What is Atheism? II

0. Introduction

Recently I wrote a blog post about different ways of thinking about the definition of the terms ‘atheism’ and ‘atheist’. I was interested in the relation between belief and degrees of belief. Does lacking a belief mean lacking all degree of belief? To me, it seemed like the answer was ‘no’; one can lack a belief that p, yet still have some small degree of belief that p. That was how I describe my own internal doxastic state with regards to the proposition that no gods exist. I don’t feel like it is correct to say that I believe that no gods exist, but I have a small degree of belief that they don’t.

Part of my reasoning behind why I am only slightly in one direction rather than the other is because it is a proposition in metaphysics, and this seems like the most one can ever really have about such propositions. For example, it seems at least conceptually possible that there exists some god who is entirely unverifiable, some sort of deist god who never intervenes in the world and has left no trace of his existence for us to find. How could I ever know if such a god existed? Obviously, I couldn’t. But this type of god would also be the sort of thing that I couldn’t get any information about at all, either for or against. For this type of thing, there couldn’t be any evidence, and so one can never be confident that it doesn’t exist. So while I have an intuition or feeling that they probably don’t exist, it is not strong – after all, I don’t think that I know what the world is like at the most fundamental lever, so I don’t place much weight in what my pre-theoretical intuitions about that sort of thing say. They do lead me in one direction, but only slightly. That’s my view anyway. (An interventionist god who cares about human suffering seems far less likely to me than this epistemologically inaccessible god, and I would have a far lower degree of belief in such a personal god).

So I would say that my degree of belief that there are no gods is more than 0.5, but not much more. It seems to me that this doesn’t qualify as a strong enough belief for saying “I believe that there are no gods”. To me, saying that requires a higher degree of belief than I have. It is a declaration of a certain level of commitment to something to say “I believe that p”, and while it is not saying that you are utterly convinced that p, it is saying more than that you are minimally convinced; belief means something like ‘somewhat convinced’. I’m not sure that there is a precise numerical value which is the cut-off point between non-belief and belief; certainly not for all possible circumstances anyway. However, I just feel like my degree of belief is not strong enough to qualify in this context.

This is just like a situation where you may feel quite sure that a given person has not enough hair to count as hirsute, even though you are not sure whether there really is a precise number of hairs that one has to have in order to count as hirsute, or if so what that number of hairs is. Yet you just feel quite sure that this amount of hair isn’t enough. That’s how I feel about the god proposition.

Before getting to that point though, I spent some time explaining how there is some controversy about the definition of ‘atheist’, and this caused a bit of discussion on the comments below my past blog post on this – quite a lot longer than the actual article itself (and is still continuing as I write this), where people continued to discuss how they saw the right and wrong ways to define ‘atheist’.

My main point in the first section of that post was actually to argue that what seemed like a significant discussion between the atheist and theist is actually just a trivial definitional exercise on which nothing of any significance hangs. By this, I mean that it doesn’t matter if people disagree about whether someone should be called an atheist, a lacktheist or a hard-atheist, etc. The doxastic stance that the person holds, and any burden of evidence that comes with it, is what is actually important, and it remains the same regardless of what definition is used for the terms involved. We should just agree on a definition at the start of the argument and then move on into the interesting stuff.

Here I want to make that point as clearly as possible. So I will visualise a set of related positions – not a comprehensive list, but a reasonably thorough and precise list – so that we can see clearly what the different definitional positions are. I also want to express how this area is actually surprisingly rich from a logical point of view, and the various combinations of positions makes for an interesting enough landscape to categorise as a purely academic exercise. However, once we have a good grasp of the different definitions we could plausibly have in mind (the ‘landscape’), we can see how a particular person’s view gets classified. As we shall see, on some views I am an atheist, on some I am an agnostic atheist. The ability to translate between the different schemes considered provides the potential for more than just an academic classification exercise. It suggests the ability to help people stop talking past one another by providing a precise translation manual. All too often people hold different but not clearly articulated ideas about what it means to be an ‘atheist’ or ‘agnostic’ when they are in discussion and hopefully by setting out the landscape clearly we can be of some help here.

  1. Degree of belief

First, let’s consider the scale of belief about some proposition p (which will remain fixed as ‘some god exists’. The scale ranges from 0 for absolute conviction that p is false, to 1 for absolute conviction that p is true, with 0.5 being the middle point:


We can consider, for some agent a which we will keep fixed, that there are various propositions about a‘s beliefs and knowledge claims that we are interested in:

  • Bp      = a believes that p
  • B~p    = a believes that not-p
  • ~Bp    = it is not the case that a believes that p
  • ~B~p  = it is not the case that a believes that not-p
  • Kp       = a knows that p
  • K~p     = a knows that not-p
  • ~Kp     = it is not the case that a knows that p
  • ~K~p   = it is not the case that a knows that not-p

We will track where these propositions go under the belief scale, and then move on to add labels for various positions in as well.

