58
INTRODUCTION TO JEAN BURIDAN’S LOGIC
Buridan almost never calls a consequent ‘true’ and ‘false’ in this
treatise. Rather, he calls them acceptable (bona) or not, though technically
an unacceptable consequence is not a consequence (this is an instance of
the laxity noted above). A consequence is acceptable if it satisfies the strict
definition for consequences as given above. In that respect consequences are
indeed like inferences, which are valid of invalid, not true or false.
83
Buridan’s division of consequences is found in TC 1.4.9, as follows:
CONSEQUENCES
Formal
Material
Simple
ut nunc
The first division is between formal and material consequences. A conse-
quence is formal if and only if it satisfies the Uniform Substitution Principle
(TC 1.4.2–3): it is acceptable for any uniform substitution for any of its
categorematic terms; otherwise the consequence is material. Formal conse-
quences are in this respect like tautologies, and so like strict implication:
they remain acceptable (true) for any uniform substitution of categorematic
terms (non-logical constants).
Material consequences fail this test, but may yet be necessary; such
material consequences are called simple. As an example Buridan points
out that the sentence “A man runs; therefore, an animal runs” is not for-
mal, for a substitution-instance is “A horse walks; therefore, wood walks”
(TC 1.4.3). The genus-species relation between ‘man’ and ‘animal’ need not
be preserved under substitution. On the other hand, it is clearly a neces-
sary consequence.
84
Material consequences come in two forms: simple and
ut nunc. A simple material consequence is a consequence which is not for-
mal but satisfies the strict definition of consequence given above. They are
acceptable only through reduction to a formal consequence, namely “by the
addition of some necessary sentence or sentences which, when assumed with
the antecedent, render the consequence formal” (TC 1.4.4). In the example
above the necessary sentence is “All men are animals,” so that “A man
(TS 3.7.41). But note that the definition is not stated by talking about the truth of
p
∗
and q
∗
; this is due to complications which arise from taking Liar-sentences into
account: what a sentence signifies to be is the case is a necessary, but not a sufficient,
condition for the truth of the sentence.
83
In a few places Buridan says “consequentia valet,” which I have translated as ‘valid.’
84
In the spirit of Carnap’s “meaning-postulates” we could modify our substitution-rules
to preserve relations among terms, as was done by the later Scholastics, but Buridan’s
refusal to do so indicates an admirable attempt to sever logic from metaphysics.
c Peter King, from Jean Buridan’s Logic (Dordrecht: D. Reidel 1985) 3–82.
INTRODUCTION TO JEAN BURIDAN’S LOGIC
59
runs and all men are animals; therefore, an animal runs” is a formal con-
sequence. Simple material consequences are thus treated as enthymematic
(TC 1.4.5–6).
The final kind, ut nunc consequences, are strictly speaking not nec-
essarily consequences: they are acceptable if we replace the ‘impossible’ in
the definition of consequence with ‘it is not the case.’ An ut nunc conse-
quence, then, is a sentence such that it is not the case that the antecedent
obtains and the consequent fails to obtain. But here again we may take
the tense of the verb in ‘it is not the case’ seriously, and so discuss conse-
quences which hold as a matter of fact at other times—as Buridan says, ut
nunc, ut tunc, or ut nunc pro tunc. Thus Buridan’s theory of tense-logic
will be confined to part of the theory of ut nunc consequences. Not that
the ut nunc consequence may behave just like the material conditional; all
that is required for it to be acceptable is the factual lack of the antecedent
obtaining with the consequent failing to obtain.
Consequences are further examined in the first seven theorems in
TC 1. Theorem I-1, Theorem I-5, and Theorem I-7 state the key character-
istics of consequences: they can never lead from truth to falsity, nor from
possibility to impossibility, nor from the necessary to the non-necessary;
equally, the necessary follows from anything, and from the impossible (such
as the conjunction of contradictories) anything follows. In Theorem I-3 the
law of contraposition is stated for consequences, and in Theorem I-4 the law
of transitivity. Together, all of these characteristics define the nature of an
acceptable consequence.
7.3 Assertoric Consequences
The rest of the theorems in TC 1 are devoted to particular asser-
toric consequences. These are two forms: equipollence, that is, sentences
which follow from each other as consequences, and conversion, in which
the subject-term and the predicate-term of a sentence are the same, but
their positions are reversed or the syncategoremata are altered (or both).
In TC 1.8.47, Buridan says that all equipollences and conversions are con-
tained in Theorem I-8 (given in 1.8.40 but as revised in 1.8.43):
[Theorem I-8 (revised)] (a) Any two sentences of which neither can
have some cause of its truth which is not a cause of the truth of the
other sentence follow from the same sentences; (b) any two sentences
of which one has, or can have, more causes of its truth that the other
sentence, although every cause of the truth of the latter is a cause of
the truth of the former, are so related that the sentence with more
causes of its truth follows from the sentence with fewer, but (c) not
conversely.
c Peter King, from Jean Buridan’s Logic (Dordrecht: D. Reidel 1985) 3–82.