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I am in the process of reading through Priest’s paper “Aristotle on the Law of Non-contradiction” (or the original and far better title “To Be and not to Be- that is the Answer…”). I am very convinced by Priest in that Western Philosophy has taken the LNC for granted so to speak. Aristotle’s original arguments for the LNC are rather difficult to work through (like any ancient text, especially for me). On the other hand, I think Aristotle’s arguments succeed more than Priest thinks they do. Priest goes along with the traditional view (since he freely admits to not being a scholar of Aristotle) that Aristotle presents 7 refutation arguments for the LNC. My best guess for what a “refutation” argument is, is an argument that tries to show that asserting a the proposition in question produces an absurdity that cannot be accepted. Priest describes it as something partly like an argument by reductio ad absurdum, but not quite because in order to have a really reductio argument one has to assume the LNC. Aristotle wants to try and avoid this type of argument because such argument would be question begging for the LNC.
(I have noticed the difficulties with arguing for or against the LNC. Patrick Grim (2004) discusses this issue, but I think Lewis’s letter in the same collections of papers is a little more direct. Lewis points out that in order to debate the LNC you have to assume it to be the case. At the very least, the LNC has to hold in the debate you are engaged in, otherwise no debate is occurring. I am worried this might be too quick of a response to people who reject the LNC like Priest. There appear to be instances where we intuitively accept a violation of the LNC (while trying to avoid calling it “a violation” like beliefs for example).)
I think I agree with Priest when discussing refutations 6 and 7. Refutation 1 is taking a long time for me to go through since it is a very long argument. I have worked through refutation 2-5 and I have arguments against Priest’s. Since I don’t want this post to be too long, I will go with my favorite one: #3.
The third refutation is “another {consequence that every contradiction being true} is that it is not necessary either to assert or deny. For if it is true that he is a man and not a man, plainly he will be neither a man nor not a man…”.[1] Priest agrees with Kirwan’s reading of this argument “…the violators of the LNC are committed to a denial of the LEM, too.”[2] I think a small problem with Priest’s wording is rather apparent since it doesn’t seem like there is such a thing as “violators” of the LNC for Aristotle. Aristotle’s resistance to contradictions goes further than many are willing to go. He claims that there are not even contradictory beliefs earlier in book G, “For it is impossible for anyone to believe that the same thing is and is not…”.[3] Even though this might seem rather counter-intuitive (Priest certainly thinks so[4]) there are reasons for accepting what Aristotle says but I will not say anything further in the topic.
If we try to write the LNC logically, one possible way to do so is ~(p Ù ~p) which by DeMorgan’s Law can be replaced with the logically equivalent ~p Ú p (the LEM). At first glance, it looks like the LNC and LEM are equivalent laws. The denial of one entails the denial of the other. Priest does not agree with this and to illustrate this he considers a contradiction p Ù ~p and if we apply double-negation to both variables then we get ~p Ù ~~p which by DeMorgan’s we get ~(p Ú ~p). For the dialetheist “…~(a Ú ~a) does not rule out ~a Ú a”[5] so Aristotle’s argument does not work against him.
Like what Priest will do in other criticisms of Aristotle, Priest is begging the question with this argument. Priest admits that he does not read negation as Boolean negation.[6] Since Aristotle is most likely thinking of negation in this way[7] then Priest needs to provide an argument for why he thinks Aristotle is wrong to think of negation in that way. If Aristotle is correct about negation and Priest is wrong then Aristotle’s argument works against the dialetheist. Within this passage Priest does not provide one.
Priest does argue that Aristotle’s argument provides good reason for rejecting the trivialist (someone who thinks everything is true). He reads Aristotle as giving a type of “what makes sense for communication” reason for accepting LNC. “If someone believed that everything was true then there would be no point in their asserting anything.”[8] When we engage in some type of discussion in which we assert something the purpose of that assertion is to try and convince those around us to believe what we assert. If we are a trivialist then there is no point to asserting anything because we already believe that the ones around us believe everything so to assert something does nothing. While Priest doesn’t think this says anything about the trivialist being right or wrong, only the absurdities that come with having such a view. I agree with Priest, but I think we should take his last point a little more seriously. The absurdity should count as evidence for the falsity of the trivialist’s view.
