## Aquinas’ Third Way argument II – Another counterexample

0. Introduction

In the previous post, I looked at Aquinas’ third way argument, as presented by apologist Tom Peeler. He proposed a causal principle, similar to what Aquinas proposed. Aquinas said:

“that which does not exist only begins to exist by something already existing”.

Peeler said:

“existence precedes causal influence”.

But basically, they are arguing for the same principle, namely:

Causal Principle) For something to begin to exist, it must be caused to exist by some pre-existing object.

From now on, let’s just call that ‘the causal principle’. Peeler was using this principle to support the first premise of his argument, which was:

“If there was ever nothing, there would be nothing now”.

The idea is that if Peeler’s principle were true, then the first premise is true as well. In the previous post, I argued that even if we accept all this, the argument does not show that an eternal being exists. Rather, it is compatible with an infinite sequence of contingent things.

In this post, I want to make a slightly different point. Up to now, we have conceded that the causal principle entails that there are no earlier empty times. However, I want to insist that this is only true if time is discrete. If time is continuous, then the causal principle dos not entail that there are no earlier empty times. I will prove this by constructing a model where time is continuous and at which there are earlier times which are empty, and later times which are non-empty, yet there is no violation of the causal principle.

1. The causal principle

I take the antecedent of this conditional premise, i.e. “there was ever nothing”, to mean ‘there is some time at which no objects exist’, which seems like the most straightforward way of taking it. Therefore, if the causal principle is to support the premise, the causal principle must be saying that if an object begins to exist, then it must not be preceded by a time at which no objects exist.

Strictly speaking, what the principle rules out is empty times immediately preceding non-empty times. Take the following model, where we have an empty time and a non-empty time, but at which they are not immediately next to one another on the timeline. Say that t1 is empty, and t3 is non empty:

In order to use the causal principle to rule this sort of model out, we need to fill in what is the case at t2. So let’s do that. Either t2 is empty, or it is not. Let’s take the first option. If t2 is empty, then t3 is immediately preceded by an empty time, and we have a violation of Peeler’s principle. Fair enough. What about the other option. Well, if t2 is non-empty, then t3 is not a case that violates Peeler’s principle, because it is not immediately preceded by an empty time. However, if t2 has some object that exists at it, then it is a case of a non-empty time immediately preceded by an empty time, because t1 is empty. Therefore, this second route leads to a violation of Peeler’s principle as well.

The point is that if all we are told is that there is some empty time earlier than some non-empty time, without being told that the empty time immediately precedes the non-empty time, we can always follow the steps above to rule it out. We get to a violation of the causal principle by at least one iteration of the sort of reasoning in the previous paragraph.

However, this whole way of reasoning presupposes that time is discrete rather than continuous. If it is continuous, then we get a very different verdict. That is what I want to explain here. If time is continuous, we actually get an even more obvious counterexample than model 2.

2. Discrete vs continuous

Time is either discrete, or it is continuous. The difference is like that between the natural numbers (like the whole integers, 1, 2, 3 etc) and the real numbers (which include fractions and decimal points, etc). Here is the condition that is true on the continuous number line, and which is false on the discrete number line:

Continuity) For any two numbers, x and y, there is a third number, z, which is in between them.

So if we pick the numbers 1 and 2, there is a number in between them, such as 1.5. And, if we pick 1 and 1.5, then there is a number in between them, such as 1.25, etc, etc. We can always keep doing this process for the real numbers. For the natural numbers on the other hand, we cannot. On the natural numbers, there just is no number between 1 and 2.

A consequence of this is that there is no such thing as the ‘immediate successor’ of any number on the real line. If you ask ‘which number is the successor of 1 on the real number line?’, there is no answer. It isn’t 1.01, or anything like that, because there is always going to be a number between 1 and 1.01, like 1.005. That’s just because there is always going to be a number between any two numbers on the real number line. So there is no such thing as an ‘immediate successor’ on the real number line.

Exactly the same thing imports across from the numerical case to the temporal case. If time is continuous, then there is no immediately prior time, or immediately subsequent time, for any time. For any two times, there is a third time in between them.

This already means that there cannot be a violation of Peeler’s principle if time is continuous. After all, his principle requires that there is no non-empty time immediately preceded by an empty time. And that is never satisfied on a continuous model just because no time is immediately preceded by any other time, whether empty or non-empty. However, even though the principle cannot be violated, this doesn’t immediately mean that it can be satisfied. It turns out, rather surprisingly, that it can be satisfied.

2. Dedekind Cuts

In order to spell out the situation properly, I need to introduce one concept, that of a Dedekind Cut. Named after the late nineteenth century mathematician, Richard Dedekind, they were originally introduced as the way of getting us from the rational numbers (which can be expressed as fractions) to the real numbers (some of which cannot be expressed as fractions). They are defined as follows:

A partition of the real numbers into two nonempty subsets, A and B, such that all members of A are less than those of B and such that A has no greatest member. (http://mathworld.wolfram.com/DedekindCut.html)

We can also use a Dedekind cut that has the partition the other way round, of course. On this version, all members of B are greater than all those of A, and B has no least member (A has a greatest member). This is how we will use it from now on.

3. Model 5

Let’s build a model of continuous time that uses such a cut. Let’s say that there is a time, t1, which is the last empty time, so that every time earlier than t1 is also empty. The rest of the timeline is made up of times strictly later than t1, and they are all non-empty:

The precise numbers on here are just illustrative. All it is supposed to be showing is that every time up to and including t1 is empty, and that every time after t1 is non-empty. There is no first non-empty time, just because there is no time immediately after t1 at all. But there is a last empty time, which is just t1.

This model has various striking properties. Obviously, because it is a continuous model, there can be no violation of Peeler’s principle (because that requires time to be discrete). However, it is not just that it avoids violating the principle in this technical sense. It also seems to possess a property that actively satisfies Peeler’s causal principle. What I mean is that on this model, every non-empty time is preceded (if not immediately) by non-empty times. Imagine we were at t1.01 and decided to travel down the number line towards t1. As we travel, like Zeno’s tortoise, we find ourselves halfway between t1.01 and t1, i.e. at t1.005. If we keep going, we will find ourselves half way between t1.005 and t1, i.e. t1.0025, etc. We can clearly keep on going like this forever. No matter how close we get to t1 there will always be more earlier non-empty times.

So the consequences can be expressed as follows. Imagine that it is currently t1.01. Therefore, it is the case that some object exists. It is also the case that at some time in the past (such as t1) no objects existed. Whatever exists now could have been brought into existence by previously existing objects, and each of them could have been brought into existence by previously existing objects, and so on forever. So, it seems like this model satisfies Peeler’s version of the causal principle, that existence precedes causal influence, and Aquinas’ version of the principle, that “that which does not exist only begins to exist by something already existing”. Both of these are clearly satisfied in this model, because whatever exists has something existing earlier than it. However, it does so even though there are past times at which nothing exists.

