Frege’s argument for platonism

0. Introduction

Contemporary platonism (with a small ‘p’) can trace its roots back to Frege, in particular to his 1918 paper Thought. There are many fascinating arguments and proposals in this paper, which is one of the richest in the early analytic tradition. In particular, I want to look at one argument, which is broadly contained within pages 298 and 302. The argument is basically that communication (and science generally) would be impossible if platonism were not true; and clearly communication is possible, so platonism is true. What needs to be defended is the first premise, which links the possibility of communication with platonism. This is what Frege explains in the section I want to focus on.

  1. Thoughts and Propositions 

Frege is setting the scene for 20th century philosophy of language by giving us a very vivid account of a the notion of a proposition. Frege does not use this word however, but uses the term ‘thought’ for this task. Here is how he introduces the notion:

“Without wishing to give a definition, I call a thought something for which the question of truth arises. So I ascribe what is false to a thought just as much as what is true. … The thought, in itself immaterial, clothes itself in the material garment of a sentence and thereby becomes comprehensible to us. We say a sentence expresses a thought.” (p. 292)

Thus a ‘thought’ is what is true or false (something for which “the question of truth arises”), and is what is expressed by sentences. He is quite clear on page 293 that only are declarative sentences express ‘thoughts’ (“Only those sentences in which we communicate or state something”). He goes on over the pages up to 298 to explain how the same ‘thought’ can be expressed by multiple synonymous sentences, (“It makes no difference to the thought whether I use the word ‘horse’ or  ‘steed’ or ‘cart-horse’ or ‘mare’ “), and that indexical sentences can express the same ‘thoughts’. Frege’s example of the latter is that the thoughts expressed when Dr. Gustav Lauben says “I have been wounded”, and when Leo Peter hears this and remarks some days later, “Dr. Gustav Lauben has been wounded”. In each case, different sentences express the same ‘thoughts’.

Thus, by ‘thoughts’, Frege means what contemporary analytic philosophers mean by ‘propositions’, and from now on I will refer to them as the latter.

The question then is what type of things these propositions are. Frege argues that there are two candidate types of thing for propositions to be, before rejecting both of them and proposing a third way.

2. The Outer World

Firstly, propositions might be thought to be ordinary material objects of some sort:

“A person who is still untouched by philosophy knows first of all things which he can see and touch, in short, perceive with the senses, such as trees, stones and houses, and he is convinced that another person equally can see and touch the same tree and the same stone which he himself sees and touches. Obviously no thought belongs to these things” (p. 298-299)

Propositions, like ‘Donald Trump is the president’, clearly often involve material objects (such as the man Donald Trump), but the propositions are not themselves objects. The proposition that Trump is president is not one of those things out there in the world alongside Donald Trump. Thus, propositions are of a different kind than ordinary material objects.

3. The Inner World

Having dismissed propositions being part of the ‘outer world’ of material objects, he considers a a second realm; the ‘inner world’. This second realm is introduced to us by Frege as follows:

“Even an unphilosophical person soon finds it necessary to recognise an inner world distinct from the outer world, a world of sense-impressions, of creations of his imagination, of sensations, of feelings and moods, a world of inclinations, wishes and decisions. For brevity I want to collect all these, with the exception of decisions, under the word ‘idea’.” (p. 299).

Thus, ‘idea’ is a technical term which refers to all the various aspects of experience, such as sensations, feelings, desires, etc. Frege then considers the question of whether propositions could be ideas or not. To do this, he outlines some distinctive features of ideas.

Firstly, in contrast to material objects, ideas are had by bearers. Take some aspect of experiential inner life, such as a desire to eat a cake. It makes no sense to consider this desire existing apart from a bearer who has the desire, such as my desire to eat this cake. It is a metaphysical impossibility for there to be an ‘unaccompanied desire’, i.e. a desire to eat cake somehow drifting through the world without someone who has the desire; desires, just like all ideas, are had by bearers. In contrast, a material object, like a cake, can perfectly easily be considered without any bearer. Indeed, there is no contradiction in imagining the cake being the only thing in existence; an unaccompanied cake is metaphysically possible.

