Molinism and the Grounding Objection, Part 1

0. Introduction 

Molinism is the view that there are true counterfactuals involving agents making libertarian free choices, and that these counterfactuals are known by God. See this for more background.

Perhaps the most common objection to Molinism is referred to as the ‘grounding problem’. The issue is just that there seems to be nothing which explains why true Molinist counterfactuals are true. They seem to be just true, but not true because of anything in particular. Here is how Craig puts it in his paper Middle Knowledge, Truth–Makers, and the “Grounding Objection” (henceforth MK, and from which all the Craig quotes will come in this post):

“What is the grounding objection? It is the claim that there are no true counterfactuals concerning what creatures would freely do under certain specified circumstances–the propositions expressed by such counterfactual sentences are said either to have no truth value or to be uniformly false–, since there is nothing to make these counterfactuals true. Because they are contrary–to–fact conditionals and are supposed to be true logically prior to God’s creative decree, there is no ground of the truth of such counterfactual propositions. Thus, they cannot be known by God.”

One way of thinking about this issue is that the grounding problem itself presupposes the ‘truth-maker’ principle. According to this principle, every true proposition is made true by something. If the truth-maker principle is correct, and if nothing makes Molinist counterfactuals true, it follows that they are not true. Hence, it follows that there are no such truths for God to know.

In response to this, a Molinist can either deny the truth-maker principle, or accept it and provide a truth-maker for the counterfactuals. As Craig makes explicit, he believes he can make the case that either strategy is plausible:

“For it is far from evident that counterfactuals of creaturely freedom must have truth-makers or, if they must, that appropriate candidates for their truth-makers are not available.”

Craig gives reasons that one might want to deny the truth-maker principle in general. He also explains how one might think about Molinist counterfactuals not having truth-makers. He also offers an account of how they could have truth-makers. If any of these works, it seems that the grounding objection has been rebutted. In this series I will look at his proposals, and argue against them. In this first post, I will just look at the positive case that Craig sets out for Molinism.

  1. The (supposedly) intuitive case

Craig mentions a comment from Plantinga that he agrees with, about how plausible it is that there should be true Molinist counterfactuals:

“No anti–Molinist has, to my knowledge, yet responded to Alvin Plantinga’s simple retort to the grounding objection: “It seems to me much clearer that some counterfactuals of freedom are at least possibly true than that the truth of propositions must, in general, be grounded in this way.””

Craig goes on to say that the grounding problem is:

“…a bold and positive assertion and therefore requires warrant in excess of that which attends the Molinist assumption that there are true counterfactuals about creaturely free actions.”

Plantinga is saying that the fact that there are Molinist counterfactuals is more plausible than the truth-maker principle. To show that we should prefer the truth-maker principle to Molinist counterfactuals, we need warrant for the truth-maker principle “in excess” of that for Molinist counterfactuals. Not an easy job, thinks Craig, who says that the warrant for Molinist counterfactuals is “not inconsiderable”.

In his ‘Warrant for the Molinist Assumption’ section of MK, Craig provides three aspects of the case which supposedly shows that Molinist counterfactuals have ‘not inconsiderable’ warrant already. These are as follows:

  1. First, we ourselves often appear to know such true counterfactuals.”
  2. Second, it is plausible that the Law of Conditional Excluded Middle (LCEM) holds for counterfactuals of a certain special form, usually called “counterfactuals of creaturely freedom.””
  3. Third, the Scriptures are replete with counterfactual statements, so that the Christian theist, at least, should be committed to the truth of certain counterfactuals about free, creaturely actions.”

In this post, I will focus on the first of these three.

2. The epistemic objection – Molinist counterfactuals are unknowable

The first one of these, along with the third and Plantinga’s quote from above, are all related. They are rebutted by what I will call the ‘epistemic objection’.  According to this objection, even if they were true, it isn’t possible for an agent to know Molinist counterfactuals.

It seems to Craig to be obvious that we “often appear to know” Molinist counterfactuals to be true. Yet, there seems to be good reason to think that we cannot know Molinist counterfactuals.

In order to help explain things, I want to make an important distinction, which is between Molinist counterfactuals and what I will call ‘probably-counterfactuals’. So, an example of a Molinist counterfactual is:

a) Had Louis been tempted, he would have given in.

An example of a probably-counterfactual is:

b) Had Louis been tempted, he probably would have given in.

The difference between a) and b) is merely the word ‘probably’. The difference it plays is huge though. I think that it makes the difference between being crucial to rational reasoning generally (like b), and being utterly useless (like a). I think that Craig’s claims about Molinist counterfactuals only really make sense if they are ultimately being made about probably-counterfactuals, and I will explain why I think this in what follows.

First of all, Craig thinks that we “often appear to know” Molinist counterfactuals, like a). But this is strange. Maybe God could know them (although, I don’t think that can be maintained either), but how could a mere mortal like me know them? All I can really know, we might suppose, is i) what I have some kind of access to empirically (a posteriori), and ii) what I can reason about abstractly (a priori). And neither of these routes can get me to the conclusion that Louis would have freely chosen to give in to the sin had he been tempted.

I don’t have empirical access to counterfactual situations, so that rules out the first epistemological route; nothing about the empirical world that I can investigate can tell me which of the two options Louis would have freely chosen to make.

But mere abstract reasoning cannot ever decide which of two options an agent with libertarian free choice would make either; it doesn’t follow logically from any purely a priori antecedent conditions. Thus, Louis’ choice seems literally unknowable to an agent like me. Not only that, but all Molinist counterfactuals become unknowable for the same reason.

On the other hand, knowing b) seems relatively straightforward, at least in principle. Let’s suppose Louis has a strong track record of giving in to sin when tempted, and that I know this because I have witnessed it personally. Perhaps he has also told me about how much he hates living in the stuffy confines of the monastery and yearns for some temptation to give into. Any number of scenarios like this could support the idea that I could come to believe with good reason that he probably would have given in had he been tempted.

Thus, a) seems literally unknowable, whereas b) is eminently knowable. They are therefore, epistemically asymmetric.

3. The utility objection – Molinist counterfactuals are useless

Craig says:

“Very little reflection is required to reveal how pervasive and indispensable a role such counterfactuals play in rational conduct and planning. We not infrequently base our very lives upon the assumption of their truth or falsity.”

He is right about the fact that counterfactuals play a “pervasive and indispensable” role in “rational conduct and planning”. But where is wrong is that it is probably-counterfactuals which are doing most of the work, and Molinist counterfactuals do none (and indeed, could not do any). The reason for this difference in utility is because of the epistemic asymmetry between probably-counterfactuals and Molinist counterfactuals.

Here is an example to play with to make this point clear. Imagine I am deciding whether or not to leave my bike unlocked or not while I go into the library. Let’s suppose that I see the well-known bike thief, Louis, lurking just round the corner. I decide to lock my bike up. When I return after finding the book I want, I am glad to find my bike is still there. I begin to unlock my bike, and at this point you ask me: “Why did you lock your bike up?” My answer is going to be something like this:

c) Had I not locked up my bike, Louis probably would have stolen it.

It is the likelihood of Louis stealing the bike that motivated me to lock it up. My reasoning process included the fact that I had good reasons to think that e) was true. The place that the probably-counterfactual plays in my reasoning is completely clear. It makes perfect sense for a probably-counterfactual to be what I am using here to come to my decision to lock the bike up.

The idea that I used a Molinist counterfactual is almost unintelligible though. Imagine my reply had been the following:

d) Had I not locked up my bike, Louis would have freely chosen to steal it.

It would be bizarre for me to say that, because there is no way for me to know that d) is true rather than false. Given that Louis has libertarian free will, he could have chosen to steal the bike, but he could have also chosen not to steal the bike. The scenario where he freely chooses to steal the bike, and the scenario where he freely chooses refrain from stealing the bike, are literally identical in every respect up to the point where he makes a decision. There is nothing at all, even in principle, that could justify my belief that one would happen rather than the other. Possibly, God knows something I don’t, but it is clear that I do not. Thus, there is no way it can be part of my (rational) decision making process, for I have no reason to think that it is true rather than false.

If this wasn’t bad enough, we can develop the worry. Imagine that standing next to Louis is Louise, who I know has never stolen a bike, or indeed anything, in her entire life. My belief is that she is unlikely to steal my bike. Her presence is therefore not a consideration I took into account when I locked my bike up. If you asked me when I got back to my why I did not consider her presence, I would have said that it was because of something like the following:

e) Had I not locked up my bike, Louise probably would not have stolen it.

I was under the belief that even if I had not locked my bike up, Louise probably wouldn’t have stolen it. While the presence of Louis plays a role in my reasoning, and the presence of Louise does not, and this is easily cashed out in terms of probably-counterfactuals.

But when we come to consider that it wasn’t probably-counterfactuals, but Molinist counterfactuals that were part of my reasoning, we run into a problem. This is because an entirely symmetric Molinist counterfactual can be created for Louise:

f) Had I not locked up my bike, Louise would have freely chosen to steal it.

Given that Louise has libertarian free will, she could have chosen to steal the bike, but she could have also chosen not to steal the bike. The scenario where she freely chooses to steal the bike, and the scenario where she freely chooses refrain from stealing the bike, are literally identical in every respect up to the point where she makes a decision. Each of Louis and Louise are perfectly symmetrical in this respect, so there is no reason for me to believe both that e) is true and f) is false. But unless I do have this (non-Molinist) asymmetric view about e) and f), my inclination to treat them differently utterly inexplicable.

The very thing that the counterfactual would need to do to be an ‘indispensable’ part of my reasoning process is inexplicable if they are Molinist counterfactuals.

4. A possible reply

There is a possible reply that could be made on behalf of the Molinist at this point though. Clearly, our Molinist friend might reply, we cannot know for sure whether a Molinist counterfactual like a) or d) or f) is true rather than false. Only God can know that for certain. However, I have set the bar too high. We can reasonably infer such counterfactuals from the truth of the probably-counterfactuals, which I already conceded are not problematic to know. So, for example, it is from the premise that Louis probably would have stolen the bike, that I infer that he would have freely chosen to steal the bike. Obviously, this is not a deductive inference (for it is not deductively valid), but it is a reasonable inductive inference.

