## 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.

## Getting an ought from an is

0. Introduction

In the Treatise of Human Nature, Hume outlined the ‘is-ought’ problem, sometimes referred to as ‘Hume’s Guillotine’. The idea is that it is not possible to argue validly from ‘descriptive’ statements (about how things are) to ‘normative’ conclusions (about how things ought to be).

Hume describes how he often notices a change that takes place when he is reading certain passages on moral philosophy:

“I am surprised to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not. This change is imperceptible; but is, however, of the last consequence” (Section 3.1.1)

Examples of this switch include the move from 1 to 2 in the following examples:

A)

1. X makes people happy
2. Therefore, people ought to do X

B)

1. God commands people to do X
2. Therefore, people ought to do X

If we want to turn A) into a valid argument, we would naturally want to add another premise, as follows:

1. X makes people happy
2. People ought to do what makes them happy
3. Therefore, people ought to do X

Now the argument is valid. But now the conclusion follows from a set of premises which are not all descriptive. Our new premise 2, needed to make the argument valid, is normative (because it is about what ought to be the case, not just what is the case). Therefore, it is not a case of getting ‘an ought from an is’; but of getting ‘an ought from an ought and an is‘. Hume’s point is that without the addition of a normative premise, like 2, an argument like A or B cannot be made valid.

We can state the is-ought problem as follows:

There is no valid argument such that the premises are purely descriptive, and the conclusion is normative.

A counterexample to this would be a valid argument with purely descriptive premises and a normative conclusion.

1. A counterexample to the is-ought problem

Consider the following example:

1. The conclusion of this argument is true
2. Therefore, we ought to do X

This inference is valid; there is no way the premise could be true without the conclusion also being true. After all, the premise says that the conclusion is true; so the only thing that makes the premise true is the conclusion being true.

The premise is seems to be quite clearly descriptive. It doesn’t include the word ‘ought’ or any synonym of the word.

On the other hand, the conclusion clearly is normative, involving the word ‘ought’ quite explicitly.

This means we have a valid argument with purely descriptive premises and a normative conclusion. This makes it a counterexample to the is-ought principle as stated above. In some sense, it shows that it is possible to derive an ought from an is, after all.

## Inspiring Philosophy and the Laws of Logic: Part 1

0. Introduction

There is a YouTube channel, called Inspiring Philosophy (henceforth IP), which is about philosophical apologetics. It has about 45k subscribers, and the videos have high visual production values. One video in particular caught my attention, as it was about the laws of logic.

Despite the relatively large audience and good production values, IP makes some pretty baffling mistakes, and a lot of them are very easy to spell out. I will try to explain the main ones here.

1. Confusions

IP’s lack of understanding about the issues involved contributes to a confusion about what is being claimed by his imagined ‘opponents’, and what he is trying to say in reply to them. This fundamental confusion is at the heart of the entire video.

In the very opening section, IP asks two general questions:

“Can we trust the laws of logic? Is logic safe from criticism, or is it just another man made construct built on sand?”

These questions are actually quite vague. What does it mean to ‘trust‘ the laws of logic? Does it just mean ‘Are the laws of logic true?’

More importantly, what exactly does he mean by ‘the laws of logic’? He never specifies what he takes the ‘laws of logic’ to actually be. Commonly in discussions like this, they are taken to be the law of excluded middle, the law of non-contradiction, and the law of identity. They are part of what is known as ‘classical logic‘, which we can think of as a group of logical systems which all share a number of principles, including those laws. We must assume that this is what IP means. Let’s refer to these three laws as the ‘classical laws of logic’

The historical development of logic shows that, in one sense, classical logic is not safe from criticism. Just like mathematics, logic has evolved over time, and it has gone through various changes (see this, and this). In particular, there are logical systems which do not include the classical laws of logic; there are systems of logic which have contradictions in, or which have exceptions to excluded middle, or where identity is treated very differently. So, suppose that IP is asking: ‘are there logical systems which do not include those particular logical laws?’ The answer is: ‘yes, there are non-classical logics‘.

Surely though IP thinks he is asking a more interesting question than this. He wants to ask whether some other non-classical logical system should be regarded as the right one. This is a much more interesting question, and much more difficult to answer. I assume that IP wants to say that the classical laws of logic are the right ones, and all the other non-classical alternatives are not right. That would be a coherent position for him to take: he is defending classical logic against rival non-classical logics.

However, this is not what IP actually articulates throughout the video. The video starts off with a claim which seems to be the target that IP wants to argue against. He says

“Many argue that the laws of logic are not true”.

Here we see the fundamental confusion right at the heart of the video. There are two distinct issues IP never distinguishes between a kind of local challenge to classical logic, and a global challenge to all logic:

Local) “Many argue that classical logic is not the right logic

Global) “Many argue that there is no right logic at all

While the first option is clearly something many people do argue, it is not quite clear whether the second option even makes sense. Are there really such people who argue that there is no such thing as logic? Who are these people? IP doesn’t ever say.

One of the main problems in what follows is that IP switches back and forth between the local and global challenge, as if he is unaware of the distinction.

2. The argument

In the first half of the video, IP offers what he calls a “simple argument” to use as a foil to respond to. He does not say where he got this argument from, but I suspect that he got it from here.

The argument goes like this:

1. Assume that the laws of logic are true
2. All propositions are either true or false
3. The proposition “This proposition is false” is neither true nor false
4. There exists at least one proposition that is neither true nor false
5. It is not the case that all propositions are either true or false
6. It both is and is not the case that all propositions are either true nor false
7. Therefore, the laws of logic are not true

We need to ignore the fact that the first premise is an example of a command, and is not expressing a proposition. We also need to ignore that the argument is not formally valid; strictly speaking, the conclusion does not formally follow from the premises. You have to assume that by ‘the laws of logic’ we mean to include the law of bivalence. If you want an argument to be formally valid, you cannot keep these sorts of assumptions implicit.

Basically, what is going on with this argument is a challenge to classical logic, or really any logic which has the semantic principle of ‘bivalence’. So it is an example of a local challenge. The principle of bivalence is expressed in premise 2, and it says that each proposition has exactly one of the following two truth values: ‘true’ or ‘false’. This principle is the target of the argument.

The liar’s paradox is notoriously difficult to give a satisfying account of within the constraints of classical logic. Therefore, some people say that the only way to account for it is to give up some aspect of classical logic. Thus, considerations of the liar’s paradox provide some reason for people who argue that classical logic needs to be rejected. In this case, the idea implicit in premise 3 is that the liar’s paradox requires bivalence to be false. They say that the Liar Proposition, i.e. “This proposition is false”, is itself neither true nor false. If they are right about this, then classical logic must be wrong. This is because classical logic says that all propositions are either true or false, but there is a proposition which is neither (i.e. the Liar Proposition).

To defend classical logic against this charge, we would expect IP to argue that the liar’s paradox is not solved by treating the Liar Proposition as neither true nor false, but that it can be solved without giving up any of the assumptions of classical logic. This would undermine the reason given here for thinking that bivalence had an exception.

However, at this point IP starts to show just what a poor grasp he has of what this argument is supposed to be showing, and what he needs to do to defend classical logic against it.

He says that “there are several problems with this argument”, but he criticises premise 2. Now, this is odd, because premise 2 is just an expression of bivalence, which is part of classical logic. If he is defending classical logic, then he should be defending premise 2; yet, he is about to offer a reason to doubt it.

IP says that the problem with premise 2 is that not all propositions are either true or false; some are neither true nor false. His example is the following:

“Easter is the best holiday”.

His reasons for thinking that “Easter is the best holiday” is neither true nor false are strange. He says that that proposition “Cannot be proven true or false” and that it is “just an expression of opinion”. “So,” he continues, “you can have propositions that are neither true nor false. Nothing in either logic or language denies this”.

Now, just hold on a minute. Let’s grant IP’s claim that the proposition “Easter is the best holiday” merely expresses an opinion. This is ambiguous between two different things.  On one hand, saying that it merely expresses an opinion might mean that it is just shorthand for:

“My opinion is that Easter is the best holiday”

If that is what IP means, then surely “Easter is the best holiday” can be true. After all, I have opinions, and sometimes they are true. In particular, the proposition “My opinion is that Easter is the best holiday” is true just so long as I really do prefer Easter to all other holidays. It would be false if I happened to prefer Halloween to Easter, etc. What is supposed to be the problem here? If such propositions are expressions of opinion in this sense, that doesn’t mean that they are not true or false.

On the other hand, “Easter is the best holiday” might not be shorthand for “My opinion is that Easter is the best holiday”. It might be taken to be something like: “Yey! Easter!” If that is what IP means, then it doesn’t have a truth-value, but then it isn’t really a proposition at all.

So, it seems like either “Easter is the best holiday” is a proposition with a truth-value, or it lacks a truth-value precisely because it isn’t a proposition. Either way round, it doesn’t seem to be any reason to doubt bivalence.

He also says that it cannot be proven. But if “Easter is the best holiday” is just taken as a proposition, then it can be proven in the same way as any other proposition:

1. If p, then “Easter is the best holiday”.
2. p
3. Therefore, “Easter is the best holiday”.

Why IP thinks we cannot enter “Easter is the best holiday” into a proof like this is a mystery.

IP concludes that the argument doesn’t work, on the basis that propositions like “Easter is the best holiday” are neither true nor false. As we have just seen, his reasons for thinking that this sort of proposition is neither true nor false are pretty unconvincing. But let’s just grant them for the sake of the argument.

He doesn’t seem to realise that if “Easter is the best holiday” is neither true nor false, then he is effectively conceding exactly the thing that the argument was supposed to be showing, i.e. that there are exceptions to classical logic. If his own example were genuinely an example of a proposition that lacked a truth value, this would be enough to undermine classical logic. So, he isn’t showing something about the argument that is wrong; he is just giving another (albeit more flawed) instance of a counterexample to classical logic.

3. Gödel

At around 2:20, IP moves on to talk about Kurt Gödel:

“The argument itself is based on Gödel’s theorems, which many think shows logic doesn’t work”.

I think what IP has in mind is that there is another type of challenge to classical logic, this time coming from Gödel’s incompleteness theorems. He gives a statement about what the incompleteness theorems show, but it crucially mistakes (and overstates) their true significance. This leaves IP drawing all the wrong consequences.

IP says that Gödel’s incompleteness theorems show that:

“No consistent system of axioms whose theorems can be listed by an ‘effective procedure’ is capable of proving all truth”

This statement stands out a bit in the video, and it sounds like IP has got it from somewhere, but he never gives any citations for this quote, so we have to guess. My first guess was Wikipedia, and I was right. What is revealing about the quote is what he leaves off. Here is how it shows on Wikipedia:

The quote in full (with the bit he missed off in italics) is:

“No consistent system of axioms whose theorems can be listed by an ‘effective procedure’ is capable of proving all truths about the arithmetic of the natural numbers“.

There is a very big difference between showing that no consistent system of axioms can prove all truth, and showing that they cannot prove all truths about the arithmetic of the natural numbers. I don’t know if he didn’t think the extra bit he left off wasn’t important, or if he did it on purpose to jazz up his point, but either way leaving it off completely changes the significance of Gödel’s incompleteness theorems.

The thing is that (when we look at it properly) Gödel’s incompleteness theorems do not pose a direct local challenge to classical logic. What they show is compatible with non-contradiction, excluded middle and the law of identity all being true (along with all the other principles of classical logic).

What the theorems show is that any system of logic that is powerful enough to express all the arithmetic propositions cannot prove all of them.

So, the result applies to a certain type of logic, called ‘mathematical logic’. This logic is built up out of first-order logic, which is itself a very basic type of classical logic (one that respects all the principles IP presumably wants to defend). If you add the right axioms to this logic, then it becomes capable of expressing things like 1+1=2, etc. Once it is able to do that, we call it mathematical logic. Gödel’s incompleteness theorems apply specifically to mathematical logic.

And because this mathematical logic itself respects the classical principles (it is a type of classical logic), this means that Gödel is just telling us something about the limits of a certain type of classical logic (classical logic that is capable of expressing arithmetic). It is pointing out a limitation in mathematical logic. That is not itself a straightforwardly a reason to think that classical logic is not the correct logic, or that the ‘laws of logic’ are not true.

Except… it might be.

The strange thing about Gödel’s proof is that it shows that arithmetic, and any more complex bit of mathematics, cannot be modelled in classical logic without having ‘blind spots’, where there is something which is true but not provable in that logic. Yet, we might just think that we obviously can prove everything in arithmetic; we might just find the limits of proof in mathematical logic to be an unacceptable consequence. Well, if you did think this, then you could use this as a reason to think that there must be contradictions.

This is because the actual theorems can be thought of as ‘either-or’ statements. They can be thought of as saying ‘either mathematical logic is consistent but has blind-spots, or it has no blind-spots but it has some contradictions in it’ – Gödel is telling us that mathematical logic is either incomplete or inconsistent – either there is something that is true but not provable, or the law of non-contradiction is false.

If you thought that the price (of denying non-contradiction) was worth it so that you didn’t have any of these weird blind-spots in your proof-theory, then you might be willing to accept the inconsistent option. Most people find contradictions more troubling than blind-spots though, and so don’t go that route. But, that is probably the most direct sort of attack you could make from Gödel against classical logic.

If you were feeling charitable, you might think that this is the sort of challenge that IP had in mind. But he dropped off the bit of the quote from Wikipedia which specifically says that Gödel’s theorems are about mathematical logic, not all logic (or even all of classical logic). I find it hard to believe that he didn’t read the end of the sentence he quoted, so either he didn’t understand that the bit he left off is crucial to understand the theorems, or he is deliberately overstating their importance. Either way, it is not great.

Now, if you know a little bit about Gödel, then you might know that in addition to his incompleteness theorems, he is also well known for his completeness theorem. This showed that the basic (classical) first-order logic is actually complete, meaning that it definitely doesn’t have any of those weird blind-spots that the extended mathematical logic has. So without the extra axioms added to first-order logic, it is capable of proving all its own truths.

And this is where we see why leaving off that bit from the Wikipedia quote was so telling. The way IP tells it, the significance of Gödel’s incompleteness theorems is that logic ‘cannot prove all truths’, which sounds like a very profound, almost mystical insight into what people can know and what they can’t. But, in reality, Gödel’s incompleteness theorems only show that some types of logic cannot prove all of their own truths. Admittedly, it is a very important class of logical systems, as it is the ones that model mathematical logic, but it is not as widespread as IP makes out. And Gödel’s completeness theorem actually proves that there are other types of logic for which this is not the case. There are also many other famous completeness theorems in logic (such as Kripke’s celebrated completeness theorem for the modal logic S5, which wouldn’t be possible if IP was right about what Gödel’s incompleteness theorems said!).

IP summarises what he thinks Gödel showed us as follows:

“All Gödel did was show that we are limited in having a total proof of something, but even without Gödel that is intuitively obvious. Many things will only be 99% probably true. But absolute certainty will always be beyond our reach”.

In reality, the significance of Gödel’s incompleteness theorems is not at all intuitive. Almost nobody expected mathematical logic to be limited in the way he showed it was. IP seems to think that Gödel just used maths to show that we can never really know anything for certain. This is demonstrably a bad interpretation of Gödel, and IP clearly has no idea what Gödel really showed us.

On the other hand, I agree that there is no particularly compelling reason to give up classical logic due to Gödel’s incompleteness theorems. I don’t find the idea of accepting contradictions just to get around incompleteness of arithmetic to be persuasive. It’s just a pity that IP wasn’t able to explain what Gödel said, how that was relevant to classical logic, and how it doesn’t mean we should reject classical logic. It’s more a case of a stopped clock accidentally showing the right time.

4. G Spencer-Brown

In the next main bit (around 3:10), IP brings up a different philosopher (or mathematician, depending on how you look at it), G Spencer-Brown, and the section he takes up is from Spencer-Brown’s book, Laws of Form. Now, this is a very strange book on logic, and not within the mainstream work on logic that philosophers usually debate. That is not to say that it is not of any value, but just to be aware that it is already a weird reference. The bit of that book that IP seems to have read is merely the preface, so it is quite easy to check for yourself (just pages ix – xii).

Anyway, IP is going back to the 3rd premise of the argument, which is the idea that the Liar Proposition is neither true nor false. He seems to be saying that Spencer-Brown advocates a solution to the problem which avoids having to postulate that the proposition is neither true nor false. This is presumably done in order to rescue the ‘laws of logic’ from the attack, and to defend classical logic.

So, the thing about the liar proposition, i.e. “This proposition is false”, is that if you assume it has a truth-value (true or false), then it sort of switches that truth-value on you. To see that, assume it is true. That would mean that what it says is the case. But what it says is that it is false. So if it is true, then it is false. The same thing happens if we assume it is false. So, we might say that any input value gets transformed into its opposite output value; true goes to false, false goes to true.

And this feature, or something similar to it, is also seen in the following mathematical example that Spencer-Brown brings up in the preface to Laws of Form. So consider the following equation:

X = -1/X

If you try to solve the equation by assuming that X = 1 (i.e. if we substitute X for 1), then we get:

1 = -1/1

However, -1 divided by 1 equals -1 (because any number divided by 1 equals itself), so: -1/1 = -1. But that means that:

1 = -1/1 = -1

The ‘input’ of 1 gets turned into the ‘output’ of -1. If we try to solve the equation by assuming that X = -1, then we get the converse result (because any number divided by itself equals 1):

-1 = -1/-1 = 1

So the assumption of X = 1, results in an output of -1. And the assumption of X = -1 results in the output of 1. This is a bit like what is going on with the liar proposition if we think of 1 being like ‘true’, and -1 being like ‘false’. In both cases, the input value gets switched to the alternative value.