In addition, ideas are metaphysically private. No two people can literally have the same aspects of experience. We can both look at the same lime tree, but the experience each of us has of the lime tree cannot be shared with one another. Frege goes so far as to say that the question of whether two people see the same thing when they both see the same green leaf doesn’t even make sense:

“Now does my companion see the green leaf as red, or does he see the red berry as green, or does he see both as of one colour with which I am not acquainted at all? These are unanswerable, indeed really nonsensical, questions” (p. 299)

These questions are unanswerable, according to Frege, because the term ‘red’ or ‘green’ is being used not to state a property of objects in the outer world of material objects, but to “characterise sense-impressions belonging to my consciousness” and, as such, “it is only applicable within the sphere of my consciousness” (p. 299). When I say ‘red’ I’m referring to a part of my inner sense experience, not an objective property out there in the world. It is metaphysically private.

So ideas are i) had by bearers, and ii) metaphysically private. However, Frege argues that these two properties are not shared by propositions.

If a proposition was an idea, then it would be had by bearers, and it would be metaphysically private. Consider the proposition expressed by the Pythagorean theorem. If that were an idea, then as Frege says:

“…one should not really say ‘the Pythagorean theorem’ but ‘my Pythagorean
theorem’, [and] ‘his Pythagorean theorem’, and these would be different” (p. 301)

The proposition would be had by bearers, just like ideas. In addition, they would be metaphysically private, and this would mean that questions over their truth and falsity would also be metaphysically private. Going back to the example of how colours are private for Frege, this was because a colour, like ‘green’ or ‘red’, is not supposed to be a quality of material objects as such, but a quality of inner mental experience instead. When I see a green field, I have a sensation of ‘greenness’ which is an inner private experience. Thus, it makes no sense to ask whether your experience of greenness is the same or different to mine. Likewise, if the Pythagorean theorem is true-for-me only, then it doesn’t even make sense to wonder if it is false-for-you:

“Then truth would be restricted to the content of my consciousness and it would remain doubtful whether anything at all comparable occurred in the consciousness of others.”  (p. 301).


“If someone takes [propositions] to be ideas, what he then recognises to be true is, on his own view, the content of his consciousness and does not properly concern other people at all. If he were to hear from me the opinion that a [proposition] is not an idea he could not dispute it, for, indeed, it would not now concern him.”” (p. 302)

This leads Frege to make the following conclusion:

“If every [proposition] requires a bearer, to the contents of whose consciousness it belongs, then it would be a [proposition] of this bearer only and there would be no science common to many, on which many could work.” (p. 301)

If propositions were thoughts, then they would be had by bearers, and metaphysically private. If they had those properties, then there could be no possibility of having a common body of science on which many could work together. In short, if propositions were ideas, then science would be impossible.

4. The Third Realm

Here is where Frege draws together the considerations into the positive vision of what a proposition is:

“So the result seems to be: thoughts are neither things of the outer world nor ideas.

A third realm must be recognised. What belongs to this corresponds with ideas, in that it cannot be perceived by the senses, but with things, in that it needs no bearer to the contents of whose consciousness to belong. Thus the thought, for example, which we expressed in the Pythagorean theorem is timelessly true, true independently of whether anyone takes it to be true. It needs no bearer. It is not true for the first time when it is discovered, but is like a planet which, already before anyone has seen it, has been in interaction with other planets”

5. Conclusion

This post obviously only just begins to scratch the surface of this idea, but the outline of the argument is hopefully somewhat clear. Frege argues that material objects have the properties of being available publicly and can exist independently of any others, unlike ideas which are metaphysically private and have to be accompanied by bearers to exist. Propositions have aspects of both, being independent of bearers, yet not being ordinary objects in the material world either.

While there are certainly lots of extremely influential attacks on Frege’s basic picture here, most notably by the later Wittgenstein, his position was extremely influential on philosophers throughout the 20th century, such as Russell, (early) Wittgenstein, Husserl, Quine, Gödel, Putnam, Dummett, etc.


How to answer the Sye-clone: Wittgenstein’s transcendental argument

0. Introduction

In a previous post, I have talked about transcendental arguments as used in philosophy. I briefly mentioned one such argument found in an aphoristic comment in Wittgenstein’s book, On Certainty. In this post, I am going explain a plausible argument which can be extracted from Wittgenstein’s aphorism. Specifically, I will say how this works as a sort of strategy for dealing with various radically sceptical challenges that could be posed to you; i.e. for dealing with the types of ‘Sye-Ten Bruggencate challenge’.

