Handbook of the History of Logic: - Fordham University Faculty

Medieval Modal Theories and Modal Logic 533
Peter Abelard tried to define the opposed relations between quantified de re modals
as well. He thought that these were the same as those between singular modal
propositions, which is completely wrong. This question was not much discussed
before its satisfactory solution in the early fourteenth century. (See p. 554 below.)
In discussing the imperfect first figure assertoric-contingency syllogisms, Aristotle
formulated the principle ‘If when A is, B must be, then when A is possible,
B must be possible’. On the basis of this, he describes syllogisms as follows: ‘If
someone were to put the premises as A and the conclusion as B, it would not only
follow that B is necessary, if A is necessary, but also that B is possible, if A is
possible’ (Prior Analytics I.15, 34a5-7, 22-4). These are the rules of inference
(4) p → q Lp → Lq
and
(5) p → q Mp → Mq.
In Metaphysics IX.9, 1047b14-20, Aristotle argues for (5) by applying the characterization
of potentiality in Metaphysics IX.3, 1047a24-6, which is based on the
definition in Prior Analytics I.13, 32a18-19: ‘I use the expressions ‘to be possible’
and ‘what is possible’ in application to something if it is not necessary but nothing
impossible will result if it is put as being the case.’ If this is applied to possibility
proper, it could be formulated as
(6) Mp &(p → q) Mq.
Aristotle did not develop his general remarks on propositional modal logic, but
these principles were dealt with in later ancient discussions of the logic of conditionals
and related issues. 94 Similar rules were often put forward in early medieval
logical treatises. 95
2.2 Modalities de dicto and de re
In dealing with modal propositions in his later commentary on De interpretatione,
Peter Abelard argues that modal terms in the proper grammatical sense of the
word are adverbs which modify the inherence between a subject and a predicate.
Terms like ‘necessarily’, ‘well’ or ‘rapidly’ are used in this manner; adverbs like
‘possibly’ or ‘falsely’ are analogously called modals, although they do not modify
actual inferences. Nominal modal terms occur in propositions like ‘Necesse est
94See W. and M. Kneale, The Development of Logic (Oxford: Clarendon Press, 1962), 128-
38; J. Vuillemin, Nécessité ou contingence: l’aporie de Diodore et les systèmes philosophiques
(Paris: Les Editions de Minuit, 1984); S. Bobzien, ‘Chrysippus’s Modal Logic and Its relation to
Philo and Diodorus’ in K. Döring and T. Ebert (eds.), Dialektiker und Stoiker (Stuttgart: Franz
Steiner, 1993), 63-84.
95See, e.g., Peter Abelard, Dialectica, 202.6-8; 219.5; Logica ‘Ingredientibus’ 429.36-7; 430.16-
17.