IP says that the ‘solution’ to this problem is to use an ‘imaginary number‘ i, which is √-1. What he means is that if we assume that X = i, then we get the following solution to the equation:

i = -1/i

Because is the square root of -1, it is already -1/i. So:

i = -1/i = i

Unlike when we assumed X was 1 or -1, where the output got switched, if we assume the input is i, then the output doesn’t get switched. Ok, got it.

The first thing to note here is that this sort of consideration is what motivated mathematicians to consider changing how they thought about mathematics. And not without some resistance. Descartes apparently used the term ‘imaginary’ as a derogatory term. Nevertheless, mathematicians were convinced that introducing imaginary numbers into their understanding of mathematics, despite being unintuitive to some extent, was warranted due to the utility that doing so brought about. What Spencer-Brown is pointing to is a reason for re-conceiving traditional mathematics.

How does this relate to the liar proposition? Unfortunately for IP, it doesn’t relate in the way he wants it to. Also, he says almost nothing about how this is supposed to relate to the liar’s paradox. He says something, it is not helpful. What he says is:

“The only problem is that we cannot epistemically understand the mathematical usage of i. And thus Gödel was proven right and not the absolute skeptic who doubts logic is true”.

Now, IP is obviously wandering off down the wrong path here. Clearly, IP finds imaginary numbers hard to think about, but it is not clear what that has to do with anything. His comment about Gödel betrays his poor grasp of his work as well. Because Spencer-Brown explained how to use i in an equation, that proves that Gödel was right? Hardly.

What is actually going on here, what IP seems unable to get, is that Spencer-Brown is not advocating for classical logic. In fact, he is quite out-there as a thinker, and proposing something quite radical. Let’s look at what Spencer-Brown says about the mathematical example that IP brought up, and how it relates to the liar paradox:

“Of course, as everybody knows, the [mathematical] paradox in this case is resolved by introducing a fourth class of number, called imaginary, so that we can say the roots of the equation above are ±i, where is a new kind of entity that consists of a square root of minus one.” (Spencer-Brown, Laws of Form, page xi, bold added by me)

Spencer-Brown is saying that the solution to the mathematical puzzle requires the addition of a “new kind of entity” to mathematics. A new kind of number. He then goes on in the next paragraph to explain how this mathematical lesson applies to logic:

“What we do in Chapter 11 is extend the concept to Boolean algebras, which means that a valid argument may contain not just three classes of statement, but four: true false, meaningless and imaginary.” (ibid)

So Spencer-Brown is playing around with a type of logic which has four truth-values, not two like classical logic has. This makes it a very exotic type of non-classical logic! IP doesn’t mention this passage, which clearly shows Spencer-Brown freely speculating on a type of logic which is very different from classical logic.

So, what we have here is an example of someone saying that the right way to solve the liars paradox is to modify classical logic in some fundamental way. IP seems to think that this example makes the point he wants to make, but if anything it points in the opposite direction completely. Far from showing that the laws of classical logic cannot be questioned, it is an example of someone questioning the laws of classical logic.

5. Conclusion

So far we have seen that IP has no real idea what the skeptical challenge to logic really consists in. He knows that sometimes people talk about reasons to doubt things like non-contradiction or the law of excluded middle, and he seems to take this to be a very radical attack on logic itself. However, we saw that he presented an argument that attempted to attack the claim that the laws of logic are true, and he hopelessly misunderstood it. It was showing that if the Liar Proposition is neither true nor false, then classical logic isn’t correct. In response, he proposed that “Easter is the best holiday” was neither true nor false, which is itself very poorly argued for, but even if it were correct would be another reason to reject classical logic. He then utterly failed to grasp Gödel, and may have deliberately misstated the theorem’s significance. Lastly, he looked at a passage from Spencer-Brown, but failed to see that if it was correct, it would be a reason to prefer a four-valued logic over the classical two-valued logic.

There is still another half of his video to go, and I will try to get round to debunking the claims made in that half as well when I get a chance.

## Logic and God’s Character

0. Introduction

Vern Poytress is professor of New Testament interpretation at Westminster Theological Seminary. He has a handy website, which he runs with John Frame, on which he has put a lot of his published work available for free. In particular, he has a copy of his book Logic: A God Centred Approach to the Foundation of Western ThoughtIn this post, I want to focus on a particular small section of the book, which is Chapter 7 (p. 62 – 68). The chapter is entitled ‘Logic Revealing God’, and in it Poytress addresses the question of whether logic is dependent on God, or if God is dependent on logic. As he says, “We seem to be on the horns of a dilemma” (p. 63).

I will go through the chapter quite closely, and it might be worth reading as it is not long (although I will provide plenty of quotes from the original). It is an instructive chapter because it highlights many of the key themes and ideas that we see presuppositionalists making in their positive arguments. It is also done by a professor in a theological seminary, with a very impressive resume, including a PhD in mathematics from Harvard, and a ThD in New Testament Studies from Stellenbosch, South Africa. Therefore, the presentation of the argument should be pretty strong. And I do think that the book is quite readable, and is packed full of great learning material for anyone wanting to study logic.

However, I think that the sections of the book which deal with the theological and metaphysical underpinnings of his view of logic, such as the one I will explore here, leave a lot to be desired. Hopefully, what I will say will be clear, and my criticisms will be justified.

1. The Dilemma

The dilemma that Poytress refers to is not spelled out explicitly, but it seems easily recoverable from what he does say. The opening line in the chapter is: “Is logic independent of God?” To start us off, it is quite natural to see logic as independent from the existence of human beings, as Poytress explains:

“Logic is independent of any particular human being and of humanity as a whole. If all human beings were to die, and Felix the cat were to survive, it would still be the case that Felix is a carnivore. The logic leading to this conclusion would still be valid … This hypothetical situation shows that logic is independent of humanity.” (p. 63)

The example that Poytress gives is slightly confusing, as the truth of the statement “Felix is a carnivore” does not seem to be merely a matter of logic, at least not a paradigmatic one. However, it is clear that the idea of independence that is in play involves the following sort of relation:

Independence X is independent of Y   iff   X would still exist even if Y did not exist

The logical relation he highlights (involving the cat) would hold even if people did not exist, and is thus independent from the existence of people. It follows that X is dependent on Y if and only if the independence condition above fails.

The cat example seems to be mixing up a few different things at the same time. The classification of Felix as a carnivore does not depend on the existence of humans, in that whether people exist or not will not change whether a cat eats meat or not. Yet this fact does not seem to be a purely logical fact, and so the independence that it establishes is not really of logic from the existence of human beings.

It seems to me that an example which makes the point he expresses with “logic is independent of any particular human being and of humanity as a whole,” would be the following. Consider the following inference:

1. All men are mortal
2. Socrates is a man
3. Therefore, Socrates is mortal.

The conclusion follows from the premises, and it does so regardless of whether Socrates exists or not. As it happens, Socrates does not exist (any longer), but this does not make the inference any less valid than when he did exist. Even if Socrates turns out to have been entirely a fictional character who never existed at all, the inference is still valid.

And indeed, the conclusion follows from the premises, regardless of whether anyone exists or not; even if everyone were to die in a nuclear war tomorrow, the above inference would remain valid. Even if there had never been any people at all, the inference would remain valid. At least, that is the thought.

Part of the reason for this thought is that we do not need to refer to the existence of any particular thing when coming to determine whether an inference is valid. We consult what it is that actually determines the validity of the inference, and in doing so we do not have to check to see if any particular thing exists. And what it is that the validity of the inference depends on is something like one of the following candidate considerations:

• An inference is valid if and only if it is possessing the correct logical form.
• An inference is valid if and only if it is truth-preserving.

Exactly how we cash this out is contentious of course, but I take it that something like these sorts of example is going to be correct. In Aristotelian logic, for example, the forms Barbara and Celerant are simply given as valid (they are the so-called ‘perfect forms’), and so is any form which is transformable into either of one of the perfect forms via the conversion rules. Different logical systems have different conceptions of what the ‘correct logical form’ is, but one thing that seems obvious is that the existence or not of any particular person, or of humanity in general, is irrelevant to the question of whether a given inference is valid or not. It is a different type of consideration that is relevant.

But if this (or something like this) is what the validity of the inference depends on, then whether it is valid or not isn’t just independent from the existence of human beings, but is independent from the existence of any existing thing – including God.

Here is how Poytress explains this idea:

“Through the ages, philosophers are the ones who have done most of the reflection on logic. And philosophers have mostly thought that logic is just “there.” According to their thinking, it is an impersonal something. Their thinking then says that, if a personal God exists, or if multiple gods exist, as the Greek and Roman polytheists believed, these personal beings are subject to the laws of logic, as is everything else in the world. Logic is a kind of cold, Spockian ideal.” (p. 62)

As I have explained, it is not just that philosophers have postulated logic as being just there, without any motivation. There are reasons, like the independence considerations I outlined, for thinking that any given inference is valid or invalid independently from the existence of any particular thing. It follows from these considerations that logic is not itself dependent on any particular thing, and ‘just is’ (as Poytress puts it).

2. Conflict

As a Christian, such a conclusion brings Poytress into conflict with his core theological doctrines. As he explains:

“This view has the effect of making logic an absolute above God, to which God himself is subjected. This view in fact is radically antagonistic to the biblical idea that God is absolute and that everything else is radically subject to him: ‘The Lord has established his throne in the heavens, and his kingdom rules over all’ (Ps. 103:19).” (p. 62)

Thus, logic seems like it is independent of God, because it seems independent of the existence of anything, yet the doctrine of God being absolute (in Poytress’ sense) requires that everything is dependent on God. I take it that this is the dilemma that he faces:

• On the one hand, logic is independent from the existence of God (as it seems independent from the existence of any entity whatsoever) but that compromises God’s absoluteness (God seems to be subordinate in some sense to logic).
• On the other hand, logic is dependent on God, which restores the absoluteness of God, but then we are owed some kind of story about how it is that the validity of an argument depends on the existence of God.

This dilemma can be put as follows:

Is God dependent on logic, or is logic dependent on God?

Poytress takes the second horn, and part of his endeavour in the chapter is to bring out how it is that we see God in logic, how logic ‘reveals God’, as a way of bolstering the claim that logic depends on God.

As a first pass, he says:

“The Bible provides resources for moving beyond this apparent dilemma.” (p. 63)

He provides three examples, which are:

1. “God is dependable and faithful in his character”
2. “the Bible teaches the distinction between Creator and creature”
3. “we as human beings are made in the image of God”

Let’s go through each of these and see what he has to say about each of them.

3. “God is dependable and faithful in his character”

With regards to 1, Poytress points to Exodus 34:6, which mentions that God is faithful, and he then explains:

“The constancy of God’s character provides an absolute basis for us to trust in his faithfulness to us. And this faithfulness includes logical consistency rather than illogicality. God “cannot deny himself” (2 Tim. 2:13). He always acts in accordance with who he is.” (p. 63)

It is not clear to me how this engages with our question, which was whether logic depends on God or God depends on logic. Poytress is identifying the faithfulness, logical consistency and inability to deny himself as three special properties that God has, but to me the possession of these properties is irrelevant to the question at hand. I will try to explain my worry with a thought experiment:

Imagine I were to build a robot. And let’s say that I build the robot in such a way that it could not knowingly lie. This would mean that I program it in such a way that it cannot provide any output which is the contradicts any of the stored data it has in its memory banks (or something like that). If so, then my robot would be analogous in some sense to this description of God. It is, in effect, programmed to be honest. Given that a robot cannot do anything which it is not programmed to do, I would be able to trust in its ‘faithfulness’, in that I could know for sure that any output it generates is consistent with its data banks. Arguably, a robot like this is also logically consistent by definition (assuming the programming is consistent), and because it cannot lie, it cannot deny itself in the relevant sense either. Thus, my robot is perfectly faithful, logically consistent and cannot deny itself. Yet, this would not establish that the validity of any given inference was dependent on the existence of the robot, however. And if not, then it is not clear why these properties being possessed by God would be relevant to establishing anything like the horn of the dilemma that Poytress is going for either.

Perhaps you have some niggling objection here. The robot case isn’t really analogous to God, you might be saying. And that is quite true. For instance, no matter how advanced, my robot wouldn’t be all-knowing. And no matter how reliable its programming is, its programming might become corrupted. Either of these indicate the possibility of some kind of error. Because of the possibility of error like this I shouldn’t trust what it tells me with 100% certainty, and this makes the two cases unalike.

However, seeing as this is just a thought experiment, imagine that (somehow) I were to make a robot which did know everything, and couldn’t have its programming corrupted. Would this mean that logic now became dependent on the existence of the robot? Would the validity of an inference now depend on the existence of this robot? I see no reason for thinking that making these imaginary improvements to my robot could possibly have this effect.

As far as I can understand, an entity’s reliability, faithfulness, or inability to self-deny, etc, can never be relevant for making its existence something upon which the validity of an inference depends. If Poytress has some reason for thinking that the possession of these properties by God makes him the thing whose existence the validity of an argument depends, he spends no time explaining them here.

There are a few options at this point.

1. By possessing these qualities, my robot becomes a thing that the validity of an inference is dependent on.
2. The possession of these properties by my robot does not qualify it for being the thing that validity depends on, but they are what qualifies God for this role.
3. The possession of these properties are not what qualifies anything for this role.

The first option seems prima facie implausible, and at the very least we have been given no reason to think that it is true. The second one leaves unanswered why it is that these qualities make God suitable for the role and not the robot, and implies that there are actually additional criteria for playing the role in question which make the difference (i.e. there must be something about God other than the possession of these qualities which distinguishes him from the robot). The third option is that these qualities are not relevant. Unless there is an additional option I cannot see, it seems like Poytress has to go with option 2, and owes us an explanation of the additional criteria.

4. The Bible teaches the distinction between Creator and creature”

So much for the first point. Let’s move on to the second one, which is about the creator/creature distinction. Poytress says the following:

“God alone is Creator and Sovereign and Absolute. We are not. Everything God created is distinct from him. It is all subject to him. Therefore, logic is not a second absolute, over God or beside him. There is only one Absolute, God himself. Logic is in fact an aspect of his character, because it expresses the consistency of God and the faithfulness of God. Consistency and faithfulness belong to the character of God. We can say that they are attributes of God. God is who he is (Ex. 3:14), and what he is includes his consistency and faithfulness. There is nothing more ultimate than God. So God is the source for logic. The character of God includes his logicality.” (p. 63)

This quote can be split into two sections. The first consists of the first five sentences (ending with “There is only one Absolute, God himself”). The first section really just affirms the doctrine of God being absolute. God alone is absolute; we are not absolute; being absolute, everything is dependent on God, including logic. This much is no help resolving the apparent dilemma we were facing earlier. It is just restating one of the two things we are trying to reconcile, i.e. the absoluteness of God. The question is how to fit this idea, of God being absolute, with the intuitive idea that the validity of an inference seems to have nothing to do with the existence of any particular thing. Simply repeating that God is absolute (in contrast to humans) does not shed any light on this issue.

The second part of the quote wanders back into the issue brought up in the previous point, by talking about the faithful character of God, and thus still seems irrelevant. Even if “[c]onsistency and faithfulness belong to the character of God”, how is the validity of an inference dependent on his existence? We are none the wiser.

Poytress does say that God’s ‘logicality’ is included in his character. And it might be thought that this is relevant somehow. After all, we are talking about logic, and ‘logicality’ is the property of being logical. Surely that is the link.

Well, I think it would be a mistake to think that. In some sense, my robot was already logical. It’s ‘brain’ is just a computer, which processes inputs and produces outputs according to some set of rules (its programming). This is a logical process; computer programming is just applied logic. It seems we are in precisely the same position we were in before. We are still left with no reason to think that if this thing did not exist, that an otherwise valid inference would be invalid. Why does being logical mean that logic depends on you? The answer, it seems, is that it doesn’t.

5. “We as human beings are made in the image of God”

On to point three. Here, Poytress is pointing to the fact that we are made in God’s image:

“God has plans and purposes (Isa. 46:10–11). So do we, on our human level (James 4:13; Prov. 16:1). God has thoughts infinitely above ours (Isa. 55:8–9), but we may also have access to his thoughts when he reveals them: “How precious to me are your thoughts, O God!” (Ps. 139:17). We are privileged to think God’s thoughts after him. Our experience of thinking, reasoning, and forming arguments imitates God and reflects the mind of God. Our logic reflects God’s logic. Logic, then, is an aspect of God’s mind. Logic is universal among all human beings in all cultures, because there is only one God, and we are all made in the image of God.” (p. 64)

The idea seems to be as follows. God makes plans, and so do we, although we only make plans on a ‘human level’. God has thoughts, and so do we, although his thoughts are ‘infinitely above ours’. So in this way, we are similar to God, without being the same as God. We are creatures, whereas he is the creator, and our likeness is only imperfect (or ‘analogical’).