Here is the inference:

  1. Had I not locked the bike, Louis probably would have stolen it
  2. Therefore, had I not locked the bike, Louis would have freely chosen to steel it

This reply has a lot going for it. Things can be known via such inductions. I think that premise 1 is true, and that it’s truth can be plausibly construed as something which increases the (epistemic) probability of 2. Thus, the inference, though inductive, seems pretty good.

I actually don’t think that 2 could be true, but that is for semantic reasons that we do not have to get into here. Let’s just say that for the sake of the argument, I accept this type of move. Where does it get us?

It might be thought that Molinist counterfactuals can indeed be known (via inductive inference from known probably-counterfactuals). Thus, the epistemic objection seems to have been countered. Indeed, once we make this move, counterfactuals like d) (i.e. had I not locked up my bike, Louis would have freely chosen to steal it) can be believed by me with justification. Thus, it is now no longer problematic to see how they might fit into my reasoning process. I believe (via inference from a probably-counterfactual) that Louis would have freely stolen my bike, and that belief is what motivates me to lock it up. Thus, the utility objection has a rebuttal as well.

5. The redundancy reply

As I said,  I think this is a good line of response. I think it is about the best there is to be had. But even if we concede it, I don’t think much has happened of any importance. Ultimately, they rescue Molinist counterfactuals at the cost of making them redundant. If they can known and can be put to work in decision making, then they necessarily do not need to be used, because there will already be something we believe (or know) which does all of their work for them.

Even if Molinist counterfactuals, like d), can be inductively inferred from probably-counterfactuals, like c), it is not clear that they can be derived from anything else. Consider the case where someone believes that Louis will freely choose to steal the bike, but does not believe that he probably will steal the bike. Such a belief can be had, but surely it is irrational. It is like holding that this lottery ticket is the winner, even while believing that it is unlikely to be the winner. Such beliefs may be commonplace (and maybe it is beneficial to believe that you will beat the odds when fighting with a disease, etc), but they are paradigmatically irrational nonetheless. Unless you believe that something is probably going to happen, you should not believe (i.e. should lack a belief) that it is going to happen.

If that is right, then it has a similar consequence for Molinist counterfactuals being used in rational processes. Unless I have inferred it from a probably-counterfactual, I cannot reasonably believe a Molinist-counterfactual. But the only way I can use a belief in a Molinist counterfactual as part of a rational decision-making process is if I reasonably believe it. Therefore, the only way I can use a belief in a Molinist counterfactual as part of a decision making process is if I already believe the corresponding probably-counterfactual.

Here is an example to make this clear.

Let’s say that I can infer that ‘Louis would freely choose to steal the bike if left unlocked’ from the premise that ‘he probably would steal the bike if left unlocked’, and from no other premise. Let’s also say that I use believe that ‘he would freely choose to steal the bike if left unlocked’, and that I use that as part of my decision process to lock the bike up. It follows that because I used that belief as part of my rational process, that I must also believe that he probably would steal the bike.

This means that even if Molinist counterfactuals played the role that Craig thinks they do in decision making, they must come with an accompanying belief about the corresponding probably-counterfactual.

And this means that, maybe Molinist counterfactuals can be known, and maybe they can be used in reasoning processes, but they can do so only if there is a reasonably believed probably-counterfactual present as well. This makes Molinist counterfactuals completely dependent on probably-counterfactuals from both an epistemic and decision theoretic point of view. You never get to rationally believe a Molinist counterfactual unless you already believe the corresponding probably-counterfactual. And you can never use your belief in a Molinist counterfactual in some reasoning process unless you also already believe the corresponding probably-counterfactual.

And as we saw, probably-counterfactuals can already do all the explanatory work in explaining why I decided to lock my bike up. I don’t need Molinist counterfactuals if I have the right probably-counterfactual, and I never have a Molinist counterfactual unless I already have the right probably-counterfactual. That makes them necessarily redundant. Maybe they can play the role Craig wants them to play, but only if the need not play it.

 

6. Conclusion

Craig’s first aspect of the warrant for Molinist counterfactuals was that we commonly know such counterfactuals. However, I showed how it seems quite hard to see how we could know such counterfactuals directly. They are not things we can experience ourselves, and they are not deducible a priori. Probably-counterfactuals, on the other hand, are eminently knowable. Craig also claimed that Molinist counterfactuals play an indispensable role in decision making, however their disconnection from our direct ways of knowing their truth-values makes them irrelevant to decision making, unlike probably-counterfactuals.

The only response to this seems to be to claim that Molinist counterfactuals can be known via inference from probably counterfactuals. While this may be true (although I still have problems with that), all it would get a Molinist would be something which can only be known because the probably-counterfactual was also known, and only does any work explaining decision making if that work could be done by the epistemically prior probably-counterfactual. They can only be saved by being made redundant.

Infinity, Hume and Euclid

0. Introduction

Can something be infinite, yet also exist, as it were, ‘in reality’? Many people say that the answer to this question is ‘no’. However, the arguments in favour of this go over terrain that is very difficult to navigate without getting lost. Here I want to look at one very small part of the issue, and what seems problematic about it to me. As I said, this stuff gets very deep very quickly, and although I know the literature a bit, I am not a specialist in this area. There plenty of discussions of this I should probably read. But these are my thoughts at the moment. Hopefully it will help me become clearer about it, and might be helpful to people trying to understand this area a bit better too.

I am thinking about the argument in relation to the ‘Hilbert’s hotel’ style of defence for the second premise of the kalam cosmological argument. In particular, imagine that an apologist uses the infinite library analogy as follows:

Suppose there were a library with infinite books in it. If you withdraw a book from the library, then there is one fewer book in the library, yet there is also the same number of books in the library. There cannot both be fewer books and the same number of books, because that is absurd.

That is the type of claim that I am looking at here. I am not looking in particular about the other difficulties which could be brought out from the infinite object examples. There are other things one might say to motivate this part of the kalam, but here I am looking at this way of motivating it. The claim I am interested in is that the truth of there being fewer books and the same number of books is itself an absurdity.

  1. Equinumerous and fewer than.

There are two principles which need to be kept in sight. They involve connecting our intuitive ideas ‘equinumerous’ and ‘fewer than’ with mathematical counterparts.

We already have a fairly clear idea about the relationship between the two concepts when we use it in casual conversation. In particular, it seems quite clear that they are mutually exclusive:

i) If the number of A’s is equal to the number of B’s, then the number of A’s is not fewer than the number of B’s.

ii) If the number of A’s is fewer than the number of B’s, then the number of A’s is not equal to the number of B’s.

In ‘real life’ examples, we say that if the number of forks in my dinner set is the same as the number of knives, then I do not have fewer forks than knives, etc. The mathematical counterparts of these terms need to preserve this highly intuitive relationship; the result must be mutually exclusive too. We will consider two linking-principles, one which links the notion of ‘equinumerous’ to a mathematical idea, and one which links the notion of ‘fewer than’ to a different mathematical idea. They are chosen because they look like they express what the intuitive ideas are getting at, and because they preserve the mutual exclusivity relationship (at least, they do to begin with).

2. Hume and Euclid

The first of these is sometimes called Hume’s Principle (HP). It can be seen as a definition of the notion of ‘equinumerous’ (or ‘the same number of elements’, or of having the ‘same cardinality’ – all three meaning the same thing here). Being the same cardinality is linked to being able to be put in a one-to-one correspondence. Imagine that I could put every fork from my dinner-set with a unique knife, and have no knives left over. According to HP, this condition is what must hold for it to be true that I have the ‘same number’ of forks as knives. The idea with HP is that “The cardinality of A is equal to the cardinality of B” means that the elements of A can be put in a one-to-one correspondence with the elements of B.

Euclid’s maxim‘ (EM) effectively says that the whole is greater than a part (this principle is advocated by Euclid in the Elements). We can see this as a definition of the notion of ‘fewer-than’. Firstly, we need to be clear about what a ‘proper subset’ is:

A is a proper subset of B if and only if

  • everything which is an element of A is also an element of B, and
  • there is an element of B which is not an element of A.

So the set of knives is a proper subset of the set of ‘items of cutlery in my dinner-set’, because there are items of cutlery which are not knives (i.e. forks). There are fewer knives than there are items of cutlery. So this the idea with Euclid’s maxim is “There are ‘fewer’ A’s than B’s” means that A is a proper subset of B.

Here they are side by side:

Hume’s principle: A is equinumerous with B if and only if the elements of A can be placed in a one-to-one relation with the elements of B.

Euclid’s maxim: There are fewer A’s than B’s if and only if A is a proper subset of B.

3. The Problem

If the number of elements that can be in a set is finite, then these principles are mutually exclusive. So if A and B are equinumerous (according to HP), then neither is fewer than the other (according to EM), and if one is fewer than the other, then they are not equinumerous. This is because, if A and B are finite, then there being something in B that is not in A entails that one could not put their respective elements in a one-to-one relation. If I am missing a knife, then I cannot place each knife with a unique fork, without having a fork left over. So far so good.

The problems come in if A and B are allowed to be sets that have infinitely many elements. When we make this move, these two intuitive principles cannot both be correct. Let A be the set of all natural numbers, [0, 1, 2, 3 … n …), and B be the set of even natural numbers, [0, 2, 4, 6 … n …). The elements of A and B can be placed in a one-to-one correspondence with one another, as Cantor showed. So by Hume’s principle, they are equinumerous. Yet, it is also clear that every element of B is an element of A, while there are elements of A that are not elements of B (i.e. the odd numbers). This means that B is a proper subset of A. So by Euclid’s maxim, B has fewer elements than A.

The problem is that A and B have the same number of elements, but B has fewer elements than A. As we saw, the intuitive relationship between being the same number and being fewer is that they are mutually exclusive. So it should never happen that A and B are both equinumerous and that one is fewer than the other. Clearly, something has to give here if we are to avoid an inconsistent result.

4. What is going on? 

The issue here is that we have a pre-theoretical idea of the terms ‘equinumerous’ and ‘fewer than’, and we have set-theoretical expressions which looked like they gave the meaning of the intuitive notions. However, our intuitions about what those terms mean differ from how their mathematical counterparts operate in certain circumstances.

