A transcendental argument for the law of non-contradiction? II

  1. Introduction

In the last post, I explained Jay Dyer’s view, that Aristotle was making a Transcendental Argument for the Law of non Contradiction in Metaphysics IV. I gave a few examples of academics who think something similar, and pointed out the tension inherent in this view, which is that it flies in the face of Aristotle’s prohibition against the possibility of directly demonstrating the LNC. I also introduced Dyer’s specific claims, which came in two varieties. The first was that Aristotle’s TA was a reductio ad absurdum argument, and the second was that it was the claim that the LNC is presupposed by making any argument at all (by making any claim whatsoever). I was at pains to insist that these are not the same thing, and gave an example of how a sentence could be proved to be true via a reductio argument, but which seemed not to be presupposed by every sentence. In what follows, I treat Dyer’s claims as distinct (even if he does not) and show problems for each of them. The problems are both philosophical and exegetical.

2. Reductio

I’ll begin with a philosophical objection to this reductio interpretation, then I’ll give an exegetical objection.

Let’s assume that the sophist (Aristotle’s opponent) has made an argument a bit like this:

  1. Everything is in flux.
  2. Therefore, “in Socrates is both age and youth together”.
  3. Therefore, Socrates is both old and young
  4. Therefore, Socrates is both old and not old.

This is an imaginary ‘Heraclitan’ challenge to the LNC (Heraclitus’ followers are some of Aristotle’s imagined opponents in Metaphysics IV). The conclusion is a contradiction. Let’s suppose that Aristotle is not interested in refuting the specifically Heraclitan premises, or the seemingly dodgy inferences used. Rather, he is interested in producing the most general form of reply to any such argument that had a conclusion like the one above.

Note that the Heraclitan here believes the conclusion of the argument; that is, he believes that Socrates is both old and not old.

Dyer’s proposal is that Aristotle is responding to such an opponent by utilising a reductio argument. But such a process doesn’t really seem to work. You can’t use a reductio to demonstrate the LNC. Let’s go back to our example from the previous post to illustrate why.

We showed that p → (q → p) is true using a reductio. We assumed its negation, ~(p → (q → p)), and showed that a logical consequence of this assumption was a contradiction, namely p & ~p. From this, we inferred that the assumption must be false. But what licensed this inference? Let’s spell it out. The inference is basically this:

  1. ~(p → (q → p)) → (p & ~p)
  2. ~(p & ~p)
  3. Therefore, p → (q → p) (via modus tollens)

We assumed our proposition was false, ~(p → (q → p)), and showed that this entailed a contradiction, p & ~p. That’s premise 1. To derive that the assumption is false (the conclusion), we needed premise 2. Yet, the only thing that motivates thinking that premise 2 is true is appealing to the LNC itself. The only reason for thinking that the consequent was false was the fact that it is a contradiction, and all contradictions are false. Thus, a reductio requires that, at some point in the derivation, we invoke the LNC explicitly. Reductio arguments require the LNC.

Thus, if our opponent denies the LNC, by making a Heraclitan argument like the one we began this section with, then using a reductio would only work if the LNC was true in the first place. And that makes its use against someone who claims that there is a true contradiction blatantly question begging. It assumes the very thing in question.

We can put the same point pragmatically. A reductio is not going to provide any motivating reason for someone who believes there is a true contradiction to change their mind. This is because the method of reductio works on the assumption that deriving a contradiction from their view is a bad thing for them to believe. Yet someone who believes there is a true contradiction will dispute whether this is a ‘bad thing’. ‘What’s wrong with contradictions?’ they may ask. The answer cannot be: ‘Because believing contradictions is bad.’ With such an opponent, we cannot simply appeal to the fact that their view leads to a contradiction as a way of motivating them to come to believe that their view is wrong. We need to do more than that. Yet that is all that a reductio has to offer.

Thus, any such appeal to a reductio, thought of as showing that the contrary leads to contradictory consequences, is straightforwardly question-begging if your opponent believes in a contradiction.

