Does the impossibility of Hilbert’s Hotel rule out an infinite past?

0. Introduction

I recently went on the Unbelievable podcast and debated with a Christian philosopher called Andrew Loke. Following our conversation, we have had a lively email exchange where we have been continuing to debate the same issues. This has been a helpful exchange for me, as it has focused my attention on one or two very specific things and made me get clear about them. Here, I want to get some of it down into a blog post.

Hilbert’s Hotel is a hotel with infinite rooms in it. For convenience, I will talk of ‘a Hilbert’s Hotel’ (or HH) to refer to any infinite object like this (such as a library with infinite books in, or a book with infinite pages in, etc). An object, x, is ‘a HH’ if and only if both of the following are true:

a) proper parts of x can be put in a one-to-one correspondence with the whole of it, and

b) each part of x concretely exists in the present.

What I want to know is, does the impossibility of a HH in this sense mean that the past is finite? In order to answer this, let us assume that there is no metaphysically possible world with an HH in it; HH’s are not possible. Let’s call that our ‘original assumption’. Now the question for this post is, if we make the original assumption, is it possible that the past is infinite?

  1. New Kalam

To put it the other way round: if a HH is impossible in the present, does that rule out an infinite past? If it did, then we could formulate a new version of the Kalam as follows:

  1. If is possible that the past was infinite, then a HH would be possible (assumption)
  2. HH’s are impossible (original assumption)
  3. Therefore, it is not possible the past is infinite. (1, 2, modus tollens)

I think that this argument is valid, but unsound; the first premise is false. I will explain why I think that here.

2. a1-worlds and a2-worlds

Before we get there, I will motivate why someone might think differently to me. Here is one reason for thinking that the argument is sound, which Andrew endorsed. It requires thinking of two different ways that the past can be infinite, which I will call a1-worlds, and a2-worlds.

Imagine that every ten minutes God creates a grain of sand ex nihilo, and adds it to a heap of sand. This seems intuitively possible. Whenever the past is infinite, and also contains a continuous cumulative process, like adding grains of sand to a heap, or adding rooms to a hotel, we shall all this an ‘a1-type infinite past’, or an ‘a1-world’ for short.

If God had been doing this once every ten minutes forever, and if time had no beginning, then there would now exist an actually infinite object, i.e. a heap of sand with an actually infinite number of grains. So there would exist an HH in the present. Thus, it seems like if it was possible that the past were infinite like this, then it would also be possible that there is an HH in the present.

This motivates premise 1.

But now premise 2 comes in. It says that HH’s are not possible. If the antecedent of premise 1 (time is infinite) is true, then the consequent is true (a HH is possible). But because of premise 2 (a HH is not possible), the consequent of premise 1 is false. Therefore, the antecedent of premise 1 is false (by modus tollens). Thus, the past is finite. This is how the new Kalam works.

However, we can think of a different type of infinite past. Imagine God creates a grain of sand ex nihilo every 10 minutes (just like above), but instead of adding them to a pile, he immediately annihilates them. In this scenario, there is no infinite heap of sand, no HH, in the present. The past is infinite, but it involves no continuous incrementally increasing cumulative process. Call such a world an ‘a2-world’.

Because an a2-world doesn’t involve an HH existing in the present, the impossibility of an HH cannot rule out the possibility of such an infinite history. This makes it different to an a1-world in a directly relevant way. The impossibility of an HH in the present is compatible with the possibility of an infinite past (contra the new Kalam), just so long as it is an a2-type infinite past and not an a1-type infinite past (i.e. just so long as there is no continuous incrementally increasing cumulative process).

 3. Disambiguating the first premise

Given that there are two types of infinite past (ones with a continuous incrementally increasing cumulative process, and ones without) the original premise was ambiguous. It said:

  1. If it is possible that the past was infinite, then a HH would be possible

It doesn’t distinguish between the two ways that the past can be infinite (i.e. a1- or a2-type). If 1 is true on both its disambiguations, then it is a distinction without a difference. However, if it is true on one but false on another, then we need to disambiguate to know which one is which.

What we need to do is consider each as a separate interpretation of the above premise, to see how it plays out on both disambiguations. So we have:

i) If the past is an a1-world, then a HH would be possible

ii) If the past is an a2-world, then a HH would be possible

If both of these are true, then the new Kalam holds. If one is false, then its first premise is also false. No surprises, it is true on the first, but false on the second. Let’s go through them one by one to see how this works.

Given that the definition of an ‘a1-type’ history includes as a clause that a HH is present, then it is obvious that whenever there is an a1-type history there is also a HH. That much is just a matter of definitions. But, because there are no possible worlds at all where there is a HH (because of premise 2 of the new Kalam), there are no a1-type worlds either. Both antecedent and consequent are necessarily false. And (somewhat counterintuitively for those not familiar with classical logic) that means that i) is a tautology, and so necessarily true.

The truth of ii) though is not so easy to determine. a2-worlds don’t themselves involve HH’s, so on their face it is not so obvious that they imply the possibility of HH’s. In order to rule out a2-worlds, the defender of the Kalam must find a way of arguing that a2-worlds are impossible. How might they do that?

Here is how. God can make a grain of sand, and he can then either destroy it or put it on a heap. Both creating grains to add to piles, and creating grains to destroy, are possible for God. If he can do one, he can also do the other. And the thought is that if he could make 1 grain and put it in a heap (which he surely can), then he could make 2, and if he could make 2 he could make 3, etc. By induction, it seems, he could make an infinite pile. So if an a2-world is possible, then so is an a1-world.

And that just means that if he can make and destroy grains of sand, then he can also do something impossible (make grains and put them in an infinite heap). In classical logic, anything that implies an impossibility is necessarily false.

If the possibility of an a2-type history implies the possibility of an a1-type history, and an a1-type history is impossible (which it is by the original assumption), then an a2-type history is also impossible (by modus tolens):

  1. If a2 is possible, then a1 is possible
  2. a1 is not possible
  3. Therefore, a2 is not possible (1, 2, modus tolens)

Premise 1 of this little argument is what I want to focus on. If it succeeds, it would collapse the disambiguation between a1-type and a2-type histories. It would mean that the first premise of the original argument (‘If the past was infinite, then a HH would be possible’) would be true. And if that were the case, then the impossibility of a HH would mean that time was finite, and it would be game over.

So the salient question becomes:

Does the possibility of an a2-type history imply the possibility of an a1-type history? In other words: if it is possible for God to spend forever creating grains of sand and then destroying them, does that mean that it is also possible for God to spend forever creating grains of sand and piling them up?

I think I can make a pretty strong case to say that the answer is ‘no’. It involves the realisation that if a HH doesn’t already exist, it cannot exist, and why that’s the case. If I’m right, then the impossibility of a HH doesn’t tell us whether past is finite or infinite. It might be, for all the premises of the new Kalam say, that the past is infinite.

4. Life in an a1-world and an a2-world 

To understand why the possibility of an a2-world doesn’t imply the possibility of an a1-world, it helps to consider what is true in a1-worlds and in a2-worlds.