2. Visualisations

Although the pair p and ~p are dichotomous (so that ‘p v ~p‘ is a tautology), this is obviously not the case for Band B~p; it is quite possible for neither of them to be true. Our agent might not believe that Kuala Lumpur is the capital of Malaysia, and might not believe that Kuala Lumpur is not the capital of Malaysia. The same goes for Kand K~p. This leaves room in the middle for a ‘belief-gap’. We will visualise this on our belief scale with a penumbra (or grey area) in which they a neither believes p nor believes that not-p. We could add this to our diagram as a shaded area underneath the section of the range to which it applies:


So in the diagram above, the grey area (the penumbra) extends beyond 0.5 degree of belief to some extent in either direction. This reflects that someone could have a degree of belief which is (say) 0.51 that p but that this would not be enough for it to be true that “believes that p“. For that to be true, a‘s degree of belief has to be greater than this. Precisely how much greater is vague and impossible to give a precise number to. All that we can (or need to) say is that for a to believe that p, their degree of belief has to be greater than the extent of the penumbra.

On either side of the penumbra, we would find the regions in which a holds a positive belief, either that p or that ~p:


We can add in knowledge claims here as well, where as we go along the scale towards 1 (or 0), we get to some (vague and impossible to make precise) point at which a doesn’t just believe that (not-p), but knows that p (not-p):


So our fully annotated diagram of the situation looks like this:


Given what I outlined in the last post on this topic, I sit just to the left of 0.5 on this scale (where p is ‘some god exists’). The green line is me:


It seems like my degree of belief is such that I don’t believe that p and I don’t believe that not-p. All this is rather uncontroversial*. The controversy, what there is of it, comes in when we decide which labels to apply to which position on the scale. We will do this by adding shaded areas to the top of the diagram. The first proposal we will consider (call it View 1) is where a ‘theist’ is someone who believes that p and an ‘atheist’ is someone who believes that not-p, which makes an ‘atheist’ a ‘hard-atheist’:


This has the consequence that theist/atheist is not an exhaustive distinction; it is possible to be in the area not covered by either (which is where I sit on this diagram). I am not an atheist on this picture (I am unclassified on this picture).

We might want to say that there is no such area, and that everyone is either a theist or an atheist. This sort of line has been put to me in the past. The idea is that if you act as if there is no god, then this makes you an atheist. Actions, it might be thought, are binary, in that you either act like you believe in a god (by going to church, praying, etc), or you act as if there is no god (by not going to church, praying, etc). This may make us think that everyone’s actions either make them an atheist or a theist (depending on how plausible we find this reasoning), in which case there should be no gap between the two positions on our scale (call this View 2):


On this view, a theist is not necessarily someone who ‘believes that p‘, but is just someone whose degree of belief that p is greater than their degree of belief that not-p. Similarly for the definition of ‘atheist’. I would count as an atheist on this view, even though I don’t believe that not-p. I count as an atheist just because my degree of belief that p is slightly less than 0.5. The definition of ‘atheist’ on this view is neither the same as ‘hard-atheist’ nor ‘lacktheist’.

As another option (call it View 3), we could think of the theist/atheist distinction as exhaustive, but draw the line between them on the point at which we switch from not believing that p to believing that p. This would make the definition of ‘atheist’ that of the ‘lacktheist’:


On this view, I count as an atheist, and interestingly so would someone whose degree of belief that was on the positive side of 0.5 but still in the penumbra; the sort of person who would say that they have a very weak degree of belief that some god exists, but not enough for them to say ‘I believe that some god exists’. That person would count as an atheist on this view.

We might think that the term ‘agnostic’ comes in here somewhere, and that it should come in to fill the gap between theist and atheist on View 1, like this (call it View 1.1):


On this picture, an atheist is someone who believes that not-p, a theist is someone who believes that p, and an agnostic is someone who does not believe either of p or not-p. On this picture, I come under the agnostic category, and not the atheist category.

We could add this version of agnostic (i.e. someone who does not believe either p or not-p) to View 2, to make View 2.1, as follows:


On View 2.1, all agnostics are also either atheists or theists; nobody is just an agnostic. On this view, I am an atheist and an agnostic.

Continuing the series, we could include the agnostic in the diagram and have ‘atheist’ pictured as a lacktheist (View 3), resulting in View 3.1, like this:


On this view, I would count as an atheist and an agnostic. On this view all agnostics are also atheists; there are no pure agnostics or agnostic theists.

However, one might think that the term ‘agnostic’ does not relate to a lack of belief but instead to lack of knowledge; it means that you don’t know if p is true, or if not-p is true. If that were the case (View 1.2), we would want to draw the agnostic area as follows:


On this view, there are ‘pure’ agnostics (and I would be one), but there are also agnostic atheists and agnostic theists (in contrast to View 1.1).

If we add this notion of agnosticism to Veiw 2, then we get View 2.2:


In this view, everyone is either an atheist or a theist (there are no pure agnostics) and there can be both agnostic atheists and agnostic theists. This only differs from View 1.1 in that the definition of agnostic is tied to knowledge, not belief.

Lastly, for completeness, I will consider View 3.2, which combines this agnosticism with the lacktheism of View 3:


On this view, an atheist is a lacktheist, and there can be agnostic atheists and agnostic theists. I am an agnostic atheist on this view.