The relevant section is when he explains that “our logic reflects God’s logic”, which is because it is us ‘thinking Gods thoughts after him’, in a process which “reflects the mind of God”. Just like with the planning and thinking examples, our grasp of logic is only analogical, which means that we have an imperfect, creaturely understanding in comparison with God’s perfect understanding. Nevertheless, we imitate of God’s thought processes.

The problem with this view is that it invites a Euthyphro-style dilemma immediately. God thinks in a particular way (a logical way) and we are to think in the same sort of way (to imitate and reflect his way of thinking). But, why does God think in this particular way? More precisely, does God think in this way because it is logical way of thinking, or is it a logical way of thinking merely in virtue of it being the way that God thinks? This is just another way of asking the same question we started with, namely: is God dependent on logic or is logic dependent on God? All we have done here is to rephrase it in terms of God’s thinking; is logic dependent on God’s thinking, or is God’s thinking dependent on logic? And there is no reason to think that rephrasing it in this manner will itself constitute any sort of solution to the initial problem.

What Poytress is actually giving us is a reason for why we (should) think in a logical way. We should think in a logical way because that’s the way that God thinks. And, whatever the merits of this point are, this plainly isn’t relevant to the initial question about the relation between logic and God. The best that can be said about this idea is that it is an answer to a different question altogether.

5. Sidebar – Logical Euthryphro

But it is also rather hopeless as a solution, when we try to run the argument to its logical conclusion. Remember, the first horn was that God thinks in this particular way because it is (independently from him thinking it) a logical way of thinking. Presumably, Poytress would find just as “radically antagonistic to the biblical idea that God is absolute” as the initial claim that God depends on logic. It really just is the same claim. It just says that logic is independent of God. So, he has to opt for the second horn, which is that this way of thinking is logical merely in virtue of being the way that God thinks.

However, there is a problem with this; it makes God’s decision to think in this way (rather than some other way) inexplicable. To sharpen up the discussion, let’s use some examples. We know that there are lots of different logical systems, including classical logic, extensions of classical logic and non-classical logics, etc. Just to take two examples, there is classical logic and intuitionistic logic. They have different fundamental principles, e.g. intuitionistic logic doesn’t have excluded middle as a general law and classical logic does. God thinks in one of these ways and not the other (presumably). Let’s say he thinks classically, and not intuitionistically. If we were to ask why he thinks in this classical way, as opposed to the intuitionistic way, the one thing we cannot say as an answer is that thinking classically is (independently of God thinking like that) the logical way to think. If we tried to say this, then we would in fact be asserting the first horn of the dilemma, which is “radically antagonistic to the biblical idea that God is absolute”.

But what else could possibly be the answer to this question? God thinks classically rather than intuitionistically because … ? It might be that God has a preference for classical logic rather than intuitionistic logic, but this preference itself cannot be based on the idea that classical logic is (independently of God thinking like that) the logical way to think, or we are right back to the initial horn again. So even if he has a preference for classical logic, it can only be based on some other type of consideration, and not that it is itself the logical way to think. But there is nothing else which could be relevant. He may prefer it because he finds it simpler than intuitionistic logic, or because he likes sound of the word ‘classical’, or because he flipped a coin and it landed heads-up rather than tails-up. But whatever the reason, it can only be something which is irrelevant. His reason can only be arbitrary (which just means that it is a decision made without relevant reason). The one thing which could be relevant is ruled out as being the first horn of the dilemma. And that is what is so pressing about this sort of Euthryphro dilemma.

So let’s say we take this horn. It means that if God thinks classically (rather than intuitionistically), and if we were to imitate the way that God thinks (as Poytress urges), then this would produce some kind of explanation for why we think classically rather than intuitionistically. However, because there is no (non-arbitrary) reason why God thinks classically rather than intuionistically, there is correspondingly no real reason why we do either.

Imagine you find me performing a series of actions, walking to and fro in my house, picking things up and putting them down again seemingly at random. If you ask me why I’m doing this, I might say that I have a reason for doing so. Maybe I say to you that these actions performed together will culminate in an effect which I desire. So, maybe I am building something, but I am in the early stages of doing so, just setting out my tools and clearing a space. To you it looks like a random set of actions, but it has a purpose. I have reasons for doing each of the things that I am doing. Maybe once I have explained my purpose, then the series of actions stops looking so random to you.

Now imagine that you come across me performing a series of actions which again seem random to you. You ask me why I am doing these things, and this time I point to the TV, where you see a figure who is performing the very same sorts of actions. I say that I am acting out this person’s actions after him, and reflecting his actions. ‘Well, why is he doing these particular actions?’, I ask. ‘Oh, no reason’, you reply.

I think that in this second situation, we would have to conclude that you are doing something which is different in type to the first example. There your actions had a reason behind them and were not arbitrary, whereas now, you are just mirroring the random actions of the figure on the TV. Really, your actions are just as random as his; there is no reason why you are doing one thing rather than another, because there is no reason why the figure on the TV is doing one thing rather than another. This is what happens if we follow through on the idea that a) we think logically because we are thinking God’s thoughts after him, and b) if logic is not independent of God. Poytress is committed to b), as the other option would be “radically antagonistic” to his idea of God, and he is also urging that we accept a) in the passage we just looked at. Thus, if we go where Poytress urges, we become like the person imitating the random actions of the figure on the TV.

But, surely, this is where God’s characteristics come into play? God is consistent, and faithful, and cannot deny himself. Surely this is relevant. He couldn’t think in an irrational way, because this would mean being inconsistent. In this way, his consistency grounds the type of logic he opts for.

This may seem like a promising rebuttal. However (no surprise), I don’t think it is. Intuitionism is consistent, and many people have found it to be rational. Michael Dummett, for example, argued strongly for intuitionism. It is not the case that someone who prefers intuitionism to classical logic is committed to any contradictions as a result (intuitionistic logic is not inconsistent). They are not necessarily going to deny themselves, or be irrational, or be ‘illogical’ (partly because they would advocate for intuitionism being the correct logic!). None of the considerations that Poytress presents give us any reason to think that God would have any real reason to prefer classical logic over intuitionism based off the character traits that he has identified.

It might even be the case that God likes paraconsistent, or even dialethic logic. If the principle of explosion really were invalid, then God would be dishonest to say that it was valid. If there really were a true contradiction somewhere (and who knows, maybe God has a morally sufficient reason to create one), then God would deny his own act of creation to say that there was not one. Thus, his honesty, truthfullness and consistency could be made to fit with there being contradictions. His characteristics could be retrofitted to be compatible with pretty much any outlandish logical or metaphysical proposal. And this is because they really just float free from, and are orthogonal to, the issues involved in the debates about non-classical logic.

6. Wrapping up

This post is already quite a lot longer than I had anticipated when I started, so I will finish up by briefly going through the final parts of the chapter we are looking at. Those are called ‘Attributes of God’, ‘Divine Attributes of Law’ and ‘The Power of Logic’. In them, Poytress makes the point that logic and God seem to share various properties:

Atemporality:

“If an argument is indeed valid, its validity holds for all times and all places. That is, its validity is omnipresent (in all places) and eternal (for all times). Logical validity has these two attributes that are classically attributed to God.” (p. 65)

Immutability:

“If a law for the validity of a syllogism holds for all times, we presuppose that it is the same law through all times … If a syllogism really does display valid reasoning, does it continue to be valid over time? The law— the law governing reasoning—does not change with time. It is immutable. Validity is unchangeable. Immutability is an attribute of God.” (p. 66)

Immaterial yet effective:

“Logic is essentially immaterial and invisible but is known through its effects. Likewise, God is essentially immaterial and invisible but he is known through his acts in the world.” (ibid)

True/truthful:

“If we are talking about the real laws, rather than possibly awed human formulations, the laws of logic are also absolutely, infallibly true. Truthfulness is also an attribute of God.” (ibid)

These properties initially do seem to be drawing a close similarity between logic and God. They seem to share a lot of properties together. And initially, this might seem to be reason to think that their doing so is significant. However, consider that the same case could be made for the rules of chess:

There is nothing in the laws of chess which refer to any times and places. If it is true that, according to the rules of chess, a pawn can move two spaces on its first move, then this is true if you play chess in Bulgaria, or in China, or on the moon. It’s truth is independent on location, which means that that rule, if true anywhere, is equally true everywhere else. But also, if we went back in a time machine to prehistoric times, and if we had taken a chess set with us, we would not have to consult the local tribe to see if they had a different set of rules for chess. It would still be true that a pawn can move two spaces on its first move, regardless of what year we are playing in. And this means that the rule’s applicability is independent of time. If it is true at one time, it is true at all times. The rules of chess, it seems, are omnipresent and atemporal as well.

Chess is also immutable. You might be thinking that chess used to be played differently. In the past, people had different rules for chess, so it isn’t immutable – chess has a history. Quite true, chess does have a history. But so does logic (trust me, I am editing a book about it). People have changed how they have thought about logical laws. For instance, the idea of existential import is present in Aristotelian logic, but not in classical logic. If we can sidestep this issue with logic, by saying that the historical development of logic is not relevant for undermining the claim that logic is immutable, then we can also do the same for chess.

The rules of chess are immaterial. We cannot touch them or measure them, etc. Yet they govern how actual pieces of material get moved about on actual chess boards. So the rules of chess are immaterial yet effective.

The rules of chess are true. It is true that a pawn can move two spaces on its first move. That is a truth.

So the rules of chess are omnipresent, atemporal, immutable, immaterial yet effective and true. Therefore, God thinks ‘chessly’? God’s nature reflects the rules of chess?

We could run the same sorts of considerations for any different (consistent) logical system, like Łukasiewicz’s three valued logic. It also has all the same sorts of properties. But can God think classically and also with three truth-values at the same time? Only an inconsistent God could do that. So if God thinks classically, rather than non-classically, there must be something about non-classical logics which means that their possession of the properties that Poytress identifies is not indicative of anything significant. Again, if there is something which makes this difference, we are not given it.

7. Conclusion

I have no doubt that Poytress is a very smart guy. I don’t have it in me to get a PhD in mathematics from Harvard. And he clearly understands logic very well. It is puzzling then that his discussions on the area I have focussed on in this post are so weak. There is really nothing he has said which helps make the case that logic is dependent on God, rather than being independent from God. I can only conclude that this part of his book was not thought through very well. The only other possibility is that he is so determined to fit together certain doctrines that he is unable to see that his arguments are weak in this area. I may look further at other aspects of the same book in later posts, but from what I have read of it so far, I don’t imagine he will change in any particularly significant way.

## Revenge

0. Introduction

Recently, I had a conversation with my friends Matt Dillahunty and Ozy about philosophy. At about the 1:25:00 mark (the link above should be timestamped), we started talking about how there may be considerations which lead philosophers to rationally question the basic ‘laws of logic’, such as the law of non-contradiction (for all p: ~(& ~p)) and the law of excluded middle (for all p: (∨ ~p)). I brought up the liar paradox, as an example of this sort of thing. Matt objected that it is actually an instance of a ‘gappy’ sentence, which is neither true nor false. At the time, I knew there was a phenomena called ‘revenge’ which poses big problems for this strategy, but annoyingly I couldn’t bring the details to the bit of my brain that makes my mouth work. Here I want to right that wrong.

1. The Liar

The liar sentence is of the following form:

a) This sentence is false.

We only have to assume what seems like a very natural assumption about how the word ‘true’ works to get there. This is that if p is true, then p. We can think of this principle like this; if I say that it is daytime, and if what I say is true, then it is daytime. Alternatively; if I say a declarative sentence, and if it is true, then what it says correctly describes the thing that the sentence is about. This seems to be at the very core of idea of truth.

A corresponding idea is there for ‘false’ as well; if p is false, then it is not the case that p. If I say that it is daytime, and if what I say is false, then it is not daytime. If I make a declarative sentence, and it is false, then it incorrectly describes the thing that the sentence is about.

So let’s apply these principles to a):

If a) is true, then a) correctly describes what it is about. But a) is about itself, and it says about itself that it is false. So if it is true, then it correctly describes itself as false. So if it is true, it is false. And that is a contradiction.

So maybe a) is false. And if a) is false, then it incorrectly describes itself; yet what it says about itself is that it is false. If its self description is incorrect, then it isn’t false; and the only other option is that it is true. So if it is false, then it is true. Contradiction again.

So if it is true, it’s false; but if it is false, it’s true. Either way you go, you run into a contradiction. This is the paradox.

2. Gaps.

Yet, maybe there is a solution here. Matt certainly proposed a solution here. His idea was that a) is neither true nor false. So, let’s run through the options and see how it works.

a) says about itself that it is false. And we are now saying that it has no truth-value at all. Well, it certainly doesn’t correctly describe itself, because it says that it is false, and it is ex hypothesi neither true nor false. If something is neither true nor false, then it is not false. So it’s own self-description fails. This seems to leave no reason to consider it true. It says about itself that it is false, but we cannot derive that it is true. So far, no contradiction.

But, it says about itself that it is false, and this is incorrect (because being neither true nor false, it is not false). And it’s hard to see why this wouldn’t count as a case of a falsity. After all, it says that it is false, yet (ex hypothesi) it isn’t (because it’s gappy). It is certainly not true that it is false; it’s own self-description fails. But does this mean that it is false? Well, only if ‘not true’ means false. And, on this assumption, where we have some sentences which are ‘gappy’ (i.e. neither true nor false), there is a difference between being not-true and being false. If we listed all the not-true sentences, it would include all the ones which had no truth-value, and all the ones which were false. Thus, being not-true does not entail being false. Thus, we seem to have got out of the trap.

It is neither true nor false, and when it says about itself that it is false we can consider it’s incorrect self-description to be a case of being not-true, rather than false.

Strictly speaking, this does work as a consistent (i.e. contradiction-free) way to think about a).

So far, so good. However, things are not over. There is a second round.

3. Revenge

Consider the ‘strengthened liar’ sentence:

b) This sentence is not true.

We have, on our assumption of ‘gappyness’, three options. Either b) is true, or it is false, or it is neither true nor false. Let’s take them one at a time:

If b) is true, then it correctly describes itself. Yet it says about itself that it is not true. So if it is true, it is not true. This is a contradiction.

If b) is false, then what it says about itself is incorrect. Yet, if it is false, then it does come under the category of not-true, which is what it says about itself. So if it is false, then what it says about itself is correct, and so it is true. And we have another contradiction.

The only other option is the one we used for a), which is that it is neither true nor false. Yet, if b) is neither true nor false, then it is in the not-true category as well (because anything which is neither true nor false is not true). But, as it says about itself that it is not true, it would seem like it has correctly described itself. If it has correctly described itself, then it is not in the not-true category, but in the true category. So if b) is neither true nor false, then it is true! This is, again, a contradiction.

So, while the gappy proposal got rid of one liar sentence (i.e. a)), it fails to help us with another one (i.e. b)). As a strategy, gappiness won a battle, but it loses the war.

4. Conclusion

The problem that the liar paradox presents is subtle, and still an open question in philosophy and logic. It may be that a solution to the generalised problem exists which involves adopting a logic which has truth-value gaps. That may be the case for all I know. But it seems clear that simply adopting truth-value gaps does not solve the underlying phenomenon. It merely pushes the problem to somewhere else. Even if a) can be got around by postulating truth-value gaps, b) cannot be. The liar paradox has had its revenge.

As the philosopher Tyler Burge put it:

“Any approach that suppresses the liar-like reasoning in one guise or terminology only to have it emerge in another must be seen as not casting its net wide enough to capture the protean phenomenon of semantical paradox.” (Tyler Burge, Semantical Paradoxes, p. 173, (1979))

## Is God’s Existence Logically Necessary?

0. Introduction

It is a common platitude in philosophy of religion to hear the claim that the existence of God is necessary, or that God exists ‘in all possible worlds’. However, it seems to me that the existence of God can only really be thought of as (at best) metaphysically necessary, but not logically necessary. Strictly speaking, God doesn’t exist in all possible worlds. In this post, I want to explain a bit about what possible worlds are, from a logical point of view, and explain how God’s existence is not logically necessary in classical propositional logic.

1. Possible worlds

Perhaps the defining idiom of 20th century analytic philosophy is the term ‘possible worlds’. The traditional story is that this terminology dates back to Saul Kripke’s early logical work on the semantics of modal logic (see here, and here). For an excellent summary of the historical lead-up to this development, see this paper by Jack Copeland. As Copeland notes, the roots of the project can be traced back further, to the development of classical propositional logic in its modern form by people like Frege, Russell and Wittgenstein. As Wittgenstein explains in the Tractatus:

“If an elementary proposition is true, the state of affairs exists: if an elementary proposition is false, the state of affairs does not exist. If all true elementary propositions are given, the result is a complete description of the world. The world is completely described by the specification of all elementary propositions plus the specification, which of them are true and which false.” (4.25-4.26)

In the second line of the Tractatus, Wittgenstein defines the world as ‘the totality of facts’. These facts, or states of affairs that exist, are what makes propositions true or false. So if we specify a truth value to each and every proposition, then this would be a way of specifying what the world is like – it specifies whether the states of affairs that correspond to the propositions exist or not. Let’s use as an example the proposition is ‘Donald Trump is the president of the USA’, which we shall refer to as p. If is true, then the state of affairs in which corresponds to p exists; i.e. if p is true, then Donald Trump really is the president of the USA.

Once the truth values of the atomic propositions are given, then this generates the truth values of more complex formulas. So if p is true and q is true, then the formula ‘p & q‘ is true as well, etc. In this way, everything that one can say about the world is said just by giving the truth values of all the atomic propositions.

