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INTRODUCTION TO JEAN BURIDAN’S LOGIC
5.4 Principles of Sentential Logic
The core of sentential logic has several elements: (i ) opposition in all
its forms, such as contradiction, contrareity, and subcontrareity; (ii ) rules
of inference or entailment, including equipollence and conversion; (iii ) prin-
ciples of well-formedness, both as criteria for the acceptability of sentences
and rules for compounding sentences. The theory of consequences is crucial
for (ii ), and will be discussed in detail in Section 7. We have generally
canvassed (iii ) in the preceding sections. The present section will explore
(i ).
In TC 1.8.1 Buridan sets forth a general principle:
For any contradiction, one of the contradictories is true and the
other false, and it is impossible that both are true together or false
together; again, every sentence is true or false, and it is impossible
for the same sentence to be true and false at the same time.
Several principles governing opposition are endorsed here: bivalence, i. e.
“every sentence is true or false”; the Law of Non-Contradiction, i. e. “it is
impossible for the same sentence to be true and false at the same time”;
and what we may call the ‘Law of Contradictories,’ i. e. contradictories must
have opposite truth-value. (See also TC 1.17 for bivalence and the Law of
Non-Contradiction.) Buridan never discusses bivalence at any length; like
most mediæval logicians, he simply assumes its truth.
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On the other hand,
the Law of Non-Contradiction and the Law of Contradictories are carefully
investigated in QM 4.11–15.
Buridan begins his discussion in QM 4.11 by asking whether con-
tradiction is the greatest form of opposition, and argues first that, strictly
speaking, sentences alone are contradictories, such as “Socrates is running”
and “Socrates is not running.” However, there are also contraries; some-
times terms themselves, or their referents, are called contraries; most prop-
erly, though, only sentences can be contraries.
There are two kinds of
contraries: (i ) sentences which are contraries due to their terms, such as
“Socrates is white” and “Socrates is black”; (ii ) sentences which are con-
traries due to their logical form, such as “Every man is running” and “No
man is running” (fol. 21ra). Technically, we can distinguish two additional
principles: the ‘Law of Contraries,’ i. e. such a pair of sentences can both
be false but cannot both be true (discussed by Buridan in TS 2.6.39), and
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Most mediæval logicians considered bivalence to be part of the very definition of
‘sentence,’ so the question did not arise. However, reflections on the problem of future
contingents stimulated discussion of bivalence, leading some (e. g. Petrus de Rivo)
to give it up. Buridan, however, in his discussion of future contingents in QM 6.5,
maintains bivalence.
c Peter King, from Jean Buridan’s Logic (Dordrecht: D. Reidel 1985) 3–82.
INTRODUCTION TO JEAN BURIDAN’S LOGIC
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the ‘Law of Subcontraries,’ i. e. such a pair of sentences can both be true
but cannot both be false.
In QM 4.13 Buridan asks “whether the sentence ‘it is impossible
that the same be present (inesse) and not present to the same at once, with
respect to the same, and so forth for other conditions’ is the first complex
principle” (fol. 22ra), that is, the Law of Non-Contradiction. His answer is
complex. Sentences may be classified according to quality, quantity, mode,
time, and the like; Buridan argues that there is no general way to state truth-
conditions applicable to all these different types of sentence.
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For example,
it is possible that the same be possibly present and possibly not present to
the same at once. In particular, there will be no general definition of ‘con-
tradictory,’ ‘contrary,’ or ‘subcontrary,’ but particular definitions for each
type of sentence. Hence the formulations given above are only schematic:
there is a form of the Law of Non-Contradiction for each type of sentence;
“all of the aforementioned principles are indemonstrable, although some are
more evident and simple [than others]” (fol. 22va).
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Equally, there will be
particular forms of the Laws of Contradictories, Contraries, and Subcon-
traries.
The traditional Square of Opposition expresses the logical relations
among a limited class of sentences: the contraries “All S is P ” (A-form) and
“No S is P ” (E-form) entail as subalternates, respectively, the subcontraries
“Some S is P ” (I-form) and “Some S is not P ” (O-form); the ‘diagonal’
pairs are contradictories, i. e. A-form and O-form on the one hand, E-form
and I-form on the other hand. If we focus on their abstract logical prop-
erties, though, we can see that the sentences on display in the Square of
Opposition are not very general: they are present-time assertoric categori-
cal sentences, the terms of which are logically simple, where the predicate-
terms are common nouns and have an implicit particular quantification,
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indefinite sentences are treated as implicitly quantified (TS 2.2.12), and sin-
gular sentences added. Buridan’s rules, however, are stated generally, so the
Square of Opposition represents only a limited amount of theory.
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Buridan’s arguments for this thesis will be discussed in Section 5.5 and Section 6.9.
49
Buridan devotes QM 4.15 to exploring whether the Law of Non-Contradiction holds
for tensed sentences, a difficult case; they will be discussed in Section 6.8.
50
We should also note here that common terms appearing in sentences are all quantified,,
explicitly or implicitly: an unquantified term may be read as involving distributive
or particular quantification, depending on the requirements of the sentence, context,
and good sense: “Every S is P ” is subordinated in Mental to either “Every S is every
P ” or “Every S is some P ” (typically the latter). Indefinite sentences are quadruply
ambiguous.
c Peter King, from Jean Buridan’s Logic (Dordrecht: D. Reidel 1985) 3–82.