Aristotle's Principle of Non-Contradiction (PNC for short)
Summary of Stanford Encyclopedia Article's main points
- three versions of the principle of non-contradiction
- semantic (i.e. about truth and statements)
- opposite assertions
be true at the
same time (Metaph IV 6 1011b13–20)
- this version is a variant of
the ontological version below
- doxastic (i.e. about our beliefs)
- It is impossible to hold
the same thing to be and not to be (Metaph IV 3
- i.e. it is impossible to hold the same
thing to be F and not to be F.
- this may not seem
- People have inconsistent beliefs
- Must one believe the consequences of one's beliefs?
one knowingly believe an outright contradiction?
- what a man says he
does not necessarily believe (Metaph IV 3
- An alternate way of understanding the doxastic claim: one
should not hold the
same thing to
be F and not to be F.
- not a descriptive, but a normative
claim, a claim about what it is rational to believe.
- one cannot rationally hold the same thing to be F and
- distinguish the possibility of believing that x is F and
F in a
particular case from the possibility of disbelieving the
of PNC in its full generality.
- ontological (i.e. about what exists)
- This one is the one referred to as PNC and is
the main version of the principle
- It is
impossible for the same thing to belong and not to belong
at the same
time to the same thing and in the same respect
- with the appropriate
qualifications: The “same thing” that belongs
must be one and the same thing and it must be the actual
thing and not
merely its linguistic expression. Also, the thing that
belong actually, and not merely potentially, to its bearer.
- Which version Aristotle intends to establish is a matter of
- PNC cannot be proven, but there are some very strong
that it is right:
- Without it, we could not do many things, Aristotle claims:
- we could not demarcate the subject matter of any of the
- we would not be able to distinguish between what something
is, for example a human being or a rabbit, and what it is
example pale or white.
- Aristotle's own distinction between essence and accident
would be impossible to draw
- the inability to draw distinctions in general would make
rational discussion impossible
- So there are indirect indications that it is right.
- BUT all of the above assumes that those who deny PNC claim
PNC applies nowhere at all: that there is no area in which It is
impossible for the same thing to belong and not to belong at
time to the same thing and in the same respect.
- Could an opponent claim that PNC works in most areas, but
- Where does Aristotle talk about it?
- Metaphysics IV (Gamma) 3–6, especially 4
- BY FAR the most important discussion
- in De Interpretatione
- chapter 11 of Posterior Analytics I
- Where is the PNC in Aristotle's system
- In the Metaphysics:
it is part of the study of being qua being
- How it belongs there is a matter of some controversy: it
not seem obvious that it belongs there
- But also one must admit that it does not seem obvious that
it does not belong there.
- It is a "common axiom"
- Common to what? to all the special sciences (i.e. biology,
mathematics, geology, etc.)
- but it is also common to the science of being qua being:
science can prove PNC: it is indemonstrable
- axiom? it is not demonstrable and so is at the bedrock of
- if it were demonstrable, the premises of that
would be more basic
- also, it cannot be demonstrated because every
relies on its being right
- you cannot prove it is right by assuming it is right
- that would be "begging the question"
- you cannot perform a reductio ad absurdum,
which would have one suppose for a moment that it is
then see what happens, because "seeing what happens"
that it is right!
- It has no specific subject matter: it applies to everything
that is (the ontological version) as well as to everything one
or imagines (doxastic version).
- Is it prior to truth, reference and identity?
- it assumes identity: it cannot be formulated without
it: "one and the same thing" involves identity
- this is pushing the Aristotelian envelope: that never
- Refuting those who deny the PNC
- Although it cannot be demonstrated, those who deny it can be
refuted on their own terms.
- The Greek term for "refutation" was "elenchus": it was used
often of Socrates' method of arguing.
- The method of elenchus is:
- "opponent" says that some claim is the case.
- "opponent" also agrees that a couple other things are the
- But the other things conflict with the first claim.
- So "opponent" is inconsistent and must give up either the
original claim or some of the subsequent claims.
- Aristotle is trying to show that the opponent of the PNC is
committed to one thing that is not contradictory.
- The opponent is not bothered by inconsistency: that is
it means to deny PNC.
- Aristotle assumes his opponent takes the very strong
that for any
for any F,
possible that F belongs to and does not belong to x at the
same time in the
- Aristotle wants to draw the opponent into saying
making a complete statement, that shows that he does
accept that x is F
and is not at the same time not F
- Aristotle seems to be arguing that the world conforms to
or that PNC is true, because it is presupposed by and
opponent's ability to say something significant.
- Aristotle may be aiming to show that the ontological
of the principle of non-contradiction is true, or he may be
show merely that it cannot be disbelieved
- Aristotle's challenge to his opponent:
- signify some one thing to both herself and another person.
- in other words, communicate just one thing: pick out one
object, "dog" or "man"
- if the communication works (i.e. if the person
and the person being communicated with both understand the
then Aristotle thinks the opponent is committed to PNC in at
- Aristotle is showing the opponent that if she wants to
PNC she must:
- pick out the same object (e.g. "human being" or "dog")
twice and say that
contradictory predicates apply to it
- if she does not mean anything definite by “human being”,
for example, then she will be unable to twice pick out a
predication, for example, a human being, and say that
- opponents cannot have any one thing to which accidents are
- Aristotle's "substance" and "essence" serves to provide
thing to which accidents adhere
- those who deny PNC cannot have such a thing to hold
together: they claim that every x is F and is not F.