Imagine, first, that an a1-world is actually the case. In that setting, an HH exists in the present. Maybe an infinite heap of sand, or an infinite hotel, whatever. The past in that world involves a continuously increasing incremental cumulative process, like God adding grains of sand to a heap.

The number of grains of sand there are in the infinite heap right now is at least equal to ω1. But how many grains were there in the pile just before God added the most recent grain? However many there were, it cannot be any finite number, because there is no finite number, x, such that ω1 – 1 = x. So before God added the latest grain of sand, there was already more than a finite number of grains of sand in the pile.

This point generalises. Before God added the latest, say, 1,000,000 grains of sand, there were still more than infinite grains of sand in the heap (because there is no finite number, x, such that ω1 – 1,000,000 = x). What this shows is that if there is a HH in the present, then there has always been a HH throughout the infinite past. If the heap is infinite now, then there is no point at which the pile had finite members.

We can also generalise this into the future too. If the heap has ω1 grains in it now, then if God started removing grains of sand from it one by one, there would always remain an infinite number of grains of sand left in the pile. It wouldn’t decrease below infinite, no matter how long he continued to take grains of sand away.

So on any a1-world, where there is a HH in the present, it follows that there must also be a HH throughout the past and throughout the future as well.

If there is a HH in an a1-world, it follows (trivially) that it is possible that there is a HH there too. Yet, our original assumption was that a HH is not possible. Thus, the a1-world is inconsistent; in it a HH is both possible (because it exists at every point in time), and also impossible (by the original assumption). They are not just inconsistent, but they are ‘full’ of contradictions, in the sense that there is a contradiction true at every time in every a1-world. Thus, a1-worlds are inconsistent, thus are not metaphysically possible worlds at all.

Now consider an a2-type history. In this case, there is no HH in the present. We know from the above reasoning that had there been a HH in the past at any point, then there would still be one now. If one existed any finite number of days ago, then not even God could have removed all the grains from it one by one at regular intervals such that there are none left today. And that is true no matter how long ago God started to remove grains.

Not only would there still be grains left, there would still be infinite grains left, no matter when he started to remove grains. Needless to say, if there was a HH at some point in the past, it would still exist now.

So, if there is no HH today, there never has been.

There is a future tense version of this too. God could start putting grains of sand in a heap now, but no matter how long he continues to do that there will never be an infinite number of them.

So, if there is no HH now, then there never will be.

So on any a2-world, there is no HH; there never was, and there never will be. This makes a2-worlds the mirror image of a1-worlds. Rather than being full of contradictions, a2-worlds are ’empty’ of them (there is no contradiction at any time in any a2-world). Unlike in the case of the a1-worlds, the original assumption that HH’s are impossible does not cause any contradiction here, because it is always false that there is a HH in all a2-worlds.

5. Objections

Imagine that an a2-world is actual. That means that there is no HH in the present, and no continuous incremental process throughout the past. However, is it possible for God to make a HH in this world? If there is no HH in the present, is it possible for there to be one?

The answer is: not if he has to start with nothing and can only add grains one at a time at regular intervals. No matter how long he does that for, he will never get to there being more than finite grains. An a2-world cannot change into an a1-world. Not even God can do that, assuming he can only add them one at a time.

That’s also true on the assumption that he adds any arbitrarily large finite number of grains to the pile at regular intervals. It doesn’t matter if he adds one grain, or a million, or a billion billion billion; so long as it is at regular intervals, like every 10 mins, there will always only be a finite number of grains, no matter how long he has been doing it. Thus, it is not possible for God to make an infinite heap of sand if there isn’t already one there.

There are two possible objections to this.

We might question the condition that God can only add finite numbers of grains to the heap at any one time. Maybe God could create all the grains at the same time, instantaneously. Maybe he could just click his fingers and make an infinite pile of sand ex nihilo. If so, then it wouldn’t be impossible for God to change an a2-world into an a1-world.

Alternatively, we could question the condition that God can only add grains at regular intervals. Maybe, God could do a supertask, and get infinite grains added in a finite amount of time. Instead of adding them at regular intervals, he could speed up the time it takes him to add them to the pile. If so, then it wouldn’t be impossible for God to change an a2-world into an a1-world.

Let’s take these one by one.

Firstly, let’s suppose that God could make infinite grains of sand appear instantly as one big heap of sand. The proposal is then that: if God existed, then a HH would be possible. But that would then contradict our assumption that a HH is metaphysically impossible. From that we could conclude that God doesn’t exist:

  1. If God could make a HH all at once, then a HH would be metaphysically possible (new assumption)
  2. A HH is not metaphysically possible (original assumption)
  3. Therefore, God cannot make a HH all at once (1, 2, modus tollens)

So the problem with the idea that God could have just made a Hilbert’s Hotel all in one go, is that it is metaphysically impossible to do that (by the original assumption). Hilbert’s Hotels are impossible, so it is impossible to make one in one go.

What about if God did a supertask? A supertask is where an infinite sequence is squeezed into a finite space or time. For example, imagine if you took one step to the door that got you half way, and your next step was only half the remaining distance, and the next half the remaining distance, etc. In this way, you could fit an infinite number of steps into a finite distance (between here and the door).

Similarly, maybe God could create the first grain of sand and add it to the heap in 10 minutes, and the next one in 5 minutes, and the next one in 2.5 minutes, etc. By the time 20 minutes had passed, he would have made an infinitely big heap of sand.

This would be an incremental process, but with an ever shorter interval between iterations. We might think that this is better than the previous idea, because it doesn’t require God doing anything infinite in extent; each iteration he adds a finite amount of sand to the heap (one grain), unlike in the last one where he did an infinite task in one jump.

However, the counter-argument is the same. If a supertask was possible (whether for God or for anyone else), then a HH would be possible. If our original assumption is that a HH is not possible, then a supertask is not possible either.

  1. If God could make a HH by a supertask, then a HH would be metaphysically possible (new assumption)
  2. A HH is not metaphysically possible (original assumption)
  3. Therefore, God cannot make a HH by a supertask (1, 2, modus tollens).

So the problem with the idea that God could have just made a Hilbert’s Hotel by a supertask, is that it is metaphysically impossible to do that (by the original assumption). Hilbert’s Hotels are impossible, so it is impossible to make one by a supertask.

The wider lesson is this: if HH’s are impossible, then there is nothing that God could do to make one.

What this shows is that if we hold fast to our assumption that HH’s are metaphysically impossible, then all God could do is add or subtract finite amounts of grains at regular intervals. He couldn’t do a supertask to build one, or make one with the snap of the fingers. And that means that if there is no HH now, then even God couldn’t make one.

And this means that if an a2-world is actual, then an a1-world (with an HH in it) is impossible.

6. Conclusion

Let’s circle back. Earlier, we had looked at the following implication:

  1. If a2 is possible, then a1 is possible

We saw a reason to think that this is plausible. After all, if it is possible for God to spend an infinite past making and then destroying grains of sand (i.e. if an a2-type history is possible), then it is possible for him to not destroy them, but pile them up instead (i.e. an a1-type history would be possible). Creating grains of sand is not beyond God’s power, and neither is putting grains of sand in a heap. Thus, surely, if an infinite past is possible, then God could have spent that whole time making an infinite heap of sand, as it just involves him doing things he can do. This is a compelling point. This is the best argument for thinking that an a1-type world is possible.