When we are thinking about this process of describing the world via the truth values of the atomic propositions, we need to specify two conditions:

• We need to ensure that we give each proposition one truth value or other; we cannot ‘forget’ to specify whether p is true or false.
• We need to ensure that we do not give the same proposition both truth values; we cannot say that p is both true and false.

Let’s call the first condition maximality, because it is saying that our description of the world is maximal in the sense that no proposition is left out. Let’s call the second condition consistency, because it is saying that our application of truth values to propositions is consistent, in the sense that no proposition is given both truth values. This allows us a nice and precise definition of a ‘world’, as used by Wittgenstein:

A world is a maximal and consistent valuation of atomic propositions

On this understanding, a different world is just a rearrangement of truth values to the basic atomic propositions. So, pretending for a moment that there are only 3 atomic propositions (p, q and r), it might be that in the actual world the following combination of propositions is true: (p, q, r). But at some other world, the following combination is true: (p, ~q, ~r).

The number of worlds is a function of the number of elementary (or atomic) propositions we have. If we have just one proposition p, then there are two worlds, because p could be true, and p could be false. If we have two propositions, p and q, then we have four worlds: one in which both are true, one in which both are false, one in which p is true and is false, and one in which p is false and is true. In this way, we can construct tables which systematically display all the combinations of truth and falsity to the basic propositions. Here is a picture, from the Tractatus, showing such a ‘truth table’ for three propositions, (pq, and r):

The method of truth tables is used by Wittgenstein in the Tractatus as a method of proving whether complex formulas are true or false independently of the values of their atomic elements, and for whether arguments are valid or invalid. In short, it is a proof theory for propositional logic, and it is both sound and complete. It is taught as an introduction to any logic class.

Given that there are two truth values on this picture, if the number of atomic propositions we have is n, then the number of worlds (i.e. maximal and consistent sets of propositions) is 2 raised to the power of n (which WordPress doesn’t seem to have a symbol for); i.e. the number of  worlds doubles for each additional atomic proposition you have.

It follows very simply from this definition of a world, that a formula of the form ‘p or ~p‘ is going to be true at each and every row of the truth table, which is to say at each world. That’s because each world has the proposition p as either true or false in it due to the maximality constraint. All ‘p or ~p‘ needs to be true is that one or other of its disjuncts is true. So, this formula is considered to be a ‘tautology’ because we do not need to look at the particular variation of truth values in the world we are considering to see if it is true or not. It is true in every world. Similarly, a proposition of the form ‘p & ~p‘ is going to be false regardless of the arrangement of the truth values to the atomic propositions. Because of the consistency condition, each world gives only one truth value to each proposition, and the formula just says that p has both truth values.

Therefore, what we might think of as the ‘law of excluded middle’ is guaranteed by the maximality condition, and the ‘law of non-contradiction’ is guaranteed by the consistency condition. That they are true in every world is a consequence of the definition of a ‘world’.

Also, the notion of validity is just that if the premises of an argument are true, then the conclusion is true. If we plug the argument into a truth table, we can give precise expression to this notion: an argument is valid if there is no row of the truth table on which the premises all come out true but on which the conclusion is also false. This just means that an argument is valid if there is no world in which the premises are true and the conclusion is false.

Although it is often not stated in terms of worlds, the ideas of tautology and validity in classical propositional logic have always made use of the notion of worlds, if construed as maximal consistent sets of propositions.

In essence, all Kripke does to this picture is to add two additional operators to the logic, which say ‘it is possible that…’ (◊p) and ‘it is necessary that…’ (□p). These operators refer, via the semantics (which I will not go into any detail over for ease of reading – though I am more than happy to at a later date) to what is true or false at other worlds, or other ‘possible worlds’. So if ◊is true, then p is true at some other possible world; and if □p is true, then p is true at every possible world. We can afford to leave out most of the details here because we are primarily focussed on logical possibility, which is handy because it gets somewhat technical otherwise.

If we ask a question about whether a given proposition, p, is logically possible, then we can see if there is a possible world where p is true. There is a world where p is true if and only if there is a maximal and complete assignment of truth-values to the atomic propositions where p is true. So is the following proposition possible?:

a) This glass of beer is full, and I am hungry.

We can formalise a) as follows, where p = ‘this glass of beer is full’ and q = ‘I am hungry’:

b) pq

Now, as it happens, this glass of beer is half empty (because I have already been drinking from it), and I am not hungry (because I have just eaten dinner), meaning that p is false and so is q. That means that b) (and thus a)) is false. But that doesn’t tell us whether it is possible or not though. What we have to consider to see whether it is logically possible or not is whether there is a contradiction in supposing that it is true. And there is no contradiction in supposing that the truth-values of the propositions are different. Though p and are both false, they could both be true. As Wittgenstein said when referring to states of affairs,

Any one can either be the case or not be the case, and everything else remain the same. (Wittgenstein, Tractatus, 1.21)

This means that we can vary the truth-values of any of our basic atomic propositions without having to change the others; all the combinations of different truth-values are possible. All that we have to watch out for is that we end up with a proposition to which we have no truth-value, or one that has both truth-values, i.e. as long as we don’t end up with an excluded middle or a contradiction.

Given that there is no contradiction in supposing that ‘This glass of beer is full, and I am hungry’, this means that there is a maximal and consistent set of propositions which contains it, and that means that it is logically possible.

Could the following be true?:

c) This glass of beer is full and the glass of beer is not full.

Now, for c) to be true, both sides of the ‘and’ would have to be true. But these are p and not-p respectively. If both were true, then p would be both true and false. This would be a contradiction, and so (because of the consistency condition) there is no possible world at which this is the case. This means that c) is not logically possible.

Ask yourself this, could the following proposition be true?:

d) This glass of beer is full and God does not exist.

Even if you think that the proposition is in actual fact false (i.e. if you are a theist with a full glass of beer), ask yourself if there is a contradiction which results from supposing that d) is true. It seems that there is not. The logical form of d) seems to just be the following:

e) r & ~g

We can show very easily that this formula is not a contradiction with a truth table:

This shows very explicitly that the formula is false if r and g are both true, and if they are both false, but the formula is true on both middle rows. So it is not always false, i.e. it is not a contradiction. I have just proved that e) is not a contradiction. If someone wants to say that d) is somehow a contradiction, then they need to provide a different logical form for the proposition than e). In propositional logic, this seems to be the only plausible rendering of the form of d), and so it seems that in propositional logic, the formula is not a contradiction. That means that God’s existence is not a logically necessary truth.

3. Conclusion

While the existence of God may be asserted as a metaphysical necessity (however that is cashed out), it cannot be asserted as a logical necessity, if the logic we have in mind is classical propositional logic. I will write a sequel to this paper where we look at the possibilities of cashing out the logical necessity of God’s existence in first order logic, and I will explain how it is not a viable claim there either.

## A New Problem for Divine Conceptualism?

0. Introduction

Divine Conceptualism (DC) is an idea about the ontological relationship between God and abstract objects, defended by Greg Welty, in his M.Phil thesis “An Examination of Theistic Conceptual Realism as an Alternative to Theistic Activism“(Welty (2000)), his Philosophia Christi paper “The Lord of Non-Contradiction” (Anderson and Welty (2011)), and his contributions to the book “Beyond the Control of God” edited by Paul Gould (Welty (2016)). Put simply, (DC) identifies abstract objects as something like ideas in the mind of God.

Welty sees his view as being quite close to that of Morris & Menzel‘s (1986) ‘theistic activism’ (TA), according to which:

“…all properties and relations are God’s concepts; the products, or perhaps better, the contents of a divine intellective activity.” (Morris & Menzel (1986), p. 166)

Morris & Menzel’s TA asserts that God created everything which is distinct from God, and that includes the divine concepts themselves. However, as Welty (2000) p.29 observes, TA is vulnerable to ‘bootstrapping’ objections. If God is supposed to be able to create his own properties, then he creates his own omnipotence (because omnipotence is a property); yet it seems that one would already have to have omnipotence in order to be able to create omnipotence. Even more forcefully: God already needs to possess the property of ‘being able to create properties’ in order to create properties. The idea of self-creation is therefore seemingly incoherent.

Welty’s DC can be seen as a modified version of TA; it is TA without the troublesome doctrine of self-creation. On DC:

“…abstract objects … are uncreated ideas in the divine mind; i.e. God’s thoughts.” (Welty, (2000), p. 43

Postulating abstract objects as uncreated divine ideas is designed to avoid the bootstrapping objections from above.

There are of course lots of different types of abstract objects, including propositions, properties, possible worlds, mathematical objects, etc. Here we will only look at propositions. One of the motivations for thinking that propositions in particular are divine thoughts is the argument from intentionality (seen in Anderson and Welty (2011), p 15-18). Propositions are intentional, in that they are about things. So the proposition ‘the cat is on the mat’ is about the cat having a certain relationship to the mat; the proposition is about the cat being in this relation to the mat. In a similar manner, thoughts are also about things. Consciousness is always consciousness of something or other. In Anderson and Welty (2011), it is argued that the laws of logic are propositions, which are necessarily true and really existing things. Given the intrinsic intentionality of propositions, these are argued to be thoughts. However, they cannot be thoughts had by contingently existing entities, like humans, as humans could have failed to exist, whereas laws of logic could not. Thus:

“If the laws of logic are necessarily existent thoughts, they can only be the thoughts of a necessarily existent mind.” (Anderson and Welty, (2011), p.19).

However, I want to point out an objection to this picture, which I have not seen in the literature (a nice summary of existing objections is found here). It is about the definition of the word ‘thought’. (It may be that this problem has been adequately documented in the literature somewhere that I have not seen. Maybe someone can let me know in the comments section.)

1. Thought

It seems to have gone unnoticed that Welty in particular oscillates between DC being construed in two different and incompatible ways. It has to do with the word ‘thought’. There is no completely standardised usage of this term in the philosophy literature. And it is a term which needs careful definition in a philosophical argument because in natural language the word ‘thought’ is sometimes used to refer to the thinking and sometimes the thought-of; it is either the token of a type of mental activity called ‘thinking’, or it is the content, or object, of the thinking. For example, we may have the intuition that my thought is private, and that it is metaphysically impossible for you to have my thought (which makes thoughts similar to perceptions in this respect). But we may also have the intuition that we can ‘put our thoughts on paper’ or ‘share our thoughts’ with other people. It seems to me that this ambiguity infects Welty’s version of DC due to his not clearly and carefully defining what he means by ‘thought’ so as to disambiguate the term between thinking and thought-of. Welty (2000), for example, doesn’t actually contain a definition of a ‘thought’ anywhere in it, even though it mentions ‘thought’ 135 times in 85 pages.

According to Anderson and Welty (2011), they seem to indicate that a thought is not the content of thinking, but the token of the act of thinking. In a footnote on page 20, they say:

We could not have had your thoughts (except in the weaker sense that we could have thoughts with the same content as your thoughts, which presupposes a distinction between human thoughts and the content of those thoughts, e.g., propositions).”

The distinction that is being made here is between thoughts, which are individualised occurrences not shareable by multiple thinkers, and the contents of those thoughts, which are generalised and shareable by multiple thinkers. I can have a thought with the same content as you, even though we cannot have the same thought. In Fregean terms, a ‘thought’ (as Anderson and Welty use the term above) is an ‘apprehension’. When one thinks about the Pythagorean theorem, one is apprehending the proposition. In order to be explicit about what I mean, I will disambiguate the term ‘thought’ by referring to the token act of thinking as an ‘apprehension’, and the content of the thought as the ‘proposition’.

2. Blurred Lines

However, in Welty (2000), this distinction is repeatedly blurred. One of the main thrusts of the position defended there is that God’s thoughts function as abstract objects:

“God and I can have the same thought, ‘2+2=4’, in terms of content. But my thought doesn’t function in the same way that God’s thought does. My thought doesn’t determine or delimit anything about the actual world, or about any possible world. But God’s thought does. Thus, it plays a completely different role in the scheme of things, even though God and I have the same thought in terms of content. Thus, God’s thought uniquely functions as an abstract object, because of his role as creator of any possible world. I am not the creator of the actual world (much less, any possible world), and thus my thoughts, though they are in many cases the same thoughts as God’s, don’t function as abstract objects in any relevant sense.” (Welty, (2000), p. 51)

Welty says that God and I can have ‘the same thought in terms of content’, which blatantly smudges the sharp distinction between the apprehension and proposition. We can each apprehend the same proposition. But can I share in God’s apprehension of the proposition? It seems that the answer would have to be: no. God’s apprehension of a proposition is surely private to God, just as my apprehension of a proposition is private to me.

Then Welty ends the passage with “my thoughts, though they are in many cases the same thoughts as God’s, don’t function as abstract objects in any relevant sense”. The only sense in which my thoughts are “the same thoughts as God’s” is in terms of the propositions that I think about being the same as the ones that God thinks about. In that sense they do function as abstract objects, precisely because they are abstract objects, namely propositions! The sense in which ‘my thoughts’ don’t function as abstract objects is in terms of the token act of thinking (the apprehension). That doesn’t function as an abstract object, but then that is not something I share with God. So Welty cannot have it that there is something, x, which is both something I share with God and which doesn’t function as an abstract object. The only reason it seems like this is possible is because of a failure to distinguish clearly between thought as apprehension, and thought as propositional content.

This confusion pops up again and again. Take the argument from intentionality, found in all three Welty publications referenced in this post. Part of the motivation for DC is that propositions are (supposedly) thoughts (because they are intentional) but that they cannot be human thoughts; a non-divine conceptualism, the doctrine that abstract objects like propositions are human thoughts, cannot do the job here. The reason for thinking that they cannot be human thoughts is as follows:

“There aren’t enough human thoughts to go around…, human thoughts don’t necessarily exist, and whose thoughts will serve as the intersubjectively available and mind-independent referents of propositional attitudes (referents that are also named by that-clauses)?”

There are three reasons given against human thoughts being able to play the role of propositions: a) there aren’t enough of them, b) their existence isn’t necessary, c) they aren’t intersubjectively available.

While these considerations look somewhat compelling when trying to think of a human conceptualism without the benefit of the distinction between apprehension and proposition, it quickly loses its force when we apply the distinction. The problem is the combination of two types of properties that propositions need. One type of property is associated with divine apprehensions, and the other type of property is associated with divinely apprehended propositions. Being of sufficient plentitude to play the role of propositions (a), and having necessary existence (b), are of one type, and being ‘intersubjectively available’ (c) is of the other. As I shall show, you cannot have both of these types at the same time, without smudging the distinction between apprehensions and propositions.

Firstly, let’s consider non-divine conceptualism, where thoughts are construed as apprehensions.

There are, of course, only finitely many human apprehensions of propositions; there are only finitely many times people have apprehended propositions. Also, human apprehensions of propositions are contingently existing things, because human minds are themselves only contingently existing things. Human apprehensions are also inherently private, and thus not intersubjectively available. So apprehensions cannot be thought of as ‘doing the job’ of abstract objects for these reasons. That much is quite clear.

On the other hand, there may be infinitely many divine apprehensions, so there would be ‘enough to go round’, and perhaps they each exists necessarily. In this sense, they seem suited to play the role of propositions. However, as apprehensions, they would not be ‘intersubjectively available’. Can I actually share in God’s apprehension of a proposition? Unless I can, they cannot play the role of an abstract object.

Thus, when considering apprehensions, although non-divine conceptualism is not suited to play the job, neither is divine conceptualism. The problem is just that apprehensions are private. So let’s compare non-divine and divine conceptualism, where we construe ‘thought’ as the contents of thoughts.

Right away it is obvious that there is no reason to think that the content of human apprehensions are limited in the same way as their apprehensions were. The contents of human apprehensions just are propositions, so of course they can play the role of propositions!

Equally, if divine thoughts are construed as divinely apprehended propositions, then there will be enough to go round, they will exist necessarily, and they will be intersubjectively available. But in both cases, this is just because propositions themselves are sufficiently plentiful, necessary and intersubjective to play the role of propositions. Obviously, propositions can play the role of propositions. Being apprehended by God, rather than humans, is not what bestows the required properties on them.

3. Begging the question?

But perhaps I have begged the question somehow. Maybe the defender of DC can stipulate that, although my apprehensions are private, God’s apprehensions are somehow intersubjectively available. Call this theory ‘divine accessibility’ (DA). So on DA, propositions are divine apprehensions (which are plentiful, and necessary existing) and crucially also intersubjectively available to humans; they can be the content of humans’ apprehensions.

So, let’s say that I am thinking about the Pythagorean theorem. Let’s say that my apprehension is A. According to DA, the content of my apprehension, what A is about, is a divinely accessible apprehension, D. But the question is, what is the content of the divine apprehension, D? What is it that God is thinking about when he has the thought which is the Pythagorean theorem? There seem to be only a few options:

Either God’s apprehension, D, has content, or it does not. If it has no content, then what is it about D which links it to the Pythagorean theorem, rather than to some other theorem, or to nothing at all? It would be empty and featureless without content.

But, if it does have content, then either the content is that ‘the square of the hypotenuse is equal to the sum of the squares of the other two sides’, or it is something else.

If it does have this as its content, then it seems like the content of D is doing all the work. It seems like the only reason God’s apprehension is linked in any way to the Pythagorean theorem is that it has the theorem as its content. If that is right, then we need to have the proposition itself in the picture for God’s apprehension to be in any way relevant.