In addition to this being a blatantly question begging way to argue against such an opponent, we also face serious exegetical problems. The claim that Aristotle was making a reductio argument runs into the prohibition against direct demonstrations, which we saw from the previous post was also the main problem for all TA interpretations. However, on Dyer’s version of things, it is particularly difficult. It’s not just that reductios involve modus tollens, but that they are explicitly included by Aristotle in his theory of the categorical syllogism; reductios are direct demonstrations in Aristotelian logic!

For example, in the Prior Analytics  (A 29), Aristotle makes the following claim:

Deductions which lead into an impossibility are also in the same condition as probative ones: for they too come about by means of what each term follows or is followed by, and there is the same inquiry in both cases. For whatever is proved probatively can also be deduced through an impossibility by means of the same terms, and whatever is proved through an impossibility can also be deduced probatively,

He is saying that ‘deductions which lead into an impossibility’ (i.e. reductio ad absurdum arguments) are of the same type (“the same condition as”) standard inferences (‘probative inferences’); “there is the same inquiry in both cases”. He is saying that there is nothing special about reductio arguments – they are of the same type as any other inference.

So, Dyer’s view is that Aristotle is giving a reductio ad absurdum argument here. Yet this interpretation runs into the problem of reconciling it with two other things Aristotle says:

i) In Metaphysics Aristotle warns that LNC cannot be given a ‘standard demonstration’, and must be shown indirectly.

ii) In the Prior Analytics, Aristotle recognises reductio ad absurdum arguments as standard types of demonstrations.

It is possible to interpret Aristotle as making a reductio argument, but a) he doesn’t actually say he was doing that, b) it would be question begging if he did, and c) he explicitly says LNC cannot be given a direct demonstration and that reductios are direct demonstrations. For these reasons then, I think this is not a good way to read Aristotle here. Whatever else he was doing, it was not giving a reductio (and even if it was, it would be question begging).

3. LNC is presupposed by everything

Dyer’s second suggestion is that “when you engage in that argument, you are assuming the existence of logic and universals, to try to refute logic and universals”. The idea here seems to be that when our Heraclitan opponent made his argument, he assumed “the existence of logic”, and part of what that means is that he assumed the truth of the LNC. If you make an argument, you need logic, and if you have logic, you need the LNC. Thus, in making an argument, you presuppose the LNC.

This has the benefit that it looks a bit like a TA. The suggestion is that the necessary preconditions for making arguments are being called into question by the very argument’s conclusion. It is a bit like if I said out loud: “I have no voice”. Having a voice is a necessary condition for saying anything (out loud), and so saying that I have no voice is self-refuting. The statement calls its own conditions into question.

And this seems to fit pretty nicely with something that Aristotle says in Metaphysics IV section 4. Just after saying that you can’t give a direct proof of LNC, he says you can give a different type of argument for it:

We can, however, demonstrate negatively even that this view is impossible, if our opponent will only say something

It seems that, whatever else is going on, Aristotle thinks that a crucial part of the ‘negative demonstration’ he has in mind involves the opponent saying something. This act itself somehow traps them. This tactic seems quite similar to what Dyer has in mind. By simply saying any argument, the opponent has presupposed logic, and with it the LNC.

However, despite the seeming attractiveness of this way of understanding Dyer, it doesn’t seem to be a good way of arguing against someone who thinks that the LNC is false.

In order to see this clearly, we need to distinguish between two types of opponent of the LNC. They are distinguished as follows. If we retain all the inferential rules of classical logic, but also hold that there is a true contradiction, then we get the consequence that every proposition is true. That is because of the principle of explosion, which is a valid inference in classical logic. It says that from a proposition and it’s negation (p & ~p), we can infer any arbitrary proposition, q. So the consequence of holding both that there is a true contradiction and that classical logic is correct is that all propositions are true. The position that all propositions are true is called ‘trivialism’, and a person who believes all propositions to be true is a ‘trivialist’. In particular, trivialists believe all contradictions are true (because they believe everything is true). So if you keep classical logic fixed, but introduce a contradiction, then you get all contradictions as a logical consequence.

