Handbook of the History of Logic: - Fordham University Faculty

Peter Abelard and His Contemporaries 145
failing this requirement, giving grounds for considering it false [Anonymous, 1983,
p. 26; Martin, 1987a, p. 397; Iwakuma, 2004, p. 326]. The Melidunenses, by
contrast, emphasize the importance of a true antecedent rather than the importance
of a relevant one; their principle is that nothing follows from a falsehood
[Martin, 1987a, p. 398; Iwakuma, 2004, p. 327]. The antecedent of (i), if taken
as a free-standing proposition, asserts that Socrates is a man and not an animal,
which cannot be true since all men are animals. So what the antecedent asserts is
false, and the principle that nothing follows from a falsehood accordingly identifies
the whole conditional as false. Both the Porretani and Melidunenses support
their analyses by counter-examples to suggest how in other instances the practice
of allowing irrelevant or false antecedents will lead to problematic results. In
both cases we have genuine propositional logic being done, the search being for
constraints on conditionals — either involving relevance conditions or truth conditions
— sufficient to demonstrate the falsity of (i). The constraints naturally
have a very limiting effect on what is accepted as a true conditional.
The response of the Parvipontani is the most forward-looking. They too deal
in genuine propositional logic, but are not concerned to constrain it in these ways
— because they consider Alberic’s argument acceptable as it stands, and feel no
incentive to demonstrate the falsity of (i) or any other premise; nor do they wish
to question the argument’s soundness. They in fact accept the principle that
from a contradiction any statement whatsoever follows, so the self-contradicting
conclusion (iv), where a proposition entails its own denial, is simply a routine
product of the initial self-contradicting antecedent of (i): “Socrates is a man and
is not an animal.” One adherent of the school states openly that “If Socrates is a
man and Socrates is not a man, then Socrates is a crab, and so on for single things,
for example a rose, a lily and the rest” [de Rijk, 1967a, p. 290; Martin, 1987a,
p. 398; Iwakuma, 2004, pp. 327–328]. The lesson, in other words, is that any
conclusion follows from a contradiction. 109 While the Parvipontani never advance
from this point to constructing a full propositional logic in the modern sense, it is
clear that they prefigure in their response to Alberic one of modern propositional
logic’s most suggestive doctrines.
109The result is motivated by this sort of argument [de Rijk, 1967a, p. 290; Martin, 1987a, p.
398; Iwakuma, 2004, pp. 327–328]:
(i) If Socrates is a man and Socrates is not a man, then Socrates is a man.
(ii) If Socrates is a man, Socrates is a man or a stone.
(iii) Therefore, if Socrates is a man and Socrates is not a man, Socrates is a man or a stone.
(iv) If Socrates is a man and Socrates is not a man, then Socrates is not a man.
(v) Therefore, if Socrates is a man and Socrates is not a man, then Socrates is a stone.
(iii) follows from (i) and (ii) by hypothetical syllogism. The inference from (iii) and (iv) to (v) is
more abbreviated. “Socrates is a man and Socrates is not a man” implies both “Socrates is a man
or a stone” and “Socrates is not a man”; these two latter propositions together imply “Socrates
is a stone” (by disjunctive syllogism). Therefore “Socrates is a man and Socrates is not a man”
implies “Socrates is a stone” — which is what (v) asserts. With the relevant substitutions, the
above argument can be made to derive any conclusion from any contradiction.