Swinburne Chapter 6: The Explanatory Power of Theism

Swinburne pauses for breath here, and goes over Bayes’ Theorem once again. He repeats the original formulation

P(h|e & k) = [P(h|k)P(e|h & k)] / P(e|k)

Where

P(h|e & k) is the probability of the existence of God, given all the various bits of evidence we will come to,

P(h|k) is what he calls the intrinsic probability of God (i.e. the probability before we consider the evidence),

P(e|h & k) is the probability that the evidence would be as it is, given that God exists,

P(e|k) is the intrinsic probability of the evidence.

He expands this last term (correctly) as follows

P(e|k) = P(e|h & k)P(h|k) + P(e|~h & k)P(~h|k)

The first term is the same as the top line of the equation. The second is the converse, i.e. the probability that the evidence would be as it is, given that God does not exist, multiplied by the intrinsic probability of God’s nonexistence.

So overall we get the following equation

P(h|e & k) = [P(h|k)P(e|h & k)] / [P(e|h & k)P(h|k) + P(e|~h & k)P(~h|k)]

Now, if it is a while since you last did any maths & algebra at school, all these letters and symbols might look a bit intimidating. Of course, they are intended to look that way. So let me show you how the theorem actually works on a more down-to-earth example.

Suppose that at a particular a school, 60% of the pupils are boys, and 40% are girls. There is a school uniform, and all the boys must wear trousers, but the girls have a choice between trousers and skirt. A quarter of all the girls choose to wear trousers. You see a child in the distance wearing the school uniform, and can see that the child is wearing trousers. What is that probability that the child is a girl?

We use precisely the same equation.

P(h|e & k) is the probability that the child is a girl, given the evidence of wearing trousers. This is what we are trying to calculate.

P(h|k) is the prior probability that the child is a girl. We know that number, it is the proportion of girls in the school, i.e. 40% or 0.4.

P(e|h & k) is the probability that any particular girl wears trousers. We know this is 0.25.

P(e|~h & k) is the probability that a boy wears trousers. This is 1, since all boys must wear trousers.

P(~h|k) is the prior probability that the child is not a girl. We know this is 0.6, because we know that 60% of the children in the school are boys.

So we have all the numbers we need, and can plug them into Bayes’ equation.

P(h|e & k) = ( 0.4 x 0.25 ) / [ ( 0.4 x 0.25 ) + (1 x 0.6 ) ]

Get the pocket calculator out (or multiply it out in your head), and it comes out at 1/7, or about 14%.

You can check this out by another method. Out of every 100 children, 60 will be boys, and 40 will be girls. Of the girls, 10 wear trousers, and 30 wear skirts. We aren’t interested in the skirt-wearing children – we can see that the child has trousers. Of the trouser-wearing children, 60 are boys, and 10 are girls. So 10 out of every 70 trouser-wearing children are girls, or 1/7.

You can extend this to cover multiple pieces of evidence. For instance, taking the above example, suppose that 80% of the girls in the school have long hair and only 10% of the boys, and you can see that the child in the distance has long hair and is wearing trousers, you can make the calculation simply by plugging the new numbers into Bayes’ theorem. First, you calculate for the trouser-wearing (as shown above). That gives you a new set of prior probabilities, for the probability that a trouser-wearer is a boy or a girl. Our new evidence is the long hair. Modify the definitions above, replacing trousers with long hair, and plug the numbers into the equation. We now get this

P(h|e & k) = ( 0.14 x 0.8 ) / [ ( 0.14 x 0.8 ) + ( 0.1 x 0.86 ) ]

0.14 is the proportion of girls among trouser-wearers, and similarly 0.86 is the proportion of boys (i.e. 1 – 0.14). 0.8 is the proportion of girls with long hair, 0.1 is the proportion of boys with long hair.

Calculate this out, and it turns out that the chance of a long-haired child wearing trousers being a girl is about 0.57, or 57%.

How can that be? The chance of any trouser-wearer being a girl was only 14%! Well, by being aware of this new piece of evidence, we can now eliminate from consideration a much larger proportion of the children i.e. all the short-haired ones.

We can check it out using the other method as well. We previously worked out that for every 100 children in the school, there were 70 trouser-wearers, of whom 10 were girls. 8 out of 10 girls have long hair, but only 6 out of the 60 boys have long hair. So among the long-haired trouser-wearers, 8 out of every 14 are girls, i.e. 57%.

And that in essence is what Bayes’ theorem is all about. There is a whole load of algebra which I shan't bother to go into here which shows why Bayes' theorem works. If you're interested in probability and statistics you shouldn't just take my word for it, I recommend that you look up the maths and understand it for yourself. But the fact is, Bayes' theorem does work.

