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

0. Introduction

Recently I debated Jay Dyer on the Non-Sequitur show. Here is a link to the debate. Dyer is a presuppositionalist YouTuber and blogger. His channel is called ‘Jay’s Analysis‘. He is an Orthodox Christian, and is known for his discussions of occultism in mainstream culture (he has a book called Esoteric Hollywood: Sex Cults and Symbols in Film).

Our debate was about Dyer’s version of the transcendental argument for the existence of God (TAG). Dyer didn’t present the argument in premise / conclusion form as such, and that limited the amount I could critique it. I hope to have a second round with Dyer, who I think was also keen to speak more about the topic. It felt like we scratched the surface of the issue. We discussed the problem of induction, and I explained my argument outlined here. We touched on the idea of divine conceptualism, and I explained my argument outlined here and here. He had replies, but we didn’t really get very far.

Quite a large section of the debate was about a claim Dyer made, which was that Aristotle made a transcendental argument (TA) for the law of non-contradiction (LNC). I want to say a few things about this here, to spell out more thoroughly what I was saying at that point in our discussion. For me, it was the most interesting bit. This is part one of two.

  1. The TA Interpretation – Academic disagreement

My view is that I don’t think Aristotle was making quite the argument Dyer says, although it is admittedly an area where scholars disagree about what Aristotle was up to (and that’s before we get to the even thornier question of whether Aristotle was correct in what he said or not). The bit of Aristotle that we are talking about is the Metaphysics, but specifically book IV (part 4).

Firstly, I want to make it clear that there are scholars who argue that Aristotle was making a TA, such as Irwin (1977), Kirwan (1996), and Walker (2006). Call this the ‘TA-interpretation’ of what is going on in Met bk IV part 4. While I disagree with this view, it is a view some academics hold.

Kirwan, for example, who I greatly admire in general, makes the following comments which are generally representative of the TA-interpretation position:

From this starting point it would seem that Aristotle proposes to argue transcendentally, demonstrating not the truth of LNC, but that without its truth the opponent could not be doing what he is doing in acceding to what is begged of him (p. 204)

Similarly, Walker says:

 What Aristotle does … is to get the sceptic to say something meaningful, and to argue that in doing so his opponent is already committed to rejecting the negation of what was asserted.

There is a big problem for the TA-interpretation of this bit of Aristotle, which is that it does too much. This is because Aristotle is at pains to point out (in the first paragraph of part 4) that LNC cannot be given a demonstration:

Some indeed demand that even this [i.e. the LNC] shall be demonstrated, but this they do through want of education, for not to know of what things one should demand demonstration, and of what one should not, argues want of education. For it is impossible that there should be demonstration of absolutely everything (there would be an infinite regress, so that there would still be no demonstration); but if there are things of which one should not demand demonstration, these persons could not say what principle they maintain to be more self-evident than the present one.

He also repeats the point in part 6:

These people demand that a reason shall be given for everything; for they seek a starting-point, and they seek to get this by demonstration, while it is obvious from their actions that they have no conviction. But their mistake is what we have stated it to be; they seek a reason for things for which no reason can be given; for the starting-point of demonstration is not demonstration.

That very much sounds like he is saying that no proof of the LNC is possible; that it has to be taken as a starting point, and cannot be the end point of a demonstration. It is something “for which no reason can be given”. Even asking for such a proof is an ignorant thing to do, and “argues want of education”.

Rather than ‘directly demonstrate’ the LNC, Aristotle says that we have to do something else. In the literature, this is commonly referred to as indirectly demonstrating it (or ‘demonstrating negatively’), following Aristotle’s lead in the first line of part 3:

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

The academic disagreement is in how to understand what Aristotle’s method is here, if not a standard direct demonstration. It is notoriously difficult to figure out what the indirect method is supposed to be however. One thing is clear, he is not offering a standard proof that the LNC is true.

Any interpretation of what Aristotle is up to, at a minimum, has to take into account the prohibition against ‘directly’ demonstrating LNC. This is why Kirwan explicitly says that his transcendental reformulation of Aristotle is “not a demonstration of LNC” (p. 204), for example.

But this is where the tension with this TA interpretation really comes in, because when we make Aristotle’s indirect demonstration into a TA, then a direct demonstration follows immediately. Kirwan says that Aristotle’s point is that “without its [i.e. the LNC’s] truth the opponent could not be doing what he is doing”. We can turn this into an argument to make the inference clear. Let A be the proposition “the LNC is true”, and let B be the proposition “the opponent can do what is begged of him”. Then Kirwan’s suggestion is that Aristotle is making the following sort of claim:

  1. If ~A, then ~B (i.e. if the LNC was not true, then the opponent cannot do what is begged of him)

Then, when Aristotle gets his opponent to make a claim (“if our opponent will only say something”), and is thus doing “what is begged of him”, we would have as a second premise that B is true. But then it follows merely by modus tollens that LNC is true.