The problem, as we saw, is that it runs into our original assumption, that a HH is impossible. That’s because if God did accumulate infinite grains of sand, like in an a1-type world, then there would be a HH now. If the original assumption is true, then there can’t be a HH now. So there can’t be an a1-world.

And that feels right to me. An a1-world has a HH; but HH’s are impossible; so a1-worlds are impossible too. The impossibility of a1-worlds just is the impossibility of the HH’s that exist in them after all. The impossibility of a1-worlds is a logical consequence of the impossibility of HH’.

We looked at an inference, which was that the possibility of a2-worlds implied the possibility of a1-worlds. But if there is no HH in the present, and HH’s are impossible, then there is nothing God can do to make one. Not even God can change an a2-world into an a1-world, if HH’s are impossible. That makes the inference false. The possibility of an a2-world does not entail the possibility of an a1-world. Nothing does (because they are impossible).

This leads us all the way back to the disambiguation we started off looking at. There, we had disambiguated the following:

  1. If it is possible that the past was infinite, then a HH would be possible

The first disambiguation was a tautology, and so true. The remaining disambiguation to consider was the second one:

ii)  If it is possible that the past was an a2-world, then a HH would be possible

We know that the consequent is false (because of the original assumption). But it seems like the antecedent is true; it is possible that the past was an a2-world. That would make ii) false.

And that means that the original premise is ambiguous, and on one disambiguation, (the second one) it is necessarily false. Here is the premise again:

  1. If it is possible that the past was infinite, then a HH would be possible

If the past is infinite, then that means that it is an a2-world (because a1-worlds are impossible). At all a2-worlds, HH’s are not possible. Therefore, the antecedent is true and the conclusion is false. This means that the impossibility of HH does not mean that the past is finite, and our new Kalam argument is unsound.


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.

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.


The limitations of Transcendental Arguments

0. Introduction

In the very first post on this blog, back in November 2015, I looked at a paper by Michael Butler, which is a first-rate examination of transcendental arguments (TAs) and their relationship to the presuppositional TAG of Van Til, Bahnsen, etc. I recommend reading Butler’s paper closely.

Part of what is interesting about that paper is the breadth of reading that Butler has done on the subject. It comes with a comprehensive literature review of the contemporary literature on TAs in analytic philosophy, and a satisfying look at the Kantian origins of this tradition. It also provides a look at Van Til’s approach, and how that was taken forwards by Greg Bahnsen. Butler, though a presuppositionalist himself, (and I think trained by Bahnsen) raises four interesting criticisms of TAG. He explains Bahnsen’s attempts to rebut these criticisms, and then criticises these rebuttals in turn. The four objections are:

(1) the nature of TAG; (2) the uniqueness proof for the conclusion of TAG; (3) the mere sufficiency of the Christian worldview; (4) the move from the conceptual necessity of God’s existence to the actual existence of the Christian God.

The last objection is probably the most interesting, and the one I paid least attention to in my previous post. As Butler says:

I consider this stricture to be the most powerful argument against TAG and the most difficult to answer.

The explanation of the objection runs as follows:

This objection revolves around the consideration that proving the conceptual necessity a worldview does not establish its ontological reality. Kant, for example, argued that the notion of causation is transcendentally necessary for thought (or at least human thought).  Without the concept of causality there could be no thought.  But just because causality is necessary for thought does not mean, so Kant argued, that the things in themselves (ding an sich) which exist independently of our conception of them, undergo causal relations.  Conceptual necessity does not guarantee ontological necessity.  In the same way, assuming that TAG is sound, all that is proved, so this objection goes, is that we must, in order to be rational, believe that God exists.

What this objection is highlighting is that transcendental arguments, such as those employed by Kant, even if thought to be successful, do not establish the existence of something in reality. Rather, they establish synthetic a priori truths, which are about the forms our experience must take. They establish, at best, only conceptual necessities, not ontological necessities.

For example, we may be forced to experience the flow of time as one event happening after another, in the past, present and future. We may be unable to conceive of how to make sense of the contrary – what would it even mean to have all your life’s experiences ‘at the same time’? Yet, even if this were so, we would still be left wondering whether this is because time is ‘in itself’ something which really does flow, or whether it is just a “stubbornly persistent illusion” as Einstein once said. It may be required for us to make sense of the world, yet not indicative of the way the world is. To put the point another way, it may be a necessary precondition for intelligibility (a phrase we hear a lot with internet presuppositionalists), yet not true.

  1. Bahnsen’s rebuttal

Butler quotes Bahnsen’s rebuttal to this objection:

…because this is an apologetical dialogue (giving reasons, expecting argument, etc.), both parties have assumed that the true viewpoint must affirm rationality.  Van Til argues that if the unbeliever’s worldview were true, rationality would be repudiated, whereas if Christianity were true, rationality would be affirmed and required.  So while the whole argument may be stated in hypothetical terms, the conclusion is actually established as true, since the hypothetical conditions was granted from the outset by both parties.  (If the unbeliever realizes this and now refuses to grant the legitimacy, demand, or necessity of rationality, he has stepped outside the boundaries of apologetics.  Furthermore, he forfeits the right to assert or believe that he has repudiated rationality, since without rationality assertion and belief and unintelligible.)

Butler is impressed with this response, though ultimately concedes that it does not address the point in hand. His positive comments are the following:

Bahnsen’s answer is that the issue is one of rationality.  If TAG establishes that Christianity is the necessary conceptual precondition of human experience (including rationality) it follows that we must hold to the Christian worldview in order to be rational. And if somebody refuses to accept the Christian worldview or God’s existence, he has no foundation for rationality and, without such a foundation, has no rational basis to object that the conclusion of TAG.

This defense carries a great deal of force.  It effectively undermines the unbelievers ability to rationally reject the Christian faith.

2. My objection

Before we go on to look at Butler’s criticism of this, and how he tries to improve on Bahnsen, I want to point out that I think we can bring up an interesting objection at this stage.

In a previous post on Stephan Molyneux, I pointed out an objection to one of his arguments. He made the following claim:

If I tell you that I like chocolate ice cream, and you tell me that you like vanilla, it is impossible to “prove” that vanilla is objectively better than chocolate. The moment that you correct me with reference to objective facts, you are accepting that objective facts exist, and that objective truth is universally preferable to subjective error.

However, this seemed to boil down to an implausible inference, namely: if you argue that p is true (“with reference to objective facts”), you are showing a preference for truth over falsity. The reason this seems implausible is that we can easily come up with examples of people engaging in arguments (even “with reference to objective facts”) who do not necessarily have any such preference for truth over falsity. My examples were:

  • Putting forwards an argument arguendo
  • Participating in debate competitions
  • A lawyer arguing because she gets paid to do so
  • A politician arguing to win votes
  • An internet troll arguing to cause irritation
  • An undergraduate arguing to impress others

What these examples show is that there are contexts in which someone can put forward an argument yet not be primarily directed towards truth, or be of the opinion that ‘truth is preferable to error’. I described these as ‘non-standard’ contexts, in contrast to what Molyneux seemed to have in mind, which should be called the ‘standard context’.

