## The Logical Form of the Grim Reaper Paradox

[Edit: it turns out that something quite similar to this is argued for in this paper by Nicholas Shackel]

0. Introduction

The Grim Reaper Paradox (GRP) comes in various different forms. Sometimes it is about the divisibility of time, and sometimes it is about whether the past (or future) is finite. Even when we fix on which of these issues it is aimed at, there are also lots of different ways it can be cashed out. It can be reapers swinging their scythes, or placing point particles on planes, etc. Much of the discussion can be on how these details are to be understood.

Here I want to highlight what the GRP is at the most abstract level. It might be that once we think about it from this rarefied perspective, without the complications about what exactly the details are supposed to be, that we can see the paradox more clearly.

1. The Schema

Any GRP has a logical form, which I shall refer to as the schema. Let’s just have as a toy example, the following:

The past has no beginning. There is an eternal machine such that each day at midnight, it checks to see if it has printed out anything yet from its printer. If it has, then it hibernates for the rest of the day. If it has not printed anything out yet, it immediately prints out the date and then hibernates for the rest of the day.

This is enough to generate our paradox. If it had not already printed anything out, this means that yesterday it would have run the same check and printed out the date. So it can’t be that the machine finds nothing printed out today. But that applies also to yesterday too, and every previous day. So although it can’t be that no date is printed out, no date could be printed on the paper.

The way to conceptualise this abstractly is as follows. There is a rule that characterises this example (and all the others). It is a universal condition that applies at all times. That condition says that some proposition p (which might be that a reaper kills Fred, or places a point particle on a plane, or that a machine prints out a date, etc) happens at a time if and only if p does not happen at any earlier time:

For all t (p at t iff for all t’ (if t'<t, then ~p at t’))

It says that p is true at t if and only if p is not true at any earlier time.

The schema on its own is not unsatisfiable. That is to say, if there are only finitely many times, then the schema can be true. In particular, the schema is true if there are only finitely many times and p is true at the first time. At that first time, p is true at t, and on the other side of the biconditional, the nested conditional (if t'<t, then ~p at t) has a false antecedent, and as it is in the scope of a universal quantifier it is vacuously true. So both sides of the schema are true. At all other times, p is not true at t (so the left side is false), and on the other side of the biconditional we have a condition that says that ~p is true at all earlier times, which is false because (as we just went through), p is true at the first time. So in all cases, the biconditional holds.

But if there is no first time, then we run into the familiar problem. If p is true at some time t, then the right side of the biconditional says that no earlier time, t’, could have p true at t’. But then take t’. It is also the case of that time that no earlier time, t”, has p true at it either. So given the left side of the biconditional, p is true at t’. Contradiction.

If p is not true at any time, then the left side of the condition is false for some arbitrary time t. But if p is not true at any time, then its not true at any time t’ earlier than t, which makes the right side of the biconditional true, which in turn implies p is true at t. Contradiction.

2. Conclusion

Now we have a purely logical version of the argument, freed from any distractions about reapers, or point particles, or eternal machines. The GRP really just says:

1. There is no first time t
2. For all t (p at t iff for all t’ (if t'<t, then ~p at t’))

As we have just seen, you can’t have both of these together. That is the GRP.

## Counting forever

0. Introduction

Here I just want to explain a simple point which comes up in the discussion of whether it is possible to ‘count to infinity’, and what that tells us about whether time must have had a beginning. Wes Morriston deserves the credit for explaining this to me properly. All I’m doing is showing two places where his point applies.

1. The targets

I have in mind two contemporary bits of philosophical literature. One is found in Andrew Loke’s work, specifically his 2014 paper, p. 74-75, and his 2017 book, p. 68. The other is found in Jacobus Erasmus’ 2018 book, p. 114. In each case, the authors are arguing that it is not possible to count to infinity because no matter how high one counts, no matter which number one counts to, there are always more numbers left to count. Here is how they express this point.

Firstly, here is Loke in his paper:

“If someone (say, George) begins with 0 at t0 and counting 1, 2, 3, 4, … at t1, t2, t3, t4, … would he count an actual infinite at any point in time? … The answer to the question is ‘No’, for no matter what number George counts to, there is still more elements of an actual infinite set to be counted: if George counts 100,000 at t100,000, he can still count one more (100,001); if he counts 100,000,000 at t100,000,000, he can still count one more (100,000,001).” (Loke, 2014, p. 74-75)

Secondly, here is Loke in his book:

“Suppose George begins to exist at t0, he has a child at t1 who is the first generation of his descendants, a grandchild at t2 who is the second generation, a great-grandchild at t3 who is the third generation, and so on. The number of generations and durations can increase with time, but there can never be an actual infinite number of them at any time, for no matter how many of these there are at any time, there can still be more: If there are 1000 generations at t1000, there can still be more (say 1001 at t1001); If there are 100,000 generations at t100,000, there can still be more (100,001 at t100,001), etc.” (Loke, 2017, p. 68)

Finally, here is Erasmus in his book:

“Consider, for example, someone trying to count through all the natural numbers one per second (i.e. 1, 2, 3, . . . ). Can the person count through the entire collection of numbers one per second? Clearly not, for no matter how many numbers the person has counted, there will always be an infinite number of numbers still to be counted (i.e. for any number n that one counts, there will always be another number n + 1 yet to be counted). Therefore, it is impossible to traverse an actually infinite sequence of congruent events and, thus, if the universe did not come into existence, the present event could not occur.” (Erasmus, 2018, p. 114)

What each is saying is that if someone starts counting now, they will never finish counting. And this is true, of course.

Think about the following mountain: it has a base camp at the bottom, but it is infinitely tall and has no highest point (for each point on the mountain, there is another one which is higher than that point). Can one start at the bottom and climb to the top of such a mountain? No, because there is no top of such a mountain.

(My dispute with Craig was not over exactly this point, but on something slightly more subtle. That was whether the following is false:

A) It is possible that George will count infinitely many numbers.

I say that A is true. All Loke and Erasmus’ sorts of considerations get you is to say that the following is false:

B) It is possible that George will have counted infinitely many numbers.

But we can leave this point here for now.)