- thus they cannot have "substance"
- the idea is that it makes no sense to say the "pale" is
- you have to say the pale "thing" is also a musical
and that requires a "thing" that just is that thing (and
not not that
- it also makes no sense to say that Jacques is just a
of accidents: without some unifying factor to make them all
they are just a heap of accidents without unity: that
must make them one thing and not not one thing.
- What if opponent refuses to take the challenge?
- opponent is no better than a vegetable
- both in that Aristotle cannot talk to a vegetable and in
the opponent cannot show that she is not a vegetable
- even if opponent refuses to try to communicate, she must
still act, and action involves commitment to the claim that
in the world is x and not not-x.
- opponent might reply that she can act "as if" there are
things in the world that are what they are and are not
what they are
- in that case, all Aristotle shows is that we have to act
"as if" PNC is correct. That is something.
- but Aristotle obviously wants more
- can he have more?
- maybe all we ever do is act "as if"
- Aristotle claims that rejecting PNC involves rejecting
that plays the role of making it the case that we can reliably
communicate AND also involves rejecting any metaphysical
things in the world that makes it the case that things are
they are and not simultaneously not what they are.
- At the end of Metaphysics IV.4,
- However much all things
be so and not so, still there is a more and a less in the
things; for we should not say that two and three are
equally even, nor
is he who thinks four things are five equally wrong with
him who thinks
they are a thousand. If then they are not equally wrong,
is less wrong and therefore more right. If then that which
has more of
any quality is nearer to it, there must be some truth to
which the more
true is nearer.
- Aristotle never works it out, but what about a world in
our views are mere approximations of truth, and even if they
be the truth, we cannot be certain of that? could there be a
"fuzzy" or "vague" essentialism?
- Aristotle's confused opponents:
- In Metaphysics
Aristotle talks about opponents who see a thing changing and
from their observations of chance that the same thing must
contrary properties: contrary properties come into existence
out of the
- Here, Aristotle introduces the distinction between
- X an be both potentially F and actually not-F
- but X cannot be both actually F and actually not-F.
- and X cannot be both potentially F and not potentially F
- but it can be both potentially F and potentially not-F.
- Other opponents had a problem because of conflicting
appearances (the following is verbatim from the Stanf. Enc. of
- There are three sorts of cases of conflicting appearances:
- Things appear different to different members of the same
species, e.g., the same thing is thought bitter by some
and sweet by
others (Metaph IV 5 1009b2–3).
- Things appear different to members of different species
(e.g., to other animals and to us) (Metaph IV 5 1009b7–8).
- Things do not always appear the same even to the senses
the same individual (Metaph IV 5 1009b8–9).
- It is not clear which appearances are true and which false
(Metaph IV 5 1009b10).
- Nothing is true (Democritus in dogmatic mood, Metaph IV
- (If something is true) a true thing is not clear to us
skeptical mood, Metaph IV 5 1009b12).
- Everything is just as true as everything else. (This is
mentioned as an explanation of premiss 2 at Metaph IV 5
is Protagoras's view, as described at the beginning of the
Another set of opponents are "Protagoras, Heraclitus, and
Theaetetus," who are characters in Plato's dialogue Theaetetus
- Aristotle agrees with 1
- Aristotle disagrees with 2: he does not think that people
really confused about which appearances are right in cases
- they trust experts
- they trust their senses for things each sense is
- many moderns take conflicting appearances seriously,
especially in ethics.
- In Metaphysics IV
Aristotle mentions Protagoras' doctrine that each individual
being is the measure of all things.
- PLATONIC detour
- At Theaetetus
151–183, Plato argues that Theaetetus, who holds that
nothing but perception, is committed to Protagoras's view
argument from conflicting appearances. If the wind appears
cold to you
but hot to me and knowledge is nothing but perception,
then we must
both be correct, as Protagoras says.
- Plato argues that Protagoras is committed to the view
nothing is anything in itself (otherwise one might be
wrong about how
it really is) and to a “secret” Heraclitean doctrine of
flux. In order
to accommodate more and more conflicting appearances,
and to avoid
violating PNC, more and more flux is needed, until we
reach a radical
version of Heraclitus's doctrine according to which
everything is “so
and not so” (Tht 183), with accompanying difficulties
language. The extended argument also contains a
“self-refutation”, where Plato draws the “exquisite”
Protagoras refutes himself if he agrees that other
people disagree with
his own view (Tht 171A-D). If they are right, then he
must be wrong!
- end of Platonic detour
- elements of Plato's discussion re-surface in Aristotle's
- Protagoras's view and the suggestion that everything is
“so and not so” go hand in hand.
- Heraclitus's followers thought that there is so much
change in the world that it is impossible to say anything
true, and so
Cratylus, one of their number, was reduced to wagging his
- At the end of chapter 6, Aristotle concludes, “Let this, then
suffice to show (1) that the firmest belief is that opposite
are not true at the same time, (2) what happens to those who
this way and (3) why people do speak in this way (Metaph IV 6
- On the first point, as we saw, it is controversial whether
Aristotle's conclusion that the firmest belief is a belief in
carries with it the presupposition that PNC is true, a
that is needed for his own project of first philosophy.
- On the second point, Aristotle shows that those who say that
they reject PNC do not really do so, or, if they do, they will
giving up intelligible discourse and action, and—one might
will be living in a world of mere sophistry and power.
- On the third point, Aristotle discusses views about
and change that lead people to say that they reject PNC.
- It is controversial how much of an essentialist or indeed
realist view one must accept if one accepts PNC, but it is
PNC is essential for the project of an Aristotelian science.
- Without it, Aristotle notes, beginners in philosophy who are
interested in the truth would be off on a wild goose chase
(Metaph VI 5