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What happens when we try to apply (C.b*) to Sorensen’s proof? Well, it would seem that the premise is a contradiction:
1. Ba[à($p)($x)(~àp & Bxp)] (Sorensen’s Premise)
2. à($p)($x)(~àp & Bxp) Î m (1, (C.b*))
3. ($p)($x)(~àp & Bxp) Î m* (2, since ($p)($x)(~àp & Bxp) is possible in m then there must be
at least one model (world) such that m has access to that model
and ($p)($x)(~àp & Bxp) is a member of that model)
4. ($x)(~àq & Bxq) Î m* (3, Existential Instantiation)
5. ~àq & Bbq Î m* (4, Existential Instantiation)
6. ÿ~q Î m* (5, Simplification and Rule of ÿ-à Interchange)
7. Bbq Î m* (5, Simplification)
8. q Î m** (7, (C.b*))
9. ~q Î m** (6, since ~q is necessary in m* then all models (worlds) are
such that if m* has access to that model then ~q is a
member of that model)
Now one might not be very convinced with the proof that I have just given. They could say that (C.b*) does not accurately apply the LNC in a belief context or the fact that within my proof that I use doxastic possible worlds and metaphysical possible worlds the same way. That (C.b*) assumes that Sorensen is wrong to begin with so my proof just begs the question.
Although I treat doxastic possible worlds and metaphysically possible worlds the same there are going to be some instances where they match up (when the beliefs are true for example). This is certainly true for cases involving the LNC. If the LNC holds then it holds everywhere, doxastic possible worlds and metaphysical possible worlds. For those who do not accept the LNC then the law will not hold equally in both types of worlds. Sorensen wants to have it both ways, he wants to maintain the LNC and maintain that people can and do have contradictory beliefs. To say that the LNC only fails with respect to belief is some how making beliefs unique. To treat beliefs in this manner appears very ad hoc and we shouldn’t let our intuitions about belief be justification for this ad hoc move.
The other problem with my proof has to do with whether employing (C.b*) is begging the question or not. Obviously, there is an attempt within the sciences and the philosophy of mind to try and explain what beliefs are. If we assume that people can and do have contradictory beliefs then, in principle, there is no way to explain what beliefs are that includes the LNC.[1] We will never be able to employ something like (C.b*) within our explanation of beliefs. If that is the case then no consistent explanation of beliefs can be given. If no consistent explanation can be given then the LNC obviously fails with respect to belief. If we are looking for consistent explanations of other parts of the world then we should look for it within our explanation of beliefs as well.
It is difficult to understand the claim “The LNC holds (there are no contradictions) and people can and do hold contradictory beliefs.” What is claimed in such a proposition is that there are no contradictions and that there are contradictory beliefs. There is nothing in the world that is a contradiction and yet there are contradictory beliefs. Such a claim is itself a contradiction.
[1] Girle 2003 also recognizes this, “One major problem with belief logics based on classical logic is that they cannot cope with inconsistent belief, other than by simply rejecting it” (p. 137).
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.