4. Conclusion

The significance of this is as follows. If we assume that time is discrete, then the causal principle entails that there are no empty earlier times than some non-empty time. So if t1 is non-empty, then there is no time t0 such that t0 is empty. So if time is discrete, then the causal principle entails premise 1 of the argument (i.e. it entails that “If there were ever nothing, there would be nothing now”).

But, things are different if time is continuous. In that case, we can have it that the causal principle is true along with there being earlier empty times. The example of how this works is model 5 above. Something exists now, at t1.01, and there are times earlier than this which are non-empty. Every time at which something exists has times earlier than it during which some existing thing could have used its causal powers to bring the subsequent thing into existence. There is never any mystery about where the causal influence could come from; it always comes from some previously existing object. However, there are also empty times on this model, i.e. all moments earlier than or equal to t1. This means that the antecedent of the conditional premise is true (“if there ever was nothing”), but the consequent is false (“there would be nothing now”). So even though the causal principle looks true, the first premise is false. So if time is continuous, then the causal principle (even if granted for the sake of the argument) does not entail the first premise, and so does not support it being true.

## Aquinas’ Third Way Argument

0. Introduction

I recently listened to a podcast, where the host, David Smalley, was interviewing a christian apologist, Tom Peeler. The conversation was prefaced by Peeler making the claim that he could prove that God existed without the use of the bible.

The first argument offered by Peeler was essentially Aquinas’ ‘Third Way’ argument, but done in a way that made it particularly easy to spell out the problem with it. In fact, Peeler gave two arguments – or, rather, I have split what he said into two arguments to make it easier to explain what is going on. Once I have explained how the first argument fails, it will be obvious how the second one fails as well. The failures of Peeler’s argument also help us to see what is wrong with Aquinas’ original argument.

1. Peeler’s first argument

Peeler’s first argument went like this (at about the 23 minute mark):

1. If there were ever nothing, there would still be nothing
2. There is something
3. Therefore, there was never nothing

As Peeler pointed out, the argument is basically a version of modus tollens, and so is definitely valid. But is it sound? I will argue that even if we grant that the argument is valid and sound, it doesn’t establish what Peeler thinks it does.

Here is the sort of consideration that is motivating premise 1. In the interview, Peeler was keen to stress that his idea required merely the fact that things exist and the principle that “existence precedes causal influence”. There is an intuitive way of spelling out what this principle means. Take some everyday object, such as your phone. This object exists now, but at some point in the past it did not exist. It was created, or made. There is some story, presumably involving people working in a factory somewhere, which is the ‘causal origin’ of your phone. The important part about this story for our purposes is that the phone was created via the causal powers of objects (people and machines) that pre-existed the phone. Those pre-existing objects exerted their causal influence which brought the phone into existence; or, more mundanely, they made the phone. The idea is that for everything that comes into existence, like the phone, there must be some pre-existing objects that exert causal influence to create it. As Aquinas puts it: “that which does not exist only begins to exist by something already existing”.

One way to think about what this principle is saying is what it is ruling out. What it is ruling out is that there is a time where no objects exist at all, followed immediately by a time at which some object exists.

Imagine that at time t0, no objects exist at all. Call that an ‘empty time’. Then, at t1 some object (let’s call it ‘a‘) exists; thus, t1 is a ‘non-empty time’. This situation violates Peeler’s causal principle. This is because a has been brought into existence (it has been created), but the required causal influence has no pre-existing objects to wield it. We can picture the situation as follows:

At the empty time, t0, there is nothing (no object) which can produce the causal influence required to bring a into existence at t1. Thus, the causal influence seems utterly mysterious. This is what Peeler means by ‘nothing can come from nothing.’ So we can understand Peeler’s causal principle in terms of what it rules out – it rules out things coming into existence at times that are immediately preceded by empty times, or in other words it rules out non-empty times immediately following from empty times. Let’s grant this principle for the sake of the argument to see where it goes.

If we do accept all this, then it follows that from the existence of objects, such as your phone, that there can never have been a time at which no objects existed (i.e. that there are no empty times in the past). That’s because of the following sort of reasoning. If this time has an object, such as your phone, existing at it, then this time must not be preceded by a time at which no objects existed. So the phone existing now means that the immediately preceding time has objects existing at it. But the very same reasoning indicates that this prior time must itself be preceded by a time at which objects existed, and so on for all times.

We can put it like this: if this time is non-empty, then so is the previous one. And if that time is non-empty, then so is the previous one, etc, etc. Thus, there can never be an empty time in the past if this time is non-empty.

This seems to be the most charitable way of putting Peeler’s argument.

2. Modelling the argument

For all we have granted so far, at least three distinct options are still available. What I mean is that the argument makes certain requirements of how the world is, for it’s premises and conclusion to be true. Specifically, it requires that a non-empty time not be immediately preceded by an empty time. But there are various ways we can think about how the world is which do not violate this principle. A model is a way that the world is (idealised in the relevant way). If the model represents a way that the world could be on which the premises and conclusion of an argument are true, then we say that the model ‘satisfies‘ the argument. I can see at least three distinct models which satisfy Peeler’s argument.

2.1 Model 1

Firstly, it could be (as Peeler intended) that there is a sequence of non-necessary objects being caused by previous non-necessary objects, which goes back to an object which has existed for an infinite amount of time – an eternal (or necessary) object. Think of the times before t1 as the infinite sequence: {… t-2, t-1, t0, t1}. God, g, exists at all times (past and future), and at t0 he exerted his causal influence to make a come to exist at t1 alongside him:

On this model, there are no times in which an object comes into existence which are immediately preceded by an empty time, so this model clearly does not violate Peeler’s principle. Part of the reason for this is that there are no empty times on this model at all, just because God exists at each time. Anyway, the fact that this model doesn’t violate Peeler’s causal principle means that there is at least one way to model the world which is compatible with Peeler’s argument. The world could be like this, for all the truth of the premises and conclusion of Peeler’s argument requires.

But, this is not the only option.

2.2 Model 2

Here is another. In this model, each object exists for only one time, and is preceded by an object which itself exists for only one time, in a sequence that is infinitely long. Each fleeting object is caused to exist by the previous object, and causes the next object to exist. On this model there are no empty times, so it is not a violation of Peeler’s principle. Even though it does not violate the principle, at no point is there an object that exists at all times. All that exists are contingent objects, each of which only exists at one time.

Think of the times before t1 as the infinite sequence { … t-2, t-1, t0, t1}, and that at each time, tn, there is a corresponding object, bn:

Thus, each time has an object (i.e. there are no empty times) and each thing that begins to exist has a prior cause coming from an object. No object that begins to exist immediately follows from an empty time. Therefore, this model satisfies Peeler’s argument as well.

2.3 Model 3

There is a third possibility as well. It is essentially the same as the second option, but with a merely finite set of past times. So, on this option, there is a finitely long set of non-empty times, say there are four times: {t-2, t-1, t0, t1}. Each time has an object that exists at that time, just like in model 2. The only real difference is that the past is finite:

In this case, t-2 is the first time, and b-2 is the first object.