What Swinburne hopes to do is to make estimates of the various probabilities regarding God, feed them into Bayes’ theorem, crank the handle and produce an overall probability for God’s existence.

In the previous chapter, Swinburne was trying to make an assessment of the prior probability of God’s existence, which he called the intrinsic probability of theism. In the example of the schoolchildren described above, we know the prior probability that any particular child is a girl. We know that because we know how many girls and how many boys are at the school, we can simply count them. Even when we can’t count an entire population, we can sample it – that is what opinion polls are all about. And with that sample we can get a fairly accurate estimate of probabilities which can be fed into Bayes’ theorem.

But we don’t have a population of gods, nor do we have a population of universes, some of which have been made by God and some of which haven’t. We have a hypothesis regarding the existence of just a single God, and evidence in the form of just one universe whose existence we know of. And yet, Swinburne wants to make use of Bayes’ theorem in order to assess the “balance of probability” of God’s existence. In order to do that, he will have to put a number on each of the following concepts.

1. The prior probability of God’s existence
2. The probability of God’s existence given each of the various “pieces of evidence” that he puts forward (the cosmological argument, the teleological argument etc)

In the previous chapter, Swinburne was working his way towards suggesting a figure for P(h|k), the prior probability of God’s existence. Even to describe it in those terms is to show the futility of the effort. Either God (by Swinburne’s definition) exists or he doesn’t, and scientific investigation should in principle be able to uncover evidence one way or the other. For instance, if God performs miracles, we might reasonably hope that one or two of them would happen while scientists have their instruments pointed in the right direction. Because Swinburne’s hypothesis is not looking for what proportion of all gods have a particular set of characteristics, there isn’t any way of expressing it as a numerical probability.

The same applies to the various conditional probabilities he is looking to assign numbers to. We know what proportion of girls at the school wear trousers, we can count them. In a larger population, we could sample. Either way, we would have a statistical method by which we could come up with a number. But how can we estimate the probability of whether God would make the universe the way it is, if he exists? There is only one way available, and that is quite simply to make the numbers up.

And that is what Swinburne does, though he disguises the fact under a huge torrent of words. By the end of this chapter, we have been through 132 pages of argument which describe Swinburne’s reasons for saying that it is justified to make up the numbers, and on what basis he will choose one number over another. In the latest chapter he describes various reasons for thinking it probable that God would want or need to create a universe containing humans. I’ll paraphrase it rather than directly quoting. He thinks that it is a good thing that humans exist, and since God is perfectly good by definition, creating humans is the sort of thing that we might reasonably expect God to do. God might also want to create animals as well, but there isn’t such a good reason for him to want to do that, so animals don’t tip the matter much one way or the other. Swinburne then for the first time in the book goes on to actually assign a number which will get plugged into Bayes’ equation. This really is worthy of direct quotation.

I have argued in this chapter that there is a modest probability intermediate between 1 and 0, to which I will give the artificially precise value of 1/2, that a God will create humanly free agents located in a beautiful physical universe, perhaps also containing animals.

If you understand probability and statistics it is possible to decode this. It means that by his own admission none of his arguments are sufficient to make any kind of estimate of probability in the matter (it's impossible by definition to have a probability outside the range 0 to 1, so he is in fact ruling nothing out at all), but because he needs a number in order to proceed to the next chapter, he's going completely arbitrarily to decide to use 1/2.

Yes, that really is how he proceeds.

Swinburne chapter 3, The Justification of Explanation

In chapter 3, The Justification of Explanation, Swinburne looks at the grounds for deciding whether a particular explanation of some phenomenon is a good one.

Swinburne is more interested in his scientifically unanalysable personal explanations, but starts out with scientific explanations. And he seems to have some very odd ideas about how scientific theories are developed.

He starts out by defining a term he uses a great deal from here on: prior probability. According to Swinburne:

The prior probability of a theory is its probability before we consider the detailed evidence cited in its support. The prior probability of a theory depends on its degree of fit with background knowledge (an a posteriori matter), and on its simplicity.

This is so wrong and backwards that it is hard even to start explaining how terribly bad this line of reasoning really is.

Firstly, the fact is quite simply that scientific theories aren’t devised this way. As a scientist, you don’t start with a theory and then go looking round for detailed evidence which might or might not end up fitting it. You start with the detailed evidence which is so far unexplained by existing theories, and see if you can work out a theory which explains it. Starting with a theory and then looking around for evidence for it is what many religious people imagine scientists do (as I have mentioned before), but I hadn’t expected somebody of Swinburne’s academic achievements to fall into this particular trap.