  1. Wittgenstein’s Aphorism

On Certainty is the last book that Wittgenstein composed. Really, it is jut the collected papers that he was working on in the final months of his life, which were published posthumously. Here is the quote that I want to focus on:

“383. The argument “I may be dreaming” is senseless for this reason: if I am dreaming, this remark is being dreamed as well – and indeed it is also being dreamed that these words have any meaning.” Wittgenstein, On Certainty.

I want to say that there are two distinct arguments in this passage, which I will call the argument from obligation, and the argument from meaning respectively. I will look at the argument from obligation here, and cover the argument from meaning in a subsequent post.

2. The argument from obligation

The key idea behind the argument from obligation is that in a dream one cannot be under any types of obligation.

Imagine a person, Scrooge let’s call him, who is miserly and mean throughout the day, every day. Each time he gets the chance to be mean to someone, he takes it. However, when Scrooge goes to sleep at night, he always has the same dream, in which he is a kind and generous man. In his dreams, whenever Scrooge gets the chance to be nice to someone, he takes it. If we come to make a moral evaluation of Scrooge, I think that we would have to say that he was an entirely miserly and mean individual. His dream-world generosity does not count at all in favour of him being a good person. Dream actions have no moral value whatsoever.

It follows from this that if one is having a dream, then one is not under any moral obligations with regards to the things in the dream. Imagine you have a dream in which you do something morally wrong, like stealing. Upon waking, although you may still feel guilty about what you did in your dream, you have not actually broken any moral obligations, because you didn’t actually do anything, let alone anything wrong. It was only the illusion of doing a morally wrong action.

To press the point, if I dream that I murder someone in cold blood, I do not need to fear going to prison when I wake up, because I have done nothing wrong. Likewise, if I am playing a one-player computer game, like GTA5, and I decide to randomly kill a passer by in the street (we’ve all been there), I have not actually violated any moral proscription against murder. Dreams, like computer games, are not real contexts as such. They are illusory contexts, ones in which moral choices are not evaluated at all. We might say that they are amoral contexts.

The reason these contexts are amoral is that there are no actual agents playing the roles of the injured parties. The utilitarian does not count as harm violent deeds done to a computer sprite, nor to a character in a dream. A deontologist does not include proscriptions about computer sprites or characters in dreams. Each of these two major meta-ethical schools concerns themselves with real agents, not characters in dreams.

What holds for moral obligation, also holds for rational obligation. Imagine someone, let’s call him Scrooge again, who spends all his time constantly debating people in chat rooms online, but constantly failing to live up to his rational obligations. So, he does things like making arguments that have their conclusions explicitly stated as one of the premises. He makes assertions, such as that p is true, but when asked to justify the claim, passes the burden to his interlocutor to prove that p is false, or he provides a deductively invalid or unsound argument, etc. Whenever he has the chance to duck a rational obligation, he takes it. However, every night Scrooge has the same dream where he is constantly debating people online, but now he is the model rational agent and always abides by the rules of rational discourse. Whenever he makes a claim, he backs it up with either a plausible looking deductively valid argument, or he provides some compelling piece of evidence, etc. However, if we come to assess Scrooge’s behaviour in terms of his rationality, we would have to say that he is actually a very irrational interlocutor. All of his actual interactions have him constantly ducking his rational obligations. Just as in the moral case, the fact that he is very well behaved in his dreams doesn’t count for anything.

So, just as dreams are contexts in which there are no moral obligations (amoral contexts), they are also contexts in which there are no rational obligations (arational contexts).

If this preceding line of argument is correct, then we have an interesting result when faced with a radical sceptical doubt, such as the doubt that one is dreaming. The insight comes out if this sceptical challenge is posed as an explicit question, in the form “how do you know you are not dreaming?”

Here it seems that there are really two options: either you are dreaming, or you are not. As we have seen from the above considerations, part of the difference between being awake and dreaming has to do with the presence of obligations, both moral and rational. So when the question is posed, there are two possibilities – either you are dreaming or you are not – and these correspond to either being under obligations (moral and rational) or not. When dreaming, you are not under any obligations. So, if, during a philosophical conversation, the sceptic asks you to to show that you are not dreaming, then on the assumption that you are in fact dreaming, you are not under any rational obligation to provide any kind of answer. It doesn’t matter if offer an invalid argument as your rebuttal, or just walk away and make a sandwich. You are not really having a philosophical conversation at all, and are not really under any rational obligation to justify your claims, or argue consistently, etc. In a dream context, these obligations are just not present. So, if you are dreaming, you do not have to worry about answering the sceptical question.