So to avoid the problem, we have to reject one of three things:

  1. The idea that equinumerous means being able to be placed in a one-to-one correspondence (i.e. HP)
  2. The idea that fewer-than means being a proper subset (i.e. EM)
  3. The idea that being ‘equinumerous-with’ is mutually exclusive with being ‘fewer-than’.

If we try to keep all three of these things, we run into the problems that give rise to the apologist’s charge of ‘absurdity’.

5. Rejecting 1

One way of proceeding is to reject HP. This means rejecting the claim that when we say that the number of A’s is the same as the number of B’s this means that the A’s can be placed in a one-to-one correspondence with the B’s. The main problem with this is that it is unclear what else being ‘equinumerous’ could mean. Possibly, it could mean something like if they were both counted, then the final number reached would be the same in each case. It is not clear whether this is actually any different however. Imagine that I count my knives by picking each one up and saying a cardinal number out loud (like the Count from Sesame Street), and then placing them off to one side in a line according to the number they received. So I put the first one down, then I place the second one next to that, and the third one next to the second one, etc. When I come to count my forks I could do exactly the same thing. If I arrive at the same number when I have finished counting each one, this just means that the two lines of cutlery would be lined up one-to-one. So this doesn’t even seem to be a different result to HP. And what else could ‘equinumerous’ mean?

The real action is about which one to reject out of 2 or 3.

6. The case for rejecting 2

One way to reject 2, but to keep 3, would be to modify the claim made in EM. At the moment, EM says that there are fewer A’s than B’s iff A is a proper subset of B. We could add another condition as follows:

Revised-EM) There are fewer A’s than B’s iff

  • A is a proper subset of B, and
  • A and B are not equinumerous

The second condition isn’t needed in the case of merely finite sets, because no finite set A can be both a proper subset of B and equinumerous with B. Thus, the original EM and the revised-EM are identical with respect to finite sets. When we move to the case where sets can be infinite, then the second condition kicks in. The set of the even natural numbers is a proper subset of the natural numbers (so the first condition is satisfied). But the set of even natural numbers is equinumerous with the set of natural numbers (in that they can be placed in a one-to-one correspondence). Because this second condition is not satisfied, this means that it is false that there are ‘fewer’ even natural numbers than natural numbers. And this means that, according to revised-EM, there is no case (finite or infinite) where A is both equinumerous-with and ‘fewer-than’ B. And thus we have resolved our problem.

According to this strategy, there is nothing wrong with equinumerous meaning being able to be put in a one-to-one relation, and there is nothing wrong with the intuitive idea that equinumerous and fewer-than are exclusive. All that is rejected is the assumption that all there is to the notion of ‘fewer-than’ is being a proper subset. In addition to this, we also need to rule out being equinumerous. Only with both in place do we have a proper mathematical equivalent of ‘fewer-than’.

7. The case for rejecting 3

On the other hand, we could proceed by rejecting 3, the mutual exclusivity of equinumerous and fewer-than. On this view, infinite sets show us clear examples of when the A’s are equinumerous with the B’s, even though the A’s are also fewer-than the B’s. One might argue that our intuitions about the relationships between these terms is based on our experience of finite things, and we mistook a property of finite things to be a logical relationship between two terms. The mutual exclusivity of equinumerous and fewer-than is not a logical truth, but is actually a contingent truth, which applies only to those cases where the sets are finite. According to this view, we should revise our notions in light of this mathematical insight.

So take some case involving infinity, such as the infinite library. The number of books left in the library after I withdraw one book is ‘fewer’ (i.e. according to the original EM) than the number of books before the withdrawal, even though there is also the same number as before the withdrawal. There is only a problem with this if you insist on the mutual exclusivity between ‘same number as’ and ‘fewer-than’. If we let go of that presupposition, and let the mathematics guide our understanding, we see that the two notions are only mutually exclusive for certain cases and not others.

This sort of revision in how we use terms guided by scientific insight is not that strange. Imagine that at some point in history we discovered androgynous frogs. Prior to that we would have said that the terms ‘male’ and ‘female’ were mutually exclusive when it came to classifying frogs; if a frog is male, it is not also female and vice versa. But after the discovery we have a choice about how to proceed. We do not, I take it, say that these are not frogs, merely because it is true that no frog is both male and female! Rather, we say that, despite what we have previously thought, ‘male’ and ‘female’ are not mutually exclusive for all frogs. We revise our understanding of ‘male’ and ‘female’, being led by the discovery.

This is what it is like in our case too, if we reject 3. We originally thought that no sets could be fewer-than and equinumerous, but this was only the case with the finite sets we had considered. Once we look at these other cases, we find out that some sets are both fewer-than and equinumerous. Once we accept this, and drop the requirement that they are always mutually exclusive, we have avoided our issue from before. Saying that the library has both the same number and fewer books is like saying that this frog is both male and female. Sometimes that is what it is like.

8. Comparison

So we have two strategies. The disagreement is over the following. Imagine A is an infinite proper subset of B, such as A being the even natural numbers and B being all the natural numbers. A is equinumerous with B. But are there ‘fewer’ A’s than B’s? The first strategy says:

No, there are not fewer A’s than B’s (because they are equinumerous)

The second strategy says:

Yes, there are fewer A’s than B’s (because one is a proper subset of the other)

It is fairly clear at this stage that if one wanted to use the Hilbert’s hotel argument as a way of bringing out an absurdity, then option 2 causes a big difficulty. This is because it denies that there is ever a case where any two sets can be equinumerous and fewer-than. In particular, the infinite case is protected from this happening by the second condition in revised-EM. In these cases, the equinumerous nature of the two sets cancels out either being fewer-than the other. The very thing the apologist wanted to point to and say ‘Look at this! It’s absurd!’ is forbidden on this view.

Indeed, the third option also causes grave issues for the apologist too. On this view we have revised our notion of ‘fewer-than’ in such a way that it is no longer mutually exclusive with ‘equinumerous’. It would be like after the discovery of androgynous frogs; if I say to you ‘This frog is male’, you could reply ‘Yes, but is it also female?’ This reply wouldn’t be ‘absurd’ at all, because these terms are no longer thought to be mutually exclusive. The same thing would apply in our case too. On this view, there being both the same number and fewer books in the library after I withdraw one is not an example of two mutually exclusive things being true at the same time. Therefore it is not absurd on this view either.

9. Conclusion

So the issue I have looked at in this post is only a very small issue in the wider context of defending the kalam. It isn’t even the only issue that is brought up in relation to the Hilbert’s hotel style of defence, or even arguably the most serious. However, it is there, and people often talk as if this issue on it’s own causes problems. People often talk about the absurdity of there being both the same number of books and fewer books after the withdrawal in this sort of setting, even if they also develop additional worries.

I think there are broadly two strategies that one can adopt in response to this line of attack. The first would be to insist that there are really no situations where there are both the same number and fewer books, and provide a precise explanation of ‘fewer-than’ according to the revised-EM above. This clearly avoids the issue. Secondly, one could embrace the presence of what seemed like two mutually exclusive terms, but explain how the mathematics shows us that the two terms are not mutually exclusive for all cases.

I find each of these approaches to be independently quite plausible, and this largely discharges the force of the attack.

The Fine-Tuning Argument and the Base Rate Fallacy.

0. Introduction

The Fine-Tuning Argument is used by many apologists, such as William Lane Craig. It is a common part of the contemporary apologetical repertoire. However, I argue that it provides no reason to think that the universe was designed. One does not need to look in too much detail about actual physics, and almost the whole set up can be conceded to the apologist. The objection is a version of the base-rate fallacy. From relatively simple considerations of the issue, it is clear that relevant variables are being left out of the equation which results in the overall probability being impossible to assess.

The Fine Tuning Argument starts with an observation about the values of various parameters in physics, such as the speed of light, the Plank constant and the mass of the electron, etc. The idea is that they are all delicately balanced, such that if one were to be changed by even a very small amount, this would radically alter the properties of the universe. Here is how Craig explains the point, in relation to the gravitational constant:

“If the gravitational constant had been out of tune by just one of these infinitesimally small increments, the universe would either have expanded and thinned out so rapidly that no stars could form and life couldn’t exist, or it would have collapsed back on itself with the same result: no stars, no planets, no life.” (Quote taken from here)

This phenomenon of ‘fine-tuning’ requires explanation, and Craig thinks that there are three possible types of explanation: necessity, chance or design.

Craig rules out necessity by saying:

“Is a life-prohibiting universe impossible? Far from it! It’s not only possible; it’s far more likely than a life-permitting universe. The constants and quantities are not determined by the laws of nature. There’s no reason or evidence to suggest that fine-tuning is necessary.” (ibid)

Chance is ruled out by the following:

“The probabilities involved are so ridiculously remote as to put the fine-tuning well beyond the reach of chance.” (ibid)

The only option that seems to be left on the table is design.

So the structure of the argument is as follows (where f = ‘There is fine-tuning’, n = ‘Fine-tuning is explained by necessity’, c = ‘Fine-tuning is explained by chance’, and d = ‘Fine tuning is explained by design’):

  1. f
  2. f → (n ∨ c ∨ d)
  3. ~n
  4. ~c
  5. Therefore, d.

1. Tuning

It seems from what we currently know about physics that there are about 20 parameters which are finely tuned in our universe (if the number is not exactly 20, this doesn’t matter – for what follows I will assume that it is 20). For the sake of clarity, let’s just consider one of these, and assume that it is a sort of range of values similar to a section of the real number line. This would make it somewhat like radio-wave frequencies. Then the ‘fine-tuning’ result that Craig is referring to has a nice analogy: our universe is a ‘radio station’ which broadcasts on only an extremely narrow range. This range is so narrow that if the dial were to be moved only a tiny amount, the coherence of the music that was being broadcast becomes nothing but white noise. That our universe is finely balanced like this is the result that has been gained from physics.

It is important to realise that this fine-tuning is logically compatible with there being other radio stations which one could ‘tune into’. Imagine I tune my radio into a frequency which is broadcasting some music, and that it is finely-tuned, so that if I were to nudge the dial even a tiny amount it would become white noise; from that it does not follow that there aren’t other radio stations I could tune into.

