Hume, D. (1739) A Treatise of Human Nature.

Mackie, J. L. (1973) Truth, Probability, and Paradox

Ramsey, F. (1928) ‘Universals of Law and Universals of Fact’

Lewis, D. (1978) Counterfactuals

Armstrong, D. A. (1983) What is a Law of Nature

Introduction

The Humean Problem of Causes

This problem was again first identified by David Hume[1], though he preferred to talk about billiard balls rather than ugly children, and we shall follow his lead. So, consider the causal relation that exists between two events involving billiard balls. In the first event, one billiard ball strikes another. In the second event, which immediately follows the first event, the struck ball moves. We say that the first ball striking the second was the cause of the second’s movement; and by this we mean that there is some power in the first event that brings the second event into existence – there is some necessary link between the two events. We say this, says Hume, because in the past we have often seen similar events linked in the same way by close succession in time, and we have concluded that the one event is always to be associated with the succeeding second event, and that the event second in time is in fact brought about by the event that precedes it. But, as Hume points out, the only induction that we can make from all these repeated observations is something like this:

One billiard ball striking another is followed by the other rolling away

A bat striking a ball is followed by the ball flying off

…

A dog running into a table is followed by the table moving over

-------------------------------------------------------------------------------------

Any object striking another is followed by the other moving off

or

If an object strikes another object then the second object will move off

And this, Hume says, is not sufficient to ground a causal claim in the strong sense that we started with. There is no observation at any time of the necessary link that we suppose to exist between ‘causes’ and ‘effects,’ and there is nothing in the inductions from experience available to us that can introduce this idea of a power. All that such evidence can establish is that there is a constant conjunction of events. Hume’s conclusion is that all talk of causes and effects is really just talk about such constant conjunctions of events, and the concept of a ‘cause’ as a necessary link between events does no more than label the fact that our minds so easily move from ideas of ‘causal’ events to ideas of ‘effectal’ events – a move that is the result of our becoming accustomed through repeated observations to one of those events succeeding the other. To be blunt, he thinks that the concept of a cause merely marks a groove worn in our mind by observations of similar events – like a bare path in a grass lawn made by people taking short cuts across it.

But Hume’s solution is hardly satisfactory, for it rejects the idea of there being anything to distinguish between a constant conjunction of events that is just accidental and one that is not accidental. Consider, for example two such constant conjunctions of my own experience. First, every time in the past that I invited people over for a barbecue it rained, and so I gave up sending out such invitations. Second, every time I left the park lights on in my previous car, the battery would be flat when I got back. We want to be able to say that there is a difference between the two conjunctions. The first is just bad luck with barbecues and tells us nothing about the way the world is. There is no necessary connection between the event of my invitation and the event of it raining; and I would not be justified in drawing the conclusion that ‘if I invite people to a barbecue, then it will rain’ because it is merely what is called an ‘accidental uniformity.’ On the other hand, the second is not just bad luck but does tell us something about the world. Leaving the lights on really did bring it about that the battery went flat; and I am justified in drawing the conclusion that ‘if a current is drawn from a battery for long enough then the battery will lose its charge.’ This is the kind of statement that we think of as describing a Law of Nature; and our problem is to discover how we make this distinction between accidental uniformities and laws – or, to say it better, how could we rationally defend the distinction that we do make?

Inductive Support

There have been a few reasonable modern efforts at making this distinction, but we’ll only look at two of them. The first of these is due to the English/Australian philosopher, J. L. Mackie[1]. The difference between accidental uniformities and laws of nature, he proposes, is that the instances of accidental uniformities just do not provide inductive support for a possible regularity, whereas the instances of a law of nature do provide such support. It’s easier to understand the point that he’s trying to make by looking at a couple of examples. Consider the accidental uniformity I mentioned above of rain occurring whenever I plan a barbecue. Mackie says that the instances of rain occurring when I make such plans just does not give us any good reason to believe that in future cases of barbecue planning rain will similarly occur. On the other hand, in the car battery example, past instances of draining batteries by leaving accessories running does give us good reason to believe that future cases of leaving accessories running will result in similar battery draining.