2. My point

All I want to highlight today is that the fact that there are ‘always more numbers left to count’ only applies to certain types of infinite count. Imagine the following three scenarios:

i) George is trying to count the positive integers, in this order: (1, 2, 3, …)

ii) George is trying to count the negative integers, in this order: (…, -3, -2, -1)

iii) George is trying to count all the negative and all the positive integers, in this order: (…, -3, -2, -1, 0, 1, 2, 3, …)

We can think of the scenarios like this:

• Scenario i) is like climbing up an infinitely tall mountain that has a bottom but no top;
• Scenario ii) is like climbing up an infinitely tall mountain that has no bottom but does have a top;
• Scenario iii) is like climbing up an infinitely tall mountain with no bottom and no top.

In each case, due to the nature of infinite sets, the tasks involve counting the same amount of numbers. Simple intuition tells you that the third involves counting more numbers than the first two (and should be the sum of the first two). However, it is actually the case that each scenario involves counting sets with the same cardinality (that is, ℵ0). Put another way, each mountain is the same height as the other two.

The key point I want to make is this. It is obviously true that the Loke-Erasmus observation (that “there will always be an infinite number of numbers still to be counted”) only applies to scenarios i) and iii). It just doesn’t apply to scenario ii).

When George is starting at 0 and counting up, he always has the same amount of numbers left to count (always ℵ0-many). The same is true for when he is in scenario iii), no matter where George is along his task.

But if he is in scenario ii) he is in a very different situation. No matter where George is in his task in this scenario, he only ever has finitely many numbers left to count. He doesn’t have infinitely more mountain to climb in scenario ii). It has a top, and no matter where he is, George is only finitely far away from the top of the mountain. Clearly, he can reach the top of such a mountain if he is already some way along his climb.

And here is where the rubber meets the road. If there is a problem with scenario ii), it is not that “there will always be an infinite number of numbers still to be counted”, because that is just false. And scenario ii) is the one that Loke and Erasmus ultimately have in their sights. That’s because it is the one where George ‘completes’ an infinite task by counting an actually infinite amount of numbers.

Put simply, the argument they are making looks like this:

1. George cannot count up to infinity, because there would always be more numbers left for him to count.
2. Therefore George cannot count down from infinity.

Put like this, the fact that the argument is invalid is plain to see.

The problem, if there is one, is not about completing, or finishing, an infinite task. It might be that the problem has something to do with starting such a task. But if so, there is really no point in talking about problems that involve the impossibility of finishing, as that is a different point.

3. Conclusion

There might be other reasons to think that one cannot count down from infinity, of course. Indeed, Loke and Erasmus both have more to say about this issue. But what one often finds in discussions like this is the following sort of move: they make an observation that applies to an endless series, and try to apply that to a beginningless series. As is simple to see, sometimes the initial observation (such as that there will always be more numbers left to count) just doesn’t apply to both. And the switch from one to the other is thereby not valid.

## Endless and Infinite

Philosopher Wes Morriston and I have coauthored a paper on the Kalam, and it has been accepted publication in the journal Philosophical Quarterly. Once it is actually available on their page access will probably be limited, unless you have an institutional subscription. However, for now you can download it (for free) via this link:

Endless and Infinite

Thanks,

Alex

## The paradox of dry eternity

0. Introduction

Like many people, I am participating in ‘dry January’, meaning that I am not drinking any alcohol during the month of January. I’m also thinking about the Grim Reaper paradox, and have spent much of the year thinking about the infinite future debate between Morriston and Craig. Interestingly, all of these things have come together, in a paradox I shall now dub the ‘dry eternity’ paradox.

Part of the inspiration for this comes from a paper I read recently by Yishai Cohen, called ‘The Endless Future: a Persistent Thorn in the Kalam Cosmological Argument‘(2015). In that, Cohen agrees with Morriston that if a beginningless past is an actual infinite, then so is an endless future, and thus if a beginningless past is impossible for being an actual infinite, so is an endless future. Cohen also argues that if the grim reaper argument shows that the past has to be finite, then a parallel version shows that the future must have an end as well. His argument is critiqued by Jacobus Erasmus in a paper called Cohen on the Kalam Cosmological Argument (2016). Erasmus’ rebuttal is that Cohen’s version of the grim reaper argument presupposes that it is possible for God to actualise an ungrounded causal chain, which can be plausibly denied.

1. Two versions of the Grim Reaper paradox

Here is how Erasmus sets up the grim reaper paradox:

Suppose that the temporal series of past events is actually infinite and that an actually infinite number of Grim Reapers exist. Suppose also that, at each past moment in time, a unique Reaper was assigned to issue a death warrant iff no previous Reaper had already issued a death warrant. (Cohen on the Kalam Cosmological Argument, p. 52)

This results in a contradictory state of affairs. Firstly, for all times tn, there must have been a warrant issued prior to tn. That’s because if no warrant had been issued by tn-1, then the reaper at tn-1 would issue their warrant (resulting in a warrant going out prior to tn).

But, this same reasoning also applies to tn-1 itself, giving us the contrary proposition. That’s because we can also say that the warrant won’t be issued at tn-1, because if it had not been issued by tn-2 it would have been issued by the tn-2 reaper (i.e. before tn-1).

Thus, we have both that the warrant must have been issued at some time prior to tnbut also that there is no time prior to tat which it could be issued. Contradiction.

Cohen applies this to the endless future. All we need to do is change the relevant tenses in Erasmus’ quote from above to get the following:

Suppose that the temporal series of future events is actually infinite and that an actually infinite number of Grim Reapers exist. Suppose also that, at each future moment in time, a unique Reaper is assigned to issue a death warrant iff no future Reaper will issue a death warrant.

But now we can derive a mirror image contradiction. Suppose that the reaper at t0 checks to see if any future reapers will issue warrants or not. At this point there are two options:

i) no future reaper will issue a warrant (in which case the treaper issues theirs)

ii) some future reaper will issue a warrant (in which case the treaper does nothing)

Suppose that at tno future reaper will issue their warrant, meaning that the reaper at t0 can issue theirs. If it were the case that no future reaper issues a warrant at t0, then, in particular, it is also true that at tno future reaper will issue a warrant (consider: if it it true today that I will never drink again, then it will also be true tomorrow that I will never drink again). But if it is true at tthat no future reaper will issue a warrant, this would mean that the reaper at tdoes issue their warrant! And, plainly enough, the reaper at tis in the future of the reaper at t0. So if, at t0, no future reaper will issue a warrant, then some future reaper (such as the one at t1) will issue a warrant! Contradiction.

