Handbook of the History of Logic: - Fordham University Faculty

170 Terence Parsons
Every A is B Every A is every B
No A is B No A is every B
Some A is B Some A is every B
Some A is not B Some A is not every B
Using William’s equipollences together with double negation, every proposition in
the expanded notation is equivalent to one of these eight forms.
Notice that throughout this extension of the scope of what counts as a categorical
proposition the quantifier signs have not been supplemented; they are ‘every’,
‘no’, and ‘some’, with the indefinite construction (which uses the indefinite article
in English, and nothing at all in Latin) treated as if it is equivalent to one with
‘some’.
The semantics of these propositions is straightforward. We can view a categorical
proposition in this form as consisting of the copula preceded by some denoting
phrases and some negations, each of which has scope over what follows it. This
can easily be given a logical form that is familiar to us by flanking the verb with
variables, supposing that the preceding denoting phrases bind these variables. Examples:
Every donkey is an animal ⇒
(Every donkey x) (an animal y) x is y
Some donkey is not an animal ⇒
(Some donkey x) not (an animal y) x is y
Some donkey an animal is not ⇒
(Some donkey x) (an animal y) not x is y
No donkey is not an animal ⇒
(No donkey x) not (an animal y) x is y
No animal is not every donkey ⇒
(No animal x) not (every donkey y) x is y
It is not difficult to formulate a modern semantical theory that would apply to
the forms that occur on the last lines of these examples, provided that we keep
in mind that an affirmative proposition is false when any of its main terms are
empty, and a negative one is true. 15
1.8 Molecular Propositions
Certain combinations of categorical propositions are called hypothetical propositions;
these include combinations that we call molecular. Their development came
15 It is often suggested that all the Aristotelian forms can be symbolized in the monadic predicate
calculus. The expansions mentioned here require the use of monadic predicate logic with
identity, and require using quantifiers within the scopes of other quantifiers.