Handbook of the History of Logic: - Fordham University Faculty

Logic in the 14 th Century after Ockham 479
3.4 Rules of inference recognized by the medieval authors
Let us now look at the logical rules of consequence recognized by our 14 th century
authors. Some of them had already been identified by earlier authors 48 (for
example the rules of opposition, equipollence and conversion for categorical propositions
described in Aristotle’s De Interpretatione); but what is remarkable in at
least some of the medieval treatises is how they attempt at a systematization of
these rules, such that from primary rules secondary rules are derived (see for example
the first chapters of Burley’s De Puritate, The Shorter Treatise — [Burley,
2000, 3-26]). Granted, other treatises are really no more than rather unsophisticated
lists of rules, with no attempt to link the logical properties of each of them
together. But quite a few of them present what we could view as the first stages
of a proof theory.
Here, I present some of these rules making use of a notation inspired by Gentzenstyle
sequent calculus. The list of rules presented here is not exhaustive in that
not all the rules studied and recognized by the medievals will be presented; the
purpose here is to give the reader an idea of the level of logical sophistication
attained by medieval treatises on consequences. A more thorough and extremely
useful listing of the rules recognized by the medieval authors can be found in
[Pozzi, 1978, 69-73]. 49
Burley, for example, lists ten main rules and several other rules that follow
from these main rules. The first four rules are indeed easily rendered within the
conceptual framework of propositional calculus, while the other six rely heavily
on the properties of terms as well (and this is why it is inaccurate to say that
medieval theories of consequences are purely propositional in nature; logical properties
of terms still play a prominent role). Boh [1982, 312-314] presents a neat
reconstruction of main rules 1 to 4, plus their derived rules, but the problem with
his reconstruction is that it implies the view that consequences are conditionals, a
view that, as already said, is rejected here.
Burley’s rule 2, for example, states that ‘whatever follows from a consequent
follows from the antecedent’, or alternatively, ‘whatever is antecedent to the antecedent
is antecedent to the consequent’ [Burley, 2000, 4]. This is basically a
formulation of the Cut-rule in sequent calculus (with the difference that no mention
is made to the contextual propositional variables that are included in sequent
calculus for the sake of generality).
Rule 2
A ⇒ B B ⇒ C
A ⇒ C
48 It is also often said that the Stoics are the genuine pioneers of propositional logic; however,
there is as of yet no evidence of direct or even of indirect influence from Stoic logic on the
development of medieval theories of consequences. That is, even if many of such rules of consequence
had already been recognized by the Stoics, it all seems to indicate that the medievals
re-discovered them independently.
49 Pozzi’s study is based on the treatises on consequences of the following authors: Ockham,
Burley, Pseudo-Scotus, Buridan, Albert of Saxony, Ralph Strode, Peter of Mantua, and Richard
Ferrybridge.