Let’s take the other horn. Suppose at tsome future reaper will issue their warrant, meaning that the reaper at t0 can stand down. Let’s suppose it is the reaper at t1. Then we are right back to the beginning of the first horn again. For the reaper at twill only issue their warrant if none of the future ones will. But if it is true at tthat none of the future ones will, then this is also true at tas well, resulting in that reaper issuing their warrant which in turn brings about another contradiction exactly like the one from above.

Cohen discusses two objections that Koons posed to him in correspondence. The first of these is that in Koons’ version of the paradox, reapers are sensitive to what past reapers have done, but in Cohen’s version they have to be sensitive to what future reapers will do; yet it isn’t possible to have causal sensitivity to future events in the same way as to past ones. The reply Cohen makes to this is that an omniscient God could communicate the future to the reapers such that they know what the others will do, thus overcoming this causal asymmetry. Koons’ second point is that it isn’t possible for God to create beings who embody his omniscience. Even if that is true, the reapers themselves do not have to be omniscient (and can be quite ignorant of, say, how many coins I have in my pocket), just so long as God ensures that they know the behaviour of future reapers. In addition, Cohen points out that Koons’ reason for thinking that the reapers cannot embody omniscience has to do with avoiding causal loops, but it is not clear that there are any causal loops as such in this story (the behaviour of reaper n+1 does not depend on the behaviour of reaper n, etc). Thus it is far from clear that Koons has a successful reply here. One could avoid this by denying the possibility of an omniscient being that knows the future and can communicate it to reapers, of course, but a theist (in particular a Christian theist) will be unlikely to pick that option.

2. Erasmus’ objection

Erasmus’ objection comes at this from a different angle. He says that Cohen’s version of the grim reaper paradox (GRP) requires the following two principles:

K1. It is possible for God to predetermine an endless future

K2. It is possible for God to actualise an ungrounded causal chain

An ‘ungrounded causal chain’ “has a non-well founded relation (xRy, zRx, zRv, wRv, … ) because the chain lacks a first cause” (Erasmus, 2016, p 53). The behaviour of the reaper at t0 is determined by (or grounded in) the behaviour of reapers that are in its future. But the behaviour of the reapers in its future, such as the one at at t1, are themselves determined by (or grounded in) the behaviour of reapers in the future of them as well. Thus there is no ‘first cause’, or grounding, for the behaviour of the reapers. Let us suppose that this is an ungrounded causal chain, and that it is also (metaphysically?) impossible for God to actualise such a causal chain.

He then goes on to show that K2 is doing all the work in generating the paradox because it also applies to ungrounded causal chains that are purely spatial in character. Here is his spatial version of the example:

For example, suppose that time had a beginning and has an end. Accordingly, the predetermined series of future events is finite. Suppose further that space is inhabited only by an actually infinite row of successive Grim Reapers such that (1) there is a first Reaper but no last Reaper, (2) each Reaper is located at a unique spatial point, and (3) all the Reapers are facing the same direction. Now, suppose that God has predetermined that, at noon tomorrow, each Reaper will swing his scythe iff no Reaper in front of him swings his scythe. Accordingly, the same contradiction as above will result at noon tomorrow, namely, regardless of whether the first Reaper swings his scythe, it is both true and false that some Reaper in front of the first Reaper swings his scythe. The contradiction disappears, however, if (K2) is false. (Ibid)

Erasmus’ conclusion then is that denying K1 is not enough to block the contradiction, as it reappears in the spatial case. But denying K2 blocks both contradictions, and as such K1 is not the offending assumption. In effect, he is saying that Cohen’s GRP doesn’t show that the future must have an end. Rather, it just shows that God cannot actualise ungrounded causal sequences.

Now is time to present my version of the paradox that does works even if K2 is false. It does not require that God actualises any ungrounded causal sequences. All that it requires in addition to K1, is one additional assumption:

K3. God can act based on his (presently available) knowledge of future events.

Suppose God has decided to undertake an infinite version of dry January. That is, he has decided to stop drinking (say) holy water forever. However, he enjoys a drop of holy water (who doesn’t?), and wants to to have one final sip. Accordingly, he determines to obey the following rule:

Every day, God will check his comprehensive knowledge of all future events to see if he will ever drink again. If he finds that he does not ever drink again, he will celebrate with his final drink. On the other hand, if he finds that his final drink is at some day in the future, he does not reward himself in any way (specifically, he does not have a drink all day).

Again, we are caught in a dilemma:

Firstly, suppose that, at t0, God consults his comprehensive knowledge of the future, and discovers that he never again drinks after t0. He immediately downs a shot to celebrate (who wouldn’t?). But in that case, when he does his check the next day, at t1, he then will (again) discover that he will never have another drink, and immediately pour himself a drink to celebrate! So even though he rewarded himself yesterday for never having another drink, he is having another drink! Contradiction.

On the other horn, suppose that, at t0, God consults his comprehensive knowledge of the future, and discovers that he does indeed have a drink at some day after t0. Accordingly, he doesn’t celebrate by having a drink on t0. But in that case, there must be some future day at which he has a drink. Suppose it is t1. In that case, it must be that at t1 God will check to see if he will have any subsequent drinks, and find that he will not, resulting in him pouring the last drink. But now we are back at the start of the first horn, because his check at at twill also reveal a dry eternity ahead, at which point he will reward himself with another final drink! Contradiction again.

So we clearly have the exact same paradox again. This time however, it is not clear that God has actualised an ungrounded causal chain. After all, at each day God knows the future, and can merely consult his own (presently available) knowledge to see what happens in the coming days. We can imagine him writing it all down in a big book and every day he consults the book. Whatever causal story that happens each day that he consults the book, it is not clear that it is an ungrounded causal chain.

4. Replies?

One might deny that God can check his own knowledge to see what he knows about the future and act on it (K3). This would be weird. Why can’t God do that? Does he not know what he foreknows? Is he repressing it? Can he not act on what he knows? He seems to act on his foreknowledge on most versions of theism (specifically any where he has a plan, or reveals the future in prophecy, etc). Denying K3 leaves only the most austere versions of deism, it seems to me. Christianity seems hard to reconcile with its denial in any case.

Objecting to the possibility of the book doesn’t help unless it is really an objection to God’s omniscience, which a theist probably isn’t going to opt for (apart from Open Theists). Denying the possibility of an omniscient being would avoid the paradox of dry eternity though.

