Probability

A Philosophical Introduction

D.H. Mellor

• Publisher: Routledge
• Year: 2004
• Print ISBN: 9780415282512
• eBook ISBN: 9781134458608
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

Probability: A Philosophical Introduction introduces and explains the principal concepts and applications of probability. It is intended for philosophers and others who want to understand probability as we all apply it in our working and everyday lives. The book is not a course in mathematical probability, of which it uses only the simplest results, and avoids all needless technicality.

The role of probability in modern theories of knowledge, inference, induction, causation, laws of nature, action and decision-making makes an understanding of it especially important to philosophers and students of philosophy, to whom this book will be invaluable both as a textbook and a work of reference.

In this book D. H. Mellor discusses the three basic kinds of probability – physical, epistemic, and subjective – and introduces and assesses the main theories and interpretations of them. The topics and concepts covered include:

* chance

* frequency

* possibility

* propensity

* credence

* confirmation

* Bayesianism.

Probability: A Philosophical Introduction is essential reading for all philosophy students and others who encounter or need to apply ideas of probability.

1
Kinds of Probability

I
Introduction

We all use probability and related ideas all the time. To explain and relate all these applications we shall need to regiment our presentation of them in one or two ways. One is by ignoring trivial verbal variations, such as the use of ‘likely’ for ‘probable’. Another and more important way is by classifying probabilities into three basic kinds: physical, epistemic and subjective. Physical probabilities will, for brevity, also be called chances and subjective probabilities credences.

Here are two examples of each kind of probability.

Chances:

Smokers have a greater chance of getting cancer than non-smokers have.

This coin toss is unbiased: it has equal chances of landing heads and tails.

Epistemic probabilities:

Astronomical data makes it very probable that our universe had a beginning.

New evidence makes it unlikely (= improbable) that the butler did it.

Credences:

I think it will probably rain tonight.

I'll bet you 4:1 that Pink Gin doesn't win the Derby.

The basis of this classification seems to be something like the following.

Chances are real features of the world. They show up, in these examples, in how often people get cancer and coin tosses land heads, and they are affected by whether people smoke, and by how coins are tossed. Chances are what they are whether or not we ever conceive of or know about them, and so they are neither relative to evidence nor mere matters of opinion, with no opinion any better than any other.

Epistemic probabilities seem not to be real features of the world in this sense. They only measure how far evidence confirms or disconfirms hypotheses about the world, for example that our world had a beginning or that the butler did it. But they are not mere matters of opinion: whether, and to what extent, evidence counts for or against a hypothesis looks like an objective matter.

Credences measure how strongly we believe propositions, like the proposition that it will rain tonight, or that Pink Gin will win the Derby. They are features of the people whose credences they are rather than features of what the credences are about. My low credence in Pink Gin's winning the Derby is more a fact about me than it is about Pink Gin or the Derby. By this I mean that my low credence that Pink Gin will win is a mere matter of my opinion, which need not be justified by any corresponding physical or epistemic probability. In particular, there is no contradiction in your credence in Pink Gin winning being higher than mine even if we both know all there is to know about the race: that after all is why you take my bet.

There is a third way in which our discussion of probabilities will need to be regimented: in terminology. As well as replacing ‘likely’ and ‘unsure’ with explicitly probabilistic terms, we shall need a general term for whatever may be called more or less probable, as in ‘It will probably rain tonight’. What is said to be probable here may be called an event, a fact or a state of affairs, but for our purposes proposition will be better, as in ‘the proposition that it will rain tonight’. The point is that there can be a probability of rain tonight even if it does not rain tonight, in which case that event, fact or state of affairs will not exist. But the proposition will: it will just not be true. (In what follows it will not matter what propositions are, nor how their truth is related to that of sentences or statements, issues which we therefore need not discuss.)

So by the probability of any proposition A, written ‘P(A)’, I shall mean the probability that A is true, for example that it really will rain tonight. The three kinds of probability I shall distinguish where necessary by writing ‘CH(…)’ for chances, ‘EP(…)’ for epistemic probabilities and ‘CR(…)’ for credences. Where I write ‘P(…)’ I mean what I say to apply to all probabilities.

Despite the initial plausibility of this division of probabilities, it remains to be seen how different our three kinds really are. A's chance might for example just be A's epistemic probability relative to all the relevant evidence we have or could get. A's epistemic probability might in turn just be the credence in A that we ought to have given our evidence about it. And my credence in A might just be what I think A's chance or epistemic probability is. These however are contentious claims, which we shall discuss in more detail later, where we could not even state them without drawing a provisional distinction between these kinds of probability.

