## 1. Three Versions of the Principle of Non-Contradiction



There are arguably three versions of the principle of
non-contradiction to be found in Aristotle: an ontological, a doxastic
and a semantic version. The first version concerns things that exist
in the world, the second is about what we can believe, and the third
relates to assertion and truth. The first version (hereafter, simply
PNC) is usually taken to be the main version of the principle and it
runs as follows: "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) (*Metaph*
IV 3 1005b19-20). The following are some of those
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. For example, it is possible for someone to be a
pitcher and not a pitcher where "pitcher" in the first
instance refers to a baseball player and in the second to a jug that
can hold beer. Also, while it is possible for *x* to be
actually *F* and potentially not *F*, it is impossible
for something to be actually *F* and actually not *F*. A
table can be actually red and potentially not red, but not actually
red and actually not red at the same time.



The second version is as follows: "It is impossible to hold
(suppose) the same thing to be and not to be (*Metaph* IV 3
1005b24 cf.1005b29-30)." Although this version is
ambiguous as it stands, it is best understood as the claim that it is
impossible to hold the same thing to be *F* and not to be
*F* &c.



As a descriptive account of human psychology, this may seem
implausible. People surely have inconsistent beliefs; indeed, most of
us have many inconsistent beliefs. This is especially true if we take
into account the consequences of our beliefs. Must one, though,
believe the consequences of one's beliefs? These remain
difficult issues in modern philosophy of language and epistemology.
Can one knowingly believe an outright contradiction? Heraclitus, for
instance, seems to say contradictory things. Here, Aristotle might
retort, and he does so retort with respect to Heraclitus, that people
can utter such words, but cannot really believe what they are saying
(*Metaph* IV 3 1005b23-26).



An alternate way of understanding the second formulation is to treat
it not as a descriptive claim about human psychology, but as a
normative claim, a claim about what it is rational to believe. On this
view, it is not that one cannot believe that *x* is *F*
and not *F* &c, but that one cannot rationally do so.



It is not completely clear how Aristotle understands the second
formulation. At the end of *Metaphysics* IV 3, Aristotle gives
a bad argument that the doxastic version rests on the ontological
version, confusing belief that not *p* with not having the
belief that *p*.



There is a further problem with this second formulation. We need to
distinguish the possibility of believing that *x* is *F*
and not *F* in a particular case from the possibility of
disbelieving the first version of PNC in its full generality.



The third version is that "opposite assertions cannot be true at
the same time" (*Metaph* IV 6 1011b13-20). As it
stands, this version is neutral about the internal structure of the
assertion, but Aristotle assumes that any assertion involves
predicating one thing of another. As with the second formulation, one
might give a psychologistic interpretation, relating to what people
actually do affirm and deny, but the idea that opposite assertions
cannot be true at the same time suggests that this third version is
better interpreted as a variant of the first formulation.



How many versions there are and which version Aristotle intends to
establish are matters of controversy. He could be establishing the
second version on the basis of the first, or the first version on the
basis of the second, or just the first version or just the second
version.


## 2. The Peculiar Status of the Principle of Non-Contradiction



Aristotle says that PNC is one of the common axioms, axioms common to
all the special sciences. It has no specific subject matter, but
applies to everything that is. It is a first principle and also the
firmest principle of all. Like *modus ponens*, as Lewis Carroll
memorably showed, PNC does not function as a premise in any argument.
Unlike *modus ponens*, PNC is not a rule of inference.
Aristotle says that it is a principle which "is necessary for
anyone to have who knows any of the things that are"
(*Metaph* IV 3 1005b15). It is no mere hypothesis.



Aristotle explains that, given its peculiar status as the firmest
first principle, PNC is not susceptible to demonstration. A
demonstration is a deductive argument, the conclusion of which is
deduced from firmer, prior premises. It follows that if PNC could be
deduced from another premise, then that premise would have to be a
firmer and prior principle, with the result that PNC could not have
been the firmest first principle. Aristotle also says that if PNC
could be demonstrated, then everything would be subject to
demonstration, which would lead to an infinite regress. Therefore
demonstration is ruled out, and it is important to be wary of
reconstructions of Aristotle's discussion in terms of ordinary
deductive arguments. Anyone asking for a deductive argument for PNC,
as Aristotle points out, is missing the point, or, rather, is asking
for something that is impossible without using PNC. You cannot engage
in argument unless you rely on PNC. Anyone who claims to reject PNC
"for the sake of argument" is similarly misguided.