One could avoid the problem by denying the possibility of an endless future. The whole point of Morriston’s original reply to Craig was to say that if the past must have a beginning, then the future must have an end. This would vindicate Morriston’s challenge against Craig. It would show that either time has both a beginning and an end, or no beginning and no end, but that there is no third option.

The seemingly only other target we can find is the rule that God undertakes to obey. Perhaps it is not a proper rule, and that somehow it isn’t possible for God to undertake to obey it. Yet this seems rather strange. Consider this similar rule:

Every day, God will check his comprehensive knowledge of all future events to see if he will ever drink again. If he finds that he does not ever drink again, will celebrate with a chocolate bar. On the other hand, if he finds that his final drink is at some day in the future, he does not reward himself in any way (specifically, he does not have a chocolate bar).

Nothing paradoxical follows from this rule. Obeying this rule means that God drinks every day up to the day when he has his final drink, after which he eats chocolate bars every day. God can obviously follow that rule.

But what could stop him from undertaking to follow the rule obtained by merely swapping out the word ‘chocolate bar’ with ‘have his final drink’? Of course, it would lead to contradiction if he were to go this route, and that is a reason to think that it is (somehow) metaphysically impossible for him to swap those words around and undertake to follow the resulting rule. On the other hand, that is just to say that this is one of the things that could be denied to avoid the paradox. It doesn’t motivate thinking that it is impossible. We could ad hoc postulate anything is metaphysically impossible to avoid any paradox.

Something has to go to avoid the paradox of dry eternity.

## The Grim Reaper Paradox, and the original solution, part 1

0. Introduction

Robert Koons has published a version of the ‘Grim Reaper paradox‘, also popularised by Alexander Pruss. One of the assumptions that sets up the paradox must be abandoned in order to avoid the contradiction. Koons and Pruss’ conclusion is that the assumption that an infinite past is possible should go.

It turns out that there is a potential solution to the paradox which was proposed by its originator Jose Benerdete, and more recently brought out in a paper by John Hawthorne (2000). I’m going to explain that solution here, and then (hopefully) develop it to the Koons/Pruss version in a part 2.

The Grim Reaper paradox was first put forwards by Jose Benardete in his 1964 book ‘Infinity‘ (p. 259). Although the developer of the paradox, Benardete was a passionate defender of the actual infinite, announcing in the blurb on the cover of his book that:

“This book is an attack on finitism in all its forms … A metaphysics of the actual infinite is offered as the solution to the contemporary crisis in the foundations of mathematics”

It is perhaps a consequence of the clarity with which he engaged with the topic was that his examples have since been able to be used to argue for precisely the opposite thesis; that is, for finitism. This is how the grim reaper paradox has been used in the contemporary philosophy of religion debate, primarily by Alexander Pruss and Robert Koons, who argue for a version of finitism (‘causal finitism’).

Here is how Benardete states the paradox:

“A man is shot through the heart during the last half minute by A. B shoots him through the preceding 1/4 minute, C during the 1/8 minute before that, &c. ad infinitum. Assuming that each shot kills instantly (if the man were alive), the man must already be dead before each shot. Thus he cannot be said to have died of a bullet wound” (p. 259)

Koons describes this story as follows:

“The story leads quickly to a contradiction, on the assumption that Fred [i.e. A] does not die unless one of the Reapers kills him. At least one Grim Reaper must act, since if all of the Reapers whose numbers are greater than 1 do nothing, then Reaper #1 will act. However, it is impossible for any Grim Reaper to act, since, for any n, Grim Reaper #n cannot do so unless Fred survives until its assigned deadline at 1/2n seconds after midnight. It is impossible for Fred to survive that long, since Fred’s surviving until Reaper #n’s deadline entails that no Grim Reaper with a number larger than (n+2) has acted, but, in that case, Reaper #(n+1) must have acted.” (Does the Universe Have a Cause?, p. 4)

The contradiction lies in the fact that Fred will surely die before the end of the minute, but that also there is no Grim Reaper who will kill him. What this paradox seems to show, at least according to the finitists, is that there could not be finite duration of time (such as a minute) that is actually divided into infinitely many sub-regions.

2. The original solution, part 1: Benerdete.

Benardete did not draw the same conclusions as Pruss and Koons from the grim reaper paradox, and remained committed to the possibility of actually infinitely divided durations and lengths. Shortly after stating the paradox, he makes the following comments about it:

As to the dead man, although he did not die of any single bullet wound, his death was certainly caused by the infinite fusillade of shots. Here, again, although he is already dead prior to each shot, he remains alive at any assigned instant which is prior to them all. Thus he cannot be said to have died at any moment of time whatever! (Infinity, p. 260)

These brief comments offer only a hint of how to understand the response. The outline can be made out however.

Firstly, Fred is alive at every moment prior to all the shots. Yet, for each shot he is already dead before that one is fired. It follows from this that “he cannot be said to have died at any moment of time whatever!” We can spell this out by saying that although there is no first moment in which it is true that Fred is dead, there is a last moment at which Fred is alive. And this is to be expected. The series of bullets (reapers) is an open infinite sequence; while there is a last bullet at 12:01, there is an endless sequence of bullets ever closer to 12:00, and in particular there is no first bullet. Given the continuity of time, we can think of the transition from 12:00 to later times as a Dedekind cut on the real numbers:

In the above diagram, the left line is time as it approaches 12:00, during which Fred is alive. This is a closed set (which is what the square join indicates), in that 12:00 is a member of that set, but it is the final member. So Fred is alive at every time up to and including 12:00. However, the line on the right is an open set (which is what the curved join indicates). So Fred is dead at every time strictly after 12:00. This is what Benardete means by “he cannot be said to have died at any moment of time whatever”. There is no first moment at which he is dead (rather, there is just a last moment at which he is alive).

This much of the solution seems fairly straightforward. It is addressing the time of death, and we have the following information. When is Fred alive? All the way up to and including 12. When is he dead? At all times later than 12. If time is continuous and actually infinitely divisible, this is what it is like to transition from one state to another (it is like a Dedekind cut on the real number line).

The second bit is where the difficulty lies. It involves the cause of death. Benardete has it that no bullet (reaper) is the individual that causes the death of Fred (“he did not die of any single bullet wound”). Yet, the totality of all of them does cause Fred to die (“his death was certainly caused by the infinite fusillade of shots”). But how can it be that the totality of the bullets causes him to die, when none of them individually causes him to die?

