Handbook of the History of Logic: - Fordham University Faculty

476 Catarina Dutilh Novaes
the notion according to which a formal (logical) consequence must satisfy both
criteria, the modal one and the substitutional one (see [Dutilh Novaes, 2005b].
This criterion for differentiating consequences had been proposed before by
Abelard, who distinguished perfect from imperfect consequences [Abelard, 1956,
253-4]: perfect consequences received their warrant from their structure (complexio)
alone, while imperfect inferences needed external warrant (for which Abelard
turned to the Topics — cf. [Abelard, 1956, 256-7]). The same criterion was later
rediscovered by Bolzano [1973] and further developed by Tarski [2002].
Intuitive though as it may seem to the modern reader, the definition of formal
consequence based on the substitutional criterion was not widely accepted in the
14 th century. As already mentioned, while the treatises on consequence influenced
by Buridan did maintain this criterion (Pseudo-Scotus, Albert of Saxony, Marsilius
of Inghen), the majority of treatises followed a different notion of formal
consequence. This alternative notion of formal consequence is what we could call
the containment notion: a consequence is formal iff the consequent is contained in
the antecedent, in such a way that whoever understands the antecedent necessarily
understands the consequent. 47
Here is Lavenham’s formulation (as quoted in [King, 2001, 133]): ‘A consequence
is formal when the consequent necessary belongs to the understanding of the antecedent,
as it is in the case of syllogistic consequence, and in many enthymematic
consequences’. Strode’s similar formulation is: ‘A consequence said to be formally
valid is one of which if it is understood to be as is adequately signified through
the antecedent then it is understood to be just as is adequately signified through
the consequent. For if someone understands you to be a man then he understands
you to be an animal.’ (as quoted in [Normore, 1993, 449]). Many other authors
held similar definitions, such as Billingham [2003, 80] and Fland [1976].
Besides the fact that this is notably an epistemic notion of formal consequence
(as opposed to Buridan’s substitutional notion), for the authors adopting the containment
notion, the extension of the concept of formal consequence is usually
wider than that of those adopting the substitutional notion of formal consequence.
Lavenham explicitly says that some enthymematic consequences are formal consequences
— that is, consequences that, with the addition of an extra premise,
acquire a syllogistic form, such as ‘Socrates is a man, thus Socrates is an animal’,
which acquires a syllogistically valid form with the addition of ‘Every man is an
animal’. For Buridan, however, enthymematic consequences are not formal consequences
because they do not satisfy the substitutional criterion, e.g. this particular
example is not valid for all substitutional instances of ‘man’ and ‘animal’. They do
become formal consequences with the addition of the missing premise, but before
that occurs they are merely material consequences (cf. [Buridan, 1976, 23]).
Another different criterion for formal consequences is presented at the very end
of the 14 th century by Paul of Venice; he characterizes formal consequences as those
47 The containment notion of (formal) consequence is not a 14 th century invention. For Abelard,
it was a necessary condition for all valid consequences; in the 13 th century, it was held by authors
such as Faversham and Kilwardby.