If you are not dreaming, then you are under all moral and rational obligations. But that means that you need to provide justifications for your positions on things to remain rational, only if you are not dreaming. Thus, being awake is a necessary condition of being under the rational obligation to respond to a potential sceptical challenge.

Here is the argument in premise-conclusion form:

  1. For all rationally obligatory actions x, one is obliged to do x if, and only if, one is awake (i.e. not dreaming).
  2. Answering the sceptic’s question “how do you know you are not dreaming?” is a rational obligation.
  3. Therefore, one is obliged to answer the question “how do you know you are not dreaming?” if, and only if, one is awake (i.e. not dreaming).

It has the form, where O(x) means ‘x is rationally obligatory’, p means ‘you are awake’, and a is ‘answering the sceptic’:

  1. ∀(x), O(x) iff p
  2. O(a)
  3. Therefore, O(a) iff p     (∀-instantiation)

3. Conclusion

What the argument from obligation argument shows is that one is not under the rational obligation to answer a sceptic who wants you to justify that you are not dreaming. If you are dreaming, then you are not under any actual challenge to defend yourself against, on pain of being irrational. In fact, nobody has challenged you at all; there is no sceptic, there is no challenge. The whole context is illusory. On the other hand, if you are in fact under the obligation to make some kind of rational response to the challenge, this must be because you are really in a conversation with someone, and not dreaming the exchange. Thus, being awake is a necessary precondition for the intelligibility of the sceptical challenge itself. We must presuppose that we are awake for the question to be something we are rationally obliged to respond to.

Infinity, Hume and Euclid

0. Introduction

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

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

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

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

  1. Equinumerous and fewer than.

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

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

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

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

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

2. Hume and Euclid

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

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

A is a proper subset of B if and only if

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

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

Here they are side by side:

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

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

3. The Problem

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

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

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

4. What is going on? 

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

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

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

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

5. Rejecting 1

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

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

6. The case for rejecting 2

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

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

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

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

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

7. The case for rejecting 3

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

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

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

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

8. Comparison

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

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

The second strategy says:

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

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

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

9. Conclusion

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

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

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

The Fine-Tuning Argument and the Base Rate Fallacy.

0. Introduction

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

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

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

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

Craig rules out necessity by saying:

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

Chance is ruled out by the following:

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

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

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

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

1. Tuning

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

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

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

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

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

2. The problem of fine-tuning 

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

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

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

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

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

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

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

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

3. The meaning of ‘life’

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


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

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

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

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

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

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

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

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

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



5. Back to the argument

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

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

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

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

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

6. Conclusion

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

Logic and God’s Character

0. Introduction

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

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

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

  1. The Dilemma 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Here is how Poytress explains this idea:

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

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

2. Conflict

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

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

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

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

This dilemma can be put as follows:

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

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

As a first pass, he says:

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

He provides three examples, which are:

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

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

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

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

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

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

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

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

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

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

There are a few options at this point.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

5. Sidebar – Logical Euthryphro 

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

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

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

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

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

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

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

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

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

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

6. Wrapping up

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


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


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

Immaterial yet effective:

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


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

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

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

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

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

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

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

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

7. Conclusion

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

Does the scientific method rest on the fallacy of affirming the consequent?

0. Introduction

There have been some rather strange suggestions from certain apologists recently about the nature of the scientific method, such as here and here. Prime among the criticisms is the claim that the scientific method rests on a fallacy called ‘affirming the consequent’. However, this is a strange claim for various reasons. Firstly, the criticism doesn’t engage with how philosophers of science actually talk about the scientific method. From around 1960, with the work of Thomas Kuhn, attempts at summing up the scientific method in a simple inferential procedure have been largely abandoned. It is now widely taken in the philosophy of science that there is no one simple pattern of reasoning that completely captures the scientific method – a phenomenon known as the ‘demarcation problem‘. So there is no simple logical model of inference which completely covers the everything in the scientific method. But this means that there is no simple model of fallacious inference which completely characterises the scientific method either. In short, the scientific method is too complex to be reduced to a simple informal fallacy.