3. The original solution, part 2: John Hawthorne

This is where Hawthorne comes in. His paper Before-Effect and Zeno Causality develops Benerdete’s solution, spelling out the principles in greater detail.

Hawthorne first considers the case of a ball rolling towards an open-infinite Zeno-sequence of walls. 2 miles away there is a wall; 1 and 1/2 miles away is another wall; 1 and 1/4 miles away is another wall; 1 and 1/8 miles away there is another wall, etc. Thus, there is an infinite sequence of walls, ever closer to the point that is exactly one mile away. There is no wall which is the ‘closest’ to the one mile point (which makes it an open sequence). Suppose the walls are impenetrable and cannot be knocked over (etc). The ball is rolled towards the walls. What happens as it arrives at the one mile mark? Hawthorne’s answer is as follows:

The ball does not proceed beyond a mile and it does not hit a wall.” (p. 625)

We are puzzled by this combination partly because we have fuzzy intuitions about what happens when there is ‘contact’ between objects and open series in this sort of setting (where space is continuous and actually infinitely divided). But we can spell it out by specifying what we mean by ‘contact’ in such a way that it makes sense. Hawthorne calls ‘open-closed contact’ what happens when “A closed surface contacts an open surface insofar as there is no unoccupied space in between the two surfaces.” When the ball arrives at the one mile point, it has achieved open-closed contact, in the sense that the closed surface of the ball has no unoccupied space between it and the infinite series of walls. Here is what Hawthorne says:

Consider the fusion of walls. Call it Gordon. On reflection it is clear that the sphere contacts Gordon. Gordon has an open surface. When the ball stops proceeding at the one mile mark, there is no unoccupied space between the sphere and Gordon. Contact occurs …  So the ball is stopped by contact: The ball hits something, though the thing that it hits is not one of the walls. (p. 626)

If we find this puzzling still, Hawthorne has the diagnosis at hand. It is because we are assuming what he calls the ‘contact principle’, namely:

If y is the fusion of x’s and z contacts y, then z contacts one of the x’s.

That principle holds for the finite case. But it is false if the x’s are infinite in number. Once we are clear about this, there is no residual puzzle, nor anything further to learn about the wall case. It is clear what happens in worlds that satisfy the original description: At a mile, the ball makes contact with the fusion of walls, which is rigid and impenetrable. As a result, it does not proceed further. The ball does not, however, make contact with any wall. (Ibid)

It is weird that the ball can be stopped by making contact with the totality (or ‘fusion’) of all the walls even though it does not contact any particular wall,  but that is partly because we are taking an intuition which is applicable to the finite cases only, and trying to apply it to the infinite case. In finite cases, there will be a first wall, and contact would be defined in relation to that wall. However, this is precisely what we do not have in this case, and relying on an intuition that presupposes that there is one will lead us into trouble.

4. From walls to reapers

When it comes to the case of the infinite sequence of bullets (or reapers), Hawthorne basically applies the same analysis. Instead of the ‘contact principle’, he identifies the analogous principle and calls it the ‘change principle’:

“If x is the fusion of y’s and y’s are individually capable only of producing effect e by undergoing change, then x cannot … produce effect c without undergoing change”

This principle is also true in the finite case but false for the infinite case. Each of the bullets (reapers) are only capable of producing the effect of killing Fred by undergoing change (by being shot, or by swinging their scythes, etc). The assumption is that this applies to the totality of bullets (reapers). And it does, in finite cases. If there were only 10 bullets, then the totality would have to change, in the sense that one of the bullets would have to be fired at Fred, for it to bring about its effect. Yet, in the case of the infinite sequence of bullets (reapers), this is not the case. The totality can bring about the effect even though none of its elements (or the totality itself) changes in any way.

Going right back to Koons, we can apply this to his comments. In the quote I gave of his from above, he begins by saying:

“The story leads quickly to a contradiction, on the assumption that Fred does not die unless one of the Reapers kills him.” (Emphasis added)

What Hawthorne’s approach questions is this assumption. It is false that one of the reapers kills Fred, but that doesn’t mean that the effect is not brought by the totality of reapers. In the infinite case at hand, that is what happens. Once again, the weird behaviour of the infinite confounds our intuitions. Yet, giving up this intuition saves the situation from contradiction, and thus avoids the paradox.

5. Conclusion

Of course, some will argue that a ball cannot be stopped by a bunch of walls unless one of those walls makes contact with the ball, and the move from finite to the infinite context should make no difference to this.

Hawthorn is sensitive to the seemingly radical nature of the conclusion, admitting that it is a “big metaphysical surprise”. He goes on to finish by saying that:

“The Contact Principle, in full generality, could be given up fairly readily on
reflection. The Change Principle has a rather deeper hold on us. It seems to us scarcely thinkable that mundane causal powers—say that of killing with a machete—could combine so as to logically entail the causal power of producing some effect without the agent of the effect undergoing change. Nevertheless, surprising as this may be, the Change Principle should be rejected. The diagnosis is complete. The logic of each case is very much in order. And our puzzlement has been traced in each case to some faulty principle relating fusions to parts. Once we discard those principles, we will have no problem in accepting the required conclusions about what happens in each case” (p. 630-631)
I intend to write a part 2 to this, where I apply Hawthorne’s analysis to the version of the Grim Reaper paradox in which the target is not the actually infinitely divided duration, but the actually infinite past.

## Darth Dawkins’ Failed Argument

0. Introduction

Recently Darth Dawkins (i.e. aggressive presup shouty man) has been running an argument according to which agnosticism is contradictory. You can see him make that argument in this short clip. The argument is fallacious, and I pointed that out to him recently. But just to make things crystal clear, I figured I would put it down in writing too.

1. Darth’s argument

Darth’s argument starts at 2:17 in that video. He says:

Another problem is that agnosticism is the claim that ‘I don’t know that any creator god exists or does not exist’. Now, if that statement is true, then it necessarily follows that Christianity is false. Now, if it necessarily follows that Christianity is false, then the agnostic knows that at least one major contender for the creator god does not exist, thereby contradicting the agnostic statement.

The way this is supposed to work is that the specific idea of the Christian God that Darth has in mind involves the notion of being ‘revelatory’, which is to say that it is part of the definition of God that he has revealed himself to you. On Darth’s conception, if this God existed, then you would know that he existed. It is this concept of God that he thinks agnosticism is incompatible with.