The standard context is something like where both parties are solemn truth-seekers, and ‘play by the rules’ of correct logical behaviour (or whatever – its not even clear what this is supposed to be). But the point is that Molyneux is effectively just presuming that all argument goes on in standard contexts, and that therefore if you make an argument you have the attitudes and beliefs of those in the standard context, i.e. you are earnestly seeking truth, and prefer truth to falsity.

However, the non-standard contexts I describe above are examples where people simply have different attitudes, and in which they are not earnestly seeking truth at all (they are doing something else). They may prefer a useful falsehood (if it wins the case, or gets them laid). This shows that you cannot make the inference that Molyneux describes; just because someone is making an argument does not establish that they prefer truth over falsity.

I might even go so far as to say that the ‘standard’ context is a kind of fantasy. Does it ever actually exist? What even are its necessary conditions? I certainly don’t know. It seems at least possible that the ‘non-standard’ contexts are in fact ubiquitous (although this is a dangerous ‘post-modern’ thought…).

Anyway, let’s tie this back in to Bahnsen’s reply. Recall, he said:

…because this is an apologetical dialogue (giving reasons, expecting argument, etc.), both parties have assumed that the true viewpoint must affirm rationality.

Here, Bahnsen is invoking the notion of the standard context, just like Molyneux, which requires that the parties “assumed that the true viewpoint must affirm rationality”.

But this seems too much of an assumption already. One could go through the motions of a debate with an interlocutor without being convinced of their rationality. I might doubt your rationality at the start, but be prepared to give you the benefit of the doubt. Maybe I took you for a rational fellow at the start, but increasingly come to doubt it as you engage with me. Am I not engaging in apologetical debate if I do that?

One could even engage in the process while being unsure of one’s own rationality. Maybe I am open to being persuaded about that question, and in the mean time I do what seems right to me in the moment of the debate. Does Bahnsen mean to say that I would not actually be participating in the debate if I was of that mindset while I did it?

And I mean, to some extent that description feels right as a description of my own mental state much of the time. My own phenomenology of rationality involves both the feeling that I am rational enough to engage in things, like conversation (and even in completing a PhD in philosophy), yet plagued with a sense of my own fallibility, seeing as I make mistakes in reasoning every day (just ask my girlfriend!). I feel aware of both the presence and the absence of my own rationality, and thus its precise extent remains somewhat unclear to me. So while I will engage in a conversation with someone like Bahnsen, he cannot assume that we both share the assumption that the truth of the matter involves my rationality, if that means to presuppose that my rational faculties are perfect. Far from it. I have a recognition that they are sometimes right and often wrong. Am I disqualified from the conversation as a result? If not, what does it even mean to say that we must both assume that the truth affirms rationality as a precondition for the debate?

But maybe all he means by the shared assumption that the truth will involve ‘affirming rationality’ is that he can assume that we both affirm enough rationality to get by in a debate. Yet, even that seems like it admits of counterexamples. Consider a variation on familiar Chinese Room thought experiments, where your interlocutor has no rationality at all.

Imagine Bahnsen is in a room and he is fed slips of paper with comments seemingly from a curious atheist about the Christian worldview, but which are in fact produced by some sophisticated mechanical algorithm. Bahnsen writes back a reply to the comments and posts them back through the slot in the door from which they came, and in this fashion enters into what he takes to be a conversation.

Imagine that Bahnsen and the algorithm debate TAG, and got into this particular bit of debate. Let’s say that the algorithm had raised Butler’s fourth objection, i.e. it had pointed out that even if TAG is sound, that only establishes that believing in God is necessary for rationality rather than establishing that God exists. At this point, imagine that Bahnsen wrote a reply on his slip of paper, which read as follows:

…this is an apologetical dialogue … [and so] both parties have assumed that the true viewpoint must affirm rationality

In this context, it is unclear whether this is even true any more. Is it still an example of ‘apologetical debate’? It is one person debating with an algorithm. Does that count? Whether we are allowed to assume all the trappings of the standard context in this setting seems rather doubtful, to say the least. What seems certain is that there isn’t really another party in the debate at all, let alone one who has assumed that ‘the true viewpoint must affirm rationality’.

This is important because the objection that the mechanical algorithm came up with is as relevant to Bahnsen as it would have been had a rational person came up with it. What difference does it make to the salience of the point itself (that TAG only establishes conceptual necessity, and not ontological necessity) if the point is made by an unthinking algorithm? It seems like Bahnsen should be just as worried about it either way.

If he left the room at this point, and then found out that there was nobody on the other side of the door, Bahnsen’s rebuttal to the fourth objection certainly seems to have been disarmed. But does he have any reason to think that the objection itself has been dealt with? I think not. What the algorithm said to Bahnsen remains true, even though it was not made by an ‘agent’ as such.

What this shows, I think, is that the problem is really his problem, not his interlocutor’s. Even if we grant that no rational agent can make the objection, it remains an objection anyway. And what this shows, I think, is that Bahnsen is really just doing a complex burden shifting exercise. Declaring victory because the interlocutor is not allowed to make the objection (while still claiming to be rational) just does not address the objection.

3. Back to Butler

Butler sees this issue in a similar way to me, although he does not put it like I do. He explains the problem like this:

The challenge is, thus, to bridge the gap between having to believe the Christian worldview because it provides the necessary preconditions of experience and showing that the Christian worldview is true.

The problem with Bahnsen’s reply to this, according to Butler, is as follows:

The problem, however, is that while TAG … demonstrates that the Christian worldview is necessary precondition for experience, it does not prove that the Christian worldview is true.  For it may be that our experience of rationality, morality, science, etc. are illusory.  Bahnsen’s reply to Montgomery, that we must make the “gratuitous assumption” that at least one worldview must be right, is without foundation. Surely, we can, for argument’s sake, conceive of the world being ultimately irrational and amoral. And if can do this, it follows that TAG, on this interpretation, fails to prove that Christianity is true.

How does Butler propose to bridge the gap between conceptual and ontological necessity? His answer involves distinguishing a conceptual scheme from the Christian worldview. The types of transcendental arguments which get us only to conceptual necessity involve conceptual schemes, whereas ones involving the Christian worldview go beyond that to ontological necessity (about how the world actually is in itself):

Stated another way, the necessity of a conceptual scheme cannot guarantee anything about the way the world must be.  For while such a scheme may organize our experience, it itself is dumb and mute and cannot, definitionally, tell us anything about the world itself.  But the Christian worldview is not a mere conceptual scheme.  It claims to do more than simply provide us with the necessary preconditions of experience. The Christian worldview posits a sovereign, creator God who is both personal and absolute in his nature. This God is, moreover, a speaking God who reveals truths to us about himself and the world. In his revelation to us he declares that he has made a world and that this world exists independently from himself and us. On the basis of his revelation, therefore, which is itself the necessary precondition of experience, we can know truths about the world and God.