However, if we pretend that this Kuhnian sea-change had not taken place, then we would most naturally associate the scientific method with the notion of inductive inference, with evidence being given in support of hypotheses (or theories).  However, induction is not guilty of the fallacy of affirming the consequent, as I shall show here.

After this wander through induction, I will try to explain what the motivations are for the apologetical critique and how that misses the mark by failing to appreciate that scientific advances are often made through falsification rather than verification.

  1. Affirming the consequent

The fallacy of affirming the consequent is any argument of the following form:

  1. If p, then q
  2. q
  3. Therefore, p

The inference from the premises to the conclusion is invalid, because it could be that the premises are true and the conclusion is false. For example, if p is false and q is true, then the premises are true and the conclusion is false. If you want a proof of this, let me know and I will provide it in the comments.

The reason it is a fallacy to use affirming the consequent is just that the argument is deductively invalid. The lesson is this: if you have a true conditional, then you cannot derive the truth-value of the antecedent from the truth of the consequent.

2. Science affirming the consequent

The idea that the scientific method commits the fallacy above can be explained very easily. We might think that theories makes predictions. This could be thought of like a conditional, where the theory is the antecedent and the prediction is the consequent; if the theory is true, then something else should be true as well. So, take a scientific hypothesis (such as ‘evolution is true’, or whatever), and a prediction that the theory makes (‘there will be bones of ancient creatures buried in the ground’, etc). Here we have the conditional:

If evolution is true, then there will be bones of ancient creatures in the ground.

Now we make a measurement, let’s say by digging in the ground to see if there are any bones there, and let’s say we find some bones. So the consequent of our conditional is true. The claim by the apologists is that when a scientist uses this measurement as support for the hypothesis, they are committing the fallacy of affirming the consequent, as follows:

  1. If evolution is true, there will be bones of ancient creatures in the ground
  2. There are bones of ancient creatures in the ground.
  3. Therefore, evolution is true.

This is the sort of reasoning that is being alleged to be constitutive of the scientific method, and, as it is stated here, it is an example of affirming the consequent.

The problem with this line of thinking is not that it isn’t fallacious (it is clearly fallacious), but it’s that it is not what goes on in science.

3. Induction

In 1620, Francis Bacon published a work of philosophy called the ‘Novum Organon‘ (or ‘new tool’), in which he proposed a different type of methodology for science than the classical Aristotelian model that came before (Aristotle’s collected scientific and logical works had been collected together under the title ‘Organon‘). One way of characterising the Aristotelian method was that one does science by applying deductive syllogistic logic to ‘first principles’ (which are synthetic truths about the world). An example of this sort of first principle in Aristotelian physics might be that all things seek their natural place. It is of the nature of earth to seek to be down, and air to seek to be up, etc. This is, supposedly, why rocks fall to the ground, and why bubbles raise to the surface of water.

Part of Bacon’s dissatisfaction with this idea is that it provides no good way of discovering what the first principles themselves are; it just tells us what to do once you have them. Aristotle’s own ideas about how one discovers first principles are not entirely clear, but it seems that he thinks it is some kind of rational reflection on the nature of things which gets us this knowledge. Regardless, Bacon’s new method was intended to improve on just that, and is explicitly designed as a method for finding out what the features of the world actually are, of discovering these synthetic truths about the world. His precise version of it is a bit idiosyncratic, but essentially he advocated the method of induction.

Without going into the details of Bacon’s method, the idea is that he was making careful observations about the phenomenon he wanted to investigate, say the nature of heat, trying to find something that was common to all the examples of heat. After enough investigation the observation of a common element begins to be reasonably considered as not just a coincidence but as constitutive of the phenomenon under question. (He famously carried out just such an investigation into the nature of heat and concluded that it was ‘the expansive motion of parts’, which is actually pretty close to the modern understanding of it.)