  1. If ~A, then ~B
  2. B
  3. Therefore, A (1, 2, modus tollens)

While this satisfies the form of a TA, it is contrary to Kirwan’s claim that it is not a demonstration that LNC is true. Modus tollens is a straightforward direct demonstration (if anything is). 

So the first and most fundamental problem with this way of reading Aristotle is that he is clear that he is not giving a direct demonstration of the LNC, yet a transcendental argument like the one Kirwan (or Walker, et al) offer is a direct demonstration (because it is really just a type of modus tollens argument).

Of course, there will be ways of reading Aristotle here which will get around the problem. We could cook up some specific meaning to the phrase ‘demonstration’ that excludes modus tollens for some reason. I bring this up merely to highlight the riddle of trying to interpret this section of Aristotle. The bottom line is that he seems to say quite clearly that you cannot prove that LNC is true, yet a TA for the LNC would be precisely that. This has lead scholars to read what he is up to differently. 

Some say that it is a type of ad hominem argument, in which you show that something the opponent has claimed is incompatible with something else they believe (see Dancy (1975)). Others see the argument being not a defence of LNC, but an attack on the principle of triviality (LNT), according to which every proposition is true (see Lukasiewicz (1971), or Priest (1998)).

To confuse things, there are various intermediary views, such as that of Elizabeth Anscombe, Christopher Kirwan (again), and Max Cresswell, that while Aristotle is giving a TA for LNC, it is only for a certain restricted form of LNC; one that deals only with essential predication (leaving out accidental predication, and also forms of sentences which are not subject-predicate at all). As Kirwin notes:

The most that could have been shown is that essential predicates are not copredicable with their contradictories. (p. 101)

If that is all Aristotle’s TA gets us, then it doesn’t even really get us the LNC at all. Whatever we want to call that principle, it is a far cry from the most general form of the LNC that Aristotle gives in Metaphysics IV, found in part 6:

“the most indisputable of all beliefs is that contradictory statements are not at the same time true

In fact, if we counted up all the various views out there, and the number of people who advocated similar views to Dyer’s, I think it would be a minority view. I also think it may have come as a surprise to Dyer that there was a wide variety of views out there, or that the TA interpretation is controversial (see 47:00 of our discussion). Dyer says that I am the first person he had come across that disagreed with his interpretation. At that point in our discussion, he said:

“As far as I know, any standard treatment of the history of transcendental arguments would go back to Aristotle’s Metaphysics

In a sense that is true; standard introductions to the history of TAs often reference Aristotle’s argument in the Metaphysics. However, if we look closely, they are often saying that this view is not the standard view. For instance, Robert Stern notes in the introduction to his book Transcendental Arguments: Problems and Prospects, that while there are people who make this suggestion, it is a ‘controversial’ claim:

“…there is clear consensus on the paradigmatic examples of transcendental arguments within epistemology, beginning with Kant’s Transcendental Deduction, Refutation of Idealism, and his Second Analogy, in the Critique of Pure Reason … Other, more controversial examples might be added, prior to Kant (such as Aristotle’s defence of his principle of non-contradiction in Metaphysics Book 4 …)” (p. 2 – 3, emphasis mine)

He is saying that there is consensus about TAs going back to Kant, but not all the way back to Aristotle.

The Stanford Article on TAs makes a similar point. It says that tracing them back to Aristotle is something one ‘may perhaps’ do, rather than being the normal thing to do:

“Prior exemplars of such arguments may perhaps be claimed, such as Aristotle’s proof of the principle of non-contradiction (see Metaphysics 1005b35–1006a28; Illies 2003: 45–6, Walker 2006: 240 and 255–6); but Kant nonetheless formulated what are generally taken to be the central examples of such arguments, so the history of the topic is usually assumed to start here, with the Critique of Pure Reason.” (emphasis mine)

So, not everyone thinks Aristotle was making a TA, and there is a controversy about arguing that they should do. Part of the problem is that the one I outlined above; it seems to do more than Aristotle says is possible to do.

But, perhaps more importantly than the view among academics, I don’t think the argument as Dyer understands it (regardless of who originally said it) is a good argument. I’ll explain why I think these things as I go along. I’ll also give my reading of what Aristotle is up to in the relevant passages, and why I don’t think (what I take to be) his argument is any good either.