Here are all 9 of the views for comparison:


3. Logical relationships

We have 9 views outlined above (View 1, 1.1 & 1.2; View 2, 2.1 & 2.3; and View 3, 3.1 & 3.2), but what are the relationships and differences between them? Here is an incomplete table showing some of the various properties of the different views and how they differ from one another:


Each view is unique in some respect or another. The difference between View 1.1 and View 1.2 (for example) is just whether agnostic means not believing either way that p or not knowing either way that p. This difference decides whether there are agnostic atheists/theists or not.

According to that summary, I am an there are two views according to which I am antheist, one according to which I am an agnostic, four according to which  I am agnostic atheist, with one where I am not classified. It is noteworthy how many different classifications one and the same doxastic attitude can come under. No wonder there is often confusion as to the usage of the terms involved.

4. Motivations

What we have is a landscape of different definitions and their various combinations. These are just the combinations I could see as being remotely justified. Each combination has something which backs it up conceptually.

  • There seem to be decent reasons for thinking about atheism and theism as being dependent on believing p and on believing not-p, which is the characteristic of View 1, 1.1 and 1.2.
  • However, the idea that belief is tied to action, and is thus binary, gives rise to the motivation for View 2, 2.1 and 2.2.
  • Then again, the definition of atheism as ‘lacktheism’ is clearly very popular among contemporary atheists, and this motivates View 3, 3.1 and 3.2.
  • There also seems to be some intuitive support for the idea that agnosticism simply fills in the space between atheism and theism, such that everyone is either an atheist, and agnostic or a theist (with no overlap), which informs View 1.1.
  • While this view of agnosticism seems fairly intuitive here, there is also something to be said for modelling agnosticism as relating to knowledge. Thomas Huxley, the person who coined the term ‘agnostic’, seems to have this association in mind when he said the following:

Agnosticism is of the essence of science, whether ancient or modern. It simply means that a man shall not say he knows or believes that which he has no scientific grounds for professing to know or believe.

Thus, there is a sort of exigetical support for the idea that agnosticism is epistemic rather than doxastic (i.e. about knowledge rather than just belief). If that is motivational for you, then you may be drawn to thinking of agnosticism as in View 1.2, 2.2 and 3.2.

5. Conclusion

It seems like a simple question, so often gone over, but so rarely gone over methodically:

What does ‘atheist’ mean?

But it has a surprisingly large number of potential, plausible-looking combinations of positions on the table. The benefit of classifying the various possible logical combinations is that we can translate between people’s usages. Here’s how:

First, one needs to assess internally what their level of belief is in the proposition being considered. Decide as best you can what your degree of belief is. Next also try to decide what you think (roughly) the thresholds are for belief and knowledge. Test yourself. For example, if you feel quite confident that you believe that p, do you also feel like you know that p? If not, then you believe without knowledge, and so you are in the middle section, etc.

In this way, we can get a feel for which region in the bottom part of the diagrams you fit into without too much need to quantify your degree of belief precisely. All you have to do is find which region of the belief scale you fit on. Once you have this in place, you can see how you are classified according to the various views. I did this in this post, and showed my results. So if someone asks me if I’m an atheist, I think my reply will now be ‘I’m an agnostic atheist on most definitions of the key terms, but on some of them I am an atheist’. This qualification doesn’t mean that I am changing my mind about what I believe, or trying to dodge any burden of proof for my claim. All the different views indicate is different ways of describing the same thing. 

There is no such thing as the ‘correct’ definition of what an atheist is. There is no such thing as the ‘correct’ definition of anything. Definitions are all arbitrary. One can use a definition in a way that fits the practices of a language using community, but other than ‘fitting in’ there is nothing else to decide whether a definition is correct or not. So there shouldn’t be a debate about what the definitions mean, from a logical point of view. We should only be interested in what people actually believe, and why.

There may be a larger political issue about the definition of ‘atheist’ due to the idea of an ‘atheist community’, but this is an issue I am not interested in. If I don’t classify as atheist enough for the atheist community, then so be it. I’m not going to change my sincerely held views just to be part of a club, and any club which is defined in terms of belief which requires people to adopt beliefs merely for the purposes of joining seems like an inherently contradictory institution.

It may be that people hold that the definitional game is more significant that I think it is for the following reason. It may be that when one joins a religion (or a new church, etc), that one sort of fits their beliefs to the community. As if someone says to themselves, ‘Now I’m part of the Calvinist community, I better figure out what beliefs I have’. This would make the beliefs follow from the belonging to a group. For all I know, this is how people view beliefs, and are happy to let themselves hold beliefs just because they are told that ‘people like us believe in such and such’. To me though, this gets the direction of travel the wrong way. First you have to have certain beliefs, and it is only because you antecedently do (or do not) hold whatever beliefs you do that you qualify for belonging to a club that is defined by beliefs. So one should say something like ‘I believe in the doctrine of predestination and original sin (etc), so I better figure out which group I belong to’. For me, beliefs come first, and labels (such as ‘Calvinist’ or ‘atheist’) follow after.