Consider what would be the case if the content of God’s apprehension was of something else entirely, like the fact that it all bachelors are unmarried men or something. In that situation, there  would be no reason to say that this apprehension was the Pythagorean theorem. The only divine apprehension that could, even plausibly, look like it is playing the role of the proposition is one which has the proposition as its content.

And if we ask what role God’s apprehension plays here it seems that the answer is that it is just a middle man in between my apprehension and the theorem. It seems to be doing nothing. When I think of the theorem, I have an apprehension, A, and all this is about is one of God’s apprehensions, D, which is itself about the theorem. If p is the Pythagorean therem, and x ⇒ y means ‘x is about y’, then we have:

A ⇒ D ⇒ p

God’s apprehension is just an idle cog which does nothing. Why not just have:

A ⇒ p

Why not just say that I have the theorem as the content of my thought? It would be a much simpler suggestion. Given that for God’s apprehension to be in any way relevant to the proposition in question it has to have the proposition as its content, we seem to require the proposition in the picture anyway. Ockham’s razor should suggest shaving off the unnecessary extra entity in the picture, which is the divine apprehension.

4. Conclusion

Thus, there are really two problems with DC. If construed as the contents of God’s thoughts, divine ‘thoughts’ just are propositions. So for DC to be in any way different from the traditional Fregean picture (where propositions are abstract objects), we have no other option but to construe divine thoughts as divine apprehensions. However, it seems that apprehensions are inherently private, and so they are unsuited to play the role of propositions. Even if we postulate that somehow divine apprehensions are accessible to everyone, they seem to become idle cogs doing nothing.

## Problems with ‘The Lord of non-Contradiction’

0. Introduction

In this post, I will not be focusing on a blog post or a non-professional apologetical argument. Rather, I will be focusing on an argument in a peer-reviewed academic journal, called Philosophia Christi (it is published by the Evangelical Philosophical Society). The paper is entitled ‘The Lord of Non-Contradiction‘, and the authors are James Anderson and Greg Welty. They are professional academics, with PhDs in respected institutions (Edinburgh and Oxford, respectively). These guys are proper academics, by any standards. I believe this to be the most philosophically rigorous version of their argument that I have come across.

The argument they present in the paper is a version of the ‘argument from logic’, in which the existence of God is argued for using the nature of logic as the motivating factor. This is a sophisticated version of the familiar presuppositionalist refrain, and is the sort of thing I imagine Matt Slick would be arguing for had he received a graduate education in philosophy as well as theology. It is an interesting paper, which certainly doesn’t fall prey to the usual fallacies that we see repeated over and over again in the non-professional internet apologetics communities. They are presuppositionalists (as far as I can gather), but this is not a presuppositional argument as such.

Despite their obvious qualities as theologians and philosophers, I still see reason to reject the argument, which I will explain here. Before we get to my reasons for criticising the argument, we should have a look at the argument as they present it.

1. The argument

The paper is divided into nine sections, the first eight of which have headings that are claims about the laws of logic; ‘the laws of logic are truths’, ‘the laws of logic are truths about truths’, ‘the laws of logic are necessary truths’, ‘the laws of logic really exist’, ‘the laws of logic necessarily exist’, ‘the laws of logic are non-physical’, ‘the laws of logic are thoughts’, and ‘the laws of logic are divine thoughts’. Here is how they summarise the argument in their conclusion:

The laws of logic are necessary truths about truths; they are necessarily true propositions. Propositions are real entities, but cannot be physical entities; they are essentially thoughts. So the laws of logic are necessarily true thoughts. Since they are true in every possible world, they must exist in every possible world. But if there are necessarily existent thoughts, there must be a necessarily existent mind; and if there is a necessarily existent mind, there must be a necessarily existent person. A necessarily existent person must be spiritual in nature, because no physical entity exists necessarily. Thus, if there are laws of logic, there must also be a necessarily existent, personal, spiritual being. The laws of logic imply the existence of God.” (p. 20)

So we see a plausible looking string of inferences from various claims, each of which has a section in the paper defending it, and often presenting citations to other papers for elaborations. We seem to be moving from simple observations about the nature of the laws of logic, that they are necessary truths, etc, to the claim that they indicate the presence of a divine mind.

Here is the argument from above in something closer to premise/conclusion form. I have had to construct this, as the authors leave the logical form of the argument informal, and in doing so, I have tried to represent the reasoning as we find it above:

1. The laws of logic are necessarily true propositions.
2. Propositions are real entities, but cannot be physical entities; they are essentially thoughts.
3. But if there are necessarily existent thoughts, there must be a necessarily existent mind.
4. If there is a necessarily existent mind, there must be a necessarily existent person.
5.  A necessarily existent person must be spiritual in nature, because no physical entity exists necessarily.
6. If there are laws of logic, there must also be a necessarily existent, personal, spiritual being.
7. A necessarily existent, personal, spiritual being is God
8. The laws of logic imply the existence of God.
9. Therefore, God exists.

The final step I have had to add in myself, as Anderson and Welty do not explicitly draw it out as such. They stop their argument at the conditional ‘logic implies God’, leaving the reader to join the dots. There are some terms that don’t quite match up properly in the above (true propositions and real entities, etc), which stop it from being formally valid.

1.1 A more formal version of the argument

Here is a more formal way of thinking about the argument, with the presentation cleaned up a bit, and as a result more stilted:

1.  If something is a law of logic, then it is necessarily true. (premise)

1a. If something is necessarily true, then it is true all possible worlds. (premise).

1b. There is something which is a law of logic. (premise)

1c. There is something such that it exists in all possible worlds. (from 1 and 1b.)

2. For everything that exists, it is either a physical thing or a thought. (premise)

2a. If something is a law of logic, then it is either a physical thing or a thought. (from 1 and 2.).

2b. If a thing exists necessarily, then it is not a physical thing. (premise)

2c. If something is a law of logic, then it is not a physical thing. (from 1 and 2b.)

2d. If something is a law of logic, then it is a thought. (from 2a. and 2c.)

2e. There is something which is a thought. (from 1a. and 2d.)

2f. There is something such that it is is a thought and that it is necessary that it exists. (from 1b and 2e)

3. If there is a thought, then there is a mind (of which it is a part). (premise)

3a. There is a thought and there is a mind (of which it is part). (from 2e. and 3)

3b. There is something such that it is is a thought and that it is necessary that it exists, and that there is a mind (of which it is part). (from 2f., 3.)

4. If something is a mind, then it is a person. (premise)

4a. There is a person. (from 3a and 4)

4b. There is something such that it is is a thought and that it is necessary that it exists, and that there is a mind (of which it is part) and this is a person. (from 3b. and 4)

5. If it is necessary that there is a person, that person must be spiritual. (premise)

5a. It is necessary that there is a person such that they are spiritual. (from 4b and 5).

6. If the laws of logic exist, then it is necessary that there is person who is spiritual. (1a and 5a)

7. If it is necessary that there is a spiritual person, that person is God. (premise)

8. Therefore, God exists (from 5a. and 7)

The argument presented above is valid. It has the advantage of showing what the various inferences are and how many assumptions need to be given in order for the argument to work. I will present two initial problems, before going into more detail about three more serious problems.

1.2 Initial problems

There are two initial problems with the argument. Firstly, the conclusion arrived at is actually weaker than ‘God exists’, and secondly there is a false dichotomy involved in one of the premises.

1.2.1 Polytheism

The first problem is in premise 3, the inference from the existence of thoughts to the existence of a mind. Take a particular law, say the law of non-contradiction. We can run through the argument up to premise 3 and show that there is a thought, then we deduce the existence of a mind from it; call that mind ‘M1’. But now run the argument again, this time with the law of excluded middle as the example. Once again, when we arrive at step 3, we deduce the existence of a mind; call it ‘M2’. The question is, does M1 = M2? It doesn’t follow logically that they are the same mind, and they could be distinct minds for all the truth of the premises entail. If so, then we would end up with two Gods at the end. Given that there are three laws of logic considered in the paper, Anderson and Welty’s argument is compatible with there being three non-identical necessarily existing minds, or Gods, which would be polytheism. The argument is not specific to laws of logic, but could use any necessary proposition, such as those of mathematics, meaning that we could be looking at an infinite number of minds.

In order to avoid this, we would have to add in as an additional premise that in all cases such as this, M1 = M2. But this seems rather implausible. Now the argument basically says, ‘laws of logic are thoughts, and so are all necessary propositions, and they are all had by the same mind, and that mind is God’. The addition of this premise is ad hoc, meaning it has no intuitive support apart from the fact that it gets us to the conclusion. For it to be considered at all plausible, there should be some independent reason given to think that it is true. Anderson and Welty consider something close to this objection:

It might be objected that the necessary existence of certain thoughts entails only that, necessarily, some minds exist.” (p.19)

However, they cash this out with a scenario in which there are multiple contingent minds, and then produce a counter-argument against this. They seem to miss the possibility that there are multiple necessary minds (i.e. polytheism), and as such their counter-argument misses my point entirely.

At the moment, even if you grant all the premises and assumptions, the argument establishes only that at least one god exists, which is presumably a lot weaker than the conclusion they intend to establish.

1.2.2 False dichotomy

Another problem with the argument above is that premise 2 (everything is either a physical thing or a thought) is a false dichotomy. In addition to arguing that laws of logic are not physical, one would have to present an argument for why the only two options are physical or thought. Anderson and Welty do not present any such argument, and as such there is no reason to accept premise 2. One might want to argue that everything has to be in one of two categories, but then one has to say something about difficult cases. We often say things like ‘there is an opportunity for a promotion’. On the face of it, we are quantifying existentially over opportunities. So opportunities exist. Are they physical things? Are they thoughts? Take haircuts as another example. Are they physical things? Are they thoughts? We could come up with some way of categorising things such that opportunities are a kind of mental entity, and haircuts are a type of physical entity, or explain away the apparent existential quantification as a mere turn of phrase, but the point is that is it is not straightforward to merely claim that everything is either mental or physical, and any argument which relies on this as a basic assumption inherits all the difficulties associated with it.

However, if I left things like that, then I think I would be seriously misrepresenting their actual argument. In reality, this premise is a product of trying to stick to the wording of what they say in the quoted section above. In the paper, they actually provide a positive argument for why laws of logic have to be considered as thoughts. So we could just change premise 2 to ‘the laws of logic are thoughts’, and have it supported independently by their sub-argument. I will come to their sub-argument, that the laws of logic have to be thoughts, in section 3 below.

In what follows, I will look at three aspects of their argument where I think there are weaknesses. These aspects will be with a) the claim that the laws of logic are necessary (part 2), b) with the inference from intentionality to mentality (part 3), and c) with a modal shift from necessary thoughts to necessary minds (part 4). They are not presented in order of importance, or any particular order.

2. The Necessary Truth Hypothesis

The first premise of the argument as stated above (‘If something is a law of logic, then it is necessarily true’) is ambiguous over the variety of necessity involved. There are several likely contenders for the type of modality involved: epistemic modality, metaphysical modality, logical modality. I consider each in turn.

2.1 Epistemic Modality

Anderson and Welty are clearly not attempting to make an epistemological claim about the status of the laws of logic. They say they are not interested in exploring the epistemological connection between the laws of logic and God (“In this paper we do not propose to explore or contest those epistemological relationships”, p. 1), so I think it is safe to assume that when they say the laws of logic are necessary, they do not merely mean epistemologically necessary.

2.2 Metaphysical Modality

More likely, when Anderson and Welty say the laws of logic are necessary, they mean the laws of logic are metaphysically necessary. They are fairly explicit about this:

“…we will argue for a substantive metaphysical relationship between the laws of logic and the existence of God … In other words, we will argue that there are laws of logic because God exists; indeed, there are laws of logic only because God exists.” (p. 1)

Nonetheless, on this reading, I find the reasons they offer for thinking the laws of logic are necessary rather strange. They say,

“…we cannot imagine the possibility of the law of noncontradiction being false” (p. 6),

And in a footnote they say that they

“…rely on the widely-shared intuition that conceivability is a reliable guide to possibility” (ibid)

The suggestion then is that the reason for thinking that non-contradiction is metaphysically necessary because they cannot imagine true contradictions. I want to bring up three issues with this methodology:

1. Conceivability is often a poor guide to metaphysical possibility
2. The falsity of non-classical laws is conceivable
3. The falsity of excluded middle is conceivable

2.2.1 Metaphysical modality and conceivability

Firstly, in contrast to their ‘widely-shared intuition’, conceivability seems to me to be a relatively poor guide to metaphysical possibility. Ever since Kripke’s celebrated examples of necessary a posteriori truths in Naming and Necessity, the epistemic and metaphysical modalities have been recognised to be properly distinct from one another. One could easily adapt those famous examples to show the independence of metaphysical possibility and conceivability.

For example, one might not be able to conceive of the morning star being identical to the evening star (if you were an ancient Babylonian astrologer, etc), but we now know that their identity is metaphysically necessary. Again, one might be able to conceive of the mind existing without the brain, but it is quite plausible their independence is metaphysically impossible. Kant famously thought Euclidian geometry was a synthetic a priori truth; one must presuppose Euclidean geometry to be true when we think about the world, which would make its falsity inconceivable. Yet our world is non-Euclidian. It took pioneering and brilliant mathematicians to imagine what geometry would be like in this case, but once their work has filtered down into mainstream educated society, this otherwise inconceivable metaphysical truth has become entirely conceivable.

A somewhat similar situation is now the case with non-contradiction. Graham Priest is a very widely respected, if controversial, logician and metaphysician who has argued for the thesis that there are true contradictions. One may disagree with his methodology and conclusions, and I am in no way asserting that dialethism is anywhere as near as well supported as non-euclidian geometry, but it seems odd to rule out all the work on dialethism and paraconsistent logic simply on the basis that one cannot conceive of it being true. It could quite easily be true regardless of your particular inability to conceive of it, as history seems to show.

To push this even further, it is worth noting that conceivability (like epistemic modality, and unlike metaphysical possibility) is agent-dependent, in the sense that what is, and is not, conceivable varies from agent to agent. I may be able to conceive of something you cannot. To take an example of an agent who cannot conceive of a thesis, and then to couple that with the claim that ‘conceivability is a reliable guide to possibility’, seems to be ad hoc. Had we started with someone else’s outlook (say Graham Priest’s), we would be using exactly the same argument to reach the opposite conclusion. The strength of the argument then would depend entirely on the choice of agent.

Anderson and Welty cannot conceive of true contradictions. But should we be consulting their notion of conceivability when trying to draw metaphysical conclusions? If we are going to use conceivability as a guide to metaphysical possibility, we had better make sure we pick an agent who’s idea of what his conceivable is suitable for the job. An agent who’s idea of what is conceivable differed radically from what is in fact metaphysically possible would be unsuitable for that purpose (a five year old child, for example). Ideally,  you would want an agent who’s idea of what is conceivable supervened perfectly on what is in fact metaphysically possible. The extent to which they differed, for some particular agent, is the extent to which conceivability, for that particular agent, is not a ‘reliable guide to (metaphysical) possibility’. Whether something is metaphysically possible could be determined by consulting whether it was conceivable for a given agent only on the assumption that what is conceivable for that agent supervenes on what is in fact metaphysically possible. But this means that what is relevant here is simply whether or not contradictions are in fact metaphysically possible, as this would itself determine whether it was conceivable for that agent; not the other way round. So we have been taken on a long and winding route, via the notion of conceivability, which ultimately is seen to be relevant only to the extent that is maps to metaphysical possibility, to get to this destination.

So, is Anderson and Welty’s inability to imagine what true contradictions would be like actually any kind of evidence that true contradictions are metaphysically impossible? The answer is: only if what they can conceive of matches perfectly (at least with respect to this issue) what is in fact metaphysically possible. We have to assume that they are right for the inference to be seen as valid. And we have been given no reason to think that this is the case. Until we are, we should draw no conclusions about what is metaphysically possible based on what they are able to conceive of. If they could produce some reason to think that what they can conceive of always tracks what is metaphysically possible, or at least successfully tracks what is metaphysically possible in this case, then we would have been given some reasonwe have been given no reason to buy the claim that true contradictions are metaphysically impossible.

There might be other reasons to think that contradictions are metaphysically impossible of course, but they are not mentioned in this paper. So the argument as stated has an unjustified premise, it seems to me.

2.2.2 Conceivability and non-classical laws

In the introduction to their paper, Anderson and Welty attempt to pre-empt a response about alternative laws of logic by saying that their argument is not dependent in any way on the  choice of these particular laws. They say:

Readers who favor other examples [of logical laws – AM] should substitute them at the appropriate points.”

I am not saying we should use any particular laws rather than the ones that they use here either. But I do want to point out that this part of the argument (about the laws being metaphysically necessary) does depend for its plausibility on the choice of laws, in contrast to the claim above. What we are being asked to accept is the inconceivability of the falsity of the laws of logic. I suggest that this far more likely to be considered true if we start with classical laws, than if we had substituted in other non-classical laws at the beginning. For example, would Anderson and Welty be prepared to defend that the falsity of the laws of quantum logic is also inconceivable? Or equally inconceivable as the falsity of the classical laws? The laws of quantum logic may well be true or false (at least from my perspective), and so their falsity is certainly conceivable to me.