It is plausible (although I don’t know enough physics to know) that if one varied only one of the 20 or so parameters, such as gravity, to any extent (not just a small amount), but kept all the others fixed, then the result would be nothing other than white noise. Maybe, if you hold all 19 other values fixed, every other possible value for gravity results in noise. However, it doesn’t follow from this fact (if it is a fact at all) that there is no combination of all the values which results in a coherent structure. It might be that changing both gravity and the speed of light, and keeping all the others fixed, somehow results in a different, but equally coherent, universe.

In mathematics, a Lissajous figure is a graph of a system of parametric equations. These can be displayed on oscilloscopes, and lead to various rather beautiful patterns. Without going into any of the details (which are irrelevant), the point is that by varying the ratio of the two values (X and Y), one produces different patterns. Some combinations of values produce ordered geometrical structures, like lines or circles, while others produce what looks like a messy scribble. There are ‘pockets’ of order, which are divided by boundaries of ‘chaos’. This could be what the various combinations of values for the 20 physical parameters are like.

Fine-tuning says that immediately on either side of the precise values that these parameters have in our universe, there is ‘white noise’. But it does not say that there are no other combinations of values give rise to pockets of order just as complex as ours. It doesn’t say anything about that.

2. The problem of fine-tuning 

It might be replied that there could be a method for determining whether there are other pockets of order out there or if it is just white noise everywhere apart from these values, i.e. whether there are other radio stations than the one we are listening to or not. And maybe there is such a method in principle. However, it seems very unlikely that we have anything approaching it at the moment. And here the fineness of the fine-tuning turns back against the advocate of the fine-tuning argument. Here’s why it seems unlikely we will be able to establish this any time soon.

We are given numbers which are almost impossible to imagine for how unlikely the set of values we have would be if arrived at by chance. Craig suggests that if the gravitational constant were altered by one part in 10 to the 60th power (that’s 10 with 60 ‘0’s after it), then the universe as we know it would not exist. That’s a very big number. If each of the 20 parameters were this finely tuned, then each one would increase this number again by that amount. The mind recoils at how unlikely that is. This is part of the point of the argument, and why it seems like fine-tuning requires an explanation.

However, this is also a measure of how difficult it would be to find an alternative pocket of order in the sea of white noise. Imagine turning the dial of your radio trying to find a finely-tuned radio station, where if you turned the dial one part in 10 to the 60th power too far you would miss it. The chances are that you would roll right past it without realising it was there. This is Craig’s whole point. It would be very easy to scan through the frequency and miss it. But if you wanted to make the case that we had determined that there could be no other coherent combination of values to the parameters, you would have to be sure you had not accidentally scrolled past one of these pockets of coherence when you did whatever you did to rule them out. The scale of how fine the fine-tuning is also makes the prospect of being able to rule out other pockets of coherence in the sea of noise almost impossible to do. It would be like trying to find a needle in 10 to the 60th power of haystacks. Maybe there is a method of doing that, but it seems like an incredibly hard thing to do. The more the apologist adds numbers for the magnitude of fine-tuning, the more difficult it is to rule out there being other possible coherent combinations of values out there somewhere.

Thus, it seems like the prospects of discovering a fine-tuned pocket of coherence in the sea of white noise are extremely slim. But this just means that it seems almost impossible to be able to rule out the possibility that there is such additional a pocket of coherence hidden away somewhere.

Think about it from the other side. If things had gone differently, and the values of the parameters had been set differently, then there might be some weird type of alien trying to figure out if there were other pockets of coherence in the range of possible values for the parameters, and they would be extremely unlikely to find ours, precisely because ours (as Craig is so keen to express) is so delicately balanced. Thus the fine-tuning comes back to haunt the apologist here.

We have a pretty good understanding of what the values for the parameters are for our universe, although this is obviously the sort of thing that could (and probably will) change as our understanding deepens. But I do not think that we have a good understanding of what sort of universe would result throughout all the possible variations of values to the parameters. It is one thing to be able to say that immediately on either side of the values that our universe has there is white noise, and quite another to be able to say that there is no other pocket of coherence in the white noise anywhere.

The fine tuning result is like if you vote for party X, and your immediate neighbours on either side vote for party Y. You might be the only person in the whole country who votes for party X, but it doesn’t follow that this is the case just because you know that your neighbours didn’t.

If the above string of reasoning is correct, then for all the fine tuning result shows, there may be pockets of coherence all over the range of possible values for the parameters. There are loads of possible coherent Lissajous figures between the ‘scribbles’, and this might be how coherent universes are distributed against the white noise. There could be trillions of different combinations of values for the parameters which result in a sort of coherent universe, for all we know. And the magnitude of the numbers which the apologist wants to use to stress how unlikely it is that this very combination would come about by chance, is also a measure of how difficult it would be to find one if it were there.

3. The meaning of ‘life’

It seems that if the above reasoning is right, then other pockets of coherence are at least epistemically possible (i.e. possible for all we know). Let’s assume, just for simplicity, that there are at least some such alternative ways the parameters could be set which results in comparably stable and coherent universes as ours. Let’s also suppose that these are all as finely tuned as our universe is. For all we know, this is actually the case. But if it is the case, then it suggests a distinction between a universe is finely-tuned, and one that is fine-tuned for life. We might think that those other possible universes would be finely tuned, but not finely tuned for life because we could not exist in those universes. We are made of matter, which could not exist in those circumstances. It might be that something else which is somehow a bit like matter exists in those universes, but it would not be matter as we know it. Those places are entirely inhospitable to us.

 

But this doesn’t mean that they are not finely-tuned for life. It just means that they are not finely-tuned for us. The question we should really be addressing is whether anything living could exist in those universes.

Whether this is possible, of course, depends on precisely what we mean by ‘life’. This is obviously a contentious issue, but it seems to me that there are two very broad ways we could approach the issue, which are relevant for this discussion. Let’s call one ‘wide’ and one ‘narrow’.

Here is an example of a wide definition of ‘life’. For the sake of argument, let’s say that living things all have the following properties:

  • The capacity for growth
  • The capacity for reproduction
  • Some sort of functional interaction with their environment, possibly intentional

No doubt, there will be debate over the conditions that could be added, or removed, from this very partial and over-simplified list, and the details do not matter here. However, just note one thing about this list; none of these properties require the parameters listed in the usual presentations of the fine-tuning argument to take any particular value. So long an entity can grow, reproduce and interact with its environment, then it is living, regardless of whether it is made of atoms or some alien substance, such as schmatoms. Thus, on such a ‘wide’ definition of ‘life’, there is no a priori reason why ‘life’ could not exist in other universes, even if we couldn’t.

On the other hand, we might define ‘life’ in terms of something which is native to our universe, such as carbon molecules, or DNA. If, for example, the gravitational constant were even slightly different to how it is, then DNA could not exist. Thus, if life has to be made of DNA, then life could not exist in any pocket of coherence in the sea of white noise apart from ours.

So there are two ways of answering the question of whether an alternative set of values to the parameters which resulted in a coherent universe could support life – a wide and a narrow way. On the wide view the answer seems to be ‘yes’, and on the narrow view the answer is definitely ‘no’.

It seems to me that there is very little significance to the narrow answer. On that view, the universe is fine-tuned for life, but only because ‘life’ is defined in terms of something which is itself tied to the physical fine-tuning of the universe. The meaning of ‘life’ piggy-backs on the fine-tuning of the physical variables. And this makes it kind of uninteresting. The same reasoning means that the universe is fine-tuned for gold as well as life, because the meaning of ‘gold’ is also tied to specific things which exist only because of the values of the variables, i.e. atoms and nucleus’, etc. Thus, if we want to say ‘fine-tuned for life’ and have that mean something other than just ‘fine tuned’, then we should opt for the wide view, not the narrow one.

But then if we go for the wide view, we are faced with another completely unknown variable. Just as we have no idea how many other potential pockets of coherence there may be in the sea of white noise, we also have no idea how many of them could give rise to something which answers to a very wide definition of ‘life’. It might be that there are trillions of hidden pockets of coherence, and that they are all capable of giving rise to life. We just have no information about that whatsoever.

 

 

5. Back to the argument

What the preceding considerations show is that the usual arguments taken to rule out the ‘chance’ explanation are missing something very important to the equation. I completely concede that our universe is extremely finely-tuned, to the extent that Craig explains. This means that if the values of the parameters were changed even a tiny amount, then we could not exist. However, because we don’t have any idea whether other combinations of values to those parameters would result in coherent universes, which may contain ‘life’, we have no way of saying that the chances of a universe happening with life in it are small if the values of these parameters were determined randomly. It might be that in 50% of the combinations there is sufficient coherence for life to be possible. It might be 90% for all we know. Even if it were only 1%, that is not very unlikely. Things way less likely happen all the time. But the real point is that without knowing these extra details, the actual probability is simply impossible to assess. Merely considering how delicately balanced our universe is does not give us the full picture. Without the extra distributions (such as how many possible arrangements give rise to coherent universes, and how many of those give rise to life) we are completely in the dark about the overall picture.

This makes the argument an instance of the base-rate fallacy. The example on Wikipedia is the following:

“A group of police officers have breathalyzers displaying false drunkenness in 5% of the cases in which the driver is sober. However, the breathalyzers never fail to detect a truly drunk person. One in a thousand drivers is driving drunk. Suppose the police officers then stop a driver at random, and force the driver to take a breathalyzer test. It indicates that the driver is drunk. We assume you don’t know anything else about him or her. How high is the probability he or she really is drunk?”

Because the ‘base-rate’ of drunken drivers is far lower than the margin for error in the test, this means that if you are tested and found to be drunk, it is a lot more likely that you are in the group of ‘false-positives’ than not. There is only one drunk person in every 1000 tested, and (because of the 5% margin for error), there are 49.95 false positives. So the chances that you are a false positive is far greater than that you are the one actually drunk person. It’s actually 1 in 50.95, which is roughly a probability of 0.02. Thus, without the information of the base-rate, we could be fooled into thinking that there was a 0.95 chance that we had been tested correctly, whereas it is actually 0.02.