However, there might be a problem with this third option. After all, object b-2 exists without a prior cause. It isn’t caused to exist by anything that preceded it (because there are no preceding times to t-2 on this model). Doesn’t this make it a violation the causal principle used in the argument?

Not really. All that Peeler’s causal principle forbids is for an object to begin to exist at a time immediately following an empty time. But because there are no empty times on this model, that condition isn’t being violated. Object b-2 doesn’t follow an empty time. It isn’t preceded by a time in which nothing existed. It just isn’t preceded by anything.

Now, I imagine that there is going to be some objection to this type of model. Object b-2 exists, but it was not caused to exist. Everything which comes into existence does so because it is caused to exist. But object b-2 exists yet is not caused to exist by anything.

We may reply that object b-2 is not something which ‘came into existence’, as part of what it is for an object x to ‘come into existence’ requires there to be a time before x exists at which it does not exist. Seeing as there is no time before t-2, there is also no time at prior to t-2 at which b-2 does not exist. So b-2 simply ‘exists’ at the first time in the model, rather than ‘coming into existence’ at the first time. Remember how Aquinas put it: “that which does not exist only begins to exist by something already existing”. There is no prior time at which b-2 is “that which does not exist”. It just simply is at the first time.

No doubt, this reply will seem to be missing the importance of the objection here. It looks like a technicality that b-2 does not qualify as something which ‘comes into existence’. The important thing, Peeler might argue, is that b-2 is a contingent thing that exists with no cause for it. That is what is so objectionable about it.

If that is supposed to be ruled out, it cannot be merely on the basis of Peeler’s causal principle, but must be so on the basis of a different principle. After all, Peeler’s principle merely rules out objects existing at times that are preceded by empty times. That condition is clearly not violated in model 3. The additional condition would seem to be that for every non-necessary object (such as b-2), there must be a causal influence coming from an earlier time. This principle would rule out the first object being contingent, but it is strictly more than what Peeler stated he required for his argument to go through.

However, let us grant such an additional principle, just for the sake of the argument. If we do so, then we rule out models like model 3. However, even if we are kind enough to make this concession, this does nothing to rule out model 2. In that model, each object is caused to exist by an object that precedes it in time, and there are no empty times. Yet, there is no one being which exists at all earlier times (such as in model 1).

The existence of such an eternal being is one way to satisfy the argument, but not the only way (because model 2 also satisfies the argument as well). Thus, because model 2 (which has no eternal being in it) also satisfies the argument, this means that the argument does not establish the existence of such an eternal being.

So, even if we grant the premises of the first argument, it doesn’t establish that there is something which is an eternal necessary object. It is quite compatible with a sequence of merely contingent objects.

2. Peeler’s second argument

From the conclusion of the first argument, Peeler tried to make the jump to there being a necessary object, and seemed to make the following move:

1. There was never nothing
2. Therefore, there is something that has always been.

The fact that the extra escape routes are not blocked off by the first argument, should give you some reason to expect the inference in the second argument to be invalid. And it is. It is a simple scope-distinction, or an instance of the ‘modal fallacy’.

There being no empty times in the past only indicates that each time in the past had some object or other existing at it. It doesn’t mean that there is some object in particular that existed at each of the past times (such as God). So long as the times are non-empty, each time could be occupied by an object that exists only for that time (as in our second and third models), for all the argument has shown.

The inference in the second argument is like saying that because each room in a hotel has someone checked in to it, that means that there is some particular individual person who is checked in to all of the rooms. Obviously, the hotel can be full because each room has a unique individual guest staying in it, and doesn’t require that the same guest is checked in to every room.

When put in such stark terms, the modal fallacy is quite evident. However, it is the sort of fallacy that is routinely made in informal settings, and in the history of philosophy before the advent of formal logic. Without making such a fallacious move, there is no way to get from the conclusion of Peeler’s first argument to the conclusion of the second argument.

3. Aquinas and the Third Way

In particular, medieval logicians often struggled with scope distinctions, as their reasoning was carried out in scholastic Latin rather than in symbolic logic. That they managed to make any progress at all is testament to how brilliant many of them were. Aquinas is in this category, in my view; brilliant, but prone to making modal fallacies from time to time. I think we can see the same sort of fallacy if we look at the original argument that is motivating Peeler’s argument.

Here is how Aquinas states the Third Way argument:

“We find in nature things that are possible to be and not to be, since they are found to be generated, and to corrupt, and consequently, they are possible to be and not to be. But it is impossible for these always to exist, for that which is possible not to be at some time is not. Therefore, if everything is possible not to be, then at one time there could have been nothing in existence. Now if this were true, even now there would be nothing in existence, because that which does not exist only begins to exist by something already existing. Therefore, if at one time nothing was in existence, it would have been impossible for anything to have begun to exist; and thus even now nothing would be in existence — which is absurd.” Aquinas, Summa Theologiae, emphasis added)

This argument explicitly rests on an Aristotelian notion of possibility. The philosopher Jaakko Hintikkaa explains Aristotle’s view:

“In passage after passage, [Aristotle] explicitly equates possibility with sometime truth, and necessity with omnitemporal truth” (The Once and Future Seafight, p. 465, emphasis added)

This is quite different from the contemporary view of necessity as truth in all possible worlds. On the contemporary view, there could be a contingent thing that exists at all times in some world. Therefore, being eternal and being necessary are distinct on the modern view, but they are precisely the same thing on the Aristotelian view. We will come back to this in a moment. For the time being, just keep in mind that Aquinas, and by extension Peeler, are presupposing a very specific idea of what it means to be necessary or non-necessary.

We can see quite explicitly that Aquinas is using the Aristotelian notion of necessity when he says “…that which is possible not to be at some time is not”. This only makes sense on the Aristotelian view, and would be rejected on the modern view. But let’s just follow the argument as it is on its own terms for now.

The very next sentence is: “Therefore, if everything is possible not to be, then at one time there could have been nothing in existence.” What Aquinas is doing is imagining what would be the case if all the objects that existed were non-necessary objects. If that were the case, then no object would exist at every time, i.e. each object would not exist at some time or other. That is the antecedent condition Aquinas is exploring (i.e. that “everything is possible not to be”).

What the consequent condition is supposed to be is less clear. As he states it, it is “at one time there could have been nothing in existence”. We can read this in two ways. On the one hand he is saying that if everything were non-necessary, then there is in fact an earlier time that is empty. On the other hand, he is saying that if everything were non-necessary, there could have been an earlier time that is empty.

Let’t think about the first option first. It seems quite clear that it doesn’t follow from the assumption that everything is non-necessary that there is some time or other at which nothing exists. Model 2 is an example of a model in which each object is non-necessary, but in which there are no empty times. If Aquinas is thinking that “if everything is possible not to be, then at one time there could have been nothing in existence” means that each object being non-necessary implies that there is an empty time, then he is making a modal fallacy. This time, the fallacy is the other way round from Peeler’s example: just because each guest is such that they have not checked into every room of the hotel, that does not mean there is a room with no guest checked in to it. Think of the hotel in which each room has a unique guest in it. Exactly the same thing applies here too; just because every object is such that it fails to exist at some time, that does not mean that there is a time at which no object exists. Just think about model 2, in which each time has its own unique object.

