David Lewis begins his discussion of causation (Lewis, 1993) by quoting Hume's second definition of causation: "...where, if the first object had not been, the second never had existed," (Hume, Section VII, Part II). Lewis, making use of Robert Stalnaker's earlier work1 on counterfactuals (Stalnaker, 1968), casts causal implication as a species of counterfactual implication. Lewis later develops a full, formal theory of counterfactuals (Lewis, 1973) in which he only elliptically addresses the question of causation during his brief discussion of natural laws. Taken with his earlier writings, it is clear he views natural laws as entailing causal implication and therefore how such would be dealt with in his full counterfactual theory.
Unfortunately, Lewis makes an outrageous claim concerning "miraculous" exceptions to natural laws in trying to drive home a point concerning the relative importance of the matters-of-fact and the natural laws of a particular possible world. This claim seems to fly in the face of the basic tenets concerning natural laws which Lewis appears to accept. Fortunately, these statements can be rejected without crippling the formal system of counterfactual logic Lewis has developed - thereby leaving us a formal system wherein natural laws, which are to be evaluated using counterfactual implication, can be rendered.
THE LANGUAGE OF CAUSALITY
Hume's definition of causation has been filled out (Nagel, 1961, pg 74) by asserting that any causal relation admits the four following traits:
LEWIS' THEORY OF CAUSATION
Lewis' theory of causal dependence can be summarized in the following fashion. Let C={c1, c2, c3, ...} and E={e1, e2, e3, ...} each be families of events, and O(ci) mean that the event ci occurs. In this formalism, it is, perhaps, best to think of the events ci and ei as things that might or could happen, i.e. as 'potential' events or events that are not realized in the actual world unless O(ci) and O(ei) are true.
Let us make this more concrete: C might be the family of events {"I stub my left big toe.", "I stub my right big toe."} and E might be the family of events {"I say, 'Ow! I just stubbed my left big toe.'", "I say, 'Ow! I just stubbed my right big toe.'"}. If we want to assert that I have actually stubbed my left big toe, we would assert that O(c1) was true - O(c1) is a proposition while c1 is an event.
Then the family of events E depends causally on the family of events C iff the family {O(ei)} depends counterfactually on the family of {O(ci)}. It is important to note that causal dependence obtains (if at all) between events, i.e. between the ci and ei while counterfactual dependence obtains (if at all) between propositions, i.e. between the O(ci) and O(ei).
COUNTERFACTUALS
The problem then is to fill out what is meant by "counterfactual dependence". A counterfactual conditional can be neatly defined as "a subjunctive conditional which implies that its antecedent is false" (Haack, 1978, pg 244). Recalling Hume's definition, casual implication can be paraphrased: 'if a were the case, then b would be the case'.
Lewis analyszes casual implications in terms of counterfactual implications, and offers as an example: "If kangaroos had no tails, they would topple over." (Lewis, 1973, pg 1). The antecedent here is clearly not true, and this cannot be treated as a material implication because a material conditional is true whenever the antecedent is false. But the material conditional, 'If kangaroos had no tails, they would not topple over' and Lewis' example obviously cannot both be true - both have false antecedents, but they lead to contrary consequents. This is sufficient to show that causal implication cannot be simply treated as a material conditional.
The question of how to evaluate this new kind of conditional leads to the concept of possible worlds4 (Stalnaker, 1968; Lewis, 1973 and 1993). A possible world is imagined in which the antecedent is true, and different from the actual world only in those aspects necessary to meet this condition. If, in this possible world, the consequent would also be true, then so is the counterfactual conditional; otherwise it is false.
The criterion of 'different only in those aspects necessary' places an important restriction on the possible worlds which can be considered in the evaluation of any counterfactual. Stalnaker uses the words: "