Handbook of the History of Logic: - Fordham University Faculty

The Assimilation of Aristotelian and Arabic Logic up to the Later Thirteenth Century 319
human being is not running’.
E-rule 2: If ‘Every’ or ‘Some’ are preceded and followed by a negation
in some proposition, then it is equivalent with its subalternate
proposition.
Example: ‘Not Every human being is not running’ is equivalent to
‘Some human being is running’.
E-rule 3: If two universal negative signs are posited in the same proposition,
and one is in the subject’s position and the other in the predicate’s
position, then, firstly, the proposition is equivalent to its contrary, and,
secondly, to its contradictory.
Example: First of all, ‘Nothing is nothing’ is equivalent to ‘Something
is something’, and, secondly, ‘Something not’ is equivalent to ‘Nothing’,
and ‘Every not’ to ‘No’.
The last rule looks a little bit strange, but it is only meant to express for example
how the quantity of the proposition is changed by adding a negation in front of
it or after it. As is clear all tree of these rules are about the way negation effect
propositions.
After these simple equipollence rules the first treatise ends with a discussion of
modal propositions. The mode is treated by these authors as adjectives and as such
they affect the way or the manner in which the proposition are taken. Adjectives
might be taken nominally or adverbially and hence modes are explained in that
way as well. They explain this in some detail, but first they note that the four
modes are necessity, possibility, contingency and impossibility. They usually add
also truth and falsity, but note that they do not change the truth value of a
proposition as the others do. 84
The mode of a proposition is either taken nominally as in ‘It is necessary that
Socrates is an animal’, or it is taken adverbially as in ‘Socrates runs contingently’.
This is the same division as Abelard had made a century earlier. 85 Peter stresses
that only the adverbial reading makes the proposition into a proper modal proposition,
since only on such a reading is the composition of the subject term and the
predicate term modified. The proper way to write a modal proposition is thus:
‘Socrates is contingently running’, and hence it is the copula that is modified in
such propositions, which means that it is really the relation between the term that
is qualified. Somewhat confusingly he insists on writing the modal propositions
like this: ‘Socrates runs is possible’. On such a reading, it looks as if the mode
‘possible’ is taken nominally and hence function as an ordinary predicate, which
84 William notes that ‘necessarily’ and ‘impossibly’ are used in two ways. Somethings are
‘necessary per se’, which cannot be false now or in the future or past, like ‘God is’, and other
propositions are ‘necessary per accidens’, which cannot be false now or in the future, but might
have been in the past, like ‘I have walked’. This distinction seems inspired by Averroes mentioned
above in section 3.2, but less ontological.
85 See above and also [Lagerlund, 2000, 35-9].