Thus, if we read Aquinas this first way, then he is committing a modal fallacy.

So let’s try reading him the other way. On this reading he is saying that the assumption that everything is non-necessary is compatible with there being an empty time. One way of reading the compatibility claim is that there is some model on which the antecedent condition (that every object is non-necessary) and the consequent condition (that there is an empty time) are both true. And if that is the claim, then it is quite right. Here is such a model (call it model 4):

On this model, there are two objects, a and b, and they are both non-necessary (i.e. they both fail to exist at some time). Also, as it happens, there is an empty time, t2; both a and b fail to exist at t2. So on this model, the antecedent condition (all non-necessary objects) and the consequent condition (some empty times) are both satisfied.

However, while this claim is true, it is incredibly weak. The difference is between being ‘compatible with’ and ‘following from’. So for an example of the difference, it is compatible with me being a man that my name is Alex; but it doesn’t follow from me being a man that my name is Alex. If we want to think about the consequent following from the antecedent condition, we want it to be the case that every model which satisfies the antecedent condition also satisfies the consequent condition, not jus that there is some model which does. But it is clearly not the case that every model fits the bill, again because of model 2. It satisfies the condition that every object is non-necessary, but it doesn’t satisfy the condition that there are some empty times.

So what it comes down to is that the claim that there are only non-necessary objects is compatible with the claim that there are empty times, but it is equally compatible with the claim that there are no empty times. Being compatible with both means that it is simply logically independent of either. So nothing logically follows from the claim that there are only non-necessary objects about whether there are any empty times in the past or not.

So on the first way of reading Aquinas here, the claim is false (because of model 2). On the second way of reading him, the claim is true, but it is logically independent of the consequent claim. On either way of reading him, this crucial inference in the argument doesn’t work.

And with that goes the whole argument. It is supposed to establish that there is an eternal object, but even if you grant all of the assumptions, it is compatible with there not being an eternal object.

4. Conclusion

Peeler set out an argument, which was that if nothing ever existed, there would be nothing now. The truth of the premises and the conclusion is satisfied by, or compatible with, model 2, and so does not require that an eternal object (like God) exists. The second argument was that if it is always the case that something exists, then there is something which always exists. That is a simple modal fallacy. Lastly, we looked at Aquinas’ original argument, which either commits a similar modal fallacy, or simply assumes premises which do not entail the conclusion.

## 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.

## Molinism and Trivial Counterfactuals

0. Introduction

I recently watched a pair of debates (which you can watch here and here) between a Molinist and a Calvinist about the idea of God’s ‘middle knowledge’. The Molinist was Eric Hernandez, and the Calvinist was Tyler Vela. The debate seemed to me to be quite imprecise, and it that both sides would have benefited from a formal framework within which they could precisely pose their various claims and counter-claims. In fairness, Tyler did give a formalised written version of his argument for the second debate (and you can see his slides here). This presented a very clear expression of  (what seems to me to be) an error that both sides were making. I wish to clear up here.

These issues have been investigated by logicians specialising in temporal logic since the late 70’s, and something of a consensus has arisen over the deficiency of the Molinist position. Neither of the participants seemed to be aware of this development. I guess that this is not surprising seeing as it is an obscure area of the literature, and requires a certain amount of technical training to read the logical and semantic details of the papers. Also, and possibly for the same reasons, the lessons do not seem to have made much of an impact on the philosophy of religion scene, never mind the theology scene. Given that I have a good knowledge of this area (having published journal articles on it) I will outline the main issues here with the hope of shedding some light on the debate.

1. Molina

Luis de Molina was a 16th century Spanish jesuit priest who formulated a position which bears his name in contemporary philosophy of religion. Molina was concerned with how to reconcile human freedom (conceived of as libertarian free will) and God’s sovereignty. However, there is a tension between God’s sovereignty and human freedom. To the extent that humans are free, they are not under the control of God (and that undermines his sovereignty); yet to the extent that God is in control of everything, humans are not perfectly free. The reformed answer to this puzzle is to repackage freedom as a variety of compatibilism. Molina was reacting to this move, and wanted to maintain the strong sense of libertarian freedom as well as the strong sense of sovereignty. It is from this mix that we get Molinism.

2. Future Contingents

The debate that Molina contributes to is one that had been going on for centuries before him. The medievals rediscovered Aristotelian texts that had been lost to western Europe during the dark ages and this contributed to the increasingly sophisticated logical debates that preceded the reformation. In particular, one topic caught the imagination of the medievals, and that was the issue of future contingents. A future contingent is a prediction, like ‘There will be a sea battle tomorrow’ (Aristotle’s example) made in a context where there could be a sea battle and there could be no sea battle. To get a feel of the modal strength of the future contingent, contrast it with an expression of possibility, and an expression of inevitability. So we might say ‘There could be a sea battle tomorrow’. This sentence can be true now even if tomorrow there is no sea battle; for often things don’t happen which were possible (a familiar fact to most people who have ever played the lottery). The modal force of this sentence is very weak. On the other hand, saying ‘There necessarily will be a sea battle tomorrow’ is much stronger. This sentence could be false even if there is a sea battle tomorrow. It may happen by accident, for example, and not of any kind of necessity. A future contingent cuts a line between these two modal extremes. Saying ‘There will be a sea battle’ is stronger than saying that there may be one, but weaker than saying that there must be one.

Aristotle argued (or at least seemed to) in his work On Interpretation (part 9) that purported examples of future contingents, if they were true now, would have to be already impossible or necessary. For if it were already true now that there will be a sea battle tomorrow, then it is going to take place regardless of what you try to do about it; its future truth seems to indicate its present inevitability. Thus, according to Aristotle’s argument, there could be no such thing as a ‘future contingent’ (i.e. a true future-tensed statement which is neither necessary nor impossible). This is a strong form of logical fatalism.

The received view of Aristotle is that his solution this this problem was to advocate that future contingents were neither true nor false, and thus to avoid the fatalism (although not everyone agrees – see this paper by Hintikka). Despite their reverence for Aristotle, the medievals found his solution to be deeply troubling, as it indicated that God could not know the contingent aspects of the future. After all, if God knows all true statements (being omniscient) and believes nothing but true statements (being infallible), then he does not know future contingents (which, being neither true nor false, are not true). Thus, God is seemingly in the dark about whether there will be sea battles tomorrow, or whether certain people will sin, etc. Aristotle’s solution is therefore incompatible with a robust conception of God’s foreknowledge. On the other hand, if God does know the truth-value of future contingent statements, then there is a theological equivalent to the problem of future contingents: God’s knowing true future contingents in advance makes them seem inevitable and thus necessary. If God knows you are going to sin tomorrow, then it is going to take place regardless of what you try to do to prevent it.