2. The two interpretations of Dyer’s Argument

Dyer makes two claims about Aristotle’s transcendental defence of LNC (see 51:17 in our debate). One is that Aristotle’s argument is a reductio ad absurdum (or, ‘impossibility of the contrary’), and the other is that Aristotle is saying that LNC is presupposed by his opponent saying anything whatsoever. In this video, at 57:00, Dyer makes the same two claims again. He says:

[The denial of LNC] is refuted by the impossibility of the contrary – Or a reductio, right? Same kind of thing. – And that is a transcendental argument. [Aristotle] says that: “You, when you engage in that argument, you are assuming the existence of logic and universals, to try to refute logic and universals.”

Clearly, the first sentence is about a reductio or impossibility of the contrary argument, and the second is that making any argument whatsoever assumes the LNC.

I’ve also found him making the second claim on his blog in this post:

Transcendental arguments are, by the nature of the case, arguments made indirectly, inasmuch as the nature of the category or concept in question is not something that can be proven directly. Aristotle first noted this with regard to the law of non-contradiction. Its truthfulness is shown by the fact that its denial presupposes its existence. To deny the law of non-contradiction presupposes a world wherein logical laws apply. Thus such transcendental categories are demonstrated indirectly.

The two claims are

  1. LNC is demonstrated by reductio (or impossibility of the contrary), and
  2. LNC is presupposed by every argument or claim (including the claim that LNC is false).

These are not the same thing, even if they have similarities.

To see why, I will give an example of a statement which is proven via a reductio ad absurdum, but which is not plausibly presupposed by every claim whatsoever. I will spend some time setting out precisely what a reductio argument is, partly to be clear, but also because this will help explaining the first philosophical problem I have below, so stick with it.

3. Being proven by reductio ≠ being presupposed by anything 

Formally, ‘reductio ad absurdum’ is a standard inferential rule in classical logic. It works like this. From a proof that A implies ~A, infer ~A. Thus, we prove ~A by showing that its contrary (A) is impossible; and ‘showing that A is impossible’ means showing that it leads to a contradiction. So when Dyer said that any denial of the LNC “is refuted by the impossibility of the contrary – or a reductio”, he is saying that denying the LNC leads to a contradiction, and from that we can infer that the LNC must be true.

Here is an example; we will prove that the following sentence is true using reductio ad absurdum:

a) If I am an egg, then if you are an elephant, then I am an egg

First of all, we will assume a) is false, and then derive a contradiction from that assumption. Let p = “I am an egg”, and q = “You are an elephant”. Then a) has the following form:

p → (q → p)

We assume the above formula is false and derive a contradiction from that. We will work with this general form, and then apply it back to a).

All we need to know to for our derivation is one thing:

The only way that a conditional is false is if the antecedent is true, and the consequent is false (see this).

All we do is apply this insight twice to derive the contradiction. Here is how.

First application:

The only way the outer conditional, p → (X), can be false is if p is true and X is false. Thus, for p → (q → p) to be false, p must be true.

Second application:

We know from the above that X is false. In our case, X is a conditional, namely: (q → p). Applying our principle for the second time, the only way that (q → p) can be false is if q is true and p is false. Thus, for p → (q → p) to be false, (q → p) must be false; and for (q → p) to be false, p must be false.

Thus, for p → (q → p) to be false, p has to be both true (first application) and false (second application). So if p → (q → p) is false, then (p & ~p), which is a contradiction, is true. Therefore, p → (q → p) is true (because its contrary is impossible).

If we apply this proof to our original example, then it shows that if the sentence “if I am an egg, then if you are an elephant, then I am an egg” is false, then the sentence “I am both an egg and not an egg” is true. Yet the latter is a contradiction, and thus false. Therefore, “if I am an egg, then if you are an elephant, then I am an egg” must be true. QED.

We just proved that a) is true by reductio ad absurdum. But Dyer claimed that proof by reductio and something being presupposed by any claim whatsoever were fundamentally the same thing (“same kind of thing” and “variations on the same point” is how Dyer put it). If so, then the fact that a) was proven by reductio should mean that a) is presupposed by any claim whatsoever. Yet, it seems hugely implausible to suppose that a) is presupposed by any claim whatsoever. Why should we think that a) is presupposed, for example, by the following?:

b) “I am a man”

There seems to be no reason at all to suppose that b) presupposes a). If there is an argument that shows that b) presupposes a), I have not heard it. It is certainly not obvious anyway.

At any rate, it is on the person who thinks that 1 is the same thing as (or a variation on the same point as) 2, to make the case that a) is presupposed by b), because we just showed that a) can be proven by reductio.

Absent any reason to think that Dyer’s claims (1 and 2) should be thought of as the same, I think we should treat them as distinct, and not run them together as he does. Accordingly, I will treat them separately in what follows.

In the next post, I will bring up philosophical and exegetical problems for each of these versions of Dyer’s claim.


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