On the other hand, an opponent of LNC may also reject classical logic, and as such avoid trivialism. Any logic that doesn’t contain the principle of explosion is called a paraconsistent logic. Someone who believed that the LNC was false could avoid trivialism by adopting a paraconsistent logic (such as Graham Priest’s logic LP). Call such a person a ‘dialteheist’. In contrast to a trivialist (who thinks all contradictions are true), a dialetheist thinks that some, but not necessarily all, contradictions are true.

Dyer’s claim is that by making any argument, one is committed to a specific type of logic, i.e. classical logic. But this is not correct. Not all logic is classical. A dialetheist could make their inferences according to some non-classical system of paraconsistent logic, such as LP, and thus construct an argument without presupposing that LNC is true.

This is just like the way that an intuitionist logician (like Brouwer) can make arguments without presupposing the law of excluded middle (LEM), or a fuzzy logician (like Lukaseiwicz) can make arguments without presupposing the principle of bivalence. Simply stating that your opponent’s argument presupposes classical logic does not make it so. A dialetheist will reject LNC but endorse some kind of paraconsistent logic (avoiding triviality). So, again, the move is question begging against a clued-in dialetheist.

Does this work if we think about a trivialist instead? Such a person does endorse classical logic. If they also hold that a contradiction is true, then they must also hold that every proposition is true. Thus, in particular, they will already believe that all arguments presuppose the LNC. So they will agree with Dyer’s claim here. Yet, they also hold that every contradiction is true, so they also believe that no arguments presuppose the LNC as well. They agree with Dyer and also disagree with him at the same time. Such a person seems to not be playing fair at this stage, but partly for this reason it is unclear whether someone making Dyer’s claim has scored a point against them or not. Trivialism is weird like that.

One thing is pretty clear though, the point being made against the dialetheist is just question begging. LNC just isn’t presupposed in paraconsistent logic, so plainly not all arguments presuppose LNC.

An additional exegetical issue is that even if we set aside the charge of question begging, this cannot be the right way to read Aristotle. We have to turn to other works of Aristotle to see this clearly. The most famous is the following comment in the Posterior Analytics book I, section 11:

The law that it is impossible to affirm and deny simultaneously the same predicate of the same subject [i.e. LNC] is not expressly posited by any demonstration except when the conclusion also has to be expressed in that form; in which case the proof lays down as its major premiss that the major is truly affirmed of the middle but falsely denied. It makes no difference, however, if we add to the middle, or again to the minor term, the corresponding negative.

The argument sounds complicated here, but the point is simple. Take this as our example: all Athenians are Greeks, all Greeks are Europeans, thus all Athenians are Europeans (that is the most basic inference rule in Aristotle’s system, known as ‘Barbara’). Aristotle’s point is that even if (somehow) some Athenians were both Greeks and also (and in the same sense) not Greeks, this wouldn’t stop the conclusion following from the premises; it would still be true that all Athenians were Europeans, even if some Athenians were both Greeks and not Greeks. If all A’s are B, and all B’s are C, then all A’s are C, even if some A’s are also not B. (See Priest, (1998), p 95).

Thus, Aristotle is quite clear that his logical system (the categorical syllogism) does not require LNC to be true in order for valid inferences to be made; in fact, he explicitly says that valid inferences could be made even if there were contradictions. He doesn’t think that there are any contradictions (he thinks LNC is true), but the point is that he doesn’t think that making a valid inference in his logic presupposes that LNC is true. The point could probably be made that Aristotle is actually a paraconsistent logician, rather than a classical logician, although there is some controversy about that. It seems right to me though, given the comments from Posterior Analytics above.

