Friday, January 29, 2016

on Aristotle's Posterior Analytics 73b10-17.

“Again, in another way what holds of something because of itself in itself, and what does not hold because of itself is incidental. E.g. if there was lighting while he was walking, that was incidental: it was not because of his walking that there was lightning—that, we say, is incidental. But what holds of itself holds in itself—e.g. if something dies while it is being sacrificed, it died in the sacrifice since it died because of the sacrifice, and it was not incidental that it died while being sacrificed.”
Aristotle, Po. An. 73b10-17, (trans.  J. Barnes), emphasis in trans.
Let ‘W’ abbreviate “William is walking” and ‘L’ abbreviate “There is lighting”. Now, suppose that
W • L
It follows that
W L
That is, from the fact that William is walking and there is lighting, it follows that if William is walking then there is lighting. But, Aristotle seems to deny this, for he says that, “it was not because of his walking that there was lighting”. Thus, Aristotle might be thought of asserting:
~(W L)
But this would be mistake, for ~(W L) implies
W • ~L
in which case Aristotle would be having us suppose, at the same time, that
L • ~L
Since Aristotle isn’t having us suppose a contradiction, how shall we represent what he’s saying? Here’s one option. When he says that “it was not because of his walking that there was lighting”, he isn’t denying W L, but rather that W L is necessary—viz.
~(W L)
So we might put it like this: Aristotle would agree that statements of the form PQ imply statements of the form P Q but not (P Q).
On the other hand, there are some cases in which statements of the form (P Q) are true. Aristotle’s example is that of a thing dying because it was sacrificed. Let ‘S’ and ‘D’ abbreviate “William is sacrificed” and  “William died”, respectively. Though Aristotle denies (W L), he affirms
(S D)
Since a statement is necessary iff its negation is impossible, (S D) implies
~~(S D)
which, via material implication, De Morgan’s law, and double negation, implies
~(S ~D)
On the other hand, since a statement is not necessary iff its negation is possible, ~(W L) implies
~(W L)
Which, via material implication, De Morgan’s law, and double negation, implies
 (W ~L)

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