*From here on out, I will assume this basic picture to be correct. One could argue that the penumbra could really just apply to the 0.5 point and extend no distance in either direction. Even if you do so, this would still mean that the grey area has some extension. The reader should feel free to imagine the grey area being larger or smaller if  they so wish if they disagree with the extent I have given it above. It should be agreed by all parties that there is some penumbra, even if it only apples to 0.5 and nowhere else.

What is Atheism?

0. Introduction

Atheism may be defined provisionally as the view according to which there are no gods. However, despite this seemingly simple idea, there is a bit of controversy about the more precise meaning of ‘an atheist’. I will spell out some of the issues involved and outline my position.

  1. Atheism and theism

One might like to think of a proposition, p, which is to be understood as follows:

0)           p = ‘Some god exists’

It seems clear that the terms ‘theism’ and ‘atheism’ have some intrinsic relationship to p. One may think that the relationship is of the following sort (where ‘iff’ means ‘if and only if’):

i)            Theism is true      iff     p is true

ii)           Atheism is true    iff     p is false

This means that theism is logically equivalent to the proposition that some god exists, and atheism is logically inequivalent to the proposition that some god exists (it is equivalent to the falsity of ‘some god exists’).

1.1 Atheist and theist

If we accept i) and ii) as the definitions of theism and atheism, then we may move on to the definition of ‘theist‘ and ‘atheist‘. Doing this means bringing the agent, a,  into the definition. The natural way to define these terms is like this:

iii)           a is a theist       iff     a believes that [p is true]

iv)           a is an atheist   iff     a believes that [p is false]

There is a direct symmetry between -ism and -ist on this view. It is a nice and easy to grasp picture. The pattern is that the definiens of iii) and iv) are just those of i) and ii) but with the words ‘a believes that…’ at the start, and that the difference between theism/-ist and atheism/-ist is just that the former has ‘p is true’, and the latter has ‘p is false’. This means that a theist is just someone who believes that theism is true, and an atheist is just someone who believes that atheism is true. Thus, we have the pleasing result that theist is to theism what atheist is to atheism.

Here is a diagram of the logical relations:


If is an atheist in the sense of iv) above, call her a ‘hard atheist‘.

2. Lacktheism

There is another way of characterising what it means to be an atheist, and this departs from the pattern we have established above. On this definition, an atheist is someone who does not believe that p is true:

v)        a is an atheist    iff      not-[believes that p is true]

This definition of atheism is well-represented in public defences of atheism. Atheists commonly claim not to have a positive belief that p is false, i.e. to believe that no god exists, but merely to lack the belief that p is true. When they are doing this, they are advocating v), and someone who does this is a ‘lacktheist’.

3. Does ‘atheist’ mean ‘hard atheist’ or ‘lacktheist’?

There is some controversy about whether ‘atheist’ means ‘hard-atheist’ or ‘lacktheist’. Often, ‘atheists’ self-describe as lacktheists, but this leads to a charge of being ad hoc by the theists. I will explain the controversy and why I think it is logically dissolvable. First, I will outline the argument by theists according to which an ‘atheist’ should be thought of as a ‘hard-atheist’.

It seems like v) (the definition of lacktheist) messes up with the symmetry we had between i) and iii) (theism and theist), and between ii) and iv) (atheism and atheist). The symmetry was that the difference between theism/-ist and atheism/-ist is that the former ascribes truth to p and the latter ascribes falsity to p. With definition v) in place of iv) though, we have switched to talking in terms of the negation of p instead. The diagram would look like this:


So, the theist is someone who believes that theism is true, but (according to v) the atheist isn’t someone who believes that atheism is true, rather they are someone who does not believe that theism is true. This seeming abnormality may be seen as reason to reject v) (lacktheist) in favour of iv) (hard-atheist). Why, we might think, should we break the symmetry? We might just insist that an atheist is to atheism as a theist is to theism. If so, then an ‘atheist’ is a hard atheist, and a lacktheist isn’t an atheist at all. Changing the definition of ‘atheist’ seems unsystematic. In this situation, it is not that atheist is to atheism what theist is to theism, so we have lost our intuitive looking principle.

Added to the feeling of oddity about breaking the symmetry of the definitions, theists may be cynical about the motives of the atheist who argues for v) rather than iv) (the lacktheist). The reason for this cynicism would be that a consequence of using v) is that the defender of it seems to have less burden of proof in an argument than the defender of iv). And a position with lighter burden is easier to defend. So, the theist may suspect the atheist is choosing definition v) over iv) for the sole reason that  it makes her position easier to defend. If that were the only motivation on behalf of the atheist, we might view her decision to do so as ad hoc. In addition, if the approach treats atheist differently from any other similar position, then there could also be the accusation of special pleading as well.