Next we have this dread word probability again. We aren’t dealing with statistical data, nor are we dealing with known causes. When we develop a new scientific theory, we are trying to elucidate previously unknown causes. The techniques of probability mathematics are entirely inappropriate here. When your causes are unknown, even if you think that you are developing a probabilistic theory, such as used in quantum mechanics, you do not and cannot evaluate the prior probability of a theory in this way.

And even in those cases where you do use probability and statistics a lot, such as in the analysis of clinical trial data to assess the effectiveness of some new drug, you can’t go back into your data and revise your hypothesis so that the data is now being used to answer a different question from the one you were asking before you collected the statistics, so that you get some kind of positive answer. Games like that make the mathematics go all wonky, even when the use of statistical techniques is appropriate.

Lastly, the scientific understanding of simplicity is quite different from Swinburne’s. A scientific theory is regarded as appropriately simple if it includes no more than is necessary to explain the phenomena in question and make predictions concerning the future behaviour of them and possibly also of other phenomena so far unobserved. So Newton’s theory of gravity is appropriately simple because it talks of a gravitational force, and describes its strength. It doesn’t make the claim that the sun exerts its force on the planets by sending out teams of invisible horses to drag the planets along their orbits. Such a claim (whether or not it happened to be true) offers no predictive power and no additional explanatory power relative to the phenomena addressed by the theory.

It is utterly meaningless to say that the theory would have been “simpler” had the gravitational force been inversely proportional to the distance between bodies rather than to the square of the distance. That greater supposed simplicity has no effect on whether the theory has a higher “prior probability” of being right before you look at the detailed evidence, because you already know that the detailed evidence doesn’t fit the simpler theory, and so you know (without any kind of evaluation of probability) that the simpler theory is simply wrong. In any scientific theory, you make our explanation as complex as is necessary to provide a generalisation which allows you to explain existing phenomena and predict future phenomena.

Swinburne then goes on to look at personal explanations and the use of prior probability. Leaving aside his dubious claim that personal explanations cannot be analysed scientifically, he is on somewhat firmer ground here, because there are lots of people in the world, and you can make statistical analyses of the sorts of things they do, and “prior probability” can return to its traditional meaning within the realms of statistical mathematics, particularly of Bayes’ Theorem. In other words, when making theories about humans behave, it is perfectly possible to create those theories in the form of P-inductive arguments.

But this doesn't help much, since Swinburne isn’t much interested (at least not in this book) in the prior probability of events caused by humans. He is interested in the “prior probability” of events of unknown ultimate cause and which he thinks might have been caused by God. At this point he is back into the realms of serious abuse of mathematics and statistics. All the equations he quotes are all perfectly good equations – when used within their appropriate context. As far as I can tell, he hasn’t made any obvious mathematical howlers, though quite frankly I haven’t looked all that hard because it really doesn’t matter whether he has the algebra right or not. The use of Bayes’ equations in this context is totally inappropriate and any conclusions based on them are completely worthless.

Swinburne on Inductive Arguments

(I haven't finished with Spong yet, but have decided that I will run the remaining chapters of Spong in parallel with some comments on Richard Swinburne's The Existence of God. Here are some comments on the first chapter of the latter.)

Richard Swinburne is one of Britain’s leading academic theologians. He is a Fellow of the British Academy. From 1985 to 2002 he was Nolloth Professor of the Philosophy of the Christian Religion at the University of Oxford.

Swinburne says in the Preface to the second edition of The Existence of God:

The Existence of God is the central book of all that I have written on the philosophy of religion. It was originally published in 1979. A ‘revised edition’ was published in 1991, but the revision consisted merely in the addition of two appendices; the main text remained intact. The present revision is a far more substantial one.

He goes on to describe the various chapters that have been changed, and the nature of those changes. The book has received good reviews, and it contains material originally published in various academic journals, including Philosophy, Religious Studies, Reason and Religion, American Philosophical Quarterly, Physical Cosmology and Philosophy, Comparative Theology, and Faith and Philosophy. The book is used as a textbook in undergraduate courses on theology and the philosophy of religion.

So this is not some ignorant backwoods preacher who wouldn’t know a philosophical argument if it hit him over the head. This is a serious theologian who sits at or near the top of his academic discipline, and The Existence of God is the book which in his own words is central to his understanding of the philosophy of religion, and which is the result of more than 30 years accumulated thought and wisdom on the subject. If Swinburne had turned up on my blog in response to my invitation to believers to produce an argument in favour of God’s existence, I could hardly have found somebody more qualified to put the case. I’m going to take this book as a kind of response to that invitation.