The answer seems to be that Christianity involves a God who tells us what the world is like, whereas a conceptual scheme cannot. It is not clear to me how this is supposed to bridge the gap though. It seems to me though that we have simply moved from one conceptual necessity to another.

Before, we had the conceptual necessity being that ‘God exists’, i.e. the conclusion of TAG could only be that it is necessary (for rationality, etc) to believe that God exists (rather than it being true that God exists). Now though, as an attempt to bridge the gap between those, Butler has expanded the proposition in question to include the explicit statement that God tells us about the world. So now we should change the conclusion to being: it is necessary (for rationality, etc) to believe that a God exists who tells us what the world is like. Notice the scope of the belief here though. It is necessary to believe that: [a God exists who tells us what the world is like]. The content of what God tells us matching what reality is like is something that is under the scope of the belief in question; the veracity of the revelation is itself an article of belief. Unless the content of the messages that God gives us were somehow known to go beyond the conceptual necessity, to ontological necessity, we have not advanced one step from where we were before. This is because Butler’s proposal is still compatible with it being false that that God exists. It is still just a conceptual necessity. What he needs to do now is bridge the gap between believing what God tells him about the world is true, and it actually being true.

But, of course, being able to bridge the gap between something being believed to be true, and it actually being true is the very thing Butler’s suggestion was supposed to be clearing up. So unless we beg the question here, and assume we have bridged the gap somehow, his proposal is ineffective. Ultimately, the belief that what God has revealed is truthful is just another synthetic a priori truth.

4. Conclusion

Butler’s response here is utterly disappointing. I genuinely enjoyed his paper, and think it is worthy of being much more widely read. The shame of it is that the ending is so flaccid.

Stephan Molyneux and UPB

0. Introduction

Stephan Mulyneux is a YouTuber with a large following. He has been described as part of the ‘alt-right’, and defends a libertarian political view. He is also associated with the men’s rights movement. On his channel he has interviews on the main page with Paul Joseph Watson, Sargon of Akkad (or ‘Carl Benjamin’), Jordan Peterson and Katie Hopkins. I will leave you to draw your own conclusions about that.

In addition to riffing on things like politics, Molyneux also publishes books on ethics, such as this one: ‘Universally Preferable Behaviour: A Rational Proof of Secular Ethics‘.

There are lots of good analyses of this book, such as this one by philosopher Dan Shahar. I’m not going to provide anything like a long, detailed look at the book. What I am going to do is highlight one issue that jumped out at me as I casually read parts of it.

  1. Speech acts

Take the following quote, from page 25 of the book:

If I say, “I do not exist,” that is an example of an idea that is inconsistent with itself, since I must exist in order to utter the sentence.

There is something wrong with saying “I do not exist”. That is due to the fact that saying something is a type of action, something that you do. As such, it is the sort of thing that is located at a particular time and place, and is done by a particular person in a specific context. The conflict is between this set of background presuppositions (one of which is that the person doing the saying exists in a specific context) and the content of the sentence itself, which is a denial of the existence of the speaker of the sentence. That is the sense of contradiction here: the content of the sentence conflicts with one of the presuppositions of the sentence. Other examples might be things like

a) ‘You are not awake’, or

b) ‘You are not reading this’, etc.

These sentences also conflict with aspects of the context that we often do not state explicitly, i.e. presuppositions.

However, there is another way to think about these where the conflict seems not so pressing. Consider b). A few minutes prior to reading b) it was true that you were not reading the sentence. So there is nothing inherently contradictory about the idea of you not reading that sentence. The conflict only appears when you actually are reading it; there is only a conflict with the presupposition of action, not with the idea expressed by the sentence itself. And when I am not performing the action of reading the sentence, there is no conflict with the content of the sentence.

This is in contrast with sentences like ‘This is both an apple and not an apple’. In this case, the sentence is internally contradictory; it expresses both p and not-p explicitly. It doesn’t matter if you say it out loud, or if we merely consider the idea expressed by the sentence. The conflict is not between the content of the sentence and some aspect of the context of the saying of the sentence. And this makes it different to the other examples.

This brings us to Molyneux’s example. It is true that I must exist in order to utter the sentence “I do not exist” (indeed to utter any sentence). But, just as with b), there is nothing inherently inconsistent with the idea of me not existing. At some point I did not exist, and at some point I will not exist. Unfortunately for me, the idea of me not existing is not internally inconsistent. The only inconsistency is between the idea of me not existing on the one hand, and the concrete situation of me saying the sentence to you at a particular place and time on the other. Another way of saying the same thing: if uttered, then the content of the sentence is in conflict with an aspect of the context of use (i.e. my existence); but the content of the sentence itself, considered independently from any context, is not inconsistent (i.e. my non-existence is not itself an inconsistent idea).

Things are a little more complicated than that even. Not all contexts are created equal. Most of the time, if I say that I do not exist, then I run into trouble (because such a saying is an action that I do at some particular place and time). But I could leave such a statement in a will, or, like a suicide bomber, record a message only to be heard after my death (“If you are watching this, then I do not exist”, etc). The idea expressed in such a situation by that sentence is perfectly intelligible. As such, Molyneux’s sentence is only in conflict with aspects of what we might call ‘standard’ contexts of use. ‘Non-standard’ contexts like the reading of wills, or the playing back of messages intended to be heard after the speaker’s death, complicate the analysis. That they are intelligible though shows that they are not inconsistent.

The fact that Molyneux says that this is “an example of an idea that is inconsistent with itself”, rather than ‘an example of a speech act that is inconsistent with some aspect of its (normal) context of use’, shows that he is not sensitive to the idea of the presuppositions involved when making speech acts. (For some further reading around speech acts and the notion of linguistic presupposition, try these links: here and here.) 

2. Premise 6

This is interesting, because it sheds some light into Molyneux’s premise 4 and 6 of his argument. He wants to argue that these premises are true because their negation would involve some kind of inconsistency with something that they presuppose, such as the example of “I do not exist” above. Thus, without using the term (or being aware of it) he is offering a sort of ‘transcendental’ argument for some of his premises.

Premise 6 is clearer than premise 4, so I will quote that here:

Premise 6: Truth is better than falsehood

If I tell you that the world is flat, and you reply that the world is not flat, but round, then you are implicitly accepting the axiom that truth and falsehood both exist objectively, and that truth is better than falsehood.

If I tell you that I like chocolate ice cream, and you tell me that you like vanilla, it is impossible to “prove” that vanilla is objectively better than chocolate. The moment that you correct me with reference to objective facts, you are accepting that objective facts exist, and that objective truth is universally preferable to subjective error. (p. 35, emphasis mine)

Molyneux is saying that if we have a disagreement about something, then I am “implicitly accepting … that truth is better than falsehood”, and that “truth is universally preferable to subjective error”.

I think that Molyneux is saying that if you argue that p is true, you are showing a preference for truth over falsity. And whatever that means to ‘prefer truth over falsity’, you have to believe that p is true for your arguing that p is true to imply that you prefer truth over falsity. Yet, it can in fact be very reasonably questioned whether arguing that p is true implies that you even believe that p is true, let alone prefer truth over falsity.