Thus, to make Dyer’s point here work, we not only have to assume that some type of classical logic is true (which makes it question begging), but it also cannot be Aristotle’s logic, because that can still be used even if there were a contradiction (as Aristotle himself tells us).

So, to summarise:

  • Dyer says only two things about how Aristotle’s comments about LNC in response to its critics count as a TA;
    • 1) that it is a reductio (or impossibility of the contrary), and
    • 2) that it shows that the opponent presupposes LNC when making any argument.
  • But against 1):
    • Aristotle says there are no direct demonstrations of LNC, yet reductio ad absurdum is a type of direct demonstration, and
    • reductios are question begging against someone who denies LNC, so it would be a bad argument on its own terms.
  • Against 2):
    • Aristotle is quite clear that his own logic does not presuppose LNC; inferences would be valid even if there were contradictions, and
    • the fact that classical logic presupposes LNC is irrelevant; anyone who rejects LNC also rejects classical logic, so this is also a bad argument on its own terms.

Thus, I contend that Dyer’s comments here fail to explain what he thinks they are explaining. There is no defence of LNC along the lines that he sketches.

As I said, his comments are only sketchy, and if he goes into more detail somewhere, I would love to see that.

4. My view

Now I have said how I think Dyer’s interpretation of Aristotle is wrong, I will outline how I read Aristotle myself. I’m not saying this is the definitive way to read him. At all. There are tons of subtle ways of reading him. But it seems fair to put my cards on the table here as well.

All we are looking at is section 4 of book IV of the Metaphysics.  Aristotle begins in paragraph 1, as we saw, by saying that the LNC cannot be demonstrated (and that those who demand a demonstration do so “through want of education”).  So far, so good.

He starts paragraph 2 with:

We can, however, demonstrate negatively even that this view [that LNC is false] is impossible, if our opponent will only say something

A ‘negative demonstration’ is not the same thing as a deductive proof (as we saw). Exactly what it is though is where scholars disagree. Dyer obviously thinks that it is a TA. I’m not so sure. I think that what follows is just a bad argument, which involves fudging the distinction between the trivialist and the dialetheist, but getting the response to both wrong.

I think what happens for the next five paragraphs is that Aristotle makes one long argument. He then makes several one-paragraph long arguments. I will not touch on the short arguments here, mainly because I think the idea that there is a TA somewhere here is due to what Aristotle says in the first five paragraphs. The general idea of the long argument is simple, but he finds spelling it out hard.

The general idea I understand to be something like this: if you get your opponent to say something, like making a claim of some kind, such as “I am a man”, then this necessarily means that he is taking a stand on that issue, i.e. he is saying that something is true (the thing said), and not also false; that it is one way and not the other.

There is a lot of ground-clearing that follows this suggestion though, and which takes up the next few paragraphs, which is what makes it hard to see what is going on. Basically, Aristotle wants to make it clear that certain ways of saying something don’t count. For instance, if someone makes a claim, but doesn’t mean anything at all by it, then this doesn’t count, as he says also in paragraph 2:

For, if he means nothing, such a man will not be capable of reasoning, either with himself or with another. But if any one grants this [i.e. that he does mean something], demonstration will be possible; for we shall already have something definite.

Other examples of ground-clearing is when Aristotle also dismisses claims that mean more than one thing, as this in paragraph 3:

…we might say that ‘man’ has not one meaning but several, one of which would have one definition, viz. ‘two-footed animal’, while there might be also several other definitions if only they were limited in number

It’s no use saying that this is a bank (meaning a river bank) and also not a bank (meaning place where money is kept), etc. In order for Aristotle’s trap to work, the claim has to be unambiguous. It has to have some meaning, but not multiple meanings. He explains what he has in mind in paragraph 4:

Let it be assumed then, as was said at the beginning, that the name has a meaning and has one meaning; it is impossible, then, that ‘being a man’ should mean precisely ‘not being a man’

The conclusion then seems to be stated in this sentence in paragraph 5:

“It is, then, impossible that it should be at the same time true to say the same thing is a man and is not a man.”