Given the impossibility of deducing PNC from anything else, one might
expect Aristotle to explain the peculiar status of PNC by comparing it
with other logical principles that might be rivals for the title of
the firmest first principle, for example his version of the law of
excluded middle--for any *x* and for any *F*, it is
necessary either to assert *F* of *x* or to deny
*F* of *x*. Instead, Aristotle defies others to find a
prior principle (*Metaph* IV 4 1006a10-11). For modern
philosophers, it is still an interesting question whether PNC is prior
to other principles of logic, or, indeed, to the notions of truth,
reference and identity. PNC assumes the notion of identity: It is
impossible for (one and) the *same* thing to belong and not to
belong to (one and) the *same* thing at (one and) the
*same* time &c. PNC is presupposed by Leibniz's law,
which states that if *x* and *y* are discernible (if one
has a property that the other lacks), then *x* is not identical
with *y*.


## 3. The Elenctic Method and Transcendental Arguments



Although PNC is not subject to demonstration, it is subject to
"elenctic refutation" according to Aristotle. The
"elenchus" refers to the Socratic method of argument. When
Socrates uses the elenchus, he gets his opponent to refute himself out
of his own mouth. The opponent makes a proposal that is shown to
conflict with other claims to which he agrees. To be consistent, the
opponent must give up one of these claims, and he usually abandons the
original proposal. This is the method of *reductio ad absurdum*
familiar to ancient Greek geometers and modern formal logicians and
mathematicians.



The idea of using an elenchus is at first sight very puzzling. The
person claiming to reject PNC is not being consistent and apparently
does not want to be. Aristotle is not trying to catch the opponent in
a formal contradiction. The opponent purportedly does not care about
that, and it would be begging the question. Instead, Aristotle's
trick is to draw the opponent into saying something, without making a
complete statement, that shows that he does accept that *x* is
*F* and is not at the same time not *F*, in spite of the
words he previously said. In other words, Aristotle needs to show that
his opponent is committed to at least one thing that is not
contradictory. The Socratic method is turned on its head.



It has long been noted that Aristotle is here assuming that his
opponent takes the very strong position that for any *x* and
for any *F*, it is possible for *F* to belong and not to
belong to *x* at the same time in the same respect &c.
Later he says that he is glad to have stamped out the view according
to which we do not have anything definite in our thinking
(*Metaph* IV 4 1009a3-5).



Aristotle's "elenctic refutation" has been
fruitfully compared to a Kantian transcendental argument.
Transcendental arguments generally run as follows: If certain aspects
of experience or thinking are possible, the world must be a certain
way. Since these aspects of experience or thinking do exist, the world
*is* a certain way. These aspects of our experience or thinking
presuppose that the world is a certain way. That the world *is*
a certain way explains these aspects of our experience or thinking and
not the other way round. On this interpretation, Aristotle would be
arguing that the world conforms to PNC, or that PNC is true, because
it is presupposed by and explains the opponent's ability to say
something significant.



Transcendental arguments are controversial. One controversy surrounds
the precise content of their conclusions and what exactly is
presupposed. Should one conclude that the world must be a certain way
or merely that we have to think that it is a certain way, in order to
have the experience and thoughts at issue? The modern debate finds its
counterpart in debate among scholars about what Aristotle is aiming to
do in his elenctic discussion. There are two main possibilities.
Aristotle may be aiming to show that the ontological version of the
principle of non-contradiction is true, or he may be aiming to show
merely that it cannot be disbelieved.


## 4. Aristotle's Challenge to the Opponent to Signify Some One Thing



Aristotle challenges the opponent to signify some one thing both to
himself and to another, for example, "human being".
Aristotle explains that the word chosen by the opponent can have
several meanings, provided one definition can be assigned to each and
each definition is assigned a different word. Aristotle says that the
word may not have infinitely many meanings. Perhaps this is because
otherwise the opponent can keep on saying that we have not yet reached
the meaning she has in mind, or perhaps Aristotle envisages infinitely
branching meanings, so that we will never arrive at anything
definite.