With the fine-tuning argument we have a somewhat similar situation. We know that our universe is very delicately balanced, and we know that we could not exist if things were even slightly different. But because we effectively lack the base-rate of how many other possible combinations of values give rise to different types of life, we have no idea how unlikely it is that some such situation suitable for life could have arisen, as it were, by chance. As the above example shows, this rate can massively swing the end result.

6. Conclusion

The fine-tuning of the universe is a fact. This does not show that the universe is fine-tuned for life though. It also does not show that the universe must have been designed. It is impossible to know what the chances are that this universe happened ‘by chance’, because we do not have any idea about the relevant base-rate of coherent and (widely defined) life-supporting universes there could be. Thus, we have no idea if we can rule out the chance hypothesis, because we have no idea what the chances are without the information about the base rate.

The Semantics of Nothing

0.   Introduction

The word ‘nothing’ has interesting semantic features. It is a ‘negative existential’, in the sense that it refers to a non-existing thing. This is perplexing, because if ‘nothing’ is a simple referring term, then the semantic role that it plays in contributing to the meaning of a sentence it features in is to point to its referent. As it has no referent, how can it play this role successfully? There are two general strategies for dealing with this puzzle; one is to treat the idea of nothing as a sort of thing, and the other is to treat it as a case of failure to refer at all.

1.   Creation from nothing

The term ‘nothing’ is deployed as part of one of the supports for the Kalam cosmological argument. The first premise of that argument is: ‘whatever begins to exist has a cause’. One of the lines of support for this premise is the familiar dictum ‘nihilo ex nihilo fit’, or ‘nothing comes from nothing’. When pressed on why this is true, a typical line of defense is that ‘nothing has no causal powers’. I say that this sentence is ambiguous, due to the word ‘nothing’. On one account the sentence treats ‘nothing’ as a referring term; something like ‘the complete lack of any object’. On the other account, the term expresses a failure to refer to any thing. The first reading (which I shall call the ‘referential sense’) is the intended sense, but it strikes me as ad hoc (and I will explain this more below). The second sense (which I shall call the ‘denotative sense’) expresses a different proposition altogether – one that fails to support the premise in any way.

2.   A Toy Example

The ambiguity can be brought to the surface if we consider the two semantic accounts of the word in more detail. Before we look at the sentence ‘nothing has no causal powers’, I want to first play with a less controversial example, to get the distinction clear. So my toy sentence is:

1) ‘Nothing will stop me getting to work on time’

First, let’s look at the referential sense of ‘nothing’, as it applies to this sentence. On this account, ‘nothing’ is just another referring term, like ‘John’, or ‘Paris’, or ‘my favourite type of ice cream’, etc. The referent of ‘nothing’ is ‘the complete absence of any things’, or something along those lines. It’s like an empty void with no contents whatsoever.

The sentence is essentially of the form ‘x will stop me getting to work on time’, where ‘x’ is an empty variable waiting to be filled by any constant (or referring term), like ‘John’ or ‘my favourite type of ice cream’, or ‘nothing’ etc. Let ‘Wx’ be a predicate for ‘x will stop be getting to work on time’. If ‘a’ is a constant that refers to my friend Adam, then the proposition ‘Wa’ means that Adam will stop me getting to work on time. I will not get to work on time, because I will be stopped by Adam from doing so. Something that Adam will do, such as physically restraining me, or hiding my keys, or just distracting me with an interesting philosophical discussion, etc, will prevent me from getting to work on time. That’s what Wa is saying.

Let ‘n’ be a constant that refers to ‘the complete absence of anything’. We could put the logical form of 1) as follows:

Ref)   Wn

Ref says that I wont get to work on time because ‘nothing’ is going to stop me. This mirrors the logical form of the sentence above where Adam prevented me from getting to work on time. But this seems wrong, as 1) doesn’t seem to say that I won’t get to work on time because of nothing (i.e. the complete absence of any thing) getting in my way. We don’t seem to be expressing the idea that ‘nothingness’ is going to hide my keys, or engage me in a philosophical discussion, etc. We are not expressing that I will not get to work on time. Rather, we are expressing something close to the opposite of that; the sentiment expressed by ‘nothing is going to stop me getting to work on time’ is that I will be on time to work, come what may. So the referential way of reading the term ‘nothing’ is not appropriate here.

Let’s look at the second account, the denotative account. On this reading 1 gets analysed out as the following (note that we still have the predicate Wx, but use a quantifier and a bound variable and so don’t need the constant ‘n’):

Den)    ~(∃x)(Wx)

On this reading, we are saying that it is not the case that there is a thing such that it will stop me getting to work on time. We could re-write Den as follows:

Den’) (∀x)~(Wx)

Den’ says that for every x, it is not the case that x will stop me getting to work on time. This captures very well the sentiment that come what may we will not let anything prevent us from getting to work on time. We would say that the denotative proposition is true in this situation, and that seems right.

Thus the two analyses are very different. They render propositions with a different logical forms and different truth-values in this case. In the referential case, we are referring to an entity, and saying of that thing that it will succeed in preventing me from getting to work on time. So the logical form of the proposition, when analysed referentially, is wrong. In addition to this semantic or logical issue, we also have a metaphysical or ontological worry. We may feel that the entity referred to in Ref is of dubious ontological status. Nothing doesn’t exist; it isn’t a thing as such. Successful reference seems to have as a presupposition that the referent exists in some sense or other. If that is right, then when we successfully refer to ‘nothing’ then there is something which is the referent for the term ‘nothing’. But if there is some referent, then ‘nothing’ doesn’t mean the complete absence of any thing. It may be that the combination of this model of reference with the insistence of ‘nothing’ meaning the complete absence of any thing is incoherent. So we can feel dissatisfied with Ref here for both ontological and logical reasons.

We may want to avoid this problem by postulating that ‘nothing’ refers to an entity, yet what it refers to is not an existing thing. Nothing is, even though it doesn’t exist. It is a something, just not an existing something. I find this way of talking almost unintelligible. It seems to me as a bedrock metaphysical principle that there are no non-existing things. There is not two types of existence; rather there is only one type of existence. If ‘nothing’ is, then it exists. The terms ‘is’ and ‘exists’ are synonymous. In this regard, I find Russell (On Denoting) and Quine (On What There Is) to be instructive.

Den, on the other hand, does not refer to any thing of dubious ontological status. When recast in the form of Den’ it clearly and explicitly quantifies over all the things that there are and says of those things that none of them are going to stop me getting to work. So it has going for it that it captures the intention behind the sentence, in that it captures that I will not be stopped. Den doesn’t require postulating two types of existence. We don’t have to say that ‘nothing’ is yet does not exist. We do not directly refer to ‘nothing’, we just refer to what there is (and say that it is none of those things).

The difference between Ref and Den could be put like this: the former is a successful reference to something that does not exist, the latter is a failure to refer to anything which does exist.

3.   The Main Case

Let’s apply this to our example of ‘nothing has no causal powers’. Let’s rewrite having no causal powers as being ‘causally inert’, and represent that as a predicate, ‘Ix’. On the referential reading, the sentence has the form:

Ref2)   In

This says ‘nothing is causally inert’. As we have seen, the model of reference used here treats nothing as a referent of the term n, which means it is the thing referred to by n. The proposition is true only if the referent of n, i.e. nothingness, is actually causally inert. And nothingness, as conceived as an empty void with no contents whatsoever, is plausibly causally inert. So the claim seems to capture well the intention behind the apologist’s assertion here. The reason that the universe couldn’t have ‘popped into being from nothing’ is that ‘nothingness’ has no abilities to make things pop into existence. It cannot do anything; it is causally inert.

The denotative reading would be as follows:

Den2) ~(∃x)(Ix)

This says that it is not the case that there is a thing such that it is causally inert. Recast in universal terms, it says:

Den2’) (∀x)~(Ix)

This says that everything is such that it is not causally inert; everything has causal powers. On this reading, we are effectively saying that abstract objects, and similar proposed causally inert entities, do not exist; there are no abstract objects, etc. This is because abstract objects are causally inert, and Den2 says that there is no causally inert thing.

One would suppose, looking at this that in the case of nothing having no causal powers, we should take the referential reading, as this makes sense of the apologist’s claims about how the universe had to have a cause. It is clearly not their intention to assert that causally inert objects don’t exist; they mean to assert that the complete absence of anything cannot itself cause something.

In the toy example, when we distinguish the referential and denotative sense of ‘nothing’, it is clear that the referential sense is incorrect. It entails something which is clearly not intended by the speaker, that I will not get to work on time, when we meant to express that come what may I will get to work on time. In the apologetical example, the analysis seems to go the other way; the denotative sense seems to entail a proposition which clearly isn’t what the apologist intends. So, while the toy example is denotative, the apologetical example is referential.

I have two worries with this conclusion:

a) If we take the referential reading of ‘nothing’ in the phrase ‘nothing has no causal powers’, then we are referring to an entity that is of questionable ontological status. It is the referent of the term n, yet it is the complete absence of any thing. So it is a thing that does not exist. We might want to follow Russell in On Denoting, and Quine in On What There Is and disallow such talk of non-existing things. Indeed, we may consider such talk of nothing as a dubious case of reification. Nothing is not a thing of any type whatsoever.

b) This is my main worry. It seems to me that most cases of the word ‘nothing’ are denotative, and almost none are referential.

Here are a few examples:

  • ‘There is nothing to split the two candidates with only days before the election.’
  • ‘There is nothing I like better than ice cream’
  • ‘Nothing pisses me off more than ice cream’
  • ‘You mean nothing to me’
  • ‘There is nothing in the fridge’

The first four cases are clearly denotative (just plug in the different readings of ‘nothing’ and see for yourself in each case). Possibly in the last example, we may want to use the referential sense, but the denotative sense seems at least as plausible. Are we expressing that there is an absence of any thing in the fridge, or that there is not any existing thing in the fridge? Neither seems preferable.

My question is: can there be an example of a sentence that uses the word ‘nothing’, and isn’t the clearly apologetical ‘nothing has no causal powers’ etc, or some other esoteric metaphysical example, for which the referential reading is clearly the correct one (and not the denotative one)?