On the positive side for this suggestion, this does give us a way of understanding the different attitude that we take to accidental uniformities and Laws of Nature. We feel that we can only really be sure that an accidental uniformity is true if we have investigated all the possible cases and found that they confirm it. I can assure you that it has been confirmed in all the cases of my barbecue planning so far, but is this enough? No, it isn’t; because there’s always the possibility that I will try just one more time – and we don’t think that the previous instances are really going to help us predict what will happen then. But the whole point of identifying a law of nature is that we should feel confident about what it says about events that have not yet occurred. We shouldn’t need to exhaust all possible instances to be assured of the truth of it. In the case of the invariably draining battery, we are very confident that if similar circumstances are created as in the previous instances, the same result will be achieved, and this is why we are inclined to associate that regularity with Laws of Nature rather than accidental uniformities.

On the other hand, merely rephrasing our intuitions is not at all the same thing as explaining how we have them or as justifying them. We saw a similar attempt to explain something by simply restating our intuitions when we were considering Goodman’s solution to his New Problem of Induction. You’ll recall that the New Problem was that the observations in any induction using ‘normal’ properties like blue or brown could also be made to feature in inductions using ‘odd’ properties like brue or blown, and those inductions would all give contradictory or inconsistent answers, and a method of inference that gives inconsistent answers is no system of inference at all. Goodman’s proposed solution was to declare that some predicates were projectible – meaning that inductions using them would work – and that some predicates were not, and inductions using them wouldn’t work. And which are the projectible predicates? Well, they’re the ones that have been shown in the past to work. (Or they were used before tomorrow and worked at any rate.) In fact, Mackie’s solution to this related problem is merely an adaptation of that unsatisfactory solution, and is subject to the same objection. Until the defenders of this proposal can come up with some independent distinction that amounts to more than ‘we can just see the difference’ this just won’t wash.

The System Solution

A different solution comes from Frank Ramsey[1] via David Lewis[2]. This solution takes not the laws themselves to be fundamental, but the system of deductions into which they fit. You’ll recall that Aristotle’s view of science – which we’ve been accepting as our own for the time being – sees knowledge as a system of deductions (demonstrations really) ultimately based on a small set of independently known truths, which we can think of as the axioms of the system. (They are like the initial definitions of ‘line’ and ‘point’ in a geometry text and all the rest of the truths of geometry, from Pythagoras’s theorem to the most complex theorems concerning conics, can be derived from them.)

The ambition of science is to include all the true statements about the world in that system of deductions. That includes all the general statements that we think of as Laws of Nature as well as all those that we want to set aside as mere accidental uniformities. Now, there are many different ways that we can do that; we can, for example, include all those true general statements as fundamental statements/axioms, or we can try to minimize the number of axioms so that the highest proportion of the truths of the system turn out to be deductions from first principles. In the first case, the system would be as strong as possible because it does include all the known truths. In the second case, minimizing the number of axioms makes the system as simple as possible, at the cost of omitting some of the known truths (perhaps temporarily) which must be included as ad hoc additions to the system. Those two approaches are two ends of a continuum, and there are an infinite number of different arrangements between them. One way of looking at the task of science is to see it as an attempt to find the systematization that is optimal in terms of strength and simplicity.

Given this view of science, Lewis’s claim is that we should treat as Laws of Nature all and only those general statements that appear as axioms and theorems in any optimal systematization. What this means is that if a general statement appears as an axiom or a theorem in all the systems that have the best balance of strength versus simplicity, then that statement counts as a law of nature. If it doesn’t appear in that way in those systems then it’s just an accidental uniformity. The general statement that batteries left with current being drawn from them will eventually be completely drained is the sort of general statement that can be derived deductively from other more basic statements about electron flow, potential differences, chemical reactions, and so on. Every optimal systematization will include those fundamental statements and thus will include as a theorem the statement about batteries being drained. On the other hand, it is probably not the case that every optimal systematization will include the statement that my barbecue invitations are followed by rain, because that doesn’t seem to be the sort of statement that is deducible from the most fundamental statements of those systems.

That might look a bit handwave-y to you, and it actually is quite vague. There are some pretty obvious criticisms that will need to be met if this proposal is to be more than a suggestion. For a start, we’re going to have to say something a good deal more precise about the two criteria ‘strength’ and ‘simplicity.’ Take ‘simplicity’ for example. This term is supposed to explain a preference for fewer rather than more axioms, but why should this be supposed to be a virtue to be weighed against the ‘strength’ of a system? Apparently it has something to do with our ability to use a set of axioms to derive further statements, and our capacity to do so is a limit on the usefulness of a system. This limit has a lot to do with our intellectual abilities – our memory, rationality, ability to concentrate, etc.; but why should membership in the set of ‘Laws of Nature’ be in any way related to our intellectual capabilities? It seems quite inconsistent with how we like to think of Laws of Nature. We wouldn’t accept that there would be different Laws of Nature for investigators who were less intelligent than ourselves; nor would we accept that super-intelligent aliens could be subject to a different set of Laws of Nature from ourselves.