Aristotle next draws a distinction between "signifying"
and "signifying about". The opponent needs to do both and
not merely signify about. One can signify about something by saying
"pale (thing)", but that does not tell you what you are
signifying, e.g., a human being. "Human being" both
signifies about and signifies a human being. If "human
being" means something definite, for example, "two-footed
animal" (Aristotle's dummy definition), then, Aristotle
concludes, it is impossible that the same thing be a human being and
not be a human being. (For a parallel discussion, see *Posterior
Analytics* I 22.)



On such an account, Aristotle is showing the opponent that if she
wants to reject PNC she must pick out the same object and say that
contradictory predicates apply, but if she does not mean anything
definite by "human being", for example, then she will be
unable to pick out a subject of predication, for example, a human
being, and say that contradictory predicates apply. Saying that an
individual human being is a human being and not a human being, where
the first means "two-footed animal" and the latter means
something different is not to reject PNC. That is why the opponent
must pick a particular word with a definite meaning.



On an alternative account, Aristotle is merely talking about kinds.
"Pale (thing)" does not signify a kind, whereas
"human being" does. Signifying one thing does not involve
signifying an individual who is a member of a kind, but simply one
kind.



The details of the argument are controversial. Is it meant to be
generalizable across the board so that it would work for things that
are not substances, for example, qualities like redness? Aristotle
elsewhere says that the sweet is necessarily so (*Metaph* IV 5
1010b24-6). Must the opponent say something that names a natural
kind? What if the opponent says "animal", naming a genus
rather than a species? What if the opponent says
"goatstag", something that does not exist? Does the
distinction between "signifying" and "signifying
about" necessarily lead to subjects that are individuals (for
example, individual human beings or horses) and their properties?
Here, we run into the issue of Aristotelian essentialism.


## 5. The Role of Aristotelian Essentialism



Aristotelian essentialism is the view that there exist what modern
philosophers would treat as natural kinds, for example, human beings,
horses and acanthus plants. Individuals belonging to these kinds have
essential natures that are definable. These individuals cannot survive
a change in essence, but they can survive a change in their accidental
properties. There is a difference between dying and dyeing one's
hair. When a human being ceases to be a human being, she dies. By
contrast, when her hair is dyed a different color, she survives. In an
earlier work, Aristotle distinguishes essence, for example the
necessary property of being rational for a human being, from accident,
for example, being asleep, and from distinctive properties, those
necessary properties that are explained by rationality but are not
part of the human essence, for example, the capacity to learn a
language (*Topics* I 5). It is a puzzling feature of the
argument of *Metaphysics* IV 4 that distinctive properties are
not explicitly mentioned.



Aristotle says that those who refuse to take up the challenge to
signify one thing "do away with substance and essence. For they
must say that all attributes are accidents, and that there is no such
thing as 'being essentially a human being' or 'an
animal'" (*Metaph* IV 4 1007a20-23).
According to Aristotle, then, those who claim to reject PNC are
rejecting Aristotelian essentialism. Aristotle connects "merely
signifying about" with denoting an accident, and
"signifying" with denoting the bearer of an accident, the
bearer being a substance. If the opponent refuses to
"signify", she is left with accidents.



Next, Aristotle tries to put Humpty Dumpty together again: Accidents
cannot be predicated of accidents, but they must be predicated of
something. For example, if there are accidents such as the musical or
the pale, these cannot be predicated of one another, so they need a
subject with an essential nature to be predicated of, for example,
Socrates, a human being. While Socrates is pale and musical, the pale
is not musical, nor the musical pale, unless all this means is that
Socrates is pale and musical. Nor does a collection of
accidents--the pale, the musical and so on--add up to one
subject like Socrates. Nor does it tell us what Socrates is, a human
being. Accidents need subjects and subjects that are substances and
have an essential nature, and, if that is the case, essentialism is
reinstated and PNC applies after all.



