Handbook of the History of Logic: - Fordham University Faculty

The Latin Tradition of Logic to 1100 17
propositions is of the form ‘It is (an) A/B/C...’, where ‘A’, ‘B’, ‘C’ stand for
predicates, and the conclusion follows — as in a categorical syllogism — because
of the relation between these terms. One moment in the treatise brings out this
point especially clearly. Boethius (II.2.3–4.6) has established that, if the major
premiss is ‘If it is not A, itisB’, then, whilst ‘It is B’ follows when the minor
premiss is ‘If it is not A’, and ‘It is A’ follows if the minor premiss is ‘It is not B’,
from the minor premisses ‘It is A’ and ‘It is B’, no conclusion follows. And yet,
Boethius points out, in the nature of things from ‘If it is not A, itisB’ and ‘It is
B’, it does follow that it is not A, and from ‘If it is not A, itisB’ and ‘It is A’, it
does follow that it is not B. The reason (III.10.4) is that the major premiss will be
true only if ‘A’ and ‘B’ are immediate contraries — they are such that everything
is either A or B. If A and B are immediate contraries, then indeed if something
is not A it is B, and if it is not B it is A. This line of thought makes sense only if
Boethius has the connections between terms, not propositions, in mind. As Chris
Martin has shown in his important study [Martin, 1991], whereas Stoic logic is
propositional, Boethius had no grasp of the very notion of propositionality; that is
to say, he was unable to see negation or consequence as propositional operations.
Boethius is not, though, entirely removed from propositional logic. There are
moments, indeed, when he seems to come close to propositional logic, as when he
gives two of the Stoic indemonstrables using, in Stoic fashion numbers to stand for
propositions: ‘if the first, then it follows that there is the second’, ‘if the second
is not, it follows necessarily that the first is not’ (I.4.6). In his commentary on
Cicero’s Topics, Boethius includes a long passage [Bk. V; Boethius, 1833, 355-9]
discussing Cicero’s seven modes of inference. This passage may be linked, not
just to Cicero’s own text, but to the presentation found in Martianus Capella
and Cassiodorus, which seems to go back to Marius Victorinus. There is, no
doubt through Cicero, genuine Stoic influence at the basis of this list. But, well by
Boethius’s time, as Anthony Speca has shown [Speca, 2001], the Peripatetic theory
of the hypothetical syllogism and Stoic propositional logic had become conflated,
to the extent that Boethius (and possibly Marius Victorinus before him), reading
Cicero’s Stoic-based list of modes of inference, took them for modes of hypothetical
syllogisms.
Indeed, there is one point here where Boethius makes clear just how far he
was from understanding Stoic logic (pace [Stump, 1989, esp. 19-22]). He knows
what Cicero’s form of the third mode is, but he deliberately changes it [Boethius,
1833, 362]. Instead of ‘Not: A and not B; A; therefore B’, Boethius puts: ‘Not:
if A then not B; A; therefore B’ [Boethius, 1833, 356-7]. If this is translated
into propositional calculus, it gives ∼ (p →∼ q); p; q, which is clearly invalid.
Boethius explains, however, that in this mode it is assumed that A and not-
B are incompatible. He understands ‘Not if A then not B’ as expressing this
incompatibility: to negate ‘If it is day, is not light’ is, for Boethius, to say that it
cannot be the case that it is day without its being light. And so it follows that,
if it is day, as the minor premiss states, it is light (cf. [Martin, 1991, 293-4]).
Boethius is therefore, as shown by his very willingness to change the formula he