[1] Metaphysics: Book Gamma 1008a 2-5
[2] Priest 2006 p. 35
[3] Metaphysics: Book Gamma 1005b 24-25
[4] Priest 2006 p. 9
[5] Priest 2006 p. 35
[6] See 2006 Priest p. 88-102
[7] Since Aristotle recognizes the a rejection of the LNC is also a rejection of the LEM.
[8] Priest 2006 p. 35 original italics.
Can I believe p and not p?
I have to admit that within my presentation I am taking a very counter-intuitive position. I claim that people don’t and can’t have contradictory beliefs. In order to argue this I must look to another debate. The Law of Non-contradiction[1] (LNC) has been rejected by many notable philosophers such as Graham Priest and J.C. Beall. The most difficult problem (as I see it) that this debate produces concerns our common-sense notion of belief. When discussing the issue with many of my colleagues, all of whom appear to claim:
(1) that the Law of Non-Contradiction holds
also maintain that
(2) people can and do have contradictory beliefs.
Many of my colleagues maintain that the acceptance of (2) does not imply a rejection of (1). Roy Sorensen is the only person I have come across that has produced a text that argues for an acceptance of (1) and (2). The rest of the texts that I have come across either by accepting (1) they have to in some way reject (2) or by rejecting (1) are more than happy to accept (2).
I have found myself agreeing with the individuals of the former type, I accept (1) and reject (2). The strong resistance that Priest has reported concerning his rejection of (1), I have found myself in a similar situation among my colleagues with my rejection of (2). My colleagues have argued along the lines of common-sense that (2) is a true claim about the world. They deal with students and friends that appear to believe contradictions. At the same time they also want to maintain that their students and friends are always wrong when they believe a contradiction. Within the first part of my presentation I show that those who try to maintain such a position are claiming something false. It cannot be the case that (1) and (2), one of them has to go (this can be done with a simple proof). Within the second part of my presentation I show that if (1) is the case, a behaviorist account of beliefs is false, and that people are not infallible with respect to their own beliefs then we have good reasons for rejecting (2).
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[1] or Law of Contradiction or Principle of Non-contradiction
Priest does something else that I agree with, he denies “explosion.” What is explosion? Explosion is the notion that a contradiction implies anything or everything (If (A and not A) then B). There are two methods in classical logic (that I am aware of) for getting explosion, truth table and direct proof.
Truth Table
(a Ù Øa) É b
F [T] T
F [T] F
F [T] T
F [T] F
Direct Proof
1. a Ù Øa (Assumption)
2. a (1, Simp)
3. a Ú b (2, add)
4. Øa (1, Simp)
5. b (3, 4, DS)
The way to avoid the truth table explosion is by introducing a new truth value which Priest has 4 (got this wrong, see comments) if I remember correctly. In order to avoid the Direct proof then you need to deny one of the rules, Priest denies the rule “Addition.” (got this wrong also, see comments)
(I am going to add some stuff onto this post later, nice thing about a blog)
I realized that I want to throw out a few things so that I don’t give the impression that I am totally opposed to Priest’s position. I have been hesitant to write this down because they are not very well thought out or I should say far less thought out than the previous posts. That being said this post will be “more bad” than my previous ones.
I think that the questions Priest has been raising about the LNC are very good questions. I think the way that Quine (Philosophy of Logic), Lewis (Letters to Priest and Beall), Stalnaker (Impossible Worlds), and others have gone about defending the LNC are very inadequate. I think it stems from the notion that failures of the LNC are trivial concerns, somehow it is easy to see that contradictions are wrong. I disagree with this, I think contradictions are much more serious than that and often difficult to catch. Contradictions, I believe, are always false, but exactly why that is the case is what I hope to work through. I think the important thing that Priest has done is drawn attention to the fact that we don’t have a good grasp on why contradictions are always false. What I disagree with is his conclusion that because we can’t quite figure out why contradictions are always false then we should conclude that contradictions are sometimes true.
I realize that is a gross oversimplification of the debate, but that is not an argument of any kind, simply an intuition about what is going on in the debate over the LNC. (I did find Shapiro’s (2006) defense of the LNC rather interesting though)
