The Semantics of Nothing

0.   Introduction

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

1.   Creation from nothing

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

2.   A Toy Example

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

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

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

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

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

Ref)   Wn

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

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

Den)    ~(∃x)(Wx)

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

Den’) (∀x)~(Wx)

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

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

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

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

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

3.   The Main Case

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

Ref2)   In

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

The denotative reading would be as follows:

Den2) ~(∃x)(Ix)

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

Den2’) (∀x)~(Ix)

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

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

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

I have two worries with this conclusion:

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

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

Here are a few examples:

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

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

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

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

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.

Creation ex nihilo

0. Introduction

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

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

  1. Setting

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

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

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

a) ‘Nothing’ has no causal powers.

b) What if universes could come from nothing?

c) A good inductive conclusion.

      3. ‘Nothing’ has no causal powers

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

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

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

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

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

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

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

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

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

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

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

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

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

¬∃x (¬Cx)

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

(¬∃x (¬Cx))  ↔  (∀x (Cx))

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

4. What if universes could come from nothing?

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

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

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

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

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

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

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

5. A good inductive conclusion.

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

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

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

6. Conclusion.

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

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

Craig’s List – Omniscience and actually existing infinities

Introduction

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

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

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

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

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

2.1. An actual infinite cannot exist.

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

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

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

Hilbert’s Hotel

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

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

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

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

Pinning down the absurdity

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

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

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

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

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

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

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

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

The infinite universe is passive

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

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

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

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

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

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

The Infinite God Objection

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

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

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

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

Call this the ‘Infinite God Objection’.

God’s knowledge is of induction schemas

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

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

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

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

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

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

Thus this escape route will not help.

God’s knowledge is non-propositional

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Craig’s God is a passive infinity

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

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

Craig’s God and Time

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

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

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

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

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

Craig’s God is an active actually existing infinity

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

Dilemma

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

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

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

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

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