The theist may insist that the situation should, in fact, be a level playing field, where each side (theist and atheist) has the same justificatory burden. The reasoning for this would be something like the following:

  1. If a says “I am a theist”, then is implicitly saying that a believes that p is true.
  2. If a (even implicitly) says “I believe that p is true”, then a has the justificatory burden of the claim “is true”.
  3. Therefore, if a says “I am a theist” then a has the justificatory burden of the claim “p is true”.    (1, 2, hypothetical syllogism)

Thus, claiming to be a theist carries with it the justificatory burden of claiming that it is true that some god exists. These justificatory relations are mirrored with our first definition of an atheist:

  1. If a says “I am an atheist” (and means definition iv), then is implicitly saying that a believes that p is false.
  2. If (even implicitly) says “I believe that p is false”, then a has the justificatory burden of the claim “is false”.
  3. Therefore, if a says “I am an atheist” this means that a has taken on the justificatory burden of the claim “p is false”.   (1, 2, hypothetical syllogism)

Thus, if we use the original definition of ‘atheist’, then the theist and atheist have the same justificatory burden. Surely, to try to change the definition of atheism here would be just to avoid this burden.

And indeed, if says “I am an atheist”, and means definition v), then a has not made an implicit claim about what a believes. Rather, a has made a claim that a does not have a belief that “p is true”. Thus, premise 1 above would be false if we used definition v) for ‘atheist’. This is why, if uses the lacktheist definition of ‘atheist’, that a‘s claim “I am an atheist” does not have the justificatory burden of the claim that “p is false”, and why the burden is avoided.

So, the claim could be that the atheist is making an illegitimate switch, from iv) (‘hard atheist’) to v) (‘lacktheist’). It could be seen as illegitimate because definition v) seems to be an otherwise arbitrary breaking of the symmetry of definitions, and seems like it is only justified through the benefit it bestows on the defender of the position (which is the root of the ad hoc complaint). We shouldn’t treat the definition of atheism differently to theism unless there is a good reason to do so  (or it would be special pleading). The atheist seems to have only selfish and illegitimate reasons for identifying as a lacktheist rather than as a hard-atheist.

4. Mirroring

However, one could in fact start the reasoning again slightly differently, and make ‘hard atheist’ look like the deviation from the pattern, and ‘lacktheist’ look like the expected one. This also transfers the charges of ad hoc and special pleading to the theist.

For example, we could stick with definition i), but define atheism as follows:

i)             Theism is true       iff                p is true

vii)           Atheism is true     iff       not-[p is true]

In a classical language, there would be nothing to distinguish between ii) and vii); ‘p is false’ is logically equivalent to ‘not-(p is true)’. Saying that atheism means that “it is not true that there are any gods”, seems just as faithful to the idea of atheism as the claim that it means that “‘there are gods’ is false”. Because they are equivalent, there is nothing one could appeal to logically which could decide in favour of i) rather than vii), and vice versa. Thus, we seem to have no real reason not to start from vii) if we want. And if we do proceed from here, then we can define theism as before, but use v) for the definition of atheism, and it looks like it is obeying the pattern of reasoning employed so far:

iii)           a is a theist       iff                a believes that [p is true]

v)            a is an atheist    iff    not-[believes that [p is true]]

Now the relations between i) and iii), and vii) and v) are just as neat and tidy as they were earlier. Here is a diagram of the logical relations:


As before, the relation between -ism and -ist is just that the -ist definition has ‘a believes that…’ added before ‘p is true’. The relation between theism and theist on the one side and atheism and atheist on the other is just that the atheism/ist side has ‘not-…’ prefixing them. On this view, a theist is to theism what an atheist is to atheism. Note that v) is the definition of a lacktheist.

So, an atheist is ‘naturally’ thought of as a lacktheist if we say that atheism means that it is not true that some gods exist. Given that starting point, it isn’t changing the pattern of definitions to get to lacktheism; instead, it looks as if insisting on hard-atheism would be unsystematic here. One can imagine a theist insisting that an atheist should still be a hard-atheist , but this time the accusation of symmetry-breaking could be levelled at the theist for doing so. Who is being unsystematic, it seems, depends on the starting point taken.

And a theist would have a selfish motive for making this demand too. We cannot ignore the fact that insisting that the atheist breaks the symmetry and uses the definition of ‘hard-atheism’ would remove the justificatory advantage that the atheist would otherwise ‘naturally’ have. However, because ii) and vii) are logically equivalent, there can be no reason to pick one over the other, and so the insistence of the theist to use the hard-atheist definition looks to the atheist as being ad hoc – being done merely for the rhetorical benefit it provides to the theist.

Thus, the two positions mirror each other perfectly. Depending on the definition given for atheism, the definition of atheist as hard-atheist or lacktheist seems unwarranted. The theist judges the atheist as trying to illegitimately lighten their own burden; the atheist judges the theist as trying to illegitimately add to the atheist burden. Whether it makes the atheist’s job harder, or the theist’s job easier, depends on whether atheism means that it is false that some gods exist, or whether it is not that some gods exist is true. And there doesn’t seem like there could be any reason for picking one over the other.

There seemed to be an observation that atheist’s were making an illegitimate move when defining atheist as lacktheist, which was being done just to get an advantage over the theist rhetorically. But if we start at another position, it would seem like the theist is the one trying to shift the burden just for their own advantage. This seems to dissolve the accusations of foul play on either side.