Various medieval philosophers, logicians and theologians offered their solutions to this problem, such as Peter Abelard, St. Anselm and William of Ockham. The Anselmian-Ockhamist solution, explained expertly by Peter Øhrstrøm here, was to hold that God knows the truth-values of future contingent statements, but to deny that this entails that the statements themselves become necessary as a result. Ockham diagnosed a ‘modal fallacy’ in the claim that his foreknowledge made them necessary; God knows that p will happen, even though it might not – these are not logically incompatible, and the modal fallacy is supposing that they are. In this sense, there can be genuine true future contingents for Ockham.

Future contingents are logically equivalent to free choices of agents with libertarian free will. A future contingent is a statement of the form ‘it will be that p‘ made in a situation where ‘it is possible that it will be that p‘ and ‘it is possible that it will not be that p‘ are both true. An agent’s choice to do is free in the libertarian sense only if they could have chosen to do and have chosen not to do x. So both concepts rely on the prediction being true (or choice being made) in a situation where it’s falsity is possible (where the choice could have not been made). Thus, libertarian free will is really just a special case of a future contingent, where the predicted content is the action of the agent.

Molina essentially accepts the Ockhamist proposal, which was that God knows the future choices of agents without this stopping the possibility of those choices being different (they could be different, but they won’t be). However, he adds to this an additional claim, which is aimed at bolstering the sovereignty consideration. God knows not just which free choices agents will make, but also those free choices they would have made were they to have been faced with different circumstances.

3. Luis

Let’s use an example to make the point clear. Imagine a medieval monk; call him Luis. He lives in a monastery high in the mountains somewhere. In this calm and peaceful environment there are seldom any opportunities for moral temptation (which is part of the point of a monastery after all). Upon entering the monastery, God knows that Luis will not sin for the rest of his life. It is still possible that he could sin (he could decide to leave the monastery and live in the sinful town at the bottom of the mountain). But God knows that though he could do this, he won’t. So far, this is just the Ockhamist picture.

We may wonder about Luis’ moral character in more detail than this though. Sure, he won’t actually sin, but this just seems to be a product of the environment he is living; he won’t be seriously tempted to sin. In a sense then, his moral character is not going to be severely tested in any way. Even though he won’t be, what would have happened if he were to be tempted? Imagine a beautiful maiden were to arrive at Luis’ bedroom one night and beg him to spend the night with her. It won’t happen (given the strict rules of the monastery), but what if it did? Would he have been able to resist, or would he have given in to temptation?

4. Middle Knowledge

Molina thought that God, in his sovereignty, had to know the answer to this sort of question. That is, God has to know the truth-value of every actual future contingent, but also of every counterfactual future contingent. Here is an example of the sort of sentence that Molina claims God would know the answer to:

a) Had it been the case that [Luis is tempted to spend the night with the maiden], then it would have been the case that [Luis will give in to the temptation].

a) is a a conditional (if…, then…), in the subjunctive mood (using the modal modifiers ‘had it been…, it would have been…’) and it has an actually false antecedent (it is not actually the case that Luis is tempted by any maiden). This makes it a counterfactual. It is important to note that the consequent (‘Luis will give in to the temptation’) is a future contingent, specifically one about his libertarian free choice. Molina’s claim is that God knows counterfactuals with future contingents as their consequents, like a).

In addition to him knowing the truth-value of these counterfactuals, the obvious supposition is that some of them are in fact true; it’s not Molinism if all such counterfactuals are false. We will come back to this at the end.

This type of knowledge that Molina claims God has is often referred to as ‘middle knowledge’. Middle knowledge is usually contrasted with two other types of knowledge that God has: natural knowledge and free knowledge. Natural knowledge concerns all the necessary, possible and impossible truths. So that 2 + 2 = 4 is necessary; that it Judas betrayed Jesus is possible; that 2 + 2 = 5 is impossible. In contrast, free knowledge concerns those facts which relate to the creation of the world. So the fact that I exist, or the fact that you are reading this blog post, is part of God’s free knowledge. Middle knowledge is usually contrasted with these two in terms of being between general facts to do with possibility, and particular facts about the contingent world; middle knowledge is supposed to concern counterfactual facts.

5. A Better Distinction Using Possible Worlds

However, this is not the best way of drawing this distinction. With the benefit of possible worlds semantics and a clear understanding of logic, we can make this distinction much more cleanly.

Possible worlds are thought of as just sets of propositions that are maximal and consistent. This just means that for every atomic proposition, p, and every world w: either p is in w or it is not, but not both.

We can then use the usual logical compositional clauses to form more complex propositional forms:

• if p is not true in w, then ~p is true in w;
• if p is true in w and q is true in w, then ‘q‘ is true in w, etc.

If there is some formula, A, which is true in all worlds, then we say that A is necessary; if it is true in no worlds then A is impossible; if it is true in some worlds but not others, then A is contingent.

All of these propositions would be items of God’s natural knowledge; he knows what is true and what is false at every world, and thus he knows what is necessary, what is contingent and what is impossible. So much for natural knowledge.

Take one world, say w1. We can designate this world as the ‘actual world’, and label it ‘@w‘. Think of it as being the world that God chose to actualise. If a proposition is true at @w, then it is simply true (or ‘true simpliciter’). (Having a special designated actual world is how Kripke originally formulated possible worlds models, though it fell out of favour with most subsequent formal treatments of possible worlds semantics). God’s free knowledge concerns what is true simpliciter (or what is true at @w).

So far, we have used possible worlds semantics to explain the contents of God’s natural and free knowledge. As noted above, the description of God’s middle knowledge is usually cashed out as concerning counterfactuals. And it is, but most counterfactuals actually come under God’s natural knowledge, a claim which we can also spell out clearly now using the benefit of possible worlds. There are two types of counterfactuals that need to be distinguished from Molinist counterfactuals, and this distinction is the counterfactual mirror of the distinction between modally weak predictions, future contingents and expressions of inevitability from above.

On the one hand, God knows ‘might’ counterfactuals of the following type:

b) If I had flipped the (fair) coin, then it might have landed heads.

This type of counterfactual uses the word ‘might’, which is analogous to the word ‘possible’; if I had flipped the coin then landing heads was possible. Equally, landing tails is also possible given the coin flip (assuming a perfectly fair coin, etc). All this means is that at at least one of the worlds which are maximally similar to the actual world at which I flipped the coin, it lands heads.

The point is that ‘might’ counterfactuals are very weak in what they claim. All that is required is that the antecedent condition is compatible with the consequent condition; that there is at least one ‘coin-flip’ world (maximally similar to the actual world) at which the coin lands heads. This just means that the flipping of the coin (in the right sort of circumstance) is compatible with it landing heads. Thus, all ‘might-counterfactuals’ come under natural knowledge. To turn the example to Luis, the following might-counterfactual is true: ‘if Luis had been tempted by the maiden, then he might have given in to the temptation’. Even though that counterfactual is true it doesn’t tell us whether Luis would give in to the temptation or not – it just tells us that he might do. This is why these might-counterfactuals don’t count as middle knowledge.

In contrast, imagine a coin which has heads on both sides (a ‘rigged’ coin). The following counterfactual, which uses ‘would’ instead of ‘might’, would be true for that coin:

c) If I had flipped the (rigged) coin, then it would have landed heads.