A bit first on the structure of the book. This is a serious academic work, the language is densely packed - it is not really intended for a popular audience. I don’t mind that - I’m prepared to put in the hard work of reading and understanding it, and explaining to you what it means to the best of my understanding.

If you are looking for simplistic arguments such as arguments from scripture, you will be disappointed. There is little or no biblical quotation, and nothing in the way of argument that “the Bible says X, therefore it must be true”. I’m sure that Swinburne is perfectly well aware that circular arguments (using your assertions as evidence of their own truth) do nothing more than go in circles. Swinburne also makes a deliberate decision to leave aside ontological arguments in this book. All his arguments have as a starting point at least one known and largely undisputed physical fact, such as the existence of the universe.

But it is quite a way into the book before he gets on to any of these arguments. He starts out in the early chapters by describing what he thinks of as being a good structure to an argument. The first six chapters have the following titles:

1. Inductive Arguments
2. The Nature of Explanation
3. The Justification of Explanation
4. Complete Explanation
5. The Intrinsic Probability of Theism
6. The Explanatory Power of Theism: General Considerations

It is only in chapter 7 (after 132 pages of introduction) that he starts on the first of his arguments in favour of God “The Cosmological Argument” using the principles he establishes in the first 6 chapters. He then goes on in subsequent chapters through Teleological Arguments (i.e. arguments from design), Arguments from Consciousness and Morality, The Argument from Providence, The Problem of Evil, Arguments from History and Miracles, and The Argument from Religious Experience. He rounds it all off with a short chapter “The Balance of Probability”.

It would be tempting to skip the first 6 chapters and go straight to the actual arguments. But unless we look first at what Swinburne thinks characterises a good argument, and whether he is right, it is not going to be possible to work out what he is talking about when dealing with the actual arguments.

So it is going to be necessary to look at these early chapters even through they don’t directly address the question of God.

The first chapter is called “Inductive Arguments”. Swinburne starts out by offering three examples of differing kind of arguments. He offers the following as an example of a valid deductive argument, where the premises (provided they are true) make the conclusion certain.

P₁: No material bodies travel faster than light.
P₂: My car is a material body.
C: My car does not travel faster than light.

The next example he gives is what he calls a P-inductive argument, where the premises, while not making the conclusion certain, render it probably true. How good the P-inductive argument is depends on how probable the premises render the conclusion.

P₁: 70% of the inhabitants of the Bogside are Catholic.
P₂: Doherty is an inhabitant of the Bogside.
C: Doherty is Catholic.

The third kind of argument Swinburne describes is what he calls a C-inductive argument.

P: All of 100 ravens observed in different parts of the world are black
C: All ravens are black.

That you have seen 100 black ravens does not by itself render the probability very great that all ravens everywhere (past present and future) all have been, are and will be black. But according to Swinburne each additional black raven you come across increases the probability that the conclusion is correct.

Swinburne states

Most of the arguments of scientists from their observational evidence to conclusions about what are the true laws of nature or to the predictions about the results of future experiments or observations are not deductively valid, but are, it would be generally agreed, inductive arguments of one of the above two kinds.

This is true, but before we see how Swinburne proceeds from here, there are a few things that need to be said about that statement.

First, a P-inductive argument can only be built if you have statistical data to work from, in the form of a population of some entities whose characteristics vary, and about which one can draw conclusions based on statistical mathematics. The mathematics of statistics has been well developed over the past 100 years or so, to the extent that a decent understanding of statistics is a necessary part of the syllabus in most scientific subjects at degree level. This is not merely the physical and biological sciences, but also all the engineering disciplines, computer science, and social sciences including sociology, geography and history.

If you want to be a mathematician in this field, it is necessary to know precisely how the mathematics works in order to make new discoveries (otherwise you can end up re-inventing old discoveries). If you merely want to make use of statistical techniques, in order to draw justified conclusions in other fields, then the vital knowledge needed is whether and under what circumstances it is valid to use a particular technique. If you use a statistical technique in circumstances it is not valid for, then your conclusions are worthless - though anybody who doesn’t understand statistics would be unaware of the fact.

Secondly, a C-inductive argument is only valid if you have not come across any contrary observations. In the example given, the observation of a 100 or any larger number of black ravens doesn’t contribute at all to the probability that all ravens are black if you are already aware of the existence of an albino raven. Therefore, conclusions that you draw on the basis of C-inductive arguments are always provisional, until they are overthrown by a contrary observation (i.e. a raven of another colour), or are explained by the discovery of some underlying natural law which explains why all ravens must be black and that it is not possible for ravens to be any other colour. If scientists have a natural law which has been established by means of a C-inductive argument, they always continue to look for that underlying explanation.