5. Post-definitional thinking

I think that the lesson of all this is just that there is nothing purely logical to appeal to which means that ‘atheist’ should be thought of as ‘hard-atheist’ rather than ‘lacktheist’. Either view is equally defensible, and any choice between them can only be ad hoc. In a sense, as it is a discussion about the nature of definitions, it is rather pointless. And this is hardly surprising.

Instead of worrying about the definition of ‘atheist’, we should rather pay more attention to the nature of the beliefs that a holds. In addition to the basic notion of a simply believing that p, we can talk about the ‘degree of belief’ that a has that p. Let’s say that the degree of belief a has that p is the following:

Da(p) = x, where 0 x ≤ 1.

Degrees of belief are real numbers between 0 and 1, rather like probabilities. They express your feeling of confidence in a proposition. 0 is maximally uncertain, 1 is maximally certain, and 0.5 is absolute indecision.

It seems to me that the proposition p, that there are any gods, is rather hard to evaluate. I find the idea of a personal loving agent quite unlikely indeed, for various reasons (it seems suspiciously like the sort of thing made up by humans, for one). However, the question of whether there are any gods seems a lot more of a difficult thing to evaluate. Perhaps some kind of being created the universe, but remains utterly divorced from the subsequent comings and goings of the world itself, or perhaps one is fascinated by the comings and goings of radically different forms of life on the other side of the universe than us, etc. These sorts of ideas are interesting, but are almost impossible to say anything about, either for or against. I kind of couldn’t have any good reasons to think that any of these sorts of hypotheses were true rather than false. What would count as evidence for or against? In this situation, my degree of belief that p (i.e. the proposition ‘some god exists’) has got to be around 0.5.

Yet, I do have a sneaking suspicion that there probably aren’t any gods like this. If you put a gun to my head and made me decide, I would opt for the no-gods option. That’s what I think is more likely, and so I my degree of belief that isn’t exactly 0.5. The following is certainly true:

Da(p) < Da(~p)

However, the imbalance seems to me to be very, very slight. I wouldn’t know how to put a precise number on it, but it seems reasonable to think that my degree of belief that ~p is between 0.5 and 0.55.

Now, does this state of mind mean that I believe that p? I certainly believe that there is an almost even probability about whether there are any gods or not, with a very slight imbalance towards no gods. My degree of belief is similarly minimally slanted towards the no-gods position. The question is the relation of these facts to the question of whether I believe that p.

Belief, as opposed to degree of belief, is an all or nothing notion. You either believe p or you don’t. Yet, my degree of belief is a scale. It ranges from 0 (definitely not belief) to 1 (definitely belief), and could be any value in between. How do the two notions relate to one another?

One idea, at one point seemingly advocated by William Lane Craig, is that the relation between belief and degree of belief is as follows:

If it is more plausible that a premiss is, in light of the evidence, true rather than false, then we should believe the premiss.” (taken from: http://www.reasonablefaith.org/apologetics-arguments#ixzz4Kp11AOyR)

The idea could be put as follows:

a believes that p    iff   Da(p) > 0.5

So long as your degree of belief that p is more than 0.5, then you believe that p is true. On this view, saying that you ‘believe that pjust means that your degree of belief is more than 0.5.

One problem with this view is that there seem to be situations in which it sounds wrong to say ‘I believe that p‘, even though they are clearly situations where our degree of belief that p is more than 0.5. Here is one.

Say I have a pack of cards, I thoroughly shuffle it and I take one out. It is the ace of spades. I discard the card, and take another one without looking at it. Let be the proposition that ‘the card is red’. Do I believe r?

The probability that r is true should be calculated as 26:51 (i.e., the remaining number of red cards:the remaining number of cards), or roughly 0.52. Given that I know this, my degree of belief that r is true should be correspondingly 0.52. Any other number would be perverse.

I think that in this situation, though my degree of belief is clearly in favour of red over black (or not-red), I still don’t think it is correct to say that I believe that the card will be red. I am sufficiently hesitant that the phrase ‘I believe that r is true’ would be misleading. It would indicate a higher degree of belief than that.

Perhaps you are thinking that you do believe that the card is red in this situation. Perhaps tilting the degree of belief 2 percentage points towards r is enough for you. If so, then consider the following version of the previous example:

Say I have a million packs of cards, I thoroughly shuffle it (somehow!) and I take one card out. It is the ace of spades. I discard the card, and take another one without looking at it. Let be the proposition that ‘the card is red’. Do I believe r?

This situation is exactly like the previous one, except that the chance that the card is red is slightly closer to being exactly 50:50 than before. The probability would be: 0.50000002. Now, the chance that the card is red is only two millionths of a percentage point more likely than that it won’t be. Do you believe that it is red? If so, then your position is probably that of Craig’s above. Any imbalance in one direction entails belief rather than disbelief.

On the other hand, you may be agreeing with my intuition that asserting belief in these situations is incorrect. If so, then this means that there is some value of degree of belief (say 0.45 – 0.55 or something) in which is is not true that you believe p or disbelieve it (or believe ~p). In this penumbra (or indeterminate area) we lack belief that p, even though we possess a positive degree of belief that p. If you think this, then it cannot be true that ‘a believes that p iff Da(p) < 0.5′.

