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 cannot 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 (suppose) the same thing to be and not to be (Metaph IV 3 1005b24 cf.1005b29–30)
  
• i.e. it is impossible to hold the same thing to be F and not to be F.
• this may not seem plausible
• People have inconsistent beliefs
  
• Must one believe the consequences of one's beliefs?
  
• Can one knowingly believe an outright contradiction?
• what a man says he does not necessarily believe (Metaph IV 3 1005b23–26)
• 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 not-F.
  
• 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.
  
• ontological (i.e. about what exists)
•  This one is the one referred to as PNC and is usually taken to be 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  (Metaph IV 3 1005b19–20).
  
• 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 belongs must belong actually, and not merely potentially, to its bearer.
• Which version Aristotle intends to establish is a matter of controversy.
• PNC cannot be proven, but there are some very strong indications 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 special sciences,
  
• we would not be able to distinguish between what something is, for example a human being or a rabbit, and what it is like, for 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 that 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 the same time to the same thing and in the same respect.
• Could an opponent claim that PNC works in most areas, but not all?
  
• 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 does 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: no science can prove PNC: it is indemonstrable
  
• axiom? it is not demonstrable and so is at the bedrock of any science
• if it were demonstrable, the premises of that demonstration would be more basic
• also, it cannot be demonstrated because every demonstration 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 wrong and then see what happens, because "seeing what happens" involves assuming 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 thinks 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 stops scholars
  
• 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 case.
• 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 what it means to deny PNC.
• Aristotle assumes his opponent takes the very strong position that for any x and for any F, it is possible that F belongs to and does not belong to x at the same time in the same respect.
• Aristotle wants 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
• Aristotle seems to 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.
• 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
• 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" for instance.
  
• if the communication works (i.e. if the person communicating and the person being communicated with both understand the same thing), then Aristotle thinks the opponent is committed to PNC in at least one instance.
• Aristotle is showing the opponent that if she wants to reject 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 subject of predication, for example, a human being, and say that contradictory predicates apply.
  
• opponents cannot have any one thing to which accidents are attributed
• Aristotle's "substance" and "essence" serves to provide the thing to which accidents adhere
• those who deny PNC cannot have such a thing to hold accidents 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 "musical"
• you have to say the pale "thing" is also a musical "thing": and that requires a "thing" that just is that thing (and not not that thing)
• it also makes no sense to say that Jacques is just a bundle of accidents: without some unifying factor to make them all "Jacques," they are just a heap of accidents without unity: that unifying factor 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 that 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 something 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 not.
• 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 anything that plays the role of making it the case that we can reliably communicate AND also involves rejecting any metaphysical analysis of things in the world that makes it the case that things are just what they are and not simultaneously not what they are.
• At the end of Metaphysics IV.4, Aristotle says:
• However much all things may be so and not so, still there is a more and a less in the nature of 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, obviously one 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 which our views are mere approximations of truth, and even if they happen to be the truth, we cannot be certain of that? could there be a "fuzzy" or "vague" essentialism?
• Aristotle's confused opponents:
• In Metaphysics IV.5, Aristotle talks about opponents who see a thing changing and conclude from their observations of chance that the same thing must have had contrary properties: contrary properties come into existence out of the same thing.
  
• Here, Aristotle introduces the distinction between potential and actual.
• 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 Phil.):
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) a true thing 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.)
• Aristotle agrees with 1
• Aristotle disagrees with 2: he does not think that people are really confused about which appearances are right in cases of conflicting appearances:
• they trust experts
• they trust their senses for things each sense is authoritative about
• many moderns take conflicting appearances seriously, especially in ethics.
• Another set of opponents are "Protagoras, Heraclitus, and Theaetetus," who are characters in Plato's dialogue Theaetetus
• In Metaphysics IV 5, Aristotle mentions Protagoras' doctrine that each individual human being is the measure of all things.
  
• PLATONIC detour
  
• At 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 argues 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!
• end of Platonic detour
  
• elements of Plato's discussion re-surface in Aristotle's Metaphysics IV.
  
• 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 finger.
  

• 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 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, and—one might add—they will be living in a world of mere sophistry and power.
  
• On the third point, Aristotle discusses views about perception 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 clear that 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 1009b36–8).