Here is  how the argument is supposed to run:

1. If Darth’s God exists, then you would know that he exists (Darth’s definition)
2. You do not know that Darth’s God exists & you do not know that he does not exist (agnosticism)
3. Therefore, you do know that Darth’s God does not exist. Contradiction!

The conclusion contradicts the second conjunct of premise 2. This is supposed to show that you cannot be agnostic about all gods, because some gods are such that if they existed you would know about them existing. Your not knowing about them existing is enough to know that they do not exist.

On the face of it, there is something fairly intuitive about this argument, and when he is aggressively shouting the premises at people it can be hard to spot where it goes wrong. But on closer inspection it is pretty clear that it is invalid, and we can bring this out very vividly.

2. The problem

So what’s wrong with the argument? Well, first of all, the argument is compressed, and there are clearly steps we haven’t made totally explicit. What exactly is the inference rule we are using to get to the conclusion? It’s not clear. So let’s make it easier. Let’s forget about the second conjunct of the second premise for a minute. Consider the following two premises:

1. If Darth’s God exists, then you would know that he exists
2. You do not know that Darth’s God exists

What follows from these premises? Well, it is basically the first two premises of modus tollens, i.e. ‘if p, then q’, and ‘not-q’. So we can apply that here and derive ‘not-p’ as follows:

1. If Darth’s God exists, then you would know that he exists
2. You do not know that Darth’s God exists
3. Therefore, Darth’s God does not exist

We can logically derive from the first two premises that Darth’s God doesn’t exist. If he did, I would know about it, but I don’t, so he doesn’t. So far, so good.

The contradiction Darth wanted to derive was using the second disjunct of agnosticism; ‘you do not know that god does not exist’. We snipped this off just to simplify the argument, but now we should bring it back in:

1. If Darth’s God exists, then you would know that he exists
2. You do not know that Darth’s God exists & you do not know that Darth’s God does not exist
3. Therefore, Darth’s God does not exist

The problem is that so far the conclusion is not the negation of this conjunct. To make it the negation, the conclusion would have to be ‘you know that it is not the case that Darth’s god exists’.

But the modus tollens we applied originally does not get us to this new conclusion. That is, the following is invalid:

1. If Darth’s God exists, then you would know that he exists
2. You do not know that Darth’s God exists
3. Therefore, you know that Darth’s God does not exist

Why is this invalid? Well, simply put, we can imagine the premises true and the conclusion false. Here is one example. Let’s just grant premise 1, as it is basically a definition anyway. Let’s suppose the subject in question does not know that this God exists (making premise 2 true). All we have to further suppose is that he doesn’t realise that this entails that Darth’s God doesn’t exist (which would make the conclusion false). This would mean that the premises are true and the conclusion is false. And there is nothing logically contradictory or incoherent about this supposition; there could easily be someone who fits the bill. Therefore, it is possible (i.e. logically consistent) for the premises to be true and the conclusion to be false, and that is what it means for an argument to be invalid.

The wider point is just that it is possible to not know everything that logically follows from what you know. When I pointed this out to Darth, I used a mathematical example. Suppose there is some conjecture in mathematics that is currently unproven. Either the conjecture is true, or it is false (it has to be one or the other). But as it is unproven, I don’t know which it is. But I do know the basic axioms of mathematics and the inference rules. So technically the truth or falsity of the conjecture (whichever it is) follows logically from stuff that I know. So this is an example of how you can not know what logically follows from what you know. And that means that you have to do more than just show that something is implied by what someone knows to conclude that they know the implication as well; often we are ignorant of the implications of what we know.

3. What about people who do know the conclusion?

But let’s suppose that if Darth walks his agnostic interlocutor through the reasoning, then he is highlighting the consequences of their belief to them. They might not have been aware of the consequences of their belief beforehand, but now they are, because Darth has helpfully demonstrated it to them.

Indulge me with a little dialogue:

Darth: ‘You believe neither that God exists, nor that he doesn’t exist, right?’

Agnostic: ‘Sure’

Darth: ‘Well, my God is such that if he existed, then you would know about him’.

Agnostic:’Ok, sure’

Darth: ‘Ok, Good. So it follows from you not currently knowing that my God exists that he doesn’t exist, by modus tollens. Right?’

Agnostic: ‘Oh yeah, I see what you mean. My mental state of not believing in him is logically incompatible with him existing.’

Now that Darth has raised to the level of consciousness how it follows from her beliefs that Darth’s God doesn’t exist, shouldn’t we say that our agnostic now knows the conclusion?

Well, maybe that’s fine. I mean, what if the agnostic person simply says something like: ‘Well, I guess I’m not agnostic about your version of God then. I am agnostic generally about the notion of god, but now you have spelled out the logical consequences of your particular God existing, I guess I am an atheist about that God; I positively believe, even know, that your God doesn’t exist.’

And once we spell it out like that, it seems perfectly reasonable. I mean, it is fine to not have exactly the same attitude towards every god concept. You might be more sceptical about the Mormon God than the Islamic God, or whatever. You might be an atheist about the Mormon God, but only agnostic about the Islamic God, etc. Why think we should have an absolutely universal attitude towards all god concepts?

Yet, this move is dismissed by Darth in this video (timestamped). Ask Yourself says that although he is generally an agnostic, he is an atheist with respect to Darth’s conception of God. Darth calls that a “childish response”, and laughs at it. But Darth’s dismissal here is itself a silly thing to say.

The problem for Darth here is that it is obviously unproblematic to take different attitudes towards different God concepts, especially if we are allowed to do what Darth does and tack on properties that God has, like ‘being such that you would know if he existed’, etc. As a particularly trivial example, consider the following:

A) The god such that it doesn’t exist

Obviously, if anyone bothered to think about this god concept, they would likely come to believe that it doesn’t exist (it doesn’t exist by definition). We can generate a slightly less trivial example as follows:

B) The god such that if I exist, then it doesn’t exist

Presuming you know that you exist, then you can easily conclude that this god doesn’t exist either. It is easy to come up with examples of this sort of thing. The god such that if I am having a sensation of blue right now (while looking at the sky or whatever) then it doesn’t exist; the god such that if I am thinking about arguments like this right now then it doesn’t exist, etc, etc.

So anyone who says they are agnostic will almost certainly caveat that claim somewhat, such as “…but obviously I do not mean that I have no opinion about trivially non-existent god concepts, such as the god such that it doesn’t exist, or the god such that if I existed then it wouldn’t exist, etc. About those types of god concept I do have an opinion, I believe that they do not exist.” It’s not childish or irrational to make that move at all.

But imagine we were to insist, as Darth seems to, that the terms ‘agnostic’ and ‘atheist’ could only be used to indicate an absolutely uniform attitude towards every god concept. Surely, then the term ‘theist’ would also fit that pattern. But if so, we could generate just the sort of trap that Darth thinks he has set. Consider the following god concept:

C) The god such that if you believe in it, then it doesn’t exist.

There is going to be something contradictory about believing in such a god. If you know C), and believe in this god, you can conclude (with a helpful interlocutor who will walk you through the steps) that such a being doesn’t exist. Therefore, there is precisely the same sort of contradiction in claiming to believe in this sort of god.

So imagine the following dialogue:

Interlocutor: So Darth, you believe in God, correct?

Darth: I am a theist, yes.

Darth: How so?

Interlocutor: Well, believing in the god such that if you believed in it then it wouldn’t exist entails that it doesn’t exist. If you grasp that inference, but continue to believe in it, then you believe in two incompatible propositions; you believe that it exists and also believe that it doesn’t exist.

Darth: But I am a Christian theist. That means I believe in the Christian God, not the God that is such that if I believed in it then it wouldn’t exist. I don’t believe that god exists at all.

In this dialogue, the attempt by the interlocutor to trap Darth into being committed to believing that the god such that it wouldn’t exist if you believe in it is obviously disingenuous. When Darth says he is a theist, he doesn’t mean he believes in that god. He means he believes in the Christian God.

This whole trap requires a kind of bait and switch, in getting Darth to commit to ‘theism’, but then to insist that he means to assert that he believes in a god concept that cannot be rationally committed to. The way out of this, which is a perfectly reasonable way out, is for Darth to insist that he has a non-universal attitude towards the family of god concepts; one of them he believes in, but the others he positively disbelieves in. That is to say, he is a theist with respect to Christian theism, but atheist about all the other god concepts (such as the one on the table). Yet, this move, nuancing the meaning of theism towards specific god concepts, is exactly the move that Ask Yourself made with respect to atheism and agnosticism. This is the very move that Darth decried as childish. But unless he allowed himself to make the same move, he would be caught by this version of his own problem.

Let’s summarise where we are:

• Darth’s argument is invalid for the simple reason that it trades off the false notion that everyone knows the logical consequences of the things they know. That is false, as the mathematical example shows. So even if you are someone who knows that if Darth’s god existed you would believe in him, and that you do not believe in him, all it takes is to be unreflective enough about the consequences of this to not form the belief that he does not exist for your mental states to be consistent. Not only is this possible, but this sort of thing happens to all of us all the time. Nobody knows all the consequences of the propositions they believe. Even Darth.
• But then if we consider someone who has actively and explicitly considered the propositions and implications in question here, they should just accept that they are not universally agnostic; about some god concepts they do actively believe that there is no such god. Not only is there nothing silly about this sort of move, because of trivial god concepts (like the god such that it doesn’t exist), unless Darth made use of the same move he would be caught in his own argument.

It might be that you believe something without realising that you believe it. Perhaps people believe things but their psychology forces themselves to deny that to themselves, like if someone witnesses something traumatic as a child and represses the memory, etc. Perhaps our sinful nature has a similar psychological effect, forcing us to repress our inate belief in the Christian God. Wouldn’t this undermine the agnostic’s claim to not believe in God?

Not really. Let’s make a distinction between gods that if they existed then you would explicitly believe that they existed, and ones where you would either explicitly believe that they exist or have a suppressed belief that they exist. The first is the version of Darth’s God that we have dealt with already. Simply not being aware of belief in this God is enough to entail its non-existence. But what about the new version? Let’s rewrite the argument from above:

1. If Darth’s God exists, then you would either explicitly believe that he exists, or have a suppressed belief that he exists
2. You do not explicitly believe that Darth’s God exists
3. Therefore, Darth’s God does not exist

This argument is invalid. It has the form ‘if p, then (q or r)’, ‘not-q’, therefore ‘not-p’. It is invalid precisely because we have to rule out both q and r in order to derive not-p; just ruling out q is not enough.

What this means is that agnosticism is not contradictory with this version of God, even if someone like Darth was to walk them through the steps to bring it to their attention. From their own lack of explicit belief, all they could conclude is that either that god doesn’t exist, or they have a belief that they are not aware of. Nothing else follows. So it makes sense to remain agnostic in such a circumstance. The trap Darth is trying to set doesn’t even spring if the type of belief involved can be suppressed. It only gets off the starting blocks if the condition associated with the god existing is such that you can determine if it holds or not. In this case, the agnostic cannot introspect and tell which of the two options (god not existing, or them having an unconscious belief) is true.

5. Conclusion

Darth’s argument here only seems like it works because he presents it in an aggressive way. It is all rhetoric and no substance. When people try to talk to Darth, it often seems like he gets the better of them. But if he were to drop the hyper-aggressive style and talk like an equal with someone, it would be clear under fair logical analysis that the argument is hopelessly flawed.

## Tom Jump’s moral theory

0. Introduction

Tom Jump is an atheist YouTuber with a prolific output, making videos several times a week, mostly debating Christian philosophers. Recently, he did a debate with Ask Yourself in which they discussed whether there are moral facts or not, with Jump arguing that there are. Jump’s position is that there is something like a ‘moral law’, which has similarities to physical laws. Their debate descended into a squabble over whether a specific statement of Jump’s expressed a proposition or not. I don’t want to follow that part of the discussion, but I do want to look at Jump’s theory as (I think) he intends it, and to point out some of the problems it has.

1. What is the theory?

As I said above, Jump thinks that morality is objective, in a similar way to physical laws. Objective in this context means that it exists independently of any minds. We take physical laws, like the law of gravitation, to obtain in the universe just as much whether there are any minds present or not. If a moral law is objective, then it too would obtain just as well without any minds in existence.

Jump’s idea is that objective morality is defined in relation to the notion of involuntary impositions. I think we can have a go at reconstructing his idea as follows:

Action x is a morally wrong iff x is an involuntary imposition on A, for some agent A.

