of terms,
Rather, Boethius wants to set out which hypothetical syllogisms are valid. Just
as the theory of categorical syllogistic lists the different combinations that are
valid, so too hypothetical syllogistic offers a tabulation of valid inference forms.
But the table is vastly more complicated. The simplest form of a hypothetical
syllogism, where the major premiss involves just two terms, is exemplified by ‘If
it is day, it is light. It is day. So it is light.’ Varying the quality of each of
the two propositions making up the major premiss yields four types of syllogism.
But the major premiss can involve three terms (for instance, ‘If, if it is A, then
it is B, itisC) or four terms (for instance, ‘If, if it is A, itisB, then if it is
C, itisD’). Taking into account all the combinations of quality, and then the
different ways of qualifying them modally, the number of combinations reaches
the ten thousands (I.8.7), and far more if the propositions are quantified (I.9.2).
But Boethius confines himself, wisely, to charting the non-modal, non-quantified
combinations — still a long and tedious task, that occupies Books II and III of
the treatise.
It is tempting to see Boethius as setting out some variety of propositional logic.
‘If it is day, it is light. It is day. So it is light’ looks very like an inference of the
form ‘p → q, p; soq’. Yet it is clear that Boethius conceives himself as doing a
sort of term logic (and the same is true for the Byzantine scholium). Each of the