Even if it turns out that they are big enthusiasts for quantum logic as well as for classical logic, finding each equally intuitive (which seems unlikely), there will surely be some far-out system of logic which has some law they find down-right implausible, for which its falsity is entirely conceivable to them. Then, their argument would not work if we substituted the laws from these logical systems instead.

This would mean that, to this extent then, their argument is only an argument for the sorts of logical systems they happen to find plausible. Thus, if a logic happens to be the one that God thinks, which also happens to be entirely implausible to Anderson and Welty (for which they find the falsity of its principles entirely conceivable), they would have failed to articulate an argument here which established a route from logic to God.

2.2.3 Excluded middle

The general argument for the laws of logic being metaphysically necessary is that their falsity is inconceivable. Here is Anderson and Welty:

Not only are the laws of logic truths, they are necessary truths. This is just to say that they are true propositions that could not have been false. The proposition that the Allies won the Second World War is a contingent truth; it could have been false, since it was at least possible for the Allies to lose the war. But the laws of logic are not contingent truths. While we can easily imagine the possibility of the Allies losing the war, and thus of the proposition that the Allies won the Second World War being false, we cannot imagine the possibility of the Law of Non-Contradiction being false. That is to say, we cannot imagine any possible circumstances in which a truth could also be a falsehood.” (p. 6, emphasis mine)

It is telling that Anderson and Welty use the law of non-contradiction as their example here, as it is admittedly rather difficult to get one’s head around the idea of it being false (none other than David Lewis famously claimed not to be able to do so).

However, this reasoning does not really work for the law of excluded middle. What we have to do to imagine that this is the case is to imagine that there is a proposition for which neither it nor its negation is true. Aristotle makes various comments in De Interpretione IX, which he (seems to) make an argument according to which statements about the future concerning contingent events, such as ‘Tomorrow there will be a sea battle’, should be considered neither true nor false. It follows from this that the law of excluded middle would be false, at least for future contingents such as this. There is controversy as to whether Aristotle was making this argument, with the issue being one of the longest logico-metaphysical debates in the history of philosophy (being discussed by Arabic logicians, medieval logicians, and modern logicians), and there is nothing like a consensus that Aristotle was correct in making this argument, if indeed he was actually making it. However, the thesis he was putting forward (that future contingents are neither true nor false) is clearly conceivable by a great many philosophers. Indeed, it is a textbook philosophical position.

So the argument was that the laws of logic are metaphysically necessary, and the support for this is that the falsity of the laws of logic is inconceivable. Yet, while it is perhaps true for the law of non-contradiction, this seems plainly false for the law of excluded middle. It is patently conceivable that it is false. Thus, the support for the laws of logic being metaphysically necessary only covers two of the three laws they themselves provide.

If we were to respond by dropping excluded middle just to get around this problem, that would be ad hoc. To respond to this, they should explain how the falsity of excluded middle is in fact inconceivable, or provide another reason for thinking that it is metaphysically necessary.

2.3 Possible worlds

Anderson and Welty attempt to provide additional support for the metaphysical necessity of the laws of logic by asserting the laws of logic are true in all possible worlds. Again, leaning heavily on the notion of conceivability, they say:

[w]e cannot imagine a possible world in which the law of noncontradiction is false…Now you may insist that you can imagine a possible world—albeit a very chaotic and confusing world—in which the Law of Non-Contradiction is false. If so, we would simply invite you to reflect on whether you really can conceive of a possible world in which contradictions abound. What would that look like? Can you imagine an alternate reality in which, for example, trees both exist and do not exist?” (p. 6).

Firstly, for the law of non-contradiction to be false, there only has to be one true contradiction, and it is not required that contradictions ‘abound’. I think I could conceive of a possible world where there is a contradiction; and it might be the actual world. Perhaps the liar sentence (‘this sentence is false’) is an example. Maybe in the actual world everything else is classical apart from the liar sentence. If so I have conceived of a world in which the law of non-contradiction is false. This does not mean that ‘contradictions abound’, and we do no have to imagine trees both existing and not existing. I seem to have met their challenge.

Remember, I do not have to show that the liar sentence is in fact both true and false at the actual world. All I have to do is be able to conceive of a world in which the law of non-contradiction is false. It seems to me that, given the work of dialethists on this area, it is conceivable.

Perhaps sensing the need for further argument, they say that contradictory worlds cannot be conceived of, because possible worlds are by definition consistent:

The criterion of logical consistency—conformity to the law of noncontradiction—is surely the first criterion we apply when determining whether a world is possible or impossible. A world in which some proposition is both true and false, in which some fact both obtains and does not obtain, is by definition an impossible world. The notion of noncontradiction lies at the core of our understanding of possibility.” (p. 6 – 7)

This passage is quite hard to interpret. However, Anderson and Welty seem to argue in a circle. They seem to think non-contradiction is necessary because inconsistent possible worlds are inconceivable. But the only reason they give for thinking inconsistent worlds are inconceivable is, by definition, we use consistency as a sort of yard-stick to ‘determine’ whether a given world is indeed possible. Thus, laws of logic are necessary because they are true in all possible worlds, but laws of logic are true in all possible worlds because the laws of logic are necessary.

I think the direction of travel from possible worlds to possibilities is misguided. Anderson and Welty appear to be under the impression there is some metaphysically significant sense in which we can check possible worlds to see if they really are possible or not; as if possible worlds were conceptually prior to possibilities. The picture painted is that there is a sort of a priori rationalistic access we have to the set of possible worlds which we can consult in order to find out about what is really possible. This idea is actually warned against by Kripke in Naming and Necessity. There he argues against the identification of a prioricity and necessity:

I think people have thought that these two things [a prioricity and necessity – AM] must mean the same of these reasons: … if something not only happens to be true in the actual world but is also true in all possible worlds, then, of course, just by running through all the possible worlds in our heads, we ought to be able with enough effort to see, if a statement is necessary, that it is necessary, and thus know it a priori. But really this is not so obviously feasible at all.” (p. 38)

It also seems to fly in the face of Kripke’s famous telescope remark:

“One thinks, in this picture, of a possible world as if it were like a foreign country. … it seems to me not to be the right way of thinking about the possible worlds. A possible world isn’t a distant country that we are coming across, or viewing through a telescope.… A possible world is given by the descriptive conditions we associate with it” (Kripke,Naming and Necessity, p 43-44).

I think, apparently in contrast to A&W, possible worlds are just a way of cashing out our notion(s) of possibility. If we are thinking about what is logically possible (with classical logic in mind), then when constructing the possible worlds we make sure to get them consistent (to keep non-contradiction) and also maximal (to keep the law of excluded middle). So a truth assignment for a formula in classical propositional logic is a ‘possible world’, so long as the truth assignment covers all cases and gives each formula only one truth value.

However, different notions of logical consequence lead to different constructions of worlds. In intuitionist logic, where we want to have mathematical propositions for which there is no formal proof to be neither true nor false, the ‘possible worlds’ (or ‘constructions’) are not maximal. They may simply leave both p and not-p out altogether. Equally, for a dialetheist who believes there are true contradictions in the actual world, where both p and not-p are true, the notion of ‘possible world’ leaves out the notion of consistency (or, if you prefer, the dialetheist includes both possible worlds and ‘impossible worlds’ in his semantics). In the actual practice of formal and philosophical logic, one normally starts with a notion of logical consequence (or of ‘laws’) and then uses logical consequence to cash out what the appropriate semantic apparatus will be like. On this understanding (the usual understanding), one cannot use the fact that maximal and consistent possible worlds do not have contradictions to tell us which logical laws to accept as true, as we need an idea of which logical laws to accept prior to accepting anything about possible worlds. So the circularity of A&W’s reasoning here is completely avoidable. They just need to appreciate the role possible worlds semantics plays in philosophical logic. If they were able to see the restrictions they put on possible worlds (maximal, consistent, etc) are not mandatory, they would be able to more readily conceive of how a possible world could be inconsistent or non-maximal. Anderson and Welty appear to resemble the 17th century geometer who cannot imagine parallel lines ever meeting and concludes the meeting of parallel lines is metaphysically impossible. Thus, Anderson and Welty’s failure to imagine what non-classical worlds would be like seems to be a limitation on their part and should not be used as a support for their argument.

In sum, Anderson and Welty provide two reasons for thinking LOL are metaphysically necessary: (i) their falsity is inconceivable and (ii) they are true in every possible world. We have shown (i) provides flimsy support for their subconclusions and (ii) is based on several confusions concerning philosophical logic and possible worlds.

2.4 Logical Necessity

Finally, the claim could instead be read as saying the laws of logic are logically necessary truths. In some sense, one cannot deny the laws of logic are logically necessary truths, but this sense is trivial. Usually, the claim that p is logically necessary, with respect to a system S, simply means the truth of p does not violate any logical law of S. When p is an instance of a logical law of S, the claim becomes vacuous. If we said ‘p is chessessary’ means ‘the truth of p does not violate any of the laws of chess’, then, provided p is one of the laws of chess, obviously, p is chessessary. The claim, while true, is trivial. The necessary truth of laws of logic, if construed as logical necessity, is not a substantive claim, such as that associated with the necessary truth of the existence of platonic objects, or of God. Logical necessity is more like the way that statements about numbers depend on which number system you have in mind; is there a number between 1 and 2? No, if you mean ‘natural number’, yes if you mean a more complex notion of number. To ask ‘but is there really a number there?’ is arguably not a sensible question at all. If this is correct, then there may be no more to the notion of logical necessity than ‘necessary given system S’, and as such each logical law is true in its own system and (in general) is not in another system.

In sum, Anderson and Welty claim that the laws of logic are necessary truths. They do not seem to be making a claim about epistemological necessity; their arguments for a claim about metaphysical necessity are highly dubious; the claim that it is about logical necessity makes it vacuous. Thus, either this part of the argument is unsupported, or trivial.

3. Propositions are intentional

The most controversial aspect of Anderson and Welty’s argument is the move from the laws of logic being propositions, through them being intentional, to them being mental (or thoughts). In order to see what is at stake here, we need to be clear about both intentionality and propositions.

Anderson and Welty’s argument at this stage seems to be of the following form:

1. All propositions are intentional.
2. Everything intentional is mental.
3. Therefore, all propositions are mental.

This little argument is clearly valid, so if the premises are also true, we would have to accept the conclusion.

I think there are reasons to doubt both premises. More specifically, there is reason to doubt that the arguments presented in Anderson and Welty’s paper support these premises.

3.1 Intentionality

The central idea behind intentionality is aboutness. Typical examples of intentional things are thoughts. So if I have a thought, it is always a thought which is about something, and it seems that there couldn’t be a thought which is not about anything. The typical philosophical authority referred to in this context is Brentano:

“Every mental phenomenon is characterized by what the Scholastics of the Middle Ages called the intentional (or mental) inexistence of an object, and what we might call, though not wholly unambiguously, reference to a content, direction towards an object (which is not to be understood here as meaning a thing), or immanent objectivity. In presentation something is presented, in judgement something is affirmed or denied, in love loved, in hate hated, in desire desired, and so on.” (Psychology from an empirical standpoint, Franz Brentano, 1874, p 68)

It has become customary to call the following claim ‘Brentano’s Thesis’:

x is intentional iff x is metnal

As this is a biconditional claim, it can be split into two conditionals:

1. Everything intentional is mental
2. Everything mental is intentional

It is standard for philosophers to argue that there are mental states which are non-intentional (Searle’s example is a vague an undirected feeling of anxiety), and thus that the second condition in Brentano’s thesis is false.

Anderson and Welty say that they are really concerned with the first of these conditions, and that

“…the argument is unaffected if it turns out that there are some non-intentional mental states” (p. 17)

What they need to do is show that there is nothing which is both intentional and non-mental. There seem to be counter-examples here though. Firstly, sentences of natural language seem to be intentional, in that they are about things. The sentence ‘Quine was a philosopher’ is about Quine. Yet that sentence is not itself mental. I can think about the sentence, of course, but the sentence itself is not mental.

The common response to this is to say that sentences are only derivatively intentional. On their own sentences are not about anything, but when read by a mind they become invested with meaning and this makes them about something. Sentences are just non-intentional  vehicles for communicating intentional thoughts. Anderson and Welty want to say that, while there may be instances of derivatively intentional phenomena (like sentences), anything which is inherently intentional is mental.

There are other approaches which hold that there are inherently intentional non-mental phenomena, such as that of Fred Dretske, according to which intentionality is best understood as the property of containing information. So an object is intentional if it contains some information. The content of the information is what makes the object about something else. So, an example is that there is no smoke without fire. In this sense, the smoke contains information about the presence of fire. Other examples stated on the Stanford page include:

A fingerprint carries information about the identity of the human being whose finger was imprinted. Spots on a human face carry information about a disease. The height of the column of mercury in a thermometer carries information about the temperature. A gas-gauge on the dashboard of a car carries information about the amount of fuel in the car tank. The position of a needle in a galvanometer carries information about the flow of electric current. A compass carries information about the location of the North pole.

All these objects are not mental, yet they carry information about things, and so are intentional in Dretske’s sense of the word. If this approach is correct, then Anderson and Welty’s inference is blocked (as there are things which are non-mental yet intentional), and with it the rest of the argument is blocked. You could not argue from the laws of logic being propositions, to them being intentional, to them being thoughts, to them being the thoughts of God. The jump from being intentional to being mental would be invalid if Dretske’s approach, or one like it, were correct.

There are problems with Dretske’s account of intentionality, as you would expect from a philosophical theory, but if Anderson and Welty want to advance the thesis that all intentional things are mental, they need to provide counter-arguments to proposals such as Dretske’s.

3.1.1 The mark of the mental

In fairness, Anderson and Welty do point to a paper by Tim Crane, about which they claim:

Following Brentano, Crane argues (against some contemporary philosophers of mind) that intentionality, properly understood, is not only a sufficient condition of the mental but also a necessary condition” (p. 17, footnote)

If this were right, then they would have some support for their claim that everything which is intentional is mental. However, I think they are using Crane to argue for a thesis that his paper does not support, and I will explain why I think this.

Crane’s main concern in his paper is to deal with intentionality being a necessary condition for being mental (i.e. that everything mental is intentional). The sufficiency claim (that everything intentional is mental), which is the only thing that Anderson and Welty are really concerned with for their argument, is only tangentially addressed by Crane in that paper. Crane’s motivation, as he explains, is to account for why Brentano would have asserted his thesis if there were so many seemingly obvious counter-examples to it:

If it is so obvious that Brentano’s thesis is false, why did Brentano propose it? If a moment’s reflection on one’s states of mind refutes the thesis that all mental states are intentional, then why would anyone (including Brentano, Husserl, Sartre and their followers) think otherwise? Did Brentano have a radically different inner life from the inner lives of contemporary philosophers? Or was the originator of phenomenology spectacularly inattentive to phenomenological facts, rather as Freud is supposed to have been a bad analyst? Or—surely more plausibly—did Brentano mean something different by ‘intentionality’ than what many contemporary philosophers mean?“(Crane, Intentionality as the mark of the mental, p. 2)

He says that he is not specifically interested in the historical and exegetical question of what Brentano and his followers actually said, but rather with the following question:

“…what would you have to believe about intentionality to believe that it is the mark of the mental?” (Crane, Intentionality as the mark of the mental, p. 2)

Thus, when Crane talks about ‘intentionality’, we should remember that he does not mean “what many contemporary philosophers mean” by the term. Rather, he has a specific aim in mind: to cash out what intentionality would be like if it was, by definition, the ‘mark of the mental’, i.e. not what intentionality is like, but what it would be like if Brentano’s thesis was true.

Most of the paper is directed at supposed examples of mental phenomena that are non-intentional, such as sense perception and undirected emotion. He gives an account of what it would mean to consider these as intentional. This effort is being addressed to defend the first part of Brentano’s thesis (that everything mental is intentional).

Although the focus of the paper is on the first part of Brentano’s thesis, Crane does directly confront the second part, i.e. the notion that everything intentional is mental:

I have been defending the claim that all mental phenomena exhibit intentionality. Now I want to return to the other part of Brentano’s thesis, the claim that intentionality is exclusive to the mental domain. This will give me the opportunity to air some speculations about why we should be interested in the idea of a mark of the mental.” (Crane, Intentionality as the mark of the mental, p. 14)

Crane addresses the Chisholm-Quine idea that sentences are intentional and non-mental phenomena. Chisholm (1957) proposed a criterion whereby we can tell if a sentence is intentional or not, which is basically if it is used in non-extensional (i.e. in intensional) contexts. Crane calls this the ‘linguistic criterion’. In response to this, Crane recommends that the position he is defending (intentionalism) should reject the linguistic criterion altogether. I will quote his reasons for recommending such a position in full:

“And given the way I have been proceeding in this paper, [the rejection of the linguistic criterion] should not be suprising. Intentionality, like consciousness, is one of the concepts which we use in an elucidation of what it is to have a mind. On this conception of intentionality, to consider the question of whether intentionality is present in some creature is of a piece with considering what it is like for that creature—that is, with a consideration of that creature’s mental life as a whole. To say this is not to reject by stipulation the idea that there are primitive forms of intentionality which are only remotely connected with conscious mental life—say, the intentionality of the information-processing which goes on in our brains. It is rather to emphasise the priority of intentionality as a phenomenological notion. So intentionalists will reject the linguistic criterion of intentionality precisely because the criterion will count phenomena as intentional which are clearly not mental.” (Crane, Intentionality as the mark of the mental, p. 15)

Thus we can see here that Crane rejects the criteria by which one says that some sentences are intentional, not because sentences are only ‘derivatively’ intentional, but “precisely because the criterion will count phenomena as intentional which are clearly not mentalUltimately, on Crane’s picture of intentionality, sentences are not intentional because they are not mental.

