Handbook of the History of Logic: - Fordham University Faculty

480 Catarina Dutilh Novaes
From this he derives a few other rules, among which: (2 ′ ) ‘whatever follows from
a consequent and from its antecedent follows from the antecedent by itself’ [Burley,
2000, 6] and (2 ′′ ) ‘whatever follows from a consequent with something added
follows from the antecedent with the same thing added’ [Burley, 2000, 7].
′ A ⇒ B A,B ⇒ C
Rule 2
A ⇒ C
A ⇒ B B,C ⇒ D
′′
Rule 2
A, C ⇒ D
Burley derives (2 ′ ) from Rule 2 plus what he takes to be a logical fact, which
the modern reader may recognize as a special case of right-weakening: ‘every
proposition implies itself together with its consequent’; similarly, he derives (2 ′′ )
from Rule 2 plus simultaneous applications of special cases of right-weakening
and left-weakening: ‘an antecedent together with something added implies the
consequent with the same thing added’. His arguments can be reconstructed as
follows:
(2 ′ )
(2 ′′ )
A ⇒ B WR
A ⇒ A, B A, B ⇒ C(hyp.) Rule 2
A ⇒ C
A ⇒ B WR, WL
A, C ⇒ B,C B,C ⇒ D(hyp.) Rule 2
A, C ⇒ D
After having shown that the medievals did know the procedure of deriving rules
from primitive rules, I now present a few other rules recognized by them (taken
from Pozzi’s very useful list, unless otherwise stated), not paying specific attention
to the deductive structure between these rules:
‘From the impossible anything follows.’
⊥⇒A
In the case of material as-of-now consequences the requirement is weaker: from
a false proposition anything follows.
‘The necessary follows from anything.’
A ⇒ T
Similarly, in the case of material as-of-now consequences the requirement is
weaker: a true proposition follows from anything (these two weaker formulations