In other words, starting with a limited number of observations of a trend, we move to the tentative conclusion that the trend is in fact indicative of a law. So the general pattern of reasoning would be that we move like this:

  1. All observed a‘s are G
  2. Therefore, all a‘s are G

The qualification of ‘all observed’ in premise 1 does most of the work in this argument. Obviously, just observing one a to be G would not count as much support for the conclusion. Technically, it would be ‘all observed’ a‘s, but it wouldn’t provide much reason to think that the conclusion is true. In order for the inductive inference to have any force, one must try to seek out a’s and carefully test them appropriately to see if they are always G’s. One must do an investigation.

So if we make a careful and concerted effort to investigate all a’s we can, and each a we come across happens to be G, then as the  cases increase, we will become increasingly confident that the next a will be G (because we are becoming increasingly confident that all a‘s are G). This is inductive inference.

With an inductive argument of this form, it has to be remembered that the conclusion does not follow from the premises with deductive certainty. Rather than establish the conclusion as a matter of logical consequence from the truth of the premises, an inductive argument makes a weaker claim; namely that the truth of the premises supports the truth of the conclusion; the truth of the premises provides a justification for thinking that the conclusion is true, but not a logically watertight one. Even the best inductive argument will always be one in which the truth of the premises is logically compatible with the falsity of the conclusion. The best one can hope for is that an inductive argument provides very strong support for its conclusion.

3. Induction affirming the consequent?

It is this inductive type of argument which the apologetical critique above is trying to address, it seems to me. They are saying that this type of scientific argument is really of the following form:

  1. If all a‘s are G, then all observed a’s will be G               (If p, then q)
  2. All observed a’s are G                                                         (q)
  3. Therefore, all a‘s are G.                                                      (Therefore, p)

Notice that the 2nd premise and the conclusion (2 and 3) is precisely the inductive argument from above; we have just added an additional premise (1), the conditional premise, onto the inductive argument. This fundamentally changes the form of the argument. Now the argument has the form of the deductively invalid argument ‘affirming the consequent’.

There are three problems with this as a critique of scientific inferences. Firstly, we have added a premise to an already deductively invalid argument, and shown that the result is deductively invalid, which is kind of obvious. Secondly, it characterises scientific inferences as a type of deductive inference, when there is good reason for thinking that they are not (at least if scientific inferences are supposed to discover synthetic truths about the world). Lastly, the addition of the first premise seems patently irrational, and obviously a perversion of normal inductive arguments. Let’s expand on each of these three problems:


All the apologetical critique has demonstrated is that one can make a fallacious deductive argument by adding premises to an inductive argument. However, inductive arguments are already deductively invalid. There is a fallacy called the inductive fallacy. It consists of taking an inductive inference to be deductively valid. So if you thought that all observed swans being white logically entailed that all swans are white, then you have committed the inductive fallacy, because you would have mistaken the relation between the premise and the conclusion to be one of deductive validity, when it is merely that of inferential support. All observed swans being white does provide some reason to think that they are all white, but the fallacy is in thinking that it alone is sufficient to establish with certainty that they are all white.

The addition of the first premise does nothing to undermine an inductive inference. It doesn’t make it more fallacious than it was in the first place. In a sense, this analysis commits the essence of the inductive fallacy, in that it says that scientific inferences are deductive when they are not; the claim that scientific inferences are guilty of affirming the consequent is itself an instance of the inductive fallacy.


We could, if we wanted to, add premises to an inductive argument to make it deductively valid, as follows:

  1. If all observed a’s are G,  then all a‘s are G        (If p, then q)
  2. All observed a’s are G                                             (p)
  3. Therefore, all a‘s are G.                                          (q)

Now the addition of the first premise has made the argument deductively valid, as it is just an instance of modus ponens.

The apologists were reconstructing scientific inferences as fallacious deductive arguments. Yet, even if we patched up the argument, as above with a deductively valid version of the inference, we still face a problem. This is that now we have a deductive argument, just like with Aristotle’s methodology. The very same reasons would remain for rejecting it, namely that as a methodology it provides no new synthetic truths; it only tells you what follows from purported first principles, not what the first principles are. We would be back to Aristotle’s dubious idea of introspecting to discover them. Thus, it isn’t desirable in principle to reconstruct an inductive argument as a deductive argument – even if the result is deductively valid. This means that the claim, that scientific reasoning is a failed attempt at being deductively valid, is implausible; even if scientific reasoning succeeded in being deductively valid that would be no help. The lesson is that they are a different type of inference, not to be judged by wether they are deductively valid or not.


