Andrew Schaeffer | Wordcount: 1,146| 2/12/2016
ARISTOTLE’S METAPHYSICS
Aristotle’s Argument for PNC
In this essay I will explain Aristotle’s Principle of Contradiction, how he
sets about proving it in his Metaphysics, and how his argument is probably
weak or could fail.
For Aristotle, Principle of Contradiction (PNC) is the most basic principle
of his Metaphysics, and a necessary condition for anyone to be able to
think or say anything. It comes in two forms: logical PNC, and
metaphysical PNC.
In this essay I will focus only on metaphysical PNC,
which I will simply call “PNC”.
PNC is the principle that “It is impossible
for the same thing at the same time both to belong and not belong to the
same thing in the same respect1.” (Met Bk.IV: Ch3 1005b18).
To illustrate
what this means, here is an example: Imagine a watermelon sitting on a
cutting board.
Would it be possible for it to be sliced in half at precisely
12:00 PM and not sliced in half at the same time?
Given the importance of PNC for Aristotle’s metaphysics, he sets out to
prove PNC in “Gamma” chapter four of his Metaphysics.
However, he
thinks he must prove PNC indirectly, because, by definition, a principle
can’t be proven.
1
In the Greek it reads: τὸ γὰρ αὐτὸ ἅμα ὑπάρχειν τε καὶ μὴ
ὑπάρχειν ἀδύνατον τῷ αὐτῷ καὶ κατὰ τὸ αὐτό
2
Beginning in “Gamma” chapter four of the Metaphysics beginning at
1007b20, Aristotle’s program is to establish PNC via reductio ad absurdum.
This means he will assume the negation of what he’s trying to prove,
generate an absurdity based off of that assumption, and as a result of this
absurdity negate the initial assumption.
Thus, if the initial assumption is
negated, Aristotle’s conclusion will be true as a result.
Aristotle’s assumption in this reductio argument is the negation of PNC;
and the absurdity Aristotle thinks he can generate in order to prove PNC
is that one and the same thing is everything, which I will call E1 2 (Met
Bk.IV: Ch4 1007b19).
Although it may appear as though the assumption
for reductio is the claim that “all contradictory statements are true of the
same subject at the same time3,” there’s a good reason to think otherwise
(Met Bk.IV: Ch4 1007b20).
For, were the negation of that claim Aristotle’s
conclusion, it would not entail PNC, the very thing he’s arguing for in this
stretch of the Metaphysics.
The negation of the claim “all contradictory
statements are true of the same subject at the same time” would entail
only that some contradictory statements are not true of the same subject
at the same time. But this doesn’t preclude a case involving some
contradictory statements being true of a subject at the same time, and for
that reason it does not entail PNC.
Given that Aristotle begins his argument by assuming the negation of
2
3
E1 stands for “1 thing is Everything.”
In the Greek it reads: …ἀληθεῖς αἱ ἀντιφάσεις ἅμα κατὰ τοῦ αὐτοῦ
πᾶσαι…
— 2 —
3
PNC, how does he get from there to the absurdity E1?
three-stage process:
I see this as a
first, he assumes the negation of PNC; second, he
shows how that implies that “all contradictory statements are true of the
same subject at the same time,” a view which I will call SPNC 4; third, he
shows how SPNC leads to E1.
Once he has E1, the game is over, and
the PNC is true.
Protagoras and Anaxagoras, though worth mentioning because Aristotle
talks about them and how their views imply the negation of PNC, play an
ancillary role in Aristotle’s thrust in arguing for how the negation of PNC
leads to SPNC.
PNC.
What is important is the assumption of the negation of
In terms of the roadmap of the argument then, Aristotle is between
the first and second stage.
He wants to show how all contradictory
statements can be made about the same thing, and begins with a single
case of contradiction: A man is both a man and not a man (assumption
for reductio, Met Bk IV Ch4, 1007b33). For Aristotle, this is like opening
Pandora’s box.
If it’s possible to predicate something of a man which
ought not to be predicated of a man, namely, that he is not a man, what
can’t one predicate of a man?
Consequently, Aristotle reasons that the
predicate ‘is not a trireme’ ought to be capable of being predicated of man
if the predicate ‘is not a man’ can.
So, the second premise is: A man is
not a trireme (premise 2). After this Aristotle reaches back to premise
one, which is the assumption for reductio, and to premise 2 in order to
4
The “S” in “SPNC” stands for “Strong.”
PNC.
After all, the claim is a stronger version of the
— 3 —
4
justify his next move. Since ‘is not a man’ is predicated of a man when it
shouldn’t, and a trireme, while being extremely different from a man is
nevertheless a negative predicate of that same man, it seems to follow
that a man is also a trireme (premise 3).
For, the negative predicate ‘is
not a trireme’ justifies predicating things of a man that have nothing to do
with a man, and the predicate ‘is not a man’ justifies predicating something
that ought not to be predicated of a man, viz., the predicate ‘is a trireme’.
Given premises 1, 2, and 3, any and all manner of things can and ought
to be predicated of a man, and these premises imply SPNC (premise 4).
Aristotle is now on the home stretch between stage 2 and 3, or
between SPNC and E1.
Within the scope of the Metaphysics “Gamma”
Chapter four from 1007b20 to 1008a3, Aristotle leaves it up to the reader
to fill in how the apparent lacuna between stage 2 and 3 should be filled.
However, the idea seems to be that if an infinite number of things are
predicated of a man, this would imply E1 (premise 5) (Met Bk IV.Ch4
1007b29).5
But, since E1 is absurd, the assumption of  (PNC) must be
false, making PNC (the conclusion) true.
Although I think the PNC is true, Aristotle’s reductio argument is likely
weak in that it probably begs the question.
For, E1 is likely to be absurd
only because it’s in violation of PNC, and not in virtue of some other type
of absurdity.
It seems to me that E1 is just an infinite collection of
violations of PNC rather than an essentially different absurdity from PNC.
5
Aristotle also talks about this in the Metaphysics “Gamma” Chapter four 1007b1.
— 4 —
5
Thus, the real essence of the absurdity seems to be simply a violation of
the PNC.
Therefore, I think Aristotle is probably begging the question in a
roundabout way.
Another reason to think Aristotle’s argument may be bad is that it may
assume that all people find E1 absurd.
However, as Cicero so nicely put
it, “There is nothing so absurd but some philosopher has said it” (De
Divinatione ii. lxviii, 120). By this, I take him to imply that there will
always be some philosopher who will think something which other people
would think fantastic.
E1 absurd.
So there are likely some philosophers who don’t find
Therefore, insofar as Aristotle takes his argument’s strength to
turn on whether other philosophers think E1 is absurd, Aristotle’s argument
is weak.
— 5 —