When it comes to propositions, it is actually quite controversial and non-standard to consider propositions to be mental (i.e. to be thoughts). Just like sentences, they are usually considered to be intentional (in the standard sense, in that they are about things) yet not mental. Anderson and Welty point to Crane as someone who has defended the thesis that everything intentional is mental. Yet, when we come to consider Crane’s special sense of intentionality, we see the author recommending that we should resist applying it to propositions just because we would end up classifying “phenomena as intentional which are clearly not mental“. Crane doesn’t deduce mentality from things that are otherwise obviously intentional; rather he ensures that everything intentional is mental by restricting the application of intentionality to only things which are obviously mental. It is a recommendation to change the meaning of intentional to get the desired result. If Anderson and  Welty want to say that the reason they have for claiming that propositions are mental is that they are intentional in Crane’s sense, then it is doubtful that this is true. It is doubtful that propositions are intentional in this sense precisely because they are not obviously mental. We could only use Crane’s sense of intentionality if we already thought that propositions were mental. Prima facie, it seems that are only as intentional as sentences, and if sentences are deemed non-intentional for Crane, then so should propositions. Thus, I see no benefit for Anderson and Welty for pointing us in the direction of Crane here.

4. Modal shift

Let’s say we grant that the laws of logic are (metaphysically/logically) necessary, and that they exist in every (metaphysically/logically) possible world. Let’s also grant that they are inherently intentional, and that they are therefore thoughts. What we would have established at this juncture is that there are some necessarily existing thoughts, which are constitutive of the laws of logic (and all other metaphysically necessary propositions). From this, Anderson and Welty draw the conclusion that this implies the presence of a divine mind:

But now an obvious question arises. Just whose thoughts are the laws of logic? There are no more thoughts without minds than there is smoke without fire … In any case, the laws of logic couldn’t be our thoughts—or the thoughts of any other contingent being for that matter—for as we’ve seen, the laws of logic exist necessarily if they exist at all. For any human person S, S might not have existed, along with S’s thoughts. The Law of Non-Contradiction, on the other hand, could not have failed to exist—otherwise it could have failed to be true. If the laws of logic are necessarily existent thoughts, they can only be the thoughts of a necessarily existent mind.” (p. 19)

So the inference from thoughts to a mind is as follows:

1. There are no thoughts without minds.
2. Necessarily there are thoughts.
3. Therefore, necessarily there is a mind.

The scope of the necessity claim in the conclusion needs to be cashed out properly, for us to be able to judge whether the inference is valid. The precise logical form of the argument is not entirely clear to me, but here is my best shot:

1. (∃x (Tx) → ∃y (My))    (If there is a thought, then there is a mind)
2. (∃x (Tx))                     (Necessarily, there is a thought)
3. (∃x (Mx))                   (Therefore, necessarily, there is a mind)

This argument follows, as it requires nothing but modus ponens, and the closure of necessity with respect to the theorems of propositional logic. The problem is that 3 is a de dicto necessity, where Anderson and Welty presumably want to have a de re necessity. They presumably want the conclusion to be that there is something that is a necessary mind (de re necessity), rather than it being necessary that there something which is a mind (de dicto necessity).

Here is an illustration of the difference between them. It is necessary that there is someone who is the oldest person alive. Say someone, let’s call them Raj, is the oldest person alive. It is not necessary of Raj that he is the oldest person, because he could die and the title of oldest person would pass to someone else. It is necessary that someone has the title (at least so long as there are people), but there is nobody of whom it is necessary that they have the title.

A&W want to say that there is a mind (God’s mind) of which it necessarily exists, which is a de re claim, and not just that it is necessary that some mind or other exists, which is a de dicto claim. The difference is between (∃x (Mx)) (‘It is necessary that there is a mind’), and (∃x (Mx)) (‘There is a necessary mind’).

If we change their argument to put the de re conclusion in that they want, it becomes the following:

1. (∃x (Tx) → ∃y (Mx))
2. (∃x (Tx))
3. (∃x (Mx))

The problem is that 3 does not follow from 1 and 2. For an illustration of the counterexample (where premise 1 and 2 are true, but this de re reading of the conclusion is false), consider the following:

It may be that each possible world has its own unique mind, which thinks the laws of logic. This would mean that premise 1 is true, as whenever there is thought, there is a mind; and it would mean that premise 2 is true, as there is thought that exists in every possible world  (specifically, the laws of logic). However, on this model, no mind exists at more than one world; each logic-thinking mind is contingent. So, ‘(∃x (Mx))’ is true, in that at every world there is a mind, but ‘(∃x (Mx))’ is false, in that there isn’t a thing which is a mind in every world.

Anderson and Welty do anticipate this response:

It might be objected that the necessary existence of certain thoughts entails only that, necessarily, some minds exist. Presumably the objector envisages a scenario in which every possible world contains one or more contingent minds, and those minds necessarily produce certain thoughts (among which are the laws of logic). Since those thoughts are produced in every possible world, they enjoy necessary existence.” (p. 19, footnote 31)

This is essentially exactly the issue laid out above. They are saying that the inference to the de dicto conclusion might be seen as invalid, on the basis of a model in which there are multiple contingent minds. This is how my counter-example above worked; it involved each world having its own unique contingent mind.

They have two responses to such a move:

One problem with this suggestion is that thoughts belong essentially to the minds that produce them. Your thoughts necessarily belong to you. We could not have had your thoughts (except in the weaker sense that we could have thoughts with the same content as your thoughts, which presupposes a distinction between human thoughts and the content of those thoughts, e.g., propositions). Consequently, the thoughts of contingent minds must be themselves contingent. Another problem, less serious but still significant, is that this alternative scenario violates the principle of parsimony.” (p. 19-20, ibid)

To begin with we have the claim that “thoughts belong essentially to the minds that produce them“. So I have this particular thought about how lovely the weather is today. While you may also be thinking that the weather is lovely today, you are not literally having the same thought as me; rather you are having a different thought, even if it has the same content. Thus, this thought is had by me (and only me) in every world in which it exists. So being a thought of mine is an essential property of that thought. Because I am a contingent being, and do not exist in every possible world, it follows that there are worlds in which my particular thought about how lovely the weather is today also does not exist. Thus, given that thoughts are essentially of the minds that think them, contingent beings can only have contingent thoughts.

I am quite sympathetic to this response. It seems right to me that contingent beings can only have thoughts that are contingent too. While the content of my thought can be necessary, the thought itself cannot be. The counterexample above does seem to require there being contingent minds. Thus, in order for the thought to have the necessity required, the mind also has to be necessary.

However, while I find all this quite agreeable, there still seems to be a problem here, although I do find this quite hard to put into words completely clearly, and maybe it is something that could be cleared up with a little more detail on the ontology of how the laws of logic relate to God’s thoughts on A&W’s part. Anyway, here is how I see it.

The distinction between the thought and the content of the thought is that the former cannot be shared across minds (I cannot have the same thought as you), while the latter can be (I can have a thought with the same content as yours). This, it seems to me, generates a little problem for the divine conceptualist. It seems like the categories of thought and content are mutually exclusive; if I think of my coffee mug, then the thought is not the content of the thought. If I think about the thought I just had about the coffee mug, then my previous thought (about the mug) is the content of a new thought (about the thought about the mug). It seems unintelligible that one and the same thought could be the content of itself. Self-reflection, it seems, is hierarchical, not circular. Call this ‘the principle of the Distinctness of Thought from Content‘ (or PDTC). If PDTC is true, then it is impossible for a thought to be the content of itself.

Of course, there is the discussion in Metaphysics about God being thought that thinks thought. The idea is that God, the pure actuality, has to be thinking which has itself as it’s own object of thought. Aristotle seems to anticipate something like the PDTC, when he says the following:

“[God’s] Mind thinks itself, if it is that which is best; and its thinking is a thinking of thinking.

Yet it seems that knowledge and perception and opinion and understanding are always of something else, and only incidentally of themselves. And further, if to think is not the same as to be thought, in respect of which does goodness belong to thought? for the act of thinking and the object of thought have not the same essence.

The answer is that in some cases the knowledge is the object. In the productive sciences, if we disregard the matter, the substance, i.e. the essence, is the object; but in the speculative sciences the formula or the act of thinking is the object. Therefore since thought and the object of thought are not different in the case of things which contain no matter, they will be the same, and the act of thinking will be one with the object of thought.” (Aristotle, Metaphysics, book 12, 1074b-1075a)

So the claim is that the divine mind thinks itself. Then in the second paragraph the objection is posed that thoughts are always about something distinct from themselves. The ‘answer’ provided by Aristotle is that “in the speculative sciences the formula or the act of thinking is the object”. Logic certainly counts as an example of a speculative science (par excellence), and so it seems that Aristotle’s claim is that when God thinks about logic, his thought is identical to the object of the thought. If this is the case, Aristotle presents no argument for it (at least not that I know of). And it seems quite strange, if taken to be the claim that when one thinks about logic, the thought is the content of the thought. It seems quite clear that when I think of the laws of logic, they are the content of my thought, and not the thought itself.

Here is an argument for my claim:

1. If p can be thought by a mind and a mind m’ , where m ≠ m’, then p is the content of their thought. (Contents of thoughts can be shared by minds)
2. If t is a thought had by m, then t cannot be had by any mind m’, where m ≠ m’. (Thoughts cannot be shared by minds)
3. Two people can both think of the law of non-contradiction.
4. Therefore, the law of non-contradiction can be the content of thoughts. (from 1 + 3, modus ponens)
5. Therefore, the law of non-contradiction cannot be a thought. (from 2 + 4, modus tollens)

The first two premises of this argument make the distinction between thought and contents of thoughts made by A&W above, and the third just says that two people can both think the LnC. It follows that the LnC cannot be a thought.

For the divine conceptualism of A&W, the law of non-contradiction is ultimately supposed to be God’s thought. So take the law of non-contradiction, ‘LnC’, and some thought had by God, T. If LnC = T, then (by the PDTC) it is not the content of T. But what is the content of T? What is God thinking about when he has the thought T which is the law of non-contradiction? The obvious answer would be that God is thinking about propositions, and how each proposition cannot be true along with its negation. But the problem with that is that it is the law of non-contradiction. That would make the LnC the content of T, and (if thoughts cannot be their own content) that would mean that T isn’t LnC. So when God thinks T, he must think about something other than the LnC.

But why is it then that T is LnC, if the content of T is something other than that propositions cannot be true with their negations? Nothing else is relevant! It seems incredible to consider that the content of T is (say) this coffee mug, while also insisting that T is the LnC. If the content of T, whatever it is, is not the mutual exclusivity of propositions and their negations, then it can only be arbitrarily connected with LnC. This makes it a mystery, ultimately, why it has anything to do with LnC, let alone being the LnC.

The question is: in virtue of what could a thought T, whose content is irrelevant to the LnC, be said to be the LnC?

There are three ways out of this problem, it seems to me.

One is to bite the bullet and say that God thinks something with completely arbitrary content, and this just is the LnC. It is a hard pill to swallow.

The next escape route would be to say that the LnC is in fact the content of T. This explains why it is that I can also think about LnC; both me and God think about the same thing. However, this option is rather like the horn of the Euthyphro dilemma that says that God likes good actions because they are good. If God has a thought which has LnC as its content, then the LnC is not to be associated with God’s thought any more than it is if I have a thought with the LnC as its content. The significance of God in the equation has been completely removed. It seems that the central claim of a divine conceptualist has been undermined if we take this route.

The only other escape route I can see here is to deny that LnC cannot be both T and the content of T. Perhaps when it comes to God’s thoughts, they can be both thought and content together. So the LnC is the content of God’s thought (i.e. he is thinking about how propositions and their negations cannot both be true) and that this thought is the law itself. It may seem unintelligible for us humans to have such a thought, but maybe this is how God thinks.

The problem with this route, it seems to me, is that it undermines the analogy between divine thoughts and mere human thoughts. When the divine conceptualist says that laws of logic are divine thoughts, we take it that the claim is saying that they are thoughts that are at least a somewhat similar to human thoughts. This seems to be required for the argument from propositions being intentional in section 3 (above). Propositions don’t seem to be mental on their face, but the idea is that they are because they are intentional, and everything intentional is mental. This last claim is undermined significantly if the extension of ‘mental’ includes things which are significantly unlike human thoughts. To the extent then that we have to broaden the category of thoughts to include the seemingly unintelligible idea of a thought being at once its own content, the universal claim is also undermined. Consider the claim spelled out in full:

“Everything intentional is mental, and and under the term ‘mental’ I include things which are very unlike human thoughts because they have properties which are unintelligible if applied to human thoughts (such as a human thought which is its own content)”

Where we have arrived at, is a destination where the central claim of the divine conceptualist is that the laws of logic are to be associated with some aspect of God, which in some sense resembles human thoughts, but that in another sense is nothing like human thoughts. Saying that the laws of logic are thoughts at all on this picture seems quite a difficult thing to maintain.

5. Conclusion

It seems to me that there are quite a few problems with the argument presented in The Lord of Non-Contradiction. Some of them are quite subtle, like the final one concerning the precise relationship between the laws and the thoughts of God, and it is entirely possible that they could be cleared up. Some of them are quite technical, such as the details of how possible worlds are cashed out in the metaphysics of modality, and A&W could be forgiven for not realising them. Some of them, I suggest, are quite a lot more serious, such as the inference from intentionality to mentality. I don’t see this being fixed up with a little revision or by spelling something out a bit more clearly. It is utterly foundational to the argument and it seems to me that it is just fallacious.

## Accounting for logic – again

0. Introduction

In this post I will be looking at a blog entry on the BibleThumpingWingnut website, entitled ‘Christianity and Logic’. The entry is written by Tim Shaughnessy, and takes a Clarkian angle. Shaughnessy’s argument is basically that Christianity can provide an ‘epistemological foundation’ for logic, using Scripture as a sort of axiomatic basis for logic, and that ‘the unbeliever’ cannot provide such a foundation, or ‘account’, for logic. If this is the first time you are encountering this Clarkian view, have a look at this article by Clark. I have written on this topic before, and I think that many of those points are directly relevant here.

For instance, here I argue that there is no binary choice between Christianity and non-Christianity; there are different versions of Christianity, different monotheistic religions, different versions of theism, and different versions of atheism. This version of Christianity is just one tiny dot on a huge intellectual landscape. To argue by elimination that this version Christianity is correct, means you have to eliminate a possibly infinite variety of systems. Pitting (this version of) Christianity against ‘the unbelieving worldview’ is already to commit the fallacy of false dichotomy. We might want to call this version of it the ‘Bahnsen fallacy’, in honour of its main witness.

More specifically with regards to the broadly Clarkian idea of deriving logical principles from the Scriptures, I have argued here that this is incoherent. Derivation requires a logical framework, which is constituted in part by logical principles (or axioms); derivation is a logical notion, and thus presupposes logical principles.

There are some new points which seem to be worth raising however, given the particular presentation by Shaughnessy, and so I will be exploring those ideas here.

1. ‘What is logic?’

Shaughnessy’s view of logic seems to be entirely gained from the study of Clark, in that he is the only author cited (rather than, say, Aristotle or Frege) on the topic of what logic is. This is unfortunate, because it seems that  Shaughnessy is unaware of the controversy surrounding the topic. So, we see him state that logic is “the correct process of reasoning which is based on universally fixed rules of thought”. This idea, that logic is about laws of thought, is a historically significant idea, coming to prominence in the 18th and 19th centuries, but it has never been a universal consensus among logicians and philosophers. These days it is not widely represented among practising logicians and philosophers at all (see this for a quick overview). The reason for this is that in the contemporary setting logic has a much broader extension, and can cover systems which deviate wildly from how we might realistically model thought (which is the preserve of logicians and computer scientists working in artificial intelligence). Logic, thought of broadly as concerning valid inference for various types of argument forms, is not considered to be tied in any special manner to how we think. There may be a logic to how we think, but logic is not just how we think. Never-the-less, Shaughnessy makes no mention of this, and simply asserts that logic has this 18th century relation to cognition.

His out-of-date description of logic becomes confounded with outright misunderstandings when he spells out what he considers to be the three laws of thought. It is utterly standard, when going down this non-modern view, to list the three laws of thought as: ‘the law of identity’, ‘the law of non-contradiction’ and ‘the law of excluded middle’. What is odd is the way these are cashed out by Shaughnessy. For instance, the law of non-contradiction is cashed out as “A is not non-A”, and the law of excluded middle is cashed out as “A is either B or non-B”. It seems to me that there is a failure of Shaughnessy to distinguish clearly between different aspects of vocabulary. There is a fundamental difference between logical vocabulary that refers to things directly (like ‘Alex’, ‘London’, ‘your favourite type of ice cream’, etc) and those which express facts (‘Alex is in London’, ‘vanilla is your favourite type of ice cream’, etc). The first are called ‘terms’, and the latter are called ‘propositions’. Propositions can be thought of as made up of terms standing in certain relations to one another. Crucially, propositions are given truth-values, true or false; terms are not. So, ‘Alex’ isn’t true or false; but ‘Alex is in London’ is either true or false. In Shaughnessy’s expression of the law of non-contradiction, we have a letter ‘A’, which seems to be a term, as it is something we are predicating something to, but then the predicate we are ascribing to it is that it is “not non-A”. The problem is that we have a negation fixing to a term, ‘non-A’. As I have pointed out before, negation is a propositional operator, and its function is to switch the truth-value of the proposition is prefixes from true to false (or vice versa). If we prefix it to a referring term, like ‘A’, then (because terms don’t have truth values), the resultant operation is undefined.