One may ask what the value is, if not 0.5? What does the value of your degree of belief that have to be in order for it to be true that you believe that p? This, I think, is a complex question. One that is so complex, in fact, that it may be malformed. There may be no answer to it as such. It may be that in certain contexts the threshold is higher than others. Perhaps this varies from person to person, from conversation to conversation, from time to time, etc. Perhaps it varies in a chaotic and untraceable manner. This may be the case, within some sort of range outside of which it doesn’t go. For example, belief is never inappropriate in the case where the agent has a degree of belief which is 0.99, for example. It seems like it also isn’t appropriate in the case where the degree of belief is 0.50000001, etc. When people say ‘I believe that p‘, they are not necessarily reporting to a precise degree of belief (0.65 rather than 0.66, say), but just that they feel that their degree of belief is sufficiently over the threshold (whatever it is). Conversely, when someone says that they believe that ~p, this means that their degree of belief that ~p is sufficiently over the threshold (whatever it is) for ~p. When one is not sufficiently over either threshold, as in the card examples above, one should not say that one believes that p, or that one believes that ~p. One simply lacks beliefs in either direction. This is perfectly compatible with the idea that the degree of belief is believed, and perhaps even known. All that matters is that the degree of belief is extremely close to 0.5.

Thus, I have made a case for my claim to lack belief, which is not ad hoc because it is motivated by a general principle about when to withhold belief either way, and is not special pleading because I apply it to any case that is relevantly similar. I do not believe that any gods exist in the same way as that I do not believe that the card is red. In each case, my degree of belief is very close to 0.5, and that is what makes it inappropriate to affirm in either direction.

5. Conclusion

I think that this characterises my views about theism. I have a degree of belief that theism is false which is marginally over 0.5, but less than enough to indicate positive belief that it is false. Whether that counts as an atheist or not probably depends on your personal choice of definitions. Definitions aside, that is my view.

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.

Skepticism, fallibilism, anti-skepticism

    0. Introduction

The following three propositions form an inconsistent triad:

P)   I do not know that I’m not in the matrix*

Q)   I know that X

R)   If I know that X, then I know that I am not in the matrix


(X is to be thought of as a proposition with content about the external world, such as ‘it is 3PM’ or ‘I am wearing trousers’, etc, rather than ‘I believe it is 3PM’ or ‘I am receiving sense-data about wearing trousers’, whose content is internal to the subject.)

We can represent the logical form of the propositions as follows, where p = ‘I know I am not in the matrix’, and q = ‘I know that X’:

P)    ~p

Q)     q

R)     q → p

There are three ways that we can formulate an argument using these propositions which generate three positions about knowledge, which I am calling ‘skepticism’, ‘falliblism’ and ‘anti-skepticism’. Each argument is derived by having two of the propositions as premises with the negation of the remaining one as the conclusion.


  1. Skepticism

The skeptic formulates her argument as follows:

R)     If I know that X, then I know that I am not in the matrix

P)     I do not know that I am not in the matrix

~Q)     Therefore, I do not know that X

This argument has the form of modus tollens:

q → p


Therefore ~q


2. Fallibilism

In contrast, the fallibilist argues from the truth of P and Q to the falsity of R, i.e.:

P)      I do not know that I am not in the matrix

Q)      I do know X

~R)     Not-(if I know that X, then I know that I am not in the matrix)


This has the form:



Therefore, ~(q → p)


What does this rearrangement say? It says that because we do know something, yet we do not know whether we are in the matrix, it follows that knowing that we are not in the matrix it is not a necessary condition for knowledge. Thus one can have knowledge without being able to rule out the skeptical hypothesis.


3. Anti-Skepticism

The final combination, the anti-fallibilist (which perhaps represents G.E. Moore?), runs as follows:R)   If I know that X, then I know that I am not in the matrix

Q)   I know that X

~P)   I know that I’m not in the matrix


The form of this argument is modus ponens:

q → p


Therefore, p


On this argument, the requirement of ruling out the matrix as a necessary condition for knowledge is accepted, and the fact of knowledge of X is affirmed, with the consequence that one knows they are not in the matrix. A refutation of the skeptical hypothesis has been achieved (hence the name ‘anti-skepticism’).



4. How to choose? 

So now we have three rival arguments, each of which picks two of the members of the triad and rejects the third. The question of which argument to pick turns then, not on logic as such, but on the question of which proposition to jettison. Which one seems the least plausible? The problem is that they all seem eminently plausible.

P says that you cannot rule out the matrix, or evil daemon hypothesis. It seems very plausible, at least to anyone who has read Descartes or watched the Matrix. Denying this premise seems to require a refutation of skepticism.

Q says that you know that X. This is obviously going to depend on your choice of X, but why not make X as plausible as you like? Let X be ‘it is now 3PM’ (if it is 3PM), or that you are wearing trousers (if you are wearing trousers), etc. It can be the most run of the mill, ordinary knowledge claim you can think of. By definition, Q should be very plausible, if any knowledge claim at all can be. And P, Q and R are all knowledge claims.

