INTRODUCTION TO JEAN BURIDAN’S LOGIC
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syllogistic Buridan takes up, he first points to the evidentness and perfec-
tion of those moods corresponding to the first-figure moods listed; no further
justification is given.
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Hence it is incorrect to think that Buridan (or any
other mediæval logician) has a formal metatheory of deductive systems.
Buridan offers two methods for showing the acceptability of a syllogism:
the Reductio-Method and the Method of Reduction. Both are founded on
Aristotle, and supplement the principles listed in the preceding section.
TheReductio-Method, traditionally required only to show the ac-
ceptability of Baroco and Bocardo, is stated as a general principle governing
consequences with a conjunctive antecedent (Theorem III-3 in TC 3.4.17–
18):
A grammatically consequential sentence of the form “p∧q; therefore,
r” is a (syllogistic) consequence if and only if for the sentence ¬r
contradictory to r (i ) the sentence “p∧r; therefore, ¬q” is acceptable
for the sentence ¬q contradictory to q; (ii ) the sentence “q ∧ ¬r;
therefore ¬p” is acceptable for the sentence ¬p contradictory to p.
Since the antecedent of such a sentence is itself equiform to a conjunctive
sentence, the theorem holds by consequential contraposition (Theorem I-3),
De Morgan’s Laws, and consequential importation. The Reductio-Method
consists in assuming that a syllogistic consequence fails to hold and showing
that clauses (i ) and (ii ) are satisfied. We can easily illustrate this method
by proving the acceptability of Bocardo: “Some M is not P, and all M is
S ; therefore, some S is not P.” Taking the contradictory of the conclusion
“All S is P ” and the minor premiss we can syllogize in Barbara “All S is P,
and all M is S ; therefore, all M is P,” and this conclusion contradicts the
original major premiss; equally, taking the contradictory of the conclusion
and the major premiss we can syllogize in Baroco “All S is P, and some M
is not P ; therefore, some M is not S,” and this conclusion contradicts the
original minor premiss. Hence Bocardo is acceptable.
vowel indicates that the premiss characterized by the vowel is to be converted simply,
that is, characterized by the vowel is to be converted per accidens, that is, the terms
transposed in the subalternate; (v ) the letter ‘m’ indicates that the premisses are to be
transposed; (vi ) the letter ‘c’ indicates that the reduction is a reductio ad absurdum.
For example, take Camestres, of the second figure. The vowels indicate that the form
is “All P is M, and no S is M ; therefore, no S is P.” To reduce Camestres we apply
(iii ) to the minor and to the conclusion, and by (v ) we transpose the premisses, giving
us “No M is S, and all P is M; therefore, no P is S ”—a syllogism in Celarent.
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Buridan discusses the perfection of the assertoric syllogism in TC 3.4.56–57 (with an
aside on ampliation and the ‘what is. . . ’ locution in 3.4.56), the oblique syllogism
in TC 3.7.9, syllogisms including identificatory relative-terms in TC 3.7.23, and the
divided modal syllogism in TC 4.2.4 and 4.2.28.
c Peter King, from Jean Buridan’s Logic (Dordrecht: D. Reidel 1985) 3–82.
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INTRODUCTION TO JEAN BURIDAN’S LOGIC
The other way to show the acceptability of a syllogistic consequence
is the Method of Reduction, stated in Theorem III-4 (TC 3.4.10–22):
If the sentence “s; therefore q” is a consequence, then a sentence
of the form “p ∧ q; therefore, r” is a consequence if and only if a
sentence of the form “p ∧ s; therefore, r” is a consequence.
This theorem holds by Theorem I-4; we may illustrate the Method of Re-
duction to show the acceptability of Datisi, which has the form “All M is
P, and some M is S ; therefore, some S is P.” We know from the preceding
sections which conversions hold, and in particular that we immediately infer
an I-form from an I-form by transposition of the terms (obversion); thus the
sentence “Some M is S ; therefore, some S is M ” is a consequence. Hence
we may replace the minor of Datisi with “Some S is M ” and syllogize in
Darii “All M is P, and some S is M ; therefore, some S is P,” and since this
is an acceptable consequence so is Datisi.
8.4 Assertoric Syllogistic
Buridan investigates four main forms of assertoric syllogistic: the
traditional syllogistic; the temporal syllogism, in which the terms or copula
are temporally ampliated; the oblique syllogism, in which the sentences
contain oblique terms; and the variation syllogism, in which the middle
term is finite in one premiss and infinite in the other.
Buridan states which syllogisms in particular are acceptable in var-
ious theorems. His theorems are a model of rigor.
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It will be useful to
investigate certain general points systematically. First, let’s review Buri-
dan’s procedure for the traditional syllogism—a distinctly non-traditional
approach.
In TC 3.4.37 Buridan notes that the semantic principles for the syl-
logism discussed above specify the traditional syllogistic, and in 3.4.38 he
approaches the problem in a combinatorial way. A conjugation is a pair of
premisses; if we simply list them as permutations of quantity and quality,
there are sixteen possibilities (numbered for further references):
(1) [AA] affirmative universal and affirmative universal
(2) [AI] affirmative universal and affirmative particular
(3) [AE] affirmative universal and negative universal
(4) [AO] affirmative universal and negative particular
(5) [EA] negative universal and affirmative universal
(6) [EI] negative universal and affirmative particular
(7) [EE] negative universal and negative universal
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For the sake of convenience, examples of each kind of syllogism Buridan finds accept-
able are given in the notes to each theorem. Some difficulties are also noted.
c Peter King, from Jean Buridan’s Logic (Dordrecht: D. Reidel 1985) 3–82.