Sunday, 16 August 2009

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.


  1. On page 16, Swinburne clarifies:

    "I shall in future call P(p|q), the logical probability of p on q. This is clearly an a priori matter. If q represents all the relevant evidence, the value of P(p|q) cannot depend on further evidence - it measures what evidence you have already got shows. It is an a posteriori matter whether, in 1,000 tosses, 505 have landed heads; but an a priori matter whether that evidence gives a probability of .505 to the next toss landing heads."

    He is working with an intepretation of probability, logical probability, which is distinct from statistical (or frequentist) probability. Unfortunately, he only implies that it is distinct, while putting his explication of it in his book Epistemic Justification, where it is distinguished from subjective probability and epistemic probability.

    Bayes' Theorem is used in historical inquiry as well, so I'm quite certain one need not assume a frequentist view of probability in order for it to be utilized.

  2. Philip
    I realise that he's working with an interpretation of probability, because he certainly isn't working with probability itself because he has no statistics to use.

    If Swinburne wanted to make it clear that he wasn't dealing with statistical probability, he should have made no mention of Bayes' Theorem anywhere in the book.

    But since he does, the book is absolutely littered with references to Bayes and mentions of terms from his equations (literally in every chapter), then either he doesn't understand that he's not dealing with data and propositions that can't be analysed using statistical tools, or he does realise it and hopes that you don't. I wouldn't like to guess which of those two possibilities is true, and since I know better than to use probability and statistics where they aren't appropriate, I won't put odds on it.

  3. Hi Jonathan, I found your blog via Oystein Elgaroy's discussion of Swinburne. Very interesting article. I thought where does he pull these numbers from? How many creations has he witnessed to judge probability or whatever? It seems he can just pluck numbers out of his bum and voila!

  4. Interesting take on things. And thanks for putting in the work of reviewing all this stuff.

    I don't think I agree that using Beyes Theorem makes no sense within "a priori probability". Assigning such probabilities *is* just a means to say "I think 60% sounds about right", but once you do so Bayes Theorem does probide you with a reasonable way to integrate lots of individual guesses into an overall estimate of the likelihood of the hypothesis. The problem is not with using Bayes Theorem, it is with estimating probabilities wildly and subjectively while pretending to do so objectively and rigorously.

    What Swinburne is doing, essentially, is assuming that one can consider an abstract population of universes, and build a probability measure based on a priori arguments about what such a population would look like under the god hypotheis or without it. This is a reasonable tool for thinking about the problem, even if it isn't backed up with any data. The problem is with constructing the abstract probability space, not with applying Bayes Theorem to it.


  5. Yair
    The point is that numbers have no meaning except within the context of a population on which you can do statistics. You can decide whether a particular drug is effective in treating some condition by trying it out on lots of people and counting up the number of people who improve. From the numbers you have available, it is possible to make a P-inductive argument as to whether the drug probably works or not. (It's actually a bit more complicated than that, but that is the basic idea.)

    There is only one universe (that we know of) and only one God whose existence is being hypothesised. To say that God 60% exists is obviously meaningless. Either he does or he doesn't, and the only way we will ever find out with some degree of certainty is to continue in our search for evidence. But in the meantime, it is meaningless to attempt to apply statistical techniques to a sample set of 1.

  6. Frequency is not the only meaning of probability. Subjectivists have been using it for centuries as a degree of belief. See Bayes' Theorem in the Standford Encyclopedia of Philosophy.

    Consider a murder investigation. The murder happened only once, and in one way. But still the detective can use probabalistic reasoning to form and eliminate suspects, and reach conclusions on who murdered and how. This is based on past knowledge, derived in a frequentist manner, yes; but the calculation of the probabilities of his beliefs about the specific murder is not based on an ensemble of these murders - there is only one murder. In effect, the detective is building a probability space consisting of all ways the guy could have died, assigning prior probabilities according to past knowledge, and then using Bayes' Theorem to narrow down the suspects and scenarios based on evidence relevant to this murder.

    Of course, the problem for Swinburne is that we don't have good frequentist knowledge to base our prior probabilities on when it comes to naturalistic vs. theistic universes. Thus he is forced to make subjective assessments, which would have been fine except that he pretends he is making objective ones. Well, that should have been Swinburne's problem - as it is, he manages to entangle himself in quite a few others...