According to Aristotle, if the opponent refuses to speak, he is no
better off than a vegetable. While this may seem to be merely
ill-tempered abuse, it has a serious side. If the opponent rejects PNC
*and essentialism*, then he cannot explain why he is not a
vegetable. Aristotelian essentialism, if correct, applies to oneself
as well as to other things in the world. At first sight, it is not
clear why the PNC-opponent is left with a picture of anything. But
perhaps that is Aristotle's point. Aristotle gives the
PNC-opponent a world in which accidents can be linked up in any way he
likes. Anything goes in such a world, or nothing goes, depending on
his taste. Alternatively, as Aristotle puts it, "all things will
be one" (*Metaph* IV 4 1007a19).



Aristotle is arguing that the rejection of PNC leads to the rejection
of Aristotelian essentialism, and the acceptance of Aristotelian
essentialism leads to the acceptance of PNC. A controversial question
is whether Aristotle is also arguing that the acceptance of PNC
necessarily goes hand in hand with the acceptance of Aristotelian
essentialism. If Aristotle is claiming that to signify something is to
signify a bearer that has an essential nature, this would lead to the
acceptance of both PNC and some form of essentialism. The argument
itself does not rule out Platonic forms as the bearers, or momentary
objects, or numbers. Nevertheless, the argument could be supplemented
by Aristotle's arguments elsewhere against Platonic forms, and
his distinctions between different types of change in the world.


## 6. The Principle of Non-Contradiction and Action



Aristotle notes that even if the opponent fails to speak, she must
still act, and if she acts in a certain way, that shows that she
thinks that things in the world are one way rather than another, and
that some courses of action are better than others. That is why people
do not walk into wells or over precipices, and it shows that people
think it better not to walk into a well or over a precipice than to do
so. Their actions show that they have beliefs that conform to PNC.



In response to Aristotle, one might wonder whether it is possible to
act merely *as if* one has certain beliefs, without having
them. In Hellenistic philosophy, the question arises whether the
skeptic can live his skepticism, and in modern philosophy the parallel
question arises whether moral anti-realists can act on their theory.
If the answer is yes, then all Aristotle will have shown is that we do
act *as if* we are committed to PNC, an answer that falls short
of Aristotle's aim in *Metaphysics* IV 4, whether this is
interpreted as to show that PNC is true or to show that PNC is
indubitable.



On this view, we are left with a skeptical Humean or pragmatic
account. Aristotle's account would be parallel to Hume's
skeptical solution to the problem of induction: We just do proceed as
if induction is correct, even though we lack any justification for so
doing. Or compare Wittgenstein: "..the end is not an unfounded
presupposition--It is an unfounded way of acting." (*On
Certainty*, Section 110).



Is such a view coherent? Here it is appropriate to ask the skeptic
what kind of justification she is seeking. The question returns us to
the problem of PNC being a first principle for which there is no
ordinary proof. Is it possible to act *as if* one has certain
beliefs even though one does not? Here the skeptic owes the
non-skeptic some account of "as-if belief" that differs
from actual belief. If this is to act following "how things
appear to one", Aristotle is justified in asking whether they
can even appear to be *F* and not *F* to the same human
being at the same time. "As-if belief" may commit one to
PNC just as much as actual belief does.



Action is the place where our beliefs collide with the world. If the
skeptic is skeptical about beliefs, it is not clear that she can
accept that there are *actions*, as opposed to reflexes and
other involuntary movements. Such a PNC-opponent would become a robot,
not just a vegetable.


## 7. The Principle of Non-Contradiction and Proximity to the Truth or the Truth-Like



At the end of chapter 4, Aristotle says that however much things
"are so and not so", there "is a more and a less in
the nature of things", for someone who thinks that four things
are five is less wrong than someone who thinks that they are a
thousand, and so there must be some truth to which the more true is
nearer. He adds that even if there is not some truth to which the more
true is nearer, "still there is already something firmer and
liker the truth, and we shall have got rid of the unqualified doctrine
which would prevent us from having anything definite in our
thought" (*Metaph* IV 4 1009a2-5). He first points
out that we can make comparative judgments, even if we do not make
absolute judgments, and that comparative judgments presuppose some
absolute standard. But even if there is no absolute standard,
something truth-like is presupposed and so we will not be prevented
from having anything definite in our thought.



One might wonder how much of a concession these points are to
Aristotle's opponent. A modern scientific realist would insist
that our views merely approximate the truth. But would it be enough
that there be something "liker the truth"? It would be
interesting to work out the implications of these ideas for
Aristotle's essentialism. Would Aristotle be sympathetic to
"fuzzy essentialism" or "fuzzy realism"? How
much vagueness can Aristotelian realism allow? These are issues about
realism that later Hellenistic philosophers and modern philosophers
consider in more detail.


