Handbook of the History of Logic: - Fordham University Faculty

168 Terence Parsons
1.7 Quantifying the Predicate Term; Equipollences
The above is mostly Aristotle’s logic with some of its consequences spelled out.
Some medieval authors in the 1200’s expanded the stock of Aristotle’s categorical
propositions by allowing quantifier signs to combine with predicate terms, and
allowing not’s to occur more widely, so as to produce propositions like ‘No A is
not every B’, or ‘Not every A some B is not’. I call Aristotle’s forms ‘standard
Aristotelian categorical propositions’, and I call the expanded forms ‘categorical
propositions’. The forms of categorical propositions will be further extended in
section 3.
Along with these expanded forms, writers added a series of equipollences —
logically equivalent pairs of propositions that differ only in having one part replaced
by an “equivalent” part. These are like contemporary rules of quantifier exchange,
such as the equivalence of ‘∀x’ with ‘∼ ∃x ∼’.
“To reduce [a syllogism] per impossibile is to infer the opposite of one of the premises
from the opposite of the conclusion together with the other premise. For suppose
we take the opposite of the conclusion of this fourth mood (namely, ‘Every stone
is a man’) together with the major premise and construct a syllogism in the first
mood of the first figure in this way:
Every man is an animal;
every stone is a man;
therefore, every stone is an animal.
“This conclusion is the opposite of the minor premise of the fourth mood. And this
is what it is to prove something [by reduction] per impossibile.”
These instructions work perfectly provided that conversion by limitation is used in the correct
order; from particular to universal in premises, and from particular to universal in conclusions.