The conventional way to express the law of non-contradiction is with a propositional variable, ‘p’, which ranges over all propositions, as follows:

¬(p ∧ ¬p)    (‘it is not the case that both p and not-p’)

If you want to express this using propositions where the relation of terms is explicit (i.e. in a first-order manner), then it would be as follows, where ‘Px’ is a predicate and ‘a’ is a term:

¬(Pa ∧ ¬(Pa))   (‘it is not the case that a both is and is not P’)

The same problem infects “A is either B or non-B”. The correct way to express this is just that for every proposition, either it is true, or it’s negation is true:

p ∨ ¬p     (‘either p or not-p‘)

It is bizarre to say that either ‘A is B or non-B’. There is no predicate ‘non-B’; rather, either B applies or it doesn’t. Take the proposition that I am 6 feet tall. Either I am 6′ or I am not. In the second case I don’t have a property, called non-6′. What would this property be? Every height other than 6′? I am not 6′, but I am also not every height other than 6′. I just am 5’11”. So the way Shaughnessy expresses excluded middle is also confused.

And it’s not like stating non-contradiction and excluded middle is extremely complicated; all it involves is: ‘p or not-p’, and ‘not both p and not-p’. He hasn’t simplified them for a non-specialist audience – he has just misrepresented them.

So we have an out-of-date view of logic, coupled with a technically incorrect presentation of the principles under discussion. It’s not a great start to an article about the nature of logic.

1.1 Logic in the Bible?

Perhaps Shaughnessy’s misrepresentation of the basic laws of thought is more understandable when we see where he is going with all of this. The ultimate point he will be driving at is that these laws are found in the Bible. Various snippets of the Bible are then presented as evidence of this, but because they don’t really fit that well with the laws when expressed properly, he has written them in such a way that the claim that they are found in the Bible becomes (slightly) easier to swallow. Here is what he has to say about it:

The law of non-contradiction (A is not non–A) is an expression of the eternal character and nature of God, “for he cannot deny [contradict] himself” (2 Tim. 2:13). The law of identity (A is A) is expressed in God’s name, “I AM WHO I AM” (Exodus 3:14), and the law of the excluded middle (A is either B or non-B) is expressed in Christ’s own words, “He who is not with Me is against Me” (Luke 11:23).

Let’s take these one at a time. It is hard to take them seriously, but I will try.

In the book of Timothy, it is said that God cannot contradict himself. I say that this is completely irrelevant to the principle of non-contradiction. There is a difference between saying things, and things being true (or false). The law of non-contradiction is about the latter, not the former. It isn’t a rule which says ‘thou shalt not contradict thy self’. It says that there is no proposition for which both it and its negation are true. It doesn’t proscribe what you can or cannot say at all.

For example, I can contradict myself, and sometimes do. Does this mean I broke the law of non-contradiction when I did so? No, of course not. Imagine I say ‘It is sunny now, at 14:07’, and then a few minutes later, ‘It was not sunny then, at 14:07’. The two sentences I uttered were expressing (from different times) that it was and was not sunny at 14:07. Obviously, it would be a contradiction if both of these were true, as p and not-p would both be true (exactly what the law of non-contradiction forbids). But were they both true? That would mean that it was both sunny and not sunny at the same time. Conventionally thinking, this is impossible. Therefore, while I contradicted myself, I didn’t break the law of non-contradiction. I expressed a true proposition, and then when I uttered the negation of that proposition what I said was false (or vice versa). Contradicting yourself isn’t a case of breaking the law of non-contradiction.

Back to the Biblical example, God cannot contradict himself. So what? The law of non-contradiction is true even though people can contradict themselves. An example of a being, even an infinite one, who cannot contradict themselves, is not an example of the law of non-contradiction. To think that it is, is to mix up the idea of saying two contradictory things with two contradictory propositions both being true.

1.1.2 Identity

Shaughnessy does manage to state the law of identity correctly, which is that (for all referring terms) A = A. Everything is identical to itself. According to the example given, the law of identity is expressed in “I am who I am”, which is the answer God gives to Moses in the book of Exodus. It has always baffled me as to why this has been seen as a profound thing for God to say here. God tells Moses to go to the Pharaoh and bring the Israelites out of Egypt. Moses basically says, ‘who am I to do that?’ God says that he will be with Moses, but Moses wants a bit more reassurance for some reason:

Moses said to God, “Suppose I go to the Israelites and say to them, ‘The God of your fathers has sent me to you,’ and they ask me, ‘What is his name?’ Then what shall I tell them?”

God said to Moses, “I am who I am. This is what you are to say to the Israelites: ‘I am has sent me to you.’” (Exodus, 3: 13-14)

One of my favourite comedy series ‘Knowing Me, Knowing You’, staring Steve Coogan, features a pathetic TV chat show host, called Alan Partridge. In episode 2, he is interviewing an agony aunt called Dannielle, played by Minnie Driver, who is listing the things she likes in men:

Dannielle: Power is attractive. Sensitivity. Sense of humour. I like a man who knows who he is.

Alan: I’m Alan Partridge.

If you think that the law of identity is expressed by Exodus 3:14, then you should also hold that it is expressed in this little bit of Alan Partridge script.

I’m just going to leave that there.

1.1.3 Excluded Middle

In the last example, Jesus saying “He who is not with Me is against Me” is an example of someone expressing something stronger than the law of excluded middle. The logical law of excluded middle says that for every proposition, p, either it or its negation is true. There are two propositions being considered in the saying above, put together in the form of a disjunction. The two propositions are:

‘x is with Jesus’

‘x is against Jesus’

The combined disjunction is universal, in that it applies to everyone:

For all x: either x is with Jesus or x is against Jesus.

We could write this in first order logic as follows:

∀x (Wx ∨ Ax)

However, this isn’t a logical truth. There is no logical reason to stop someone being neither with nor against Jesus. The following is not a logical contradiction:

∃x (¬Wx ∧ ¬Ax)      (‘there is an x such that it is not with Jesus and it is not against Jesus’)

If Jesus had said ‘Either you are with me or not with me’, then he would have said something which would have been logically true (because of the law of excluded middle). It would have the following form:

∀x (Wx ∨ ¬Wx)

Therefore, when Jesus says that everyone is either with him or against him, something which goes beyond the law of excluded middle, and it is not a logical truth. Why this has been picked to be an instance of this law can only be put down to either the author not understanding what the law actually states, or being so determined to find something that fits the pattern that they wilfully ignore the fact that it doesn’t.

1.2 The problem

If we are thinking of the examples of someone not contradicting themselves, or of everyone being split into the ‘with’ or ‘against’ categories, then we have (at best) particular instantiations of these rules, but not examples of the rules. Consider the difference between:

a) A sign which said ‘do not step on the grass’.

b) Someone walking along the path next to the grass.

With regards to a), we would say that it had the rule, ‘do not step on the grass’, written on it. On the other hand, b) would just be an instance of the someone following the rule.

Finding Jesus saying ‘Either you are with me or you aren’t’ would be like finding someone walking next to the grass. Sure, it instantiates what the law of excluded middle is about, but it isn’t the rule. The rule is general. It says ‘nobody walk on the grass’, not just this guy in particular; excluded middle says ‘for all propositions, either p or not-p‘. The Bible nowhere makes generalised statements about language, reasoning or validity.

So the examples fail in that they aren’t actually instances of the rules (as the laws themselves are muddled by Shaughnessy), but they also fail because (even if we pretend that they do instantiate the rules) they aren’t examples of the rules. The Bible doesn’t have the law of excluded middle stated in it. It instantiates it, in that every proposition expressed in the Bible is either true or false, but that is not important at all. Every proposition expressed in any book is either true or false! Exactly the same goes for non-contradiction. There is nothing special about the Bible such that you can find the three rules of thought in it. If you want to see what a book looks like which explicitly has the rule of non-contradiction in it, read Aristotle’s Metaphysics, book IV, section 3:

“...the most certain principle of all is that regarding which it is impossible to be mistaken; for such a principle must be both the best known (for all men may be mistaken about things which they do not know), and non-hypothetical. For a principle which every one must have who understands anything that is, is not a hypothesis; and that which every one must know who knows anything, he must already have when he comes to a special study. Evidently then such a principle is the most certain of all; which principle this is, let us proceed to say. It is, the same attribute cannot at the same time belong and not belong to the same subject and in the same respect.

For Aristotle, the basic declarative sentence (the basic proposition) is the ascription of an attribute (or property) to a subject, and this is explored explicitly by him at great length. So ‘Alex is happy’ is this type of sentence. When he says “the same attribute cannot at the same time belong and not belong to the same subject and in the same respect”, this is simply to say that there cannot be any proposition, such as ‘Alex is happy’, for which it is true that ‘Alex is happy’ and it is also true that ‘Alex is not happy’, i.e. we cannot have both p and not-p. In contrast to the Bible then, Aristotle does not just give an instance of a sentence of the same form as the law of non-contradiction, like ‘it is not that Alex is both happy and not happy’ – he reflects on this and states the general proposition in its generalised form. It is explicit. With the case of the Bible, we have shoddy eisegesis going on, where Aristotelian principles are being read into a text that doesn’t have them.

So far, not great. Shaughnessy makes the following claim:

It is precisely because the laws of logic are embedded in Scripture that the Christian is able to establish from an epistemological standpoint that they are fixed and universal laws. Without this epistemological foundation, we cannot account for the laws of logic

Well, given what I’ve written above, it should be pretty obvious that I disagree with that. The laws of logic are not in the Bible. Given this, by his own standards, Shaughnessy doesn’t have an ‘epistemological foundation’ and ‘cannot account for’ these laws. Too bad.

2. An epistemological foundation for logic

Shaughnessy then presents the standard presuppositional line, the one we all knew was coming, where they brag about how great their ‘account’ of logic is, and how rubbish ‘the other account’ is.

The unbeliever cannot account for logic in his own worldview and therefore cannot account for his ability to think rationally. The challenge has been made many times to unbelievers to account for logic in their own worldview and it has always fallen short or gone unanswered. Never has an adequate response been given. In formal debates, the challenge is often ignored by the unbeliever, yet the challenge demands an answer because debates presuppose logic. The unbeliever is required to use logic in order to make his argument against Christianity consistent and intelligible, but only the Christian worldview can account for logic. He is therefore required to rob the Christian worldview in order to make his argument against Christianity intelligible.”

Ok, well we’ve all seen this over and over again. So I am going to meet the challenge head on, and provide a few different ‘accounts’ of logic, which could be ‘epistemological foundations’ for it.

First of all, what do we mean by and ‘epistemological foundation’ for something? Well, I take it to mean something in virtue of which we can come to know something. So, an epistemological foundation for x could be thought of an an answer to the question, ‘how is it that we are able to know about x?’

Given that, our question is: ‘How is it that we are able to know about logic (and in particular those logical laws)?’. In order to play the game right, I shall not appeal to God in any way, I will just go along with the idea that logical laws are things that have some kind of ontology capable of allowing reference to them, and I will just pretend that the three principles cited by Shaughnessy (identity, non-contradiction and excluded middle) really are ‘logical laws’, even though it is a clumsy and out-dated way to talk about logic. I will play the game anyway, just to be a good sport.

2.1 They are self-evident.

Here is the first way of answering that question: we are able to know about logical laws because they are self-evident truths. This just means that to think about them is to know that they are true. They don’t need anything else to support my knowledge of them, because they are self-evident. This is a really simple answer, and there isn’t much more to be said about it.

The response might be something like: “that’s rationalism! You are saying that all knowledge is rationally determined based on self-evident truths, like Spinoza!” Before we get into the standard disputes about rationalism and empiricism, I want to point out that I don’t need to also say that this is how I get knowledge generally. The question is about logical laws only. Maybe these are the only self-evident truths, and I gain knowledge about other parts of the world through empirical access, or mystical intuition, or because a ghost illuminates the right answer for me. Who cares? The point is that this plainly is an answer to the question ‘how could we know about logical laws?’. It doesn’t require a God of any type, so is available to an atheist (or a theist, or really anyone apart from those people who for some reason are committed to the view that there are no such things as self evident truths). They are pretty good candidates for self-evident truths if you ask me, and I would dispute the claim that there are candidates that are more plausible (is ‘cogito ergo sum’ more plausible as a self-evident truth than non-contradiction? They seem even, if anything). If anything is self evident, its the law of non-contradiction. So this view is plausible, at least on first blush.

If there is a secret cheat-card answer to this that presuppositionalist apologists have, I’ve never heard it. Remember the challenge: “The challenge has been made many times to unbelievers to account for logic in their own worldview and it has always fallen short or gone unanswered.” Well, that’s one account. Here is another one.

2.2 They are synthetic a priori knowledge

Here is my second proposal: we are able to know about logical laws because they are synthetic a priori truths. In the Critique of Pure Reason, Immanuel Kant summarises his views on this type of knowledge as follows:

“…if we remove our own subject or even only the subjective constitution of the senses in general, then all constitution, all relations of objects in space and time, indeed space and time themselves would disappear, and as appearances they cannot exist in themselves, but only in us. What may be the case with objects in themselves and abstracted from all this receptivity of our sensibility remains entirely unknown to us. We are acquainted with nothing except our way of perceiving them, which is peculiar to us, and which therefore does not necessarily pertain to every being, though to be sure it pertains to every human being.”

Synthetic a priori knowledge has the property that it is integral to how we see the world. It is subjective, in the sense that Kant explains above (that is, if we were to remove the subject, then it would also disappear), but it is also universal, in the sense that it applies to “every human being”. So, space and time may be known a priori, yet the knowledge is not simply analytic (i.e. true in virtue of the meaning of the words used), but synthetic (true because of more than just the meaning of the words used). What we know is the form of our intuition, which is a non-trivial fact about the way things are, but is also directly available to us, as subjects, a priori. We are programmed to see the world in a spatio-temporal way.

Kant has his own ways of demonstrating that this is the case, using transcendental arguments which inspired Van Til and should be familiar to all presuppositionalist apologists. Essentially you show that the contrary leads to a contradiction. So we have to see the world in terms of space and time, because the contrary view (where we do not see the world in such a way) leads to complete incoherence. Space and time are necessary presuppositions of the intelligibility of experience (a phrase presuppositionalists love to use). As such, we have transcendental proofs for them. Presuppositionalists, like the gang at BibleThumpingWingnut.com, should welcome this methodology, as it is basically the sophisticated version of the Van Tillian method they endorse themselves, only directed squarely at epistemological issues.

I say that we just point the synthetic a priori machinery at the laws of logic, and there we go, an epistemological foundation for the laws of logic. We know excluded middle, non-contradiction and identity as forms of intuition. Everyone has them (which explains their apparent universal character). If we try to conceive the world without them, we get incoherence (which shows their necessity).

On this view, we are not suggesting that these principles have metaphysical necessity. As good Kantians, we simply say that we cannot know about the numenal realm. But this should be perfectly acceptable to those presuppositionalists who throw the gauntlet of providing an epistemological foundation for the laws of logic. They are the ones, after all, who think that these principles are the ‘laws of thought’. On this reading of what they are, the Kantian line seems perfectly suited.

It would be really hard to imagine a presuppositionalist mounting a successful attack against this view, which didn’t also backfire and undermine their own transcendental arguments. You can’t have it both ways. If you are going to use transcendental arguments for God, I’m going to use them for what I want as well.

2.3 They are indispensable

Here is one last attempt. How do we know about the laws of logic? Well, they are indispensable to our best theories of science, so it is reasonable to believe in them. This is a version of the Quine-Putnam indispensability argument for the existence of mathematical entities. Here is how I see the argument going:

1. We are justified to believe in all the entities that are indispensable to our best scientific theories.
2. Laws of logic are indispensable to our best scientific theories.
3. Therefore, we are justified to believe in the laws of logic.

I’m not personally that convinced by premise 2, but presumably Shaughnessy and all those who throw down the presup gauntlet are. Premise 1 says that we have justification to believe in those things which are indispensable to our best theories, and I think this is going to be accepted by most people. We believe in viruses because our best science tells us that they exist. It is reasonable to hold the belief in viruses on this basis.

This argument doesn’t say that we have conclusively established that the laws of logic exist, but it provides justification. Presuming a broadly fallibilist idea of justification (as most contemporary professional epistemologists do), then even though the indispensability argument doesn’t ensure the laws of logic exist, it provides sufficient support for the belief that they do to be justified. So it allows us to have justified belief in the laws of logic existing. If that belief is also true, then we know that they exist. Thus, this is an explanation of how we come to know (as in ‘justified true belief’) that the laws of logic exist. Thus, it is an answer to how we can have knowledge of them, and ultimately part of an epistemic foundation, and an ‘account’, of them.

3. Conclusion