## 8. The Argument from Conflicting Appearances



In chapter 5, Aristotle distinguishes two types of opponent, those who
claim to reject PNC for the sake of argument, and those Pre-Socratics
who are genuinely perplexed. He now addresses the second type.
Anaxagoras and Democritus are led to say that contradictions are true
at the same time, because they are confused by change. They see
contraries coming into existence out of the same thing, and infer that
the same thing must have had contrary properties. Aristotle introduces
his distinction between the potential and the actual to dispel their
confusion. An object can be potentially *F* and potentially not
*F*, but it cannot be actually *F* and actually not
*F* at the same time.



Other philosophers are led by the argument from conflicting
appearances to accept conclusions that violate PNC or lead to general
scepticism. Aristotle presents the argument as follows:


1. There are three sorts of cases of conflicting appearances:
	1. 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).
	2. Things appear different to members of different species (e.g., to
	other animals and to us) (*Metaph* IV 5 1009b7-8).
	3. Things do not always appear the same even to the senses of the
	same individual (*Metaph* IV 5 1009b8-9).
2. It is not clear which appearances are true and which false
(*Metaph* IV 5 1009b10).
3. Conclusions:
	1. Nothing is true (Democritus in dogmatic mood, *Metaph* IV 5
	1009b11-12).
	2. (If something is true) it is not clear to us (Democritus in
	skeptical mood, *Metaph* IV 5 1009b12).
	3. Everything is just as true as everything else. (This is mentioned
	as an explanation of premiss 2 at *Metaph* IV 5
	1009b10-11. It is Protagoras's view, as described at the
	beginning of the chapter.)



In response to this argument, Aristotle concedes most of the first
premise. He agrees that things do appear different to different
members of different species, to different members of the same
species, and even to the same individual, although he denies that the
same thing can appear differently to the same sense of the same
individual at the same time (*Metaph* IV 5 1010b18).



Unlike modern philosophers, Aristotle does not attack the inference
from premises 1 and 2 to the conclusions in 3. Instead, he attacks
premise 2. If the attack is successful, he will have knocked down all
three conclusions at the same time. Aristotle's attack on
premise 2 does not rest on accepting a majority verdict. Aristotle
appears to accept his opponents' argument that, in cases like
1a, where things appear different to different members of the same
species, majority decision is not an appropriate criterion for truth,
because if the majority were ill or mad, the minority would be thought
to be in that condition. He argues instead that people are not really
confused as to whether magnitudes and colors are such as appear to
those at a distance or those nearby, or whether what appears to the
weak or strong is heavier or whether what appears to one asleep or
awake is true. Once awake, a person in Libya is not confused about
whether his dreams of Athens or his waking experiences are true; he
does not start out for the odeon. Therefore, according to Aristotle,
people do not really find it unclear which appearances to take as
true.



Aristotle extends his discussion to opinions, arguing that not all
opinions are equally authoritative. Aristotle points out that when it
comes to questions about our future health, the opinion of the
physician is not on a par with a lay person. Nor are our senses
equally authoritative on the same subject matter. Each sense is
authoritative about its own special objects. For example, sight, and
not taste, is the authority on color, but taste, and not sight, is the
authority on flavor (*Metaph* IV 5 1010b11-17). According
to Aristotle, then, it is far from unclear which appearances, or whose
opinions, are to be trusted in cases of conflict. Those who profess to
deny this, show, by their own actions, presumably by trusting only
their waking appearances when they are awake, and by consulting a
physician when they are ill and so on, that they believe quite the
reverse. Therefore, according to Aristotle, the second premise of the
argument from conflicting appearances is false, and so the argument
fails.



Aristotle has not had the last word. The skeptics of Hellenistic times
made much of arguments from conflicting appearances, and modern
philosophers continue to discuss their efficacy, especially in the
field of ethics.