Consider the phrase ‘for the sake of the argument’, or the synonymous Latin phrase ‘arguendo’. The function of these phrases is to indicate that something is being postulated provisionally, for the purposes of exploring the implications that come with it. Its function is to indicate explicitly that there is no presupposition that what is being argued for is true. Yet it is being argued for nonetheless.

Consider debate competitions, where the debaters are given topics that they have to come up with a good line of argument for. In such an activity, the purpose of the argument is to perfect the skill of arguing, not for pursuing the truth. It is rather like exercising in a gym, rather than playing a sport. One can become good at running without running any races. Similarly, one can meaningfully engage in debate without thereby presupposing that one believes in what one argues for (or presupposing that one ‘values truth’).

As a additional few examples, consider:

  • A lawyer who argues on behalf of her client because she gets paid to do so
  • A politician who argues what he believes his constituents want to hear
  • An internet troll who argues what she believes will irritate her audience the most
  • An undergraduate who argues what he believes will impress the girl he fancies

What these examples show is that the action of engaging in argument does not need to imply that the arguer believes the propositions for which they argue in favour. They could hold a mercenary-like attitude towards truth-telling, only ever saying what they believe will increase their power with those around them. Who is to say that this isn’t in fact the most common form of arguing?

All of these are examples where the person arguing need not also hold the attitude of ‘valuing truth over falsity’.

3. Conclusion

Unlike other cases of linguistic presupposition, Molyneux’s example isn’t a clear case where it has the presupposition he needs it to have for the transcendental argument to be plausible. Consider a plausible case from the Standford page on presupposition:

  1. The dude released this video before he went on a killing spree
  2. Therefore, the dude went on a killing spree

In contrast, Molyneux’s argument is something like this:

  1. You say (as part of an argument) that the world is not flat, but round
  2. Therefore, you believe that the world is not flat, but round

Or possibly:

  1. You make an argument
  2. Therefore, you value truth over falsity.

Yet, all of the examples of contexts given above show that this does not follow. Thus, the implicit transcendental argument contained in Molyneux’s argument here is invalid. Making an argument does not automatically validate the ‘universally preferable behaviour’ of valuing truth over falsity.


Successive addition

0. Introduction

One of the two philosophical arguments which is supposed to show that the history of the universe must be finite is the impossibility of forming an actual infinite by successive addition. I think this argument begs the question, because there is one premise which can only be true if we assume that the conclusion is true.

  1. The argument

The argument, which can be seen here, looks like this:

1. A collection formed by successive addition cannot be actually infinite.
2. The temporal series of past events is a collection formed by successive addition.
3. Therefore, the temporal series of past events cannot be actually infinite.

I am going to grant premise 2 for the sake of the argument, although I think it could be questioned. All that I want to focus on is premise 1. This premise, it seems to me, can only be regarded as true if we assume that the conclusion is true.

First, what is ‘successive addition’? It means nothing more than continually adding one over and over again, 1 + 1 + 1 …, which is itself akin to counting whole numbers one at a time, 1, 2, 3 … . The idea is that such a process can never lead to anything but a finite result, as Craig explains:

“…since any finite quantity plus another finite quantity is always a finite quantity, we shall never arrive at infinity even if we keep on adding forever. Infinity in this case serves merely as a limit which we never attain.”

2. The counterexample 

There is obviously a close connection between numbers and our concept of time. Exactly what that relationship is, doesn’t matter too much here. One thing that seems obvious though is that we routinely associate sequences of whole numbers with durations of time. Consider the convention which says that this year is 2019. What this means is that if there had been someone slowly counting off integers one per year since year 0, by now he would have counted up to the number 2019.

Adapting this familiar idea, we can postulate that there is some metronomic person counting off whole integers one every minute. After three minutes he will have counted up to the number 3, after one hundred minutes he will have counted up to the number 100, etc.

Let’s make this very simple and intuitive idea slightly more formal. Let us think of a counting function for this person. It takes an input, x, and returns an output, y. The value of x will be some amount of time that has passed (three minutes, one hundred minutes, etc), and the value of y will be whatever number has been counted to (3, 100, etc).

This counting function is therefore akin to asking the question:

‘After x units of time have passed, which number have they counted to?’

The value of y will be the answer to the question.

When Craig says “any finite quantity plus another finite quantity is always a finite quantity”, we can cash this out in our function as saying something like:

If the value of x is finite, then the value of y is finite. 

No matter how much time has passed, so long as it is a finite amount of time, then the number that has been counted to must be merely some finite number.

However, what happens if the value of x is not finite (i.e. if it is transfinite)? Let’s suppose that the amount of time that has passed is greater than any finite amount, i.e. that an infinite amount of time has passed. Will the value of y still remain finite? Clearly, the answer here is no. After all the finite ordinal numbers comes the first transfinite ordinal number, ω. If the amount of time that has passed is greater than any finite amount, but less than any other transfinite amount, then the number that will have been counted to will be ω. That is, the function we have been using so far returns this value if we set the value of x to the right amount. But what this seems to say is that if you had been counting for an infinite amount of time, then the number you would have counted to would be greater than any finite number.

At no point have we specified that we are not using successive addition, i.e. counting. WE have explicitly said that this is what we are doing. All we have varied is how long we have been doing it for. The lesson seems to be that if you only count for a finite amount of time, then you cannot construct an actual infinite by successive addition, but if you do it for an actually infinite amount of time, then you can.

Thus, in order for premise 1 to be considered correct, we have to restrict the amount of time we spend counting to arbitrarily high finite amounts of time. If we place that restriction on, then the premise looks true. But if we take this restriction off, then the premise is false, as we just saw.

This means that whether the premise is true or false depends on whether we think that the value of x can be more than any finite number or not. And that just means whether the extent of past time can be infinite or not. If it can be, then we have enough time to have counted beyond any finite number. Yet, that is the very question we are supposed to be settling here. The conclusion of the argument is that the past is finite. Yet we need to suppose precisely this proposition in order to make the first premise true. Without it, the first premise is false.

Thus, the argument seems to simply beg the question here.

More on the FreeThinking Argument

0. Introduction

In the last post, I explained the ‘FreeThinking Argument Against Naturalism’ (FAAN). I criticised premise 3, which was that “If libertarian free will does not exist, rationality and knowledge do not exist”. In this post, I will look at a reply Stratton gave to an objection that is somewhat similar to mine, in a post he wrote called Robots and Rationality. In the end, nothing he says helps at all.

  1. Robots

Stratton says that some people object to premise 3 of his argument by saying that “computers seem to be rational and they do not possess libertarian free will”. Nevertheless, he thinks that he has a reply to this such that the “deductive conclusions of the Freethinking Argument remain unscathed”.

Right off the bat, Stratton begs the question against the view I outlined in the last post. Consider the very next line, in which he says:

“…simply by stating that computers are, or robots of the future could be, rational in a deterministic universe *assumes* that the determinist making this claim has, at least briefly, transcended their deterministic environment and freely inferred the best explanation (the one we ought to reach) via the process of rationality to correctly conclude that computers are, in fact, rational agents.”