Thirdly, to use the term “probability” in the context of C-inductive arguments is misleading, because it suggests you are dealing with statistical mathematics when in fact you aren’t.

This is all very important. Swinburne in the first chapter states that he believes there to be no valid deductive arguments either for or against the existence of God which are based on premises universally accepted to be true. Therefore, the entire book consists of an estimate of the balance of probability of God’s existence, based on the strength of the various P-inductive and C-inductive arguments he works his way through.

The main tool that Swinburne uses is Bayes’ Theorem. In brief summary, Bayes’ Theorem is a mathematical tool that allows you to work backwards when calculating conditional probabilities. Even without Bayes' Theorem, it is easy to work out the probability of some overall outcome consisting of some combination of individual events by multiplying the probabilities of individual events.

Bayes’ Theorem allows you to work in the opposite direction. If that you know that various combinations of events of different probabilities that can lead to a particular outcome, Bayes’ Theorem allows you to calculate the relative probability of the various possible routes to that outcome, given that it has actually happened.

Swinburne doesn’t mention Bayes’ Theorem by name until Chapter 3 and page 66, but he starts using concepts and terms from it right in the first chapter. This is clear from the fact that the index contains the term “confirmation theory” giving page ranges in the first chapter, and then also says under the term “see also Bayes’s Theorem”. Let’s let Swinburne take up the narrative in his own words.

My strategy will be as follows. Let h be our hypothesis - ‘God exists’. Let e₁, e₂, e₃, and so on be the various propositions that people bring forward as evidence for or against his existence, the conjunction of which form e. Let e₁ be ‘there is a physical universe’. Then we have the argument from e₁ to h - a cosmological argument. In considering this argument I shall assume that we have no other relevant evidence, and so k will be mere tautological evidence [i.e. all other irrelevant knowledge]. Then P(h|e₁ & k) represents the probability that God exists given that there is a physical universe - and also given mere tautological evidence, which latter can be ignored. If P(h|e₁ & k) > 1/2 then the argument from e₁ to h is a good P-inductive argument. If P(h|e₁ & k) > P(h|k), then the argument is a good C-inductive argument. But when considering the second argument, from e₂ (which will be the conformity of the universe to temporal order), I shall use k to represent the premiss of the first argument e₁; and so P(h|e₂ & k) will represent the probability that God exists, given that there is a physical universe and that it is subject to temporal order.

You should be able immediately to spot the problem. Since we aren’t dealing with statistical propositions, Swinburne has no statistical data to work from, and so has no numbers to plug into Bayes’ equations.

Moreover, Bayes’ Theorem works only on known causes with known probabilities, (known at least to some degree of precision). Swinburne’s situation is that we don’t know the causes of the universe. Because we don’t know, we have no means of allocating probabilities to this or that hypothesis. All we can do is keep looking for more evidence that enables us to form a hypothesis that goes further than our present knowledge. And when we do so, we still won’t be able to assign a probability to it, since we don’t have a population of universes of differing characteristics from which we can draw statistical conclusions. Any attempt to put a number on it is entirely arbitary, nothing more than saying "I think this is 60% probable because I think that number sounds about right."

This to me seems to be a misuse of statistical techniques so basic that it ought to go into statistics textbooks as a classic example of how not to do it.

I find it hard to account for this remarkable state of apparent ignorance on Swinburne’s part. It seems that he really doesn’t understand the limits of Bayes’ Theorem and the purposes to which it can be put. This would require that in the 17 years in which he was Nolloth Professor at Oxford, surrounded by some of the world’s leading academic experts on almost every topic under the sun, no academic with any knowledge of statistics was invited to read the first chapter of his book (for instance when he was preparing the Revised Edition while at Oxford) and advise whether the use of Bayes’ Theorem in this context was appropriate, and that Swinburne, though surrounded by all these experts, never consulted any of them in order to ensure that he had a correct understanding of Bayes’ Theorem.

Furthermore it would appear that none of the journals which published material subsequently incorporated into this book noticed anything amiss, and likewise none of those who conducted peer review of any of his papers noticed any problem.

If this is the case, then this doesn’t merely reflect extremely poorly on Swinburne himself, but on the whole academic study of theology, and the extent to which it has walled itself off from any other field of knowledge.

There is another possibility: that I’m mistaken in my own understanding of Bayes’ Theorem, and that Swinburne’s use of it is in fact justified. I’ve given my reasons for thinking I’m right. Bayes’ Theorem is well enough documented. You can look it up and judge for yourself.