Are there ever cases where the referential sense is the correct one, apart from the use in things like supporting the Kalam? If the answer to this question is ‘no’, then the use by the apologist is ad hoc in the support for the Kalam case. This is an open question (feel free to suggest candidate sentences in the comments section). If there is a plausible looking case, then the charge of ad hoc-ness can be deflated.

Creation ex nihilo

0. Introduction

I have recently come across a blog written by Richard Bushey, which has lots of typical apologetical arguments summarised by the author. As such, it is an interesting place to look around to find typical bad arguments to straighten out.

Here I want to look at one in particular, not because there is anything original about it, but really because there is nothing original about it. The post is an example of the sort of regurgitation of arguments made by people like William Lane Craig that one often encounters on the internet. Here is the post, entitled ‘Can a universe emerge from absolutely nothing?‘. In it, Bushey explores the idea of the creation of the universe ex nihilo (or ‘from nothing’), and rehearses some of the common arguments for why this isn’t possible.

  1. Setting

The setting for the topic discussed in the post is ultimately the cosmological argument (probably specifically the kalam cosmological argument popularised by William Lane Craig, on which I have written before). The idea is that one of the arguments put forward to prove the existence of God is that the existence of the universe requires a causal explanation, which could only be God, as a necessarily existing being. The response to this that Bushey is addressing here is to basically call into question whether the universe requires causal explanation. As he explains:

Many people seem to take it for granted that things do not just appear with absolutely no cause. But it would be quite convenient for the atheist if it were the case that this were a possibility. Atheism would then be able to deflect one of the seminal arguments for the existence of God. We need to be able to provide some justification for thinking that universes cannot emerge from absolutely nothing.

Bushey offers five distinct points, and I want to look at three of them (I have nothing to say of any note about quantum vacuums, and am happy to grant that God doesn’t need a cause to exist, at least for now). The three points I will address here are labelled by Bushey as:

a) ‘Nothing’ has no causal powers.

b) What if universes could come from nothing?

c) A good inductive conclusion.

      3. ‘Nothing’ has no causal powers

As the title of this section suggests, Bushey is arguing here that the reason the universe has to be caused by something, such as God, is that nothing is itself not able to cause anything. As an intuition pump to get you in the mood to agree with him, Bushey offers the following examples:

If your co-worker was taking a day off, the boss would naturally ask, “Who is going to cover your shift?” If the coworker said, “Nobody,” the boss would be concerned. ‘Nobody’ has no causal powers. They cannot perform the function of the job because ‘nobody’ designates the absence of somebody. Similarly, if I said that “There is nothing to eat,” my stomach would be empty. If I said that there was nothing that could stop the invasion of a particular army, I would be expressing that the military force would go unchallenged. 

Now we have the idea of what it means to say that ‘nothing’ lacks causal powers. ‘Nothingness’ cannot play the role of a co-worker, satisfy an empty stomach, or impede an oncoming army. Nothingness can’t do anything. Given that primer, here comes the beef:

So when atheists tell us that a universe could emerge from absolutely nothing, or attempt to provide accounts of how nothing could have produced the universe, they are expressing an incoherent thought. If ‘nothing’ designates the absence of anything at all, then it follows that there are no causal powers. If there are no causal powers, then it lacks the capacity to produce universes.

Given that nothingness cannot fill-in for an absent waiter’s shift in a cafe, it seems perfectly reasonable to extend this to think that it cannot manufacture universes either.

So, what is wrong with this? Well, we might already be suspicious of the first example. The boss might be concerned with the fact that nothingness has no causal powers, but I would suggest that it is more likely that he is really concerned about the lack of something to fill in which has the relevant causal powers. And these are not two ways of saying the same thing. It is not like the co-worker said ‘Don’t worry boss – nothingness will fill in for me’, to which the boss replied ‘Oh no, not bloody nothingness again! It’s complete lack of causal powers always ends up causing me grief when it comes time to tidy up at the end of the evening!’ By saying that nothing (or nobody) is going to fill in for you at work, you are saying that there is no thing about which it is true that that thing is going to fill in for you at work; you are not saying that there is this thing called ‘nothing’, about which it is true to say that it is going to fill in for you at work. We must keep these two subtly different understandings entirely distinct when we think about this, or else we are led down a garden path of confusion by Bushey here.

Consider Russell’s treatment of negative existentials in On DenotingI might want to express the fact that I don’t have a sister by saying ‘my sister does not exist’. On face value, we might think that the best way to think about the semantic value of such a phrase is as a referent about which it is true that she doesn’t exist; as if I refer to a non-existent entity. However, says Russell, far better would be to think about it like this: we are simply saying that for all the things that do exist, none of them are my sister. The propositional function ‘x is my sister’ is false for all existing things.

Let’s apply this to the boss example. Is the boss worried that a) there is a non-existent entity, who has no causal powers, filling in for a shift, or is he worried that b) for all the things that there are with the relevant causal powers, it is false that any of them is filling in for the shift? I see no reason at all to suppose that the best way of reading that situation would be by stipulating a), and every reason to suppose that it would be b). Unless Bushey has some additional argument as to why this reading is not acceptable, we at least seem to have an unproblematic rendering of this example here.

Let’s apply this to the universe example. If an opponent of the cosmological argument (who may or may not be an atheist) suggested that maybe nothing caused the universe to exist, which of the following would be be better to render this as:

a) Before the universe existed, there was nothingness, and that caused the universe to come into being.

b) For all the things that there have ever been (in any sense whatsoever), none of them caused the universe to exist.

Again, I see no reason to think that a) would be the intended meaning of such a suggestion, and every reason to think that it would be b). When someone says that ‘nothing caused the universe to exist’, they just mean the propositional function ‘x caused the universe to exist’ is false for all values of x, not that there is a value of x, called ‘nothing’ about which it is true.

Even saying that ‘nothing lacks causal powers’ is already wrong. ‘Nothing’ isn’t a thing. It is shorthand for ‘it is not the case that there is a thing’, i.e. the negation of an existential quantifier: ¬∃. So, taken literally, the phrase ‘nothing lacks causal powers’, would be rendered as follows (where ‘Cx’ is ‘x has causal powers’):

¬∃x (¬Cx)

Using nothing but the definition of the universal quantifier, we can prove the following equivalence in classical logic:

(¬∃x (¬Cx))  ↔  (∀x (Cx))

This just shows that the phrase ‘nothing lacks causal powers’ logically just means the same as ‘everything has causal powers’. Reifying ‘nothing’ to the status of an abstract object, with no causal powers, is just to misuse language; a crime which is unforgivable when there is a logically straightforward, and existentially unproblematic, analysis available.

4. What if universes could come from nothing?

Bushey has another go at providing some reason for thinking that the universe could not have come from nothing. This time he picks up on another well rehearsed argument from William Lane Craig. The idea this time is that if someone wants to hold that the universe might have come into being out of nothing, then why think that only universes could come into being out of nothingness? Here is how Bushey puts it:

Suppose for a moment that it were true that things could appear without any cause at all. If that were the case, then our rational expectations for the universe would seem to be unjustified. It would become inexplicable why anything, and everything did not emerge without a cause at all. This point was charmingly made by Dr. William Lane Craig in his debate with Dr. Peter Slezek. He pointed out that nobody is concerned that as they are sitting in this debate, a horse may have appeared uncaused out of nothing in their living room and is currently defecating on the carpet as we speak.

The idea seems to be that if we grant special exemption to universes being able to come from nothing, we would be rationally compelled to extend this to cover everything. We should expect random things popping into existence all the time, yet we don’t. We implication is that we don’t have this expectation because we know that things require causes to come into being, and cannot come into being in the absence of causes.

So, should we give a special pass to universes? Isn’t that special pleading if we do so? I say it isn’t, and that again there is a subtle but powerful misunderstanding about nothingness which is driving this line of argument.

Take the idea of a horse just appearing in front of you and defeacting on the floor. We know this isn’t going to happen (setting quantum probabilities to one side). But why do we know this? I say that the reason for this isn’t because we know that things cannot come from nothing. That idea isn’t even relevant. If you are at home in your front room wondering if a horse is about to suddenly appear, that isn’t an example of nothingness! What you know is that the relevant causal properties of what exists around you isn’t sufficient to produce a horse. You know that a horse cannot be produced by this particular type of something.

Let’s turn to the idea of the universe. Given the understanding gained from section 3 above, we do not have to think of ‘nothingness’ as preceding and causing the existence of the universe. We could just say that there is no thing (in any sense) that preceded and caused the universe. The beginning of the universe is the beginning of everything. So, the context which was not conducive to a horse popping up in front of you in the previous example has no counterpart here. There is no ’empty space’ into which the universe pops. There is no ‘nothingness’ waiting to be filled with a universe.

Could an infinite empty void of nothingness suddenly give rise to a universe? I don’t know. Could the universe simply be all that there is? I don’t see why not. Pointing out that horses don’t suddenly appear in front of us randomly is completely irrelevant.

5. A good inductive conclusion.

This last point is quite similar to the previous one, and has a similar root of misunderstanding with it. Here is Bushey again:

Common experience indicates that things have an explanation. They do not just appear, uncaused, out of absolutely nothing. The entire project of science is predicated upon this premise. Science is the search for causes within the natural world. If we were to establish the premise that things appear without a cause, then the project of science would be wholly undermined. Scientists who searched for causes of natural phenomenon would be engaging in a fruitless endeavor. It may just be that their specimen emerged without a cause. Why does a fish have a particular gill? Perhaps it appeared, uncaused, out of nothing.

It is quite easy to spot the error here. Take the fourth sentence in that quote: “Science is the search for causes within the natural world”. I don’t think this is the best definition for science one could find, but it is particularly bad that it is the one Bushey uses in this context. If science is the search for causes within the natural world, then there is no reason to think that it applies to things beyond the natural world. Just because things in the universe behave a certain way, doesn’t mean that the universe itself has to display those behaviours. Say everything in the sea floats, would it follow that the sea floats? If there is no causal explanation for the universe, which simply is all that there is, it would not follow that things that actually exist could not be described by science, or that we would have no reason to think that every particular fact in the universe had a causal explanation.

6. Conclusion.

There is no reason provided in Bushey’s post to think that the universe has to have a cause. One should resist the temptation to reify nothingness into an amorphus blob lacking in certain properties. Don’t slide from a failure of reference to an existent thing, to a successful reference to a non-existent thing. The universe didn’t pop into existence from a pre-existent state of nothingness. It just has a finite past.