Pending our later discussion, we must not read too much into the above labels for kinds of probability, and the cursory explanations of them. It is not easy to say, for example, what kind of entities chances are: different theories tell different stories, as we shall see, and some theories deny that any such entities exist at all. Similarly with credences, whose existence is also disputed, since some philosophers deny that belief comes by the degrees that credences are supposed to measure. Similarly with epistemic probabilities, which are no less contentious. Some philosophers deny, for example, that evidence which fails to falsify a scientific theory confirms it to any positive degree; while others, who admit that such confirmation is possible, decline to use probabilities to measure it.

These are among the controversies that call for caution in reading our labels, which are not meant to define the probabilities they stand for, still less to show that such probabilities exist. Their purpose is simply to draw some provisional distinctions without begging too many of the questions we shall discuss later. Meanwhile the best way to show what the labels are meant to stand for is to give some more, and more serious, examples.

II
Chances

The role of chance in modern physics is hard to overstate. Theories of microphysics now ascribe chances to (propositions about) almost all small-scale events. These include the interactions of subatomic particles, and the transformations of atoms on which both nuclear power and nuclear weapons depend. How similar all these physical applications of numerical probability are is indeed a moot point, although most philosophers of physics would agree that they all fit the minimal assumptions I have made about chances, which are all we need to distinguish chances from credences and epistemic probabilities.

This is why, to avoid raising controversial but irrelevant questions about, e.g. the role of probability in quantum theory, I shall stick to its simplest and least contentious microphysical application, to radioactivity. On the well-established theory of this, all unexcited atoms of a given radioelement have an equal chance of decaying in any given interval of time. Atoms of radium, for instance, have a 50/50 chance of decaying within a definite period, of about 1600 years, which is, for that reason, called its half life.

My calling chances physical probabilities must not however be taken to imply that they figure only in physics. On the contrary, they figure in almost all modern sciences. There are the chances of the mutations on whose natural selection the generation and evolution of plant and animal species depends. Then there are the chances needed by natural selection itself: the chances of individuals with various genetic traits surviving to produce offspring with those traits.

Chances are also essential to explanations of how epidemics of infectious diseases like influenza start, spread and die away. What explains the course and duration of these epidemics is the way the chances of infection rise with the proportion of those already infected, and are reduced by vaccination and by immunity produced by the disease itself.

Psychology too involves chances, some of them with obvious practical implications. Take the way our chances of misremembering arbitrary numbers rise with the number of digits in them. Our chances of remembering four digit numbers correctly are high, which is why PINs have only four digits. Thereafter our chances of misremembering numbers rise rapidly, which is one reason whycredit card numbers are so long.This example, like that of epidemics, shows how chances can be as importantin practice as they are in theory. Reducing the chances of illness, and raisingthose of recovery from it, are the core concerns of medicine and public health. Making road, rail and air travel safer is a matter of reducing the chance of death and injury it involves. Insurance premiums are based on assessments of the chances of whatever is insured against. And so on and so on.

Chances, in short, are everywhere, and not only in the games of chance to which much of the modern theory of probability was originally applied. It may therefore seem surprising that anyone should deny, as many philosophers have done, that there is in reality any such thing as chance. That denial will become less surprising when we see how hard it is to say what chance is, and how many rival theories of it there are. That however is the business of later chapters. Here I aim only to show the apparent ubiquity of chance, which is what makes understanding it so important.

III
Epistemic Probabilities

If the probabilities postulated by scientific theories are chances, those used to assess these and other theories are epistemic. Take one of the examples given in section I, only put the other way round: the low probability that our universe has existed for ever as opposed to having a beginning. That probability is not a chance, in the sense of being a feature of the world, like the chances of radium atoms decaying within intervals of future time. If the universe did have a beginning, there was never a time when it had some chance, however small, of not having one. The probabilities of these rival ‘big bang’ and ‘steady state’ theories are not physical but epistemic. That is to say, they measure the extent to which the astronomical a...

Table of contents

• Cover
• Half Title
• Full Title
• Copyright
• Dedication
• Contents
• Preface
• Introduction and Summary
• 1 Kinds of Probability
• 2 Classical Probabilities
• 3 Frequencies
• 4 Possibilities and Propensities
• 5 Credence
• 6 Confirmation
• 7 Conditionalisation
• 8 Input Credences
• 9 Questions for Bayesians
• 10 Chance, Frequency and Credence
• References
• Index