Because the coin is rigged, its landing heads is inevitable once it is flipped (assuming of course that it cannot land perfectly on its side, etc). This just means that in every (maximally similar) ‘coin-flip world’, the coin lands heads. And we can immediately see that this is the case, because no matter which way it lands, it will land heads. To make the example relevant to Luis again, we can easily think of consequents which are inevitable given the truth of the antecedent. For example: ‘If Luis had been tempted by the maiden, then he would have been tempted’. The consequent is (in a particularly trivial way) necessitated by the truth of the antecedent. In every (maximally similar) temptation world, Luis is tempted. This sort of example would also not count as middle knowledge, as it does not tell us what free choice Luis would make in the counterfactual situation. This example, like the one above, is also an example of natural knowledge.

So far, we have seen two types of counterfactuals, ‘might-counterfactuals’ and ‘would-counterfactuals’ and neither of them count as middle knowledge (they are both just natural knowledge). What we need to get there is a sort of Goldilocks modality, which is between ‘would’ and ‘might’. There is no natural locution for this in ordinary English, so I will use the somewhat stilted phrase ‘actually-would’. So in contrast to b) and c), the Molinist counterfactual, the real example of middle knowledge, is:

d) If I had flipped the (fair) coin, then it actually-would have landed heads

When we hear d), we need to remember that it doesn’t mean that the coin might land heads, and it doesn’t mean that the flipping of the coin necessitates it landing heads. It means that, though it is possible that it land tails, if it were flipped would in fact happen to land heads. To make the example relevant to Luis, consider the counterfactual: ‘If Luis had been tempted, then he actually-would have given in.

6. Red Line

Now we have clearly and precisely stated the thesis that Molina argues for. He is saying that at least some counterfactuals of type d) are true, and God knows them – they constitute God’s ‘middle knowledge’. The question is how to model this claim. With the previous two types of counterfactuals, we were able to use the standard ideas from the literature on possible worlds semantics (which come from David Lewis, see this and this). Put simply, ‘would’ counterfactuals rely on what is true at every maximally similar antecedent world, whereas ‘might’ counterfactuals rely on what is true at at least one maximally similar antecedent world. These are what grounds these two types of counterfactuals, they are what makes them true, which is to say that they are the semantics for those counterfactuals. But what is it that grounds the truth of the Molinist counterfactual? What is its semantics? There is reason to think that at the moment there is nothing to appeal to – nothing to hang our metaphysical hat on, as it were.

Here is one way of thinking about the situation which makes it clear that as things stand there is no obvious candidate. Consider a simple model, we have three worlds, w1w2 and w3. Let’s say that w3 is the actual world, @w (which we will draw in red). In @w, Luis is not tempted by the maiden (she does not go to the monastery at all). In w1 and w2 Luis is tempted. He gives in in w1 but not in w2. We can picture this as three worlds which ‘branch’ from one another as follows (worlds ‘overlap’ when they share all the same atomic propositions, and ‘branch’ from one another when they differ over the truth of a proposition):

We can ‘hang our hat’ on a feature of this model to ground the truth of the counterfactual that Luis might have given in: there is at least one of the tempted-worlds in which Luis gives in (i.e. w1). We can also hang our hat on a feature of this model to ground the falsity of the counterfactual that Luis would have given in: it is not the case that he gives in on all of the tempted worlds (i.e. in w2 he does not give in). Each of these types of counterfactual receive a truth-value in a straightforward way. What is unclear is how one could ground the claim that, had he been tempted, Luis actually-would have given in. He gives in on one tempted-world but not the other; w1 and w2 have nothing to distinguish one from the other. Why say that he would give in rather than not give in?

6. Trivial Counterfactuals

It is at this point that I saw both contributors to the podcast making a move which is mistaken. I am quite prepared to believe that Tyler was lead astray by Eric’s lack of clarity at this point (after all, Eric is the Molinist and he should have been able to explain his own position clearly). However, the both made the same move, which obscured the rest of the conversation.

Here is what happened. Tyler was asking Eric if it was possible for God to create the world such that everybody freely chooses to believe in God. They both agreed that it was logically possible for this to happen, in the sense that there was no logical contradiction in the supposition that it happens. However, Eric insisted that though it was logically possible, it was not ‘feasible’ for God to do this. Unfortunately, no definition was given for ‘feasibility’. Tyler wanted to demonstrate that if feasibility has no metaphysical content, then the appeal to it was ad hoc here, being added without any motivation other than avoiding the problem. How he went about framing his argument demonstrated that a clear framework for the semantics for Molinist counterfactuals was lacking. Here is how he presented his argument on the second show:

Now, the actual details of Tyler’s argument are not important here for my purposes. Just note that the first premises of each of the three arguments are conditionals, and the antecedents talk about worlds being possible and God actualising worlds. They are effectively little counterfactuals about what would be the case had God actualised different worlds.

The implicit idea here is that in a Molinist counterfactual, one changes which world is the actual world; that we move the red line on the picture. Tyler’s whole argument is about what would be the case if a different world were actualised. It is as if this is how the model would look to make the Molinist counterfactual ‘If Luis had been tempted, he would have given in’ true:

There are two immediate problems with this idea though. Firstly, what we have is no longer really a counterfactual situation. A counterfactual has to have an actually false antecedent, yet on this model the antecedent (Luis is tempted) is actually true (because w1 is now the actual world). Secondly, and more importantly, it is trivial. The problem is that if we move the designation of the actual world to a different place in the ‘tree’, then this change settles the matter of the truth of the consequent of the conditional. And the antecedent of Tyler’s conditionals have the mention of which world is being actualised explicitly as stated in the antecedent. To make this clear, consider the following two questions one may ask:

e) Had God actualised world w1, would Luis have given in to temptation?

f) Had Luis been tempted, would he have given in?

There is a big difference between e) and f). Firstly, nobody would ever say e), outside a contrived philosophy seminar-room example. The reason is partly because in real life possibilities are not labelled neatly like w1 and w2, etc. But let’s suppose that we can get around this somehow (that a magic world-labelling dictionary is available to everyone who introspects hard enough). The problem now is that stating the world by name implicitly includes all the propositions that are true at that world. That’s all a world is! It is like saying:

g) Had God actualised world w1, in which Luis is tempted and gives in, would he give in?

Nobody would ask a question like g) because it contains its own answer, and is thereby trivial. e) is trivial because it is just an elliptical way of asking g). Likewise, the following counterfactual (which is similar to Tyler’s) is trivial:

h) If God had actualised w1, then Luis would have given in to temptation.

This provides us reason to think that h) cannot be any part of the semantics of the Molinist counterfactual d). The reason h) is not equivalent to d) is that h) is trivial, whereas d) is not. Just like the way that e) is trivial and f) is not. What this means is that the correct semantics for the Molinist counterfactual, d), is not just moving the red line in the model for d) to a different place. Tempting though it may be, the analysis of a Molinist counterfactual is not to conceptualise a counterfactual about what would be the case if God had actualised a different world. To do so is to misunderstand Molinism. As I said, I think this mistake was being made by both parties in the debate, although Eric bears the responsibility for articulating his own position correctly.