At least, maybe it does. I don’t know whether the universe was created or not. Maybe a loving personal god made it in order to teach me about morality. Maybe it popped into existence from a pre-existing state of nothingness. Maybe it is just all there is. My point is that you don’t get to prove the first of these by undermining the second, given that there is a coherent third. That would be a fallacy of false dichotomy.

Craig’s List – Omniscience and actually existing infinities

Introduction

William Lane Craig has famously argued for the ‘Kalam cosmological argument’ (in many places, but for example in Craig & Sinclair [2009]). Here is the argument:

  1. Everything that begins to exist has a cause.
  2. The universe began to exist.
  3. Therefore, the universe had a cause (Craig & Sinclair [2009], p 102).

The argument is clearly valid, as it is a version of modus ponens. Thus, in order to deny the conclusion, one must argue that the first or second premise is not justified.

Most people have argued against premise one, disputing whether all things which begin to exist have causes for their existence, or the fact that a fallacy of composition may be at play with the generalization from all things in the universe to the universe as a whole. I will not be pursuing this line of argument here, but will instead look at premise two.

Premise two seems to be supported by physics, specifically cosmogony, which some say indicates that the spacetime we exist within came into existence at the big bang. People who know more about this than I do tell me that this is actually a misconception of this theory, and that it is not really a theory about the origin of spacetime at all. However, we can avoid delving any further into the details of the physics, because Craig does not rest his argument on the interpretation of the big bang theory. There is a logical argument Craig spends time going into, according to which the universe must have had a beginning – that it is impossible for the universe to have always existed. Here is that argument:

2.1. An actual infinite cannot exist.

2.2. An infinite temporal regress of events is an actual infinite

2.3. Therefore, an infinite temporal regress of events cannot exist. (ibid, p 103)

It is on this supporting argument that I wish to focus. Specifically, it is the first premise of this argument that I will be spending time going into here. If we can knock this premise out, then it undermines the entire supporting argument, and with it the credibility of the main argument. If we can deny 2.1, we can avoid having to assent to 3.

Hilbert’s Hotel

In order to motivate 2.1 (that an actual infinite cannot exist), Craig uses the example of ‘Hilbert’s Hotel’. In this imagined hotel there is an infinite number of rooms. Infinity has a distinctive property, according to which a proper subset of it can be equal in cardinality to the whole, there are various counter-intuitive consequences, which Craig uses to motivate the idea that this could not actually exist. For example, if the hotel is full but a prospective guest arrives asking for a room, the hotel manager can simply ask each occupant to move into the next room, thereby making room number one free. Because there is an infinite number of rooms, there will be room for every occupant, thus making a newly free space for the new guest to stay in, even though the hotel was full. Even if infinite new guests turn up, the hotel manager can make room by getting everyone in the hotel to move into the room with the room number that is twice the number of their current room (so room number two gets room number four, room number four gets room number eight, etc.). This frees up an infinite number of rooms, even though the hotel was full. Craig comments:

“Can anyone believe that such a hotel could exist in reality? Hilbert’s hotel is absurd. But if an actual infinite were metaphysically possible, then such a hotel would be metaphysically possible. It follows that the real existence of an actual infinite is not metaphysically possible” (Craig & Sinclair [2009], p. 109-110).

If this is correct, then because a universe with no first moment would constitute an actually existing infinity, it follows that the universe had a first moment. Thus, the idea is that it is no objection to simply say that maybe the universe always existed. It couldn’t have always existed, says Craig.

However, it is not clear to me that his objection really applies to the universe, and I will spell this out in more detail now.

Pinning down the absurdity

One might wonder what specifically it is about Hilbert’s hotel that Craig finds absurd. It seems that the sheer scale of the hotel, the fact that it has infinite rooms, is not itself absurd to Craig. If it was, then the example would simply have been:

‘Imagine that there is a hotel with infinite rooms in – that’s absurd!’

Given that the example was more complex than this, it seems that just saying that the hotel is infinite is not enough for Craig to bring out the absurdity. Nor does simply adding that the hotel actually exists constitute the absurdity, otherwise the example would have been:

‘Imagine that there is a hotel with infinite rooms in, and that it actually exists – that’s absurd!’

Surely, when picturing Hilbert’s hotel, one pictures it as actually existing. Adding that it actually exists is somewhat empty as a property, and surely not enough on its own to make the difference between not absurd and absurd. So what is it that pushes us over this threshold?

It seems to me, given the examples used to illustrate the absurdity of Hilbert’s hotel, that Craig’s idea is as follows. The factor that gets us across the line is what we might call the behavior of the hotel. With an infinite hotel, given certain conditions obtaining, contradictions can be manifested, and contradictions are absurd. So it took the new guest to arrive, and for everyone to shuffle up one room, for an absurdity to become manifested; namely, the hotel is full, but also has a room available for a new guest. If the guest does not arrive, or arrives but is turned away by the manager, then where is the absurdity? How do we generate a contradiction without interacting with the hotel? It seems like the only way we could imply an absurdity in that case would be simply pointing out that the hotel has infinite rooms. But if this was on its own enough to constitute absurdity, why bother with the example of the guest arriving? Is it just for rhetorical effect? It seems to me that the answer is that without the guest arriving and the creation of the new free room, Craig thinks that nothing absurd is present.

If this right, then we could employ a distinction between active and passive infinities. An active infinity is one that manifests absurd behavior (like being full but also making room for a new guest), whereas a passive infinity is one that does not (like a Hilbert’s hotel which never admits new guests). Now, it should be noted that a passive infinite retains the potential to manifest absurdity; it is passive just so long as it doesn’t actually do so.

This makes the distinction between ‘actually existing’ and ‘not actually existing’ slightly wide of where the beef is here. It seems we could have an actually existing Hilbert’s hotel, which remains passive, and for all Craig has said, this would not be absurd. The absurdity only kicks in when an actually existing infinity becomes active.

The infinite universe is passive

The problem with Craig spelling out the nature of the absurdity associated with actually existing infinities like this, is that it doesn’t apply to the eternally existing universe. There are models where we could make his objection apply, but the most natural way of cashing it out avoids his problem, as I will explain.

Imagine a number line that contains all integers running from minus infinity, through 0 all the way up to positive infinity. Now think of 0 marking out this very moment now. This is a bit like the most natural way of thinking about the eternal universe; each moment has infinitely many earlier moments and infinitely many later moments. If this is how Craig is characterizing the eternally existing universe, then it is a passive infinity. There is no corresponding example to making a free room, or withdrawing a book. One cannot add a moment to time, nor take one away. It is a ‘closed’ infinity. In fact, it is arguably metaphysically impossible to add a time or take one away. Thus, Craig may be correct that active infinities are metaphysically impossible, but because the eternal universe is not one of these, then he has no objection to the eternal universe.

As I said, there are ways of cashing out the eternally existing nature of the universe according to which Craig’s point holds. For example, consider the ‘growing block’ theory of time. According to this theory, the past is a fixed set of facts, which is growing as time moves forwards. We continually add new truths to the stock of settled past truths. If this were the model, then we would have an infinite list of past truths, but we would be able to add to it. In a sense, this would resemble Hilbert’s Hotel and thus make the universe an active infinity.

It should be noted that even on this growing block theory, there is room to doubt whether this really counts as an absurdity. With the hotel example, we can derive a sort of contradiction, in the sense that the hotel was full, but had room for a new guest. If being full means that there is no room, then this is a contradiction. But it is not clear what is the contradictory sentence we are supposed to be able to make out of the growing block theory here. Sure, there are infinite past moments, and then a new one gets added to the pile as time moves forward. The only contradiction I can see here is that the cardinality of the past moments is the same, even after a new one is added to the block. If so, then we have our candidate.

It is a weak candidate, as it seems to me that we ought to simply accept that this is what an infinite block would be like. However, let’s assume that Craig has scored his point here, and that the growing block theory is absurd for that reason. No such account can be leveled at the eternal universe outlined above. It has an infinite number of moments, but there is no possibility of adding new moments or taking them away, so it is passive. It seems like we can block Craig’s argument by simply explaining clearly what an eternal universe looks like, and that while it is infinite in extent, it manifests no absurdity.

In fact, this will form one horn on a dilemma I wish to place Craig in. As we shall see, if there is a problem with the growing block theory, then it also affects Craig’s version of God. The dilemma will be that either the universe is infinite in temporal extension, or God doesn’t exist.

The Infinite God Objection

Craig’s God is omniscient. This means that ‘God knows only and all truths’. Watch him commit to this position here:

It is uncontroversial that there are mathematical truths, like that it is true that 2 + 2 = 4. God knows all these truths as well (Craig explicitly makes this point at 6:20 in the video above). To make the point as simple as possible, God knows the solution to every equation of the form x + y = z, where the variables are natural numbers. As there is an infinite number of such solutions (with a cardinality equal to the smallest infinity, ℵ0), it follows that God’s knowledge is correspondingly at least as infinite as the cardinality of the natural numbers (and obviously greater if he also knows all real number solutions as well).

Let’s consider Craig’s God’s knowledge of these arithmetic solutions as a list of truths, which we could call ‘Craig’s List’. It would be an infinitely long list. So Craig’s God’s knowledge is infinite.

But, according to the Hilbert’s Hotel argument from above, the infinite cannot actually exist. Therefore, an omniscient God cannot actually exist. Craig’s God is omniscient. Therefore, by his own argument, Craig’s God cannot exist.

Call this the ‘Infinite God Objection’.

God’s knowledge is of induction schemas

It could be objected here that God does not need to know every arithmetic truth, such as 2 + 2 = 4, because as long as he knows the base case and all relevant induction schema, he would know enough to deduce the answer to any similar equation. If this were the case, then it would drastically limit the amount of propositions God would need to know, from infinite to a mere handful.

My response to this is that if this were all that were required to know all mathematical truths, then I know all mathematical truths. After all, I know the base case (that 0 is a number) and the relevant induction schema. God and I both have the same resources at hand, and if this is all it takes to know all mathematical truths, then we both know all mathematical truths. This is an awkward consequence, to say the least.