But that’s wrong. The act of stating “computers seem to be rational and they do not possess libertarian free will” can be done in a deterministic universe, no problem. It doesn’t require ‘transcending the environment’. You could even make that statement in a deterministic, naturalistic universe, and have a justified true belief about it while you are doing it.

Here’s how. Assume internalism about justification, so that justifications are beliefs. That means that for me to have a justified true belief that p, the true belief needs to be supported by other beliefs, say q and r, which are the justifications for believing that p. There are two things we can also say about how the justifications need to be related to p. This is not the only way you could cash this out, but it will do for our purposes:

  • They have to be related to p in the ‘right sort of way’ (arbitrary beliefs cannot be justifications), and
  • It needs to be that my believing q and r is why I believe p (it’s no good for me to believe q and r, but believe p because the coin landed heads, etc).

What does it mean for q and r to be related to p in the ‘right sort of way’? This is obviously a very complicated question to answer. We don’t have to settle it here though. Let’s just consider clear cases. The relationship between them just needs to be such that q and r provide significant support for p; they raise the probability, at least the subjective assessment of the probability, of p by the person with the beliefs. A very clear case of this would be if q and r logically entailed p. Other examples would be if q and r were raised the probability of p far above 0.5, to something like 0.9. We don’t need to worry here about exactly where the line is though, because we will consider just a clear example, one where we have logical entailment, because that is clearly a justification. And we only need one example to show that the principle that Stratton appeals to is false, after all. So here it is.

Assume I have two beliefs:

A) My laptop seems to be rational

B) My laptop does not have LFW

I could have those beliefs in a deterministic, naturalistic universe, no problem. These beliefs would be brain states that I have (on assumption). Let’s say that they deterministically cause me to have another brain state, which is:

C) (at least some) computers seem to be rational and they do not possess LFW

Because this belief is caused by the first two, it’s true that I only believe it because I believe the first two. Yet those two paradigmatically justify the third. They logically entail the third. Anyone who believes that their laptop seems to be rational and that it does not have LFW, is thereby justified in believing that (at least some) computers are rational and do not possess LFW.

So on this proposal, I believe C, it is true, and I possess beliefs, A and B, which significantly raise the probability of C (by logically entailing it), and the having of A and B is why I believe C. Thus, it meets the criteria I gave above for counting as a justification for the true belief that I have.

Now, obviously, Stratton would object here. He thinks that the criteria for justifications of p should be that they are beliefs, that are related to p in the right sort of way, you believe p because you have the justifications, and also that believing p was a freely chosen action. But what is the reason that we should accept this additional criteria?

2. Coercion

Assume that some agent A punches some other agent B in the face. Suppose also that A has a desire to hurt B. A natural answer to the question for why A punched B (i.e. what the reason was for A’s action) is that he desired to hurt B. The action could be regarded as free, and the reason is part of why he did the free action.

However, imagine that we learned subsequently that A was a Manchurian candidate, and had been brainwashed, or hypnotised, such that given a certain trigger (maybe by seeing a woman in a polka-dot dress), then he would instinctively take a swing at B. Now, we might think, his antecedent desire to hurt B cannot really be the reason why he punched B. Given that he was compelled to do it (we might say caused to do it), by seeing the woman in the dress, that really isn’t the reason at all. Because he was coerced to do it by the brainwashing, he was not doing it because of the reason he had (his antecedent desire to hurt B). This seems plausible. And if it is right, then being coerced (or being caused) is incompatible with doing something because of having a motivation (like a desire). This could be questioned, but let’s grant it, for the sake of the argument.

Stratton does not give this sort of argument, but you could imagine that it is the sort of thing he has in mind to support his claim that a belief cannot be justified unless it is freely chosen. There does seem to be an analogy here. If the reason for an action cannot be a desire unless it is free of coercion, then maybe the justification for a belief cannot be another belief unless it is free of causation. Maybe each has to be free for it to count.

Even though there is some plausibility to the analogy to begin with, I think it is easy to start to see that the two cases are really quite different. Even if we grant that coercion completely rules out freely acting due to motivations (desires), the case where I come to believe something without willing to believe it is far less clearly problematic.

Consider a case where someone really wants to believe, say, that their son is innocent of a murder. They may, nevertheless, come to believe that the son is guilty during the trial, where all the evidence is presented. We could describe this situation as her being compelled to believe that her son was guilty despite her firm will to not believe this. Of course, this is not a mandatory reading, and no doubt other ways of describing this situation could be given as well. The point is just that this description seems far less problematic than assuming that A was both a brainwashed Manchurian candidate, and also acted with a desire as his reason. Being compelled to believe p, because the evidence caused you to do so, doesn’t seem incompatible with believing p with justification in the same way. Thus, the analogy is clearly questionable.

Stratton did not offer the coercion analogy as an argument against my position. I offered it on his behalf, because I don’t think he has an argument. But to me the analogy is not plausible, because even if you grant the action case, the belief case doesn’t seem problematic in the same way. What’s true about reasons for actions is not necessarily the same as what’s true about justifications for beliefs. And because of that we would need to see an argument to the effect that the claim about beliefs is true, and not just an appeal to the action case.

3. Luck

Stratton makes the following comments a few lines later on, where he appeals to the notion of luck:

“…if determinists happen to luckily be right about determinism, then they did not come to this conclusion based on rational deliberation by weighing competing views and then freely choosing to adopt the best explanation from the rules of reason via properly functioning cognitive faculties. No, given determinism, they were forced by chemistry and physics to hold their conclusion whether it is true or not.” 

So the idea here is that I could believe that determinism is true, and be correct about that (I could be “right about determinism”), but that this is just a matter of luck. He is saying that, in general, on determinism, one could believe p simply because the causal history of the world happened to be such that I hold that belief. If so, then my holding the belief is unrelated to whether p is true, or what the justifications are for holding that p is true; it’s all a matter of what the causal history of the world is like and nothing more.

Three things.

Firstly, luck implies contingency. If an event is lucky, it has to be possible for it to have happened differently, or not at all. For it to be lucky that I won the lottery, it has to be actually possible that I could have lost. If I rigged the lottery so that it had to show my ticket number, then my winning is no longer a matter of luck. But on determinism, all events are necessary, because they couldn’t have happened differently. So while things might look as if they were lucky (in the sense that the rigged lottery result might look lucky), they weren’t really. And if so, then no belief that I hold is lucky.

In order to have lucky events on determinism, then we need contingency. The only way to get that is if the initial conditions of the universe are themselves contingent. But if the initial conditions themselves could have been different, then, since all subsequent events depend on that contingent event, it makes all events contingent, because for each event it could have gone differently. That is, the initial conditions could have been different, leading to the event being different. And if that is all that is required for an event to be considered as ‘lucky’, then all events are lucky, even on determinism. And that means that even if I was caused to believe that p, and p was true, and I was caused to believe it on the basis of justifications, this would still be lucky. The question then is if luck is cashed out like this, just how this undermines the claim that p is a justified true belief. It seems that it doesn’t.