R says that if you know that you are walking down the street, or that it is 3PM, or that you are wearing trousers, or whatever, then you know you are not in the matrix. This is also very difficult to deny. If I know that I am walking down the street, then it is true that I am walking down the street. If it is true that I am walking down the street, then I am not in the matrix. Thus, if I know I am walking down the street then I am not in the matrix. Therefore, it seems that if I know that X, then I know I am not in the matrix.

Where is the weak link here?


*for ‘in the matrix’, feel free to substitute in ‘am a brain in a vat’, ‘am being deceived by a Cartesian daemon’, etc.

The Infinite Regress for Revelational Epistemology

[This idea is inspired by a very similar regress problem as set out in a draft version of ‘On Knowledge Without God: Van Tillian Presuppositionalism and Divine Deception by Daniel Linford and Jennifer Benjamin.]

Traditionally, it is held that there are two ways of gaining knowledge; either through the senses, or through the use of pure reason. These carry the names of ‘a posteriori’ and ‘a priori’ knowledge respectively. While a priori knowledge can be known with certainty, it is also devoid of any content about the world; one can deduce that the interior angles of a triangle sum to 180º, but not whether any actual triangles exist. In contrast, a posteriori knowledge provides genuine content about the world, but can always be doubted; my senses are telling me that it is daytime, but perhaps I am dreaming. So one has a sort of certainty but no content, one has content but no certainty.

Some presuppositional apologists try to have the best of both worlds, with a third type of epistemological category; revelation. This has the content of a posteriori knowledge, but with the certainty of a priori knowledge; one can know that God exists ‘in such a way that they can be certain’. It is an impressive claim, but one which I think is susceptible to an infinite regress.

There is a simple apologetic mantra, often used by presuppositionalists, about the impossibility of having this type of knowledge unless you are on the right side of the creator of the universe. It says that ‘unless you knew everything, or were told by someone who did, it would be impossible to be certain about any matter of fact’. The obvious implication is that only by being directly revealed something by God can we come to know it for certain. Let’s try to put this clearly:

Revelation)    x can know p for certain if and only if God has revealed to x that p.

I claim that there is a problem for this idea; that it faces an infinite regress. The problem has to do with the possibility of mistaken claims of revelation.

So imagine a person, let’s call him Sye, who thinks that they have had a revelation from God that p is true. In addition, let’s also imagine that some other person, let’s call him Ahmed, thinks that he has had a revelation from God that ~p is true (i.e. that p is false). Now, if we asked him about this, Sye is clearly going to say that only he is correct in this matter. Sye would say that poor old Ahmed mistakenly thinks he has had a revelation when he has not.

But the question would become ‘how can Sye know this?’ Imagine that Sye offers up something about his revelation that he claimed made the difference, and according to which he could tell that his revelation was genuine, and not a mistake. This could only be something relating to the way in which Sye experienced the revelation. But no extra experience could make this difference. If Sye said that in his revelation God told him with a really loud booming voice, or with a golden shimmer around the page, etc, and this is how he knew the message was genuine, we could always postulate that Ahmed’s revelation was delivered in a similar manner. The internal experiences of both agents could be exactly similar in all relevant respects, and it is still conceptually possible for at least one of them to be suffering from a false impression. There cannot be a foolproof experience that confers certainty, or else the empiricists would have had this in the first place, and we would have had no need for revelation at all. Thus, nothing about the experience of the revelation would mark it out as being reliable rather than mistaken.

There could be no a priori explanation for this either, as they are devoid of content, and can never tell us about what is true in the world. They only relate ideas to one another, and so could never say whether, in this actual case, Sye was mistaken or not.

The revelationalist has a natural go-to answer here though, which he will find very tempting, but which I urge is going to lead to the regress. He has a third epistemological route, and he may well be tempted to bring it into action on this question. So Sye may well say that the reason he knows that God’s revelation that p was correct, was that God revealed to him that he had revealed to him that p. Call this a ‘second-order’ revelation; a revelation about a revelation. This would sure-up the worry over whether had been revealed or not. God has not only told Sye that p, but he also tells Sye that he has told Sye that p.

But then we could run the argument all over again. Imagine now that Ahmed also thinks he has received a similar second-order revelation from God; not only that he has revealed that ~p, but also that he has revealed to him that he has revealed to him that ~p. How can Sye know that he is the correct one, and that Ahmed is incorrect? Again, the only thing he can do is refer once more to the notion of revelation, so that God reveals to him that he had revealed to him that he had revealed to him that p! Thus, Sye would need to appeal to a third-order revelation to sure up the second-order revelation.

But we can run the argument all over again, where Ahmed gets the same third-order revelation, etc, etc. This process clearly goes on forever. At no point in the iterative process can Sye ever lay claim to the type of certain knowledge he is looking for, because at every point there is a possible Ahmed who could have exactly the same experience. The possibility of error over the revelation is a sort of un-holy ghost which can never be banished.

My conclusion from this is that revelational epistemology, as conceived here, is vulnerable to an infinite regress problem, from which it can never escape. It provides no new route to knowledge at all.