7. Red Lines

If that is not the answer, then what is? This is where we find the logic literature that I referenced in the introduction to be very helpful. As far as I know, the first systematic logical account of a Molinist branching model was put forward by McKim and Davis in 1976. It was made into a much more elaborate theory by Thomason and Gupta in 1980. A similar theory was also developed by Brauner, Ohrstrom and Hasle in 1999. The way these theories work is to postulate not just one red line, but multiple red lines. In the case of the actual world, what makes a future contingent true is that what it predicts is true in the actual future; the sentence ‘there will be a sea battle’ is true if and only if there is a sea battle in the actual future. The idea of the ‘actual future’ is what breaks the symmetry between all the various possible futures of that moment. In the counterfactual situation, there is no such symmetry breaker, and this is what leaves us only able to ground ‘would’ and ‘might’ counterfactuals, but not Molinist counterfactuals. There is nothing for God to hang his hat on, as it were. What these authors above all have in common is the idea that they need to break the symmetry in the counterfactual situations by adding in actual futures at each counterfactual branching point. At each counterfactual situation where there is a future contingent (like a monk being tempted and deciding whether to give in to it or not) there needs to be an counterfactual ‘actual’ future (a ‘counteractual’ as it were). So our little model would have to be modified to make it a proper Molinist moodel:

Now we can say that the semantics of a Molinist counterfactual is as follows:

i) ‘Had it been the case that A, then it actually-would have been the case that C’ is true if and only if it is true in the counteractual future future of the most similar A-point that it will be that C’.

So, in the actual world, the Molinist counterfactual ‘had Luis been tempted, he would have resisted’ is true because in the maximally similar tempted situation, the actual future has him resisting the temptation. In the counterfactual situation in which Luis was tempted, he actually resists temptation.

So a technical addition to our models, a specification of counteractual futures at each branching point, provides the required metaphysical feature for us to hang our semantic hat on. God’s middle knowledge is just that he knows where all the red lines are in the overall tree of branching worlds.

8. Problems

Despite its seemingly attractive solution, there are some widely recognised and severe problems for this Molinist semantics. These come in two categories; technical and conceptual.

The technical difficulties are explained in Belnap and Green (1995) and in Belnap et al (2001)  (chapter 6), and also a in chapter of my PhD thesis which is available here (p. 7 – 9). The issue has to do with the semantics of tenses. The problem has to do with iterated tenses, like “It will be that it was that p“, etc. These do not operate properly on the Molinist account, with the result that various tense-logical tautologies are violated by the Molinist logic. Consider the ‘tempted’ point in our Molinist model above. At that time, it is true that Luis is being tempted. Now, it is usually considered a tautology that if something is presently true, then in the past it was going to be true. That ‘you are reading this blog post’ is true now, so before you started reading it the sentence ‘you will read this blog post’ would have been true. This seems to be an elementary fact about how tenses work. Yet, at the ‘tempted’ point in our model, if we go back in the past to the trunk of the tree we find ourselves in a situation where the actual future leads to Luis not being tempted at all.  So even though he is being tempted, it was not the case that he was going to be tempted. In fact, because the actual future of the trunk leads to him not being tempted, we have it that Luis is being tempted, even though in the past it was true that he will never be tempted. This is an odd result. Thomason and Gupta, and Brauner, Ohrstrom and Hasle do make modifications which avoid this issue, but only at a cost. Each time they modify the model it leads to a different intuitive tautology not being true, which led Belnap to describe  the process of constructing ever more complicated Molinist models as ‘mere idle filigree’.

The conceptual problem is just that it is hard to make any sense out of the idea of counteractual futures. Unless one is a full-on modal realist (in the vein of David Lewis), you will think that there is a pretty big ontological difference between what is actual and what is merely possible. What is actual concretely exists, and what is merely possible does not. Yet, when the Molinist posits actual futures of merely possible situations, we find that this intuition gets lost. Are we saying that these situations are sort of concrete and existing? How can something be sort of concrete and existing? If they are fully concrete and existing, then what distinguishes the actual world from them? There is a big metaphysical question mark over this way of conceptualising counterfactuals which makes many think that it is wrong in principle to posit actual futures of counterfactual moments.

9. Alternatives

Instead of going the Molinist route, a more promising proposal is to just abandon the idea of middle knowledge altogether. Molinist counterfactuals are just inherently problematic. What sounds like an initially plausible proposal (that God could know what you would freely do in counterfactual situation) just comes out both technically and conceptually flawed.

Instead, we should embrace the idea that there are only ‘would’ and ‘might’ counterfactuals. In addition, we should be prepared to countenance the prospect that, strictly speaking, most ‘would’ counterfactuals are false. Have a look at these papers (here and here) for some philosophers giving weight to this proposal. When we say ‘Had I flipped the fair coin, then it would have landed heads’, this is just false. So is ‘Had I flipped the fair coin, then it would have landed tails’, although ‘Had I flipped the fair coin, then it would have landed either heads or tails’ is true. Only if the consequent is necessitated by the antecedent is a ‘would’ counterfactual true.

Why think this? Well, I suggest that the most natural way to think about what is grounding counterfactuals (and most metaphysical modality claims) is the natures of actual objects. The reason that this coin could land heads but doesn’t have to is because of the nature of the coin itself. It is because it has heads on one side but not on the other. These facts are what we are hanging our hat on. These facts are what allows us to draw the tree of possibilities in the first place. Nothing about the actual coin picks it landing heads over tails in a counterfactual situation, so there is no metaphysical fact about that. Molinism asks for God to have knowledge about facts which don’t exist.

10. Conclusion

All I wanted to do in this post was explain why a certain way of talking about Molinism is wrong, but to do that as clearly as possible, I have gone through the background of Molinism, explained the basic ideas in the semantics of counterfactuals, and outlined the main thrust of the objections to the Molinist semantics found in the logic literature on this topic. I have also provided a quick sketch of my view, which is a species of Ockhamism.

## 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.

## The Compatibility of Omniscience and Freedom

I say even if God knows what you are going to do tomorrow, this does not stop you being free to act otherwise. You won’t act otherwise, but you could.

Let’s set out a few definitions. You are free to do an action if it is possible that you do it, and if it is possible that you don’t do it. If either of these options is removed, you are no longer free. So if ‘p’ is ‘you will do x’, then you are free to do x if and only if (iff):

It is possible that p, and it is possible that not-p

Alternatively, we will write this as follows (where ‘◊’ means ‘possibly’):

◊p &  ◊~p

The problem is that this freedom condition seems to be ruled out by the idea of God’s foreknowledge. The reasoning is that if God already knows that p, then it is necessary that p. We can write this as follows, where K = God knows, and □ = necessarily:

If Kp, then □p

And if it is necessary that you are going to do x, then it is not possible that you will not do x. If necessarily p, then it is not possible that not-p:

If □p, then ~◊~p

So let’s put this into an argument that seems to show that freedom and omniscience are incompatible by deriving a logical contradiction:

Premise 1) I’m free to do x     (i.e. ‘it is possible that p and it is possible that not-p’)

Conclusion 1) Therefore, it is possible that not-p.