Secondly, putting it in terms of the belief being ‘unrelated to the truth of p’ seems to beg the question against the view I have been defending here. It could be the case that the causal history of the world also includes me having the right types of beliefs, the sorts of beliefs that count as justifiers for p (such as ones which logically entail, or raise the probability a great deal that p is true), and these would be directly related to why I believe that p is true (they are part of the cause of me believing that p is true). If that is right, then it isn’t the case that my holding the belief is unrelated to whether p is true, even if it is lucky (in the sense described above).

Thirdly, Stratton says:

“…given determinism, they were forced by chemistry and physics to hold their conclusion whether it is true or not”

It seems to me that all this is saying is that on the determinist picture, it is possible to believe something false. But I could construct a parody of this, and say:

‘Given LFW, they freely choose to believe their conclusion, whether it is true or not’

After all, you could freely choose to believe something false. That shows that it is also possible on Stratton’ view to believe something false. And that means that regardless of whether the view is determined or freely chosen, it is possible for the belief to be false. So whether you are caused to believe p, whether it is true or false, or freely choose to believe p, whether it is true or false, we are in the same position.

Again, this shows how irrelevant it is to bring up the freeness of the belief. What is important is the justification for the belief. If the justifications are there, then the belief can be JTB, regardless of whether it is determined or freely chosen.

4. Liars

Stratton makes anther appeal:

“If you have reason to suspect a certain man is a liar, why should you believe this individual when he tells you that he is not a liar? Similarly, if we have reason to suspect we cannot freely think to infer the best explanation, why assume these specific thoughts (which are suspected of being unreliable) are reliable regarding computers?”

Thinking that someone is a liar is reason to not trust what they say to you. Fair enough. The problem now is that trusting someone who is a liar, which means someone with a track record of often lying, is not relevantly analogous to thinking that the inferences made by someone is determined are not reliable. It would be, of course, if you considered someone who was determined and who had a track record of making incorrect inferences. But then, the track record is doing all the work, and the determinism is doing none of the work. I wouldn’t trust someone with LFW who had the same track record of lying either.

The problem here is that even if you “have reason to suspect we cannot freely think to infer the best explanation”, that isn’t itself reason to conclude that they are “suspected of being unreliable”. That is, even if you have reason to think that we cannot freely infer the best explanation, that doesn’t on its own mean we cannot infer the best explanation.

What matters is if the process of belief formation takes into account the justifications for holding the belief. Whether it is a determined process or one that involves a free choice is irrelevant.

For example, think of a robot which is equipped with a mechanism that analyses a target at a firing range and processes the information it receives in such a way that it reliably hits a bullseye nine times out of ten. Even though its mechanism is deterministic, that doesn’t mean it is unreliable.

Compare the robot with a free individual, with LFW, who also hits the target nine times out of ten. The reliability of their shooting is something you evaluate by looking at their record of success, and by examining the process by which they came to hit the target. If everything else is equal (they hit the same number of targets, and the internal mechanism of the robot is relevantly similar to the way the person’s eye and brain allow them to determine where to aim the gun), then the freedom itself doesn’t play any role in our assessment of which one is more reliable.

Yet, Stratton makes the assumption that the lack of freedom is a reason to doubt reliability. He says that “if we have reason to suspect we cannot freely think to infer the best explanation” then we have reason to suspect that they cannot infer the best explanation. This seems to me to be false. A sufficiently advanced robot could reliably draw the right inferences yet not have LFW.

5. Self-refutation

Stratton says:

“…the naturalist who states that he freely thinks determinism is true is similar to one arguing that language does not exist, by using English to express that thought.”

But here, surely, the problem is that anyone who states that he (LFW-) freely thinks determinism is true is uttering a contradiction. They are saying both that they believe they are free (in the LFW sense) and the determinism is true, and surely they cannot both be true. Making such a statement would be contradictory.

But, as should be clear by now, the determinist need not make such a statement. Rather than saying that they freely think that determinism is true, they should say that their belief that determinism is true is also determined. When said like that, there is no hint of self-refutation here.

He goes on:

“Until naturalists demonstrate exactly how a determined conclusion, which cannot be otherwise and is caused by nothing but physics and chemistry, can be rationally inferred and affirmed, then the rest of their argument has no teeth in its bite as it is incoherent and built upon unproven assumptions.”

I hope that by now the general idea of how this would work is clear. What has to be made explicit is that beliefs can be caused, but so long as they are caused by other beliefs (brain states, if you like), then they can still stand in the same relation to justifiers as they do on any other JTB view. So the question of ‘Why do you believe that (at least some) computers seem to be rational and they do not possess LFW?’ is answered by saying ‘Because I believe that my computer is rational and it does not have LFW’. That answer is true, even though there is also a story we could tell about how some bit of brain chemistry lead to some other bit. If those bits of brain chemistry are beliefs, then both ways of talking are true.

This is a familiar line. Why did the allies win world war two? Because Hitler overreached by invading Russia. That’s true. But, of course, there is a much more detailed story involving the precise movements of every regiment across the whole of the world. There is another story that involves the movement of all the atoms across the whole world too. All three of these are true. The fact that the much more detailed story about atoms is true doesn’t mean the others are not. It doesn’t mean the others are reducible to the story about atoms either (maybe they are, but maybe they aren’t).

The same sort of thing is going on here. There is a story about what happens at a chemical level in my brain, and another one about what beliefs I believe on the basis of other beliefs. If naturalism is true, beliefs are something like brain states. If determinism is true, then they can cause other brain states to exist. So long as this causal chemical set of reactions is correlated reliably with inferring the best explanations, then it is as good as the LFW account.

But are they correlated in that way? Well, not by default. The actions we engage in train them up. Learning to speak, going to school, reading philosophy, etc. These sorts of things  make us better at inferring the right things from our beliefs. But that can be told as a chemical causal story too. When I study I am causing my brain to make more reliable connections more often. The pathways in my brain become intrenched in certain ways, leading to me more often getting it right. Not always, but often enough to count as being rational (rationality comes in degrees, after all). Nothing about this requires LFW. All of those actions can be deterministically caused.

6. Conclusion

Stratton ends with this:

“If all is ultimately determined by nature, then all thoughts — including what humans think about the rationality of computers — cannot be otherwise. We are simply left assuming that our thoughts (which we are not responsible for) regarding computers are good, the best, or true. We do not have a genuine ability to think otherwise or really consider competing hypotheses at all.”

Firstly, note that now he is insisting that on determinism, our thoughts cannot be otherwise. If that’s right, then they should not be regarded as being lucky, or unlucky.

But regardless of that, he says that in that situation, we “are simply left assuming that our thoughts … regarding computers are good, the best, or true.” But as I showed here, we are not left in that situation. I could come to that conclusion because I also have other beliefs, which are relevant to that conclusion. He is saying that if we are determined, then we are left with nothing but assumptions. He is saying that if we are determined, then we cannot think about competing hypotheses and weigh options against each other. This is clearly incorrect. All we cannot do if we are determined is freely do those things. What we can do, if we are determined, is do those things.

Thus, the third premise of his argument fails. Nothing he says in his Robots and Rationality article helps, at all.