Our original inductive argument went from the premise about what had been observed to what had not been observed. The whole point of inductive arguments is to expand our knowledge of the world, and so this movement from the observed to the unobserved is crucial. It is essentially of the form:

The observed a‘s are G ⇒ All a‘s are G

However, the first premise of the affirming the consequent reconstruction gets this direction of travel the wrong way round. They have it as:

All a‘s are G ⇒ the observed a‘s are G

If we keep clearly in mind that the objective of the scientific inference is to expand our knowledge, the idea of starting with the set (all a‘s) and moving to the subset (the observed a‘s) is weird. How could it expand our knowledge to do so? It is an inward move. This conditional though has been added to an inductive inference by our apologetical friends as a way of forming the ‘affirming the consequent’ fallacy out of an inductive inference.  But given that it gets the direction of travel exactly backwards, why on Earth would anyone ever accept this as a legitimate characterisation of a scientific pattern of reasoning?

This last concern highlights the cynicism inherent in the affirming the consequent critique. It isn’t a way of honestly critiquing a problem in science, but just an instance of gerrymandering an inductive inference, i.e. the change has been made just for the purposes of making the inference look bad, rather than as a way of highlighting a genuine issue. There is no independent reason for adding it on.

4. Or is it?

It might be claimed that I am pushing this objection too far. After all, there is reason to add the conditional premise on to the inductive inference. This is because theories make predictions. If a theory is true, then the world will have certain properties. And we do find examples of experiments being done in which the positive test result is used as a way of confirming the theory. And if this is right, then it looks like a conditional, and we are saying that the antecedent is true because the consequent is. So are we not back at the original motivation for the affirming the consequent critique?

Well, no. We are not. Here’s why. Let’s take an example. The textbook example. In Einstein’s general relativity, one of the many differences with classical Newtonian physics is that gravity curves spacetime. That means that there would be observable differences between the two theories. One such situation is when the light from a star which should be hidden behind the sun is bent round in such a way as to be visable from Earth:


We already knew enough about the positions of the stars to be able to predict where a given start would be on the Newtonian picture, and the details of the Einsteinian theory provided ways to calculate where the star would be on that model. So, the Newtonian model said the star would be in position X, and the Ensteinian theory said it would be in position Y.

These experiments were actually done, and the result was that the stars were measured to be where Einstein’s theory predicted, and not where Newton’s theory predicted.

Is this an example of the affirming the consequent fallacy? It might look like it. After all, it may well look like we were making this sort of argument:

  1. If general relativity is correct, then the star will be at X  (if p, then q)
  2. The star is at X                                                                            (q)
  3. Therefore, general relativity is correct.                                (p)

However, the real development was not that general relativity was confirmed when these measurements were made, but that Newtonian physics was falsified. Corresponding to the above argument we have a different one:

  1. If Newtonian physics is correct, then the star will be at Y (If p, then q)
  2. The star is not at Y                                                                       (~q)
  3. Therefore, Newtonian physics is not correct.                        (~p)

The first argument is a logically invalid deductive argument; it is affirming the consequent. But the second argument is just modus tollens (If p, then q; ~q; therefore, ~p), and that is deductively valid.

What we learned with the measurement of light bending round the sun was not that general relativity was true as such, but that Newtonian physics, and any theory relevantly similar to it, was false. General relativity may still be false, for all the experiment showed us, but it showed us that whatever theory it is that does correctly describe the physics of our universe is going to be more along the lines of general relativity than Newtonian physics. We learned something about the world, even if we did not confirm with complete certainty that relativity was true. And this is what scientific progress is like.

5. Conclusion

It would be affirming the consequent if someone thought that the positive measurement deductively entailed that general relativity was true. If any scientist has gone that far, then they are mistaken. It doesn’t mean that the scientific method itself is mistaken however.


0. Introduction

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

  1. The Liar

The liar sentence is of the following form:

a) This sentence is false.

The issue with a) is that it leads to a contradiction.

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

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

So let’s apply these principles to a):

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

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

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

2. Gaps.

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

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

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

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

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

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

3. Revenge

Consider the ‘strengthened liar’ sentence:

b) This sentence is not true.

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

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

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

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

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

4. Conclusion

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

As the philosopher Tyler Burge put it:

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