Now, at this stage, I feel like I am ready to point out the problem with this. It has to do with trivialiam vs dialetheism again. If you could get a trivialist to make the sort of claim that Aristotle wants to get his opponent to make, then they might be in trouble. If a trivialist says “I am a man”, and takes that statement to be true and not also false, then they have contradicted themselves. After all, they think it is both true and false (because they believe all contradictions).

And this is how the trap is supposed to work, it seems to me. Someone who thinks there is no distinction between true and false (because everything is the same, both true and false) goes against that when they make the sort of statement Aristotle is focusing on, where they take a stand and commit to something being one way and not the other. That’s Aristotle’s point here, it seems to me. You can’t make that sort of statement if you don’t think there is any distinction between what is true and what is false.

However, as it stands, this is not a good argument against a trivialist. Aristotle has got the trivialist to say something, and commit to meaning that it is one way and not the other. But what is the consequence of that? It seems to me that what this has achieved is bringing out that the trivialist is doing something contradictory; they believe everything is true, and thus that their statement is both true and false, but in virtue of making the sort of statement that Aristotle is insisting on, they also take it to be true and not also false. If it is true and not false (because of the type of statement it is), it can’t be true and also false (yet the trivialist thinks it is).

But this is just a contradiction, and the trivialist already believes all contradictions are true. Remember, trivialists also believe complex contradictions, such as:

C) (“I am a man” is true and false) and (“I am a man” is true and not false)

Trivialists already believe C, because they believe everything is true. Aristotle’s argument only manages to bring out this second type of contradiction, which a trivialist will agree they already believe. So, while the trap Aristotle sets will definitely catch any stray trivialists in its jaws, they already believe the consequence he is highlighting anyway (because they believe everything is true), so the argument is ineffective against them. There is no point showing that a trivialist is committed to a contradiction, because they actively believe every contradiction to be true.

So the argument seems ineffective against a trivialist. Yet, it is even less effective against someone who believes some but not all contradictions, a dialetheist. Such a person might hold that there is only one true contradiction, such as the liar paradox. If so, then they do not fall into Aristotle’s trap when they say “I am a man”. That sentence (“I am a man”), they can happily agree, is true and not also false. You have to do more to catch a dialetheist in the trap than make them say something.

A dialetheist might also like the sound of a restricted LNC; maybe one that talks about all material objects and their properties. All statements like that are classical, they might say, but maybe some other class of statements (which includes the semantic paradoxes for instance) is not classical, and is contradictory. Such a person might endorse the restricted version of LNC while rejecting the more general one.

Thus, it just seems wrong for Aristotle to say “We can, however, demonstrate negatively even that this view [that LNC is false] is impossible, if our opponent will only say something”. A dialetheist can say something (like “I am a man”), and this is not itself enough to show that LNC is true. They can even do so while holding that certain versions of LNC are true. All that is compatible with some contradictions being true.

Thus, as I read it, Aristotle’s argument is a sort of failed attempt to trap a trivialist into an ad hominem argument ( by exposing that they believe something contradictory), which is ineffective because they will just grant that they do, and an even more ineffective argument against a diatheleist, who can simply agree to say the thing in the way Aristotle wants them to say it without any consequence whatsoever resulting from it.

Anyway, I said there are other ways of looking at this, and I am not an Aristotle scholar, so I am happy to be corrected. But that’s how I see it.

5. Conclusion

Dyer claimed that Aristotle’s position is that the LNC is demonstrable via a TA. His comments are helpful because lots of people tend to say similar things. Yet, I have shown here why I think that he is not only wrong to make such claims about Aristotle (through a bit of examination of the text), but that these arguments are bad arguments regardless of who originally made them. If you think that LNC is true, then you might have to just settle for the fact that it cannot be proven at all (even negatively). Candidate TAs, such as Dyer’s, seem to get us nowhere.

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