Handbook of the History of Logic: - Fordham University Faculty

554 Simo Knuuttila
propositions are necessary, possible, or impossible depending on whether they can
or cannot be true or false. Only those propositions were counted as modal, however,
which included modal terms connected to the copula (divided modals) or
connected by the copula to propositions or to dicta (compound modals). 178
The conversions of compound necessity and possibility propositions with respect
to the dictum were not considered problematic. As in Campsall, these were said to
hold by the rules that if the antecedent of a valid consequence is possible then the
consequent is possible, and if the antecedent is of necessity then the consequent
is of necessity or, as with simple conversions, if one of the convertibles is possible
(necessary) then the other is possible (necessary). Of compound contingency
propositions only those with simply convertible dicta are converted; the rule that if
the antecedent is contingent then the consequent is contingent is not valid. 179 The
whole logic of compound modal propositions was in fact based on the Aristotelian
principles for propositional modal logic which, as noted above, were widely known
in early medieval logic as well. When modal syllogisms were regarded as syllogisms
with respect to the dicta, connecting the mode ‘necessary’ with the premises and
conclusions of valid assertoric syllogisms yielded valid modal syllogisms. This was
based on (4) and the rule that if the conjuncts are necessary, the conjunction is
necessary. Uniform syllogisms consisting of compound contingency or possibility
modals were not considered valid, because the compossibility of two possible
premises was not assured. Ockham and Pseudo-Scotus remark that because de
dicto necessities are compossible with any de dicto possibilities or contingencies,
mixed compound necessity and possibility or contingency syllogisms with possible
or contingent conclusions are valid. Buridan did not mention this. 180
The main object of the fourteenth-century modal logic was the theory of divided
modals. Some treatises include discussions in which the logical relations
between various divided modal propositions were codified in the same way as
the relations between the types of assertoric propositions in the square of opposition.
Buridan taught that there were two types of copula, the affirmative ‘is’
and the negative ‘is-not’, and that modality was part of the copula in divided
modal propositions. Combining the equivalences between quantifying words with
negations with equivalent modalities, Buridan arranged divided modals into eight
groups of nine equivalent formulae. In the Summulae, these groups are presented
in a diagram showing the relations of contradiction, contrariety, sub-contrariety,
and sub-alternation between them. 181
178BC II.1, 56; PS I.25, 309; William Ockham., Expositio in librum Perihermenias Aristotelis,
ed. A. Gambatese and S. Brown, Opera philosophica II (St. Bonaventure: St. Bonaventure
University, 1978), 459-66.
179OSL II.24-5, 27, 327-8, 330-31, 334; BC II.7.12-14, 72-4; PS I.25, 310; I.30, 319.
180PS I.27, 313; I.33, 323; OSL III-1.20, 412-13; III-1.23, 419; III-1.44, 474; III-1.47, 479; BC
IV.1.1, 113.
181For this octagon of opposition, see Summulae de dialectica I.8.4-7 and the diagram in Questiones
longe super librum Perihermeneias II.9, 87; see also Hughes 1989, 109-10; E. Karger, ‘Buridan’s
Theory of the Logical Relations between General Modal Formulae’ in H.A.G. Braakhuis
and C.H. Kneepkens (eds.), Aristotle’s Peri hermeneias in the Latin Middle Ages, Artistarium
supplementa 10 (Groningen – Haren: Ingenium Publishers, 2003), 429-44.