## 9. Protagoras, Heraclitus, and Plato's *Theaetetus*



At the beginning of *Metaphysics* IV 5, Aristotle says that PNC
stands and falls with the doctrine of Protagoras, that each individual
human being is the measure of all things. In his *Theaetetus*
151-183, Plato argues that Theaetetus, who holds that knowledge
is nothing but perception, is committed to Protagoras's view via
an 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 goes on to argue that Protagoras is committed to the view that
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 for ordinary language. The extended
argument also contains a mini-argument, a
"self-refutation", where Plato draws the
"exquisite" conclusion that 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!



The details of Plato's argument, the fairness and the success of
his strategy against Theaetetus and Protagoras are matters of some
controversy. A recent point of contention is whether Plato's
Protagoras is committed to a view of "relative truth"
instead of flux, and how far Aristotle's Protagoras is similar.
What is less controversial is that elements of Plato's
discussion re-surface in Aristotle's *Metaphysics* IV.
Aristotle's opponents are said to believe that knowledge is
perception because they think that what they perceive is all that
exists, and they are also said to be impressed by arguments from
conflicting appearances. The thesis that knowledge is perception and
an argument from conflicting appearances lead to Protagoras's
view in Plato's *Theaetetus*. In *Metaphysics* IV
5, Aristotle agrees with Plato that Protagoras's view and the
suggestion that everything is "so and not so" (an
expression echoed by Aristotle earlier at *Metaph* IV 4
1006a30-31) go hand in hand. The Protagoras of Plato's
*Theaetetus* and the PNC-opponent in *Metaphysics* IV 4
are said to be committed to the thesis that nothing is anything in
itself, understood by Aristotle to mean that everything is
accidental.



While Aristotle does not saddle Heraclitus himself with the rejection
of PNC, he notes that 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
finger. Cratylus was mistaken, according to Aristotle, because there
is no radical flux. When things change, something persists, and even
if the quantity of a thing is not constant, we know each thing by what
sort of thing it is. Presumably, even if its water is constantly
flowing, we can still identify a river. Aristotle points out that if
there were radical flux, this would be tantamount to everything being
at rest, so the idea of radical flux is contradictory.



One way to think of Aristotle's strategy in *Metaphysics*
IV is as the reverse of Plato's. If the PNC-opponent says
something significant, for example, the beginning of
Protagoras's doctrine of the measure, "human being",
then she is committed to denying the thesis that nothing is anything
in itself, and to accepting a non-Protagorean view. On this
interpretation of Aristotle's strategy, at least two puzzling
features in Aristotle's discussion are resolved, why he appears
to be addressing someone who claims the contrary, and not the
contradictory of PNC, and how Aristotle can argue from the second
version of the principle of non-contradiction to the first. Since
Plato's Protagorean is forced to reject PNC across the board,
Aristotle's Protagorean PNC-skeptic would be an opponent who
claims the contrary, not merely the contradictory, of PNC. In
addition, a Protagorean PNC-skeptic, who accepts that "as things
appear to one, so they are" would have to be impressed by an
argument that shows that things cannot appear to him in a way that
would violate PNC. On his own view, showing that PNC is indubitable
would also show that it is true.



At this point, one might wonder if Protagoras can turn the tables on
Aristotle. Certainly, to a Protagorean, showing that PNC is
indubitable would show that it is true, but it also assumes his own
view, that how things appear is how they are! Does Aristotle or
Protagoras win? The same can be asked of the "exquisite"
argument of Plato's *Theaetetus*. Is Protagoras's
view refuted or confirmed? The Stoic Chrysippus apparently wrote a
whole book, now lost, on such table-turning arguments.



A further question surrounds the formulation of Protagoras's own
view. In *Metaphysics* IV 6, Aristotle explains how a
Protagorean PNC-skeptic ought to present his view so as to avoid
violating PNC: "those who seek to be compelled by argument, and
at the same time demand to be called to account for their views, must
guard themselves by saying that the truth is not that what appears
exists, but that what appears exists for him to whom it appears, and
when, and to the senses to which, and under the conditions under which
it appears." Otherwise, they will find themselves contradicting
themselves (*Metaph* IV 6 1011a20-25). However, Aristotle
still finds fault with this view because it makes everything relative
to perception, including the perceiver. Not everything can be relative
to perception, according to Aristotle. As he explains at the end of
the previous chapter, there is something beyond perception that causes
the perception and is prior. Aristotle uses his discussion of PNC and
Protagoras to stake out a realist position.