You might, nevertheless, consider that such questions are beside the point – you might think that the system solution is not supposed to provide actual criteria for application, but simply to describe/define what it is to be a law of nature. A different approach is, however, also available, and that is to deny that the Humean analysis of causation and Laws of Nature is correct.

Contra Hume

Remember that the whole point of Hume’s analysis was to show that since we could not distinguish through our senses that any two particular events were connected in any other way than by one immediately succeeding the other in time, therefore no connection other than that appeared in any of our inductive premisses, and therefore no connection but that could appear in the inductive conclusion. We were stuck with the following argument as the only way to draw conclusions from the given data:

Object A striking object B is followed by B moving off

Object C striking object D is followed by D moving off

…

Object Y striking object Z is followed by Z moving off

-------------------------------------------------------------------------------------

If an object strikes another object then the second object will move off

which we recognized as both invalid (being an inductive and not a deductive argument) and incapable of introducing a concept of necessary connection. The fact that we did, actually, conclude that there was a connection – a causal connection – was explained away as a mere trick of the mind. But how convincing should this be as an argument against there actually being some other connection that is not apparent to our senses? Stated in that way, it would seem pretty obvious that Hume’s argument is rather a weak one by which to support such a remarkable conclusion.

The fundamental problem is that Hume assumes that the only way that we can come up with a general statement as a conclusion to several particular observations is through the sort of enumerative induction that we’ve been describing. But Hume only insists upon this because he is advocating a particular theory of the mind. We aren’t bound by any such restrictions as to theory, so we shouldn’t be bound by his restrictions as to the allowable forms of induction. If we can accept that the concept of necessity is possible to us independently of the notion of causation (And why wouldn’t we? There are many forms of ‘necessity’ other than causal necessity) then there is a perfectly acceptable form of inductive argument known as the Inference to Best Explanation which will justify the conclusion that there is a necessary connection between two events such that the one is always followed by the other. [1]

In an inference to best explanation we have a set of phenomena and we choose from a selection of possible explanations the one that we think is the best according to some reasonable criteria. A simple example will make this clearer:

You return home to find your door broken and some valuable items missing.

Possible explanations include:         

                1.             A meteorite struck your door and vaporised your valuables.

                2.             Friends are playing a joke on you.

                3.             A police Tactical Response Group entered your house mistakenly.

                4.             You were robbed.

                                Explanation 4 seems the best, so you conclude

                                -------------------------------------------------------------------------------------------------------

                                You were robbed.

More generally, inferences to best explanation take the following form:

                Phenomenon C is observed

                A explains C and does so better than any rival explanation

                -------------------------------------------------------------------------

                A

The underlying assumption is that the ‘best’ explanation of a phenomenon is most likely to be true. We won’t question that assumption at this time, and neither will we go into the details of the criteria by which we judge one proposed explanation to be better than another – they are largely the sorts of common sense things that you’d expect. Instead, let’s see how it works for our current problem:

Object A striking object B is followed by B moving off

Object C striking object D is followed by D moving off

…

Object Y striking object Z is followed by Z moving off

These observations could be explained if the event of an object of a certain type striking another object of that type was necessarily connected to the event of the second object moving off.

This is the best explanation on offer for those observations

-------------------------------------------------------------------------------------

The event of an object of a certain type striking another object of that type is necessarily connected to the event of the second object moving off.

This would certainly explain how it is that we see such a constant conjunction of such events – it’s because they have this relationship of necessity existing between them. This is why my car lights being left on will result in the battery discharging. And at the same time it allows us to accept that there may be accidental uniformities (things that do in fact always occur together) without that in any way being a puzzle. Those are just accidents, they don’t occur because of some deep relationship between those kinds of events. My issuing of invites to a barbecue does not have that necessary connection to the occurrence of rain. The fact that we may have difficulty identifying the events that have that connection and those that don’t would be neither here nor there.

Of course you might well doubt that this explanation actually explains anything: simply saying that two things are necessarily connected doesn’t tell you much until you know something (anything) about this necessity. And it doesn’t help much to say that it’s what we call ‘causation.’ We were trying to explain causation, not just give it a range of different names. For this reason, this fairly common-sense solution has not yet convinced everyone.