Premise 2) God knows that p.

Premise 3) If god knows p, then p is necessary.

Conclusion 1) Therefore, p is necessary.

Premise 4) If p is necessary, then it is not possible that not-p.

Conclusion 3) Therefore, it is not possible that not-p

Conclusion 4) Therefore, it is possible that not-p, and it is not possible that not-p.

We can write exactly the same argument in symbols as follows (in the right I give whether each line is an assumption or how it follows from something previously assumed):

Premise 1) ◊p &  ◊~p                         (assumption)

Conclusion 1) ◊~p                              (from pr. 1, and conjunction elimination)

Premise 2) Kp                                      (assumption)

Premise 3) If Kp, then □p                 (assumption)

Conclusion 2) □p                                (from pr.2 and pr.3, and modus ponens)

Premise 4) If □p, then ~◊~p             (definition of □ and ◊)

Conclusion 3) ~◊~p                            (from con.2 and pr.4, and modus ponens)

Conclusion 4) ◊~p & ~◊~p                (from con.1 and con.3, and conjunction introduction)

So we have derived a contradiction; it is possible that I will not do x, and it is not possible that I will not do x. This means we have to either reject the truth of one of the premises, or reject the validity of the argument form. Now the validity is easy to address, as it uses nothing but inference rules from classical propositional logic and the duality of necessity and possibility (i.e. □ = ~◊~ and ◊ = ~□~). There is nothing controversial at all here. So we must reject the truth of at least one of premises 1, 2 or 3, on pain of having to accept a contradiction.

Premise 1.

We said that being free to do x requires that it is possible to do both x and to not do x. Not all definitions of freedom require this. In fact this is a strong condition, and ‘compatibilists’ (like Spinoza, or Frankfurt) will contend that one can be free even if only one option is possible, just so long as that option is chosen. So the prisoner is free to stay in the cell, even though it is not possible to leave, for example. So it is possible to reject this premise. I think we can keep it however, and still avoid the consequence. We do not have to be ‘compatibilists’ to argue that God’s foreknowledge is compatible with freedom.

Premise 2.

This says that God knows what will happen tomorrow. To deny this means either giving up on God’s omniscience, or on the fact that there is a truth about the future (i.e. giving up on the principle of bivalence). We could go the second route, and retain omniscience, given that there is no truth about the future for him to not know. It should be noted that if we go this route, we have to also also hold that God is located in time. In this case, he would find out what happens tomorrow with the rest of us. A timeless God cannot ‘find out what happens’, as this would be a temporal activity. Anyway, we do not have to reject bivalence or require God to be in time, as I say we can avoid the contradiction even if premise 2 is true.

Premise 3.

This, as I see it, is where the confusion sets in. It says that ‘If God knows that p, then it is necessary that p’. Why would we think this premise is true? One reason is as follows. If you know something, anything, then it has to be that it is true. After all, you can’t know something false. It’s part of the definition of knowledge that it is of something true. God, who is infallible, only makes this force stronger; he couldn’t be wrong about anything. So if he knows something is going to happen, it is definitely, necessarily, going to happen. How could he be wrong?

Well, we need to be careful about the logical form of what we are saying. It is necessary that God knows p, and truth is a necessary component of knowledge; but this doesn’t mean that what God knows is necessary. Here is the sentence that is doing all the heavy lifting conceptually:

If God knows that you will do x, then it is necessary that you will do x.

I agree that everything God knows is true, i.e. he is infallible, and that everything true is known by God, i.e. that he is omniscient. But this only amounts to the following:

Kp iff p

This says that ‘God knows that p  if and only if p’. I can even go all the way and say that this is a necessary truth:

□(Kp iff p)

Now, we can derive a conditional which is very similar to premise 3 (which I will call 3.1) from this, namely:

3.1) □(If Kp, then p)

But it is important to note that this is as far as we can go. There is no way to go from 3.1 to 3:

□(If Kp, then p), therefore (If Kp, then □p)

So premise 3 does not follow from 3.1. Moreover, I say that 3.1 is actually the correct logical form of: ‘If God knows that you will do x, then it is necessary that you will do x.’

Admittedly, the word ‘necessarily’ is in the consequent in the sentence, and that seems to count against my claim. But then we systematically leave it there when we express both de re (of the thing) and de dicto (of the word) modalities, which should have it in different places. This means we fail to distinguish between the scope of the modality in natural language. Getting the scope of the modality right will solve the problem.

Quine’s example in Word and Object (p120) is that about cyclists being necessarily two-legged (and mathematicians being necessarily rational). To adapt his example, we would say:

If x is a cyclist, then it is necessary that he has two legs.

This sentence also has the word ‘necessary’ in the consequent, when it should be prefixing the whole conditional. It expresses only that under the description of the word cyclist, x has two legs. It is possible that x falls and gets one of his legs somehow cut off, and then x would not have two legs. It is not a necessary truth about x that he has two legs, only a necessary requirement for being a cyclist. So it is necessary de dicto that x has two legs, but not necessary de re. If we speak carefully, we would say:

It is necessary that (if x is a cyclist, then x has two legs)

The above formulation is compatible with the fact that x could fall off his bike and lose a leg, because he would then stop being a cyclist. Neither him being a cyclist nor having two legs is necessary; what is necessary is the connection between being a cyclist and having two legs.

This shows that we regularly fail to state the correct logical form when expressing de dicto modal claims. Sometimes, even though the word ‘necessarily’, etc, is in the consequent, it should be prefixing the whole conditional. And I say that premise 3 is one of these cases.

So if 3.1 was used in place, it would say that it is necessary that if God knows you will do x, then you will do x, just like it is necessary that if x is a cyclist, then he has two legs. Just like with the cyclist example, you do not have to do x (and he could fall off his bike). x doesn’t have to have two-legs, its just that it is necessary that if he is a cyclist then he does. You don’t have to do x, it just that it is necessary that if God knows that you will, then you will. In each case, the conditional is necessary, meaning that the one condition is never true without the other, but the other can be false. If it is false, then the antecedent condition would be false too.

One way of putting this is that it God knows contingent truths, like that you will do x. It is necessary that he knows them, but he only knows them if they are true. It is contingent that you will x, so it is contingent that p is true, and thus God might not know it. All that is necessary is that if it is true, he knows it; and if he knows it, it is true.

If we plug 3.1 in place of 3 in our argument from above, it stops us being able to move from it being true that p, to it being necessary that p. We needed that to get our contradiction, so we have blocked the contradiction. Thus foreknowledge and freedom are compatible.

Conclusion.

So I gave an argument for the incompatibility between divine foreknowledge and freedom, making it as strong as possible, which showed logically that the two concepts lead to a contradiction. However, I suggested the the logical form of the third premise was incorrect, allowing us to keep all the strong assumptions and show that no contradiction is forthcoming.