## 10. Aristotle's Conclusion and Implications for Ethics and Politics



At the end of chapter 6, Aristotle concludes, "Let this, then
suffice to show (1) that the firmest belief is that opposite
assertions are not true at the same time, (2) what happens to those
who speak in this way and (3) why people do speak in this way"
(*Metaph* IV 6 1011b13-15).



On the first point, as we saw, it is controversial whether
Aristotle's conclusion that the firmest belief is a belief in
PNC carries with it the presupposition that PNC is true, a
presupposition that is needed for his own project of first philosophy.
On the third point, Aristotle discusses views about perception and
change that lead people to say that they reject PNC. 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 be giving up
intelligible discourse and action. It is controversial how much of an
essentialist or indeed realist view one must accept if one accepts
PNC, but it is clear that PNC is essential for the project of an
Aristotelian science.



Without PNC, Aristotle notes, beginners in philosophy who are
interested in the truth would be off on a wild goose chase
(*Metaph* IV 5 1009b36-8). This applies to both
metaphysics and science, and to ethics and politics. Without PNC, and
without the truth, we would be living in a world of mere sophistry
and, although Aristotle fails to spell this out, power.
Aristotle's discussion of PNC, then, has broader ethical and
political implications.


## 11. Dialetheism, Paraconsistency, and Aristotle



Aristotle's account of the PNC has been challenged anew by
modern-day dialetheists, who hold that there are some true
contradictions, and that Aristotle's discussion fails to show
otherwise. One approach is to argue that some of Aristotle's
discussion of PNC begs the question in favor of PNC. (Of course,
Aristotle would agree that any argument in favor of PNC must beg the
question.) Another approach is for dialetheists to refuse to let the
Aristotelian reformulate their supposed counterexamples to PNC by
adding qualifications so that these supposed counterexamples do not
violate PNC after all. The Aristotelian can counter that without those
qualifications the dialetheist has not said anything meaningful at
all. Finally, the dialetheist may ask how Aristotle would deal with
the liar's paradoxical sentence "I am lying". Why is
this not both true and false at the same time as the dialetheist
contends?



Besides dialetheists, some modern logicians, who need not be
dialetheists, think that logic can be paraconsistent, i.e., that,
contrary to classical logic, one contradiction need not lead to an
explosion where anything goes. While Aristotle is obviously not a
dialetheist, it is not clear where he stands on the issue of
paraconsistency in *Metaph* IV. Although Aristotle does claim
that if his opponent rejects PNC across the board, she is committed to
a world in which anything goes, he never argues that if (*per
impossibile*) his opponent is committed to one contradiction, she
is committed to anything, and he even considers that the
opponent's view might apply to some statements but not to others
(*Metaph* IV 4 1008a10-12). However one understands these
passages, in the *Prior Analytics*, Aristotle does commit
himself to the view that syllogistic is paraconsistent (*APr.*
II 15 64a15).


## 12. *Posterior Analytics* I 11



This is an intriguing and relatively neglected text. At first sight it
looks as if Aristotle is presenting a valid argument that includes
contradictions as some of the premises, which would be surprising
given his account of PNC in *Metaphysics* IV. However, the text
is even more obscure than usual.



There are two basic interpretations. According to one interpretation,
Aristotle does indeed include contradictions, but these are idle and
play no real logical role in the argument he presents. According to
the other interpretation, following Aquinas's commentary
(lecture 20), Aristotle is not claiming that Callias both is and is
not Callias or that he is and is not a human being, for example, but
that the term "animal"covers Callias and those who are not
Callias and not human beings, as Aristotle himself explains when he
says that the major term ("animal") is more extensive than
the middle term ("human"). (In the syllogism All
*A*s are *B*s; All *B*s are *C*s;
therefore All *A*s are *C*s, "A" is the
major term and "*B*" is the middle term.)



There are difficulties with both interpretations. Aristotle starts by
saying that no demonstration assumes PNC unless it concludes that
*x* is *F* and not not-*F*. According to the
first interpretation, Aristotle's explanation about the major
and middle term is unnecessary. However, according to the second
interpretation, the relationship between Aristotle's main point
and his comments about PNC is obscure. The text invites further
elucidation.