Part of the problem here is that we need to get clearer on what it means for something to be involuntary. This is because on various ways of understanding this term we run into trouble.

2. First go.

On one way of thinking about it, involuntary means something like ‘not actively consented to’. When things happen to someone but they have not specifically chosen that they happen, these are immoral. And sometimes that is right; sometimes things we haven’t actively consented to are immoral.

But this definition of ‘involuntary’ cannot be  what it means to be immoral, because if it were then it would classify things as immoral that are obviously not. For example, surprise birthday parties are not immoral, yet the recipient has ‘not actively consented to’ them happening. So it isn’t the case that the definition immoral is an involuntary imposition on someone, if involuntary means ‘not actively consented to’.

3. Second go

One might think that the problem with the surprise birthday case is that it isn’t involuntary unless you have actively stated that you do not want a surprise party. So maybe we could improve things by changing the definition of ‘involuntary’ from ‘not actively consented to’ to something like ‘against stated preference’. So before, the surprise party was immoral just because you didn’t say anything about the party, but now it would only be immoral if it went against your actively stated preferences. Assuming you have never actively expressed a preference for not having any surprise parties, it would not be an involuntary imposition on you for your friends to throw one for you, and so not immoral. This is an improvement, because it doesn’t misclassify surprise birthday parties.

And there is something fairly intuitive about this idea. Certainly sometimes things that go against our actively stated preferences are immoral. If someone tells you they do not want to have sex with you, but you continue to try to have sex with them then this would be a case of sexual assault, and clearly immoral.

But again, despite this partial alignment, this definition of involuntary cannot be the what it means to be immoral. That’s because there are obvious cases where things go against our stated preference, and are thus ‘involuntary’ in that sense, but that are not immoral. Imagine I go into a bar and order a beer. After I have finished it I state that my preference is for it to be on the house. If the bartender insisted that I have to pay for it, this would make it an ‘involuntary imposition’, because it is against my stated preference. But it is not an immoral thing for the bartender to do; he is quite within his moral rights to charge me for the beer, regardless of whether I stated that I would prefer not to pay for it. So there are obvious cases of things that are against my stated preferences which are not immoral.

4. Third go

As another try, we might say that something is involuntary if it is ‘against my desires’. We might think that the problem is that our previous two tries to define ‘involuntary’ were about whether we do, or no not, say something in particular. In contrast, we might think that it is about whether we have a desire or not, and not about what we say at all. So let’s define ‘involuntary’ as ‘against our desires’.

This would help with the surprise party example, as follows. Assuming I am the sort of person who enjoys surprise parties, then even though I haven’t actively stated that I consent to it, it wouldn’t be involuntary as such, because it wouldn’t be against my desires. It is the sort of thing I would have consented to had I known about it, because I desire that sort of thing to happen. So it is not involuntary, and so not immoral. So far, so good.

However, this is no help in the bartender case. I might just order a drink and desire for it to be on the house, but not say anything out loud. Is the bartender doing something immoral by charging me for the beer? No, clearly he is not. So this has the same problem here. Sometimes things happen that we don’t want to happen which are not immoral. Too bad.

This version also has problems from the other direction too. The problem is that sometimes people have immoral desires. Take a heroin addict who asks for his doctor to prescribe him some heroin. Clearly, the addict desires the heroin. Prescribing it to the addict wouldn’t be an involuntary imposition on him. But it is at least of dubious moral value for the doctor to do, if not outright immoral, even if the doctor wants the money being offered. Take another example: maybe some unstable (North Korean?) dictator asks an advanced country (the UK?) if he can buy some tanks from them. Clearly, it wouldn’t be an involuntary imposition on him to sell him the weapons, and maybe the other country wants the money. Still, just because both parties desire it doesn’t mean it is not immoral. We can easily iterate these examples.

5. Fourth go

One standard way to respond to these sorts of objections is to retreat from what people actually desire, to what they would desire if they were in some idealised state; if they were perfectly rational, etc. We might think that the heroin addict happens to desire another hit, but that he is just suffering from a lack of rationality. If he were being perfectly rational, then he would not desire to have more heroin; he would desire to get clean instead. And there is something intuitive about this particular example.

However, I think it is not so straightforward. The connection between rationality and desires is surprisingly complicated, and something debated at length by philosophers. One simple view, known as ‘Humeanism’ in the literature, is that someone is rational when their actions efficiently realise their desires. If I desire not to get wet, then knowingly walking into the rain without an umbrella is irrational. But, change the desire and the very same action becomes rational – if I want to get wet, then leaving my umbrella behind is rational.

The problem with this simple theory is that if you change the desire to, say, wanting to do something immoral, then the rational thing becomes whatever efficiently realises that desire, which would be to do something immoral. So if you want to kill someone, it might be rational to hit them over the head with a spade. Clearly, there is no guarantee that a perfectly rational person would have no immoral desires on this theory.

We could avoid this problem by abandoning the simple Humean theory. Instead of saying that only desires can motivate us, we could include beliefs too. Being rational might mean something like doing whatever realises your desires but is not believed to be immoral. So take someone who believes that it is wrong to murder people, but desires to kill you. He would be irrational if he hits you on the head with a spade because, although his actions realised his desires, they contradicted his beliefs about what is immoral.

But if someone had Jump’s starting point, then this option would collapse the whole project into circularity. We would have been led down the following path: the definition of immorality involves voluntariness, which in turn involves rationality, which itself involves the notion of beliefs about immorality, and the whole thing becomes a circle. We were offered a definition of immorality which in turn used the notion of immorality.

Thus, Jump is left with a dilemma: either tacitly include the notion of immorality in the definition of rationality, leading to circularity, or stick with Humeanism, and the problem of immoral desires.

6. Final thought

Even if this huge problem were somehow avoided, there is another one that is perhaps even more pressing. The whole point of this theory was supposed to be that it was a theory of objective morality. That means that the moral law that Jump was trying to express (which was supposed to be a bit like a physical law), doesn’t depend on minds to be true. But that is not the case here. If something is immoral when it is involuntary, then it depends on the person having some kind of intentional state, some desire or ‘will’, for it to be in contrast to. In a world where there were no people, there would be no wills for any action to be in contrast with, and so nothing would be immoral. There would be no true proposition, such as ‘x is immoral’, just because there would be no person on whom x would be an involuntary imposition. Thus, this theory is blatantly a variety of subjectivism and not a version of objectivism at all.

## 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.

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.