But this consequence is not just awkward. It is intuitively true that there are lots of arithmetical equations that I do not know the answer to, even though I could work them out given my knowledge of the induction schema. It seems more natural to say that I do not know the answers to these questions, but I know how to work out the answers. This makes the response in the God case inadequate though. To concede that God does not know the answer to any mathematical question, but knows how to work out the answer, is just to concede that there are things he does not know. The fact that he could work it out it not a defeater to the claim that he does not know it.

On the other hand, perhaps the similarity is only apparent, and that due to my limited nature, as compared to God’s unlimited all-powerful nature, there is a meaningful difference between the two cases. Perhaps it is the case that I slowly lumber through, applying the schema to the case at hand to derive the answer, and with the possibility that I could always go wrong on the way. In contrast, God applies it at lightening speed, without the possibility of getting it wrong on the way. In this case, there is no arithmetic question you could ask God to which the answer would be ‘I don’t know, but I will work it out for you’; as soon as you have asked the question he has already worked it out. Therefore it is never true that there is something he does not know.

But I could just stipulate an equation, without asking God directly. Even though, were he to think about it he would get the answer immediately, given that he is not currently applying the schema to the case, it is not true that he knows it. So there is something he doesn’t know. So he is not omniscient.

And if we avoid this by saying that he is constantly applying the schema to all cases, then we are right back to the original case, where he knows an infinite number of truths.

Thus this escape route will not help.

God’s knowledge is non-propositional

Craig could say that God’s knowledge is non-propositional, as in the Thomist conception. On this idea, God does not know lots of individual propositions, but rather has one unified knowledge of himself, which is perfectly simple.

To begin with, this contradicts his statements in the video above, where Craig explicitly states that God knows all propositions. Perhaps we can let this slide, as it is him talking somewhat informally.

In a paper entitled ‘A Swift and Simple Refutation of the “Kalam” Cosmological  Argument?‘ (1999), Craig considers a very similar objection, namely that if mathematical truths are just divine ideas, then God’s mind has infinitely many ideas. In defense of the divine conceptualist, Craig offers the following reply:

“[T]he conceptualist may avail himself of the theological tradition that in God there are not, in fact, a plurality of divine ideas; rather God’s knowledge is simple and is merely represented by us finite knowers as broken up into knowledge of discrete propositions and a plurality of divine ideas.” (Craig, (1999), p 61 – 62).

This theological tradition goes back to Thomas Aquinas, and as an explanation of this, Craig cites William Alston’s paper ‘Does God have beliefs?’ (1986). In that paper, Alston says the following:

“[C]onsider the position that God’s knowledge is not propositional. St Thomas Aquinas provides a paradigmatic exposition of this view. According to Aquinas, God is pure act and absolutely simple. Hence there is no real distinction in God between his knowledge and its object. Thus what God knows is simply His knowledge, which itself is not really distinct from Himself. This is not incompatible with God’s knowing everything. Since the divine essence contains the likenesses of all things, God, in knowing Himself perfectly, thereby knows everything. Now since God is absolutely simple, His knowledge cannot involve any diversity. Of course what God knows in creation is diverse, but this diversity is not paralleled in the intrinsic being of His knowledge of it. Therefore ‘God does not understand by composing and dividing’. His knowledge does not involve the complexity involved in propositional structure any more than it involves any other kind of complexity” (Alston, (1986), p. 288).

Thus, if the divine conceptualist can avail himself of this Thomistic tradition of God having non-propositional knowledge, then Craig himself could make the same move to avoid the charge that God knows an infinitely long list of arithmetical truths.

There is a problem of going the Thomist route here, as Aquinas himself is very explicit about whether God knows infinite things:

“Since God knows not only things actual but also things possible to Himself or to created things, as shown above, and as these must be infinite, it must be held that He knows infinite things” (Aquinas, Summae Theologica, Q14, A12).

Alston is perhaps trying to spell out a Thomist inspired view, rather than a Aquinas’ actual views. Even if Aquinas insisted that God knows an infinity of things, perhaps a non-propositional knowledge model can be adopted whereby God knows all mathematical truths without knowing an infinite list of truths. Indeed, Alston turns to F. H. Bradley’s idealism to spell out this possible model. Aston says that on Bradley’s view, the ‘base of our cognition is a condition of pure immediacy’, in which there is no distinction between different objects of knowledge. It is like taking in a painting as a whole, without focusing on any one particular bit of the painting. We can ‘shatter this primeval unity and build up ever more complex systems of propositional knowledge’, which would be like focusing on a particular brush stroke rather than the scene as a whole. This second mode of understanding is more discursively useful, but lacks the ‘felt oneness’ of the primeval apprehension. In contrast to these modes is the nature of the ‘Absolute’ itself – the world beyond our comprehension, which ‘includes all the richness and articulation of the discursive stage in a unity that is as tight and satisfying as the initial stage’. God’s knowledge, says Alston, could be modelled like this.

Wes Morriston, in his paper ‘Craig on the actual infinite’ (2002) considers this move by Craig, and concludes that Alston’s idea is of no help here:

“On Alston’s proposal, then, God’s knowledge is certainly not chopped up into a plurality of propositional states. On the other hand, it is said to have ‘all the richness and articulation’ of discursive thought. Even if this ‘richness and articulation’ does not consist in a multiplicity of propositional beliefs, it must surely involve some sort of distinction and variation and multiplicity within the divine intellect. However ‘tight and satisfying’ the unity of God’s knowledge, it must be thought of as a unity within a multiplicity – a one in a many” (Morriston, (2002), p. 159).

Ultimately, Alston’s idea is just that a God’s knowledge is a sort of synthesis of multiplicity and unity, and Morriston’s reply is that this does not eliminate the multiplicity. So it is not really any help to Craig.

Thus it seems that the non-propositional nature of God’s knowledge is not really a way of getting out of the claim that God is infinite.

Craig’s God is a passive infinity

Given that we now have the distinction between the active and passive infinity at hand, it could be that Craig’s reply would just be that God’s knowledge of arithmetic truths is a ‘closed totality’ of knowledge, and as such is passive. Just as no new moments can be added to the timeline, no new arithmetic truths can be added or subtracted from the totality of mathematical truths. As such it is infinite, but can never manifest absurdities as a result. As such, God can be infinite in this regard and not get chewed up in the teeth of Craig’s argument.

This would be a satisfactory response by Craig, but for one thing. Craig’s God has a very distinctive relationship to time, because Craig has a very particular theory of time. This makes Craig’s God particularly vulnerable to the actively infinite God objection.

Craig’s God and Time

Craig has a fairly nuanced view about God’s relationship to time. Roughly, God existed in an atemporal manner before he created the universe, but then entered into time and became temporal.

“God exists changelessly and timelessly prior to creation and in time after creation” (Craig [1978], p 503).

Craig also believes that the correct theory of time is the ‘A-theory’, according to which the fundamental temporal relations are tensed (like ‘it is now raining’, or ‘it will be sunny’, etc), rather than tenseless (like ‘raining at t1 is earlier than sunny at t2’, etc). For Craig, there is a fact about what is happening now which is metaphysically basic, and continually changing as time rolls forwards. God, being a temporal entity in time, has knowledge of this now, of ‘where he is’ on the timeline so to speak, and consequently what is presently happening:

“As an omniscient being, God cannot be ignorant of tensed facts. He must know not only the tenseless facts about the universe, but He must also know tensed facts about the world. Otherwise, God would be literally ignorant of what is going on now in the universe. He wouldn’t have any idea of what is now happening in the universe because that is a tensed fact. He would be like a movie director who has a knowledge of a movie film lying in the canister; he knows what picture is on every frame of the film lying in the can, but he has no idea of which frame is now being projected on the screen in the theater downtown. Similarly, God would be ignorant of what is now happening in the universe. That is surely incompatible with a robust doctrine of divine omniscience. Therefore I am persuaded that if God is omniscient, He must know tensed facts” (taken from http://www.reasonablefaith.org/god-time-and-eternity, which is a transcript of a paper given in Cambridge in July 23rd 2002)

This makes Craig’s God an ‘temporal epistemic agent’, that is one who is continually updating his knowledge set with new facts about reality as time passes; namely what is presently true. He doesn’t just know that at t1 it is raining – he knows that it is now raining.

Craig’s God is an active actually existing infinity

According to Craig then, God comes to know new things as time moves forwards. But he already knows an infinite number of truths, all the mathematical truths etc, and then he adds to his knowledge as time passes. However, the cardinality of his knowledge, how many truths he knows, stays the same – it is still infinite. So he knows more things, but also the same number of things. This is a manifestation of absurdity, just like Craig complained about with Hilbert’s Hotel, and at least as convincing as the growing block problem. Thus, by his own arguments, Craig’s God cannot exist.

Dilemma

It could be that Craig objects to the distinction between active and passive infinities. Perhaps it was made for rhetorical force only. If so, then his objection should be characterized as:

‘Imagine a hotel with infinite rooms, that’s absurd, therefore it couldn’t actually exist’.

If so, then I find it very implausible. In order to accept it, we would need to have something to justify it, and all Craig offers is that one can derive ‘absurd’ consequences from it, by which he means something contradictory. I agree that if we can derive contradictions from something, then it is to be rejected. However, we have seen that the only way we can get anything absurd from Craig’s examples is if we interact with the infinity, by getting the manager to free up a room for us, etc. Craig has never offered an example of any absurd consequences from thinking of actually existing infinities that are passive. Thus, if he wants to take this option, he still has all his work ahead of him for motivating the first premise of his supporting argument. Until he has provided this motivation, we are free to refrain from assenting to it, and consequently refrain from assenting to the conclusion of the Kalam argument.

But then if Craig accepts the active/passive distinction, then he has a pair of serious problems. Given the eternal universe model, it is infinite but passive. So not absurd. So it can exist. In addition, Craig’s A-theoretic nature of God means that God manifests absurd behavior. Therefore, he cannot exist.

The conclusion, then, is that either Craig has a lot of work to do explaining why actually existing infinities cannot exist, or he has in fact argued himself into a corner where an eternal universe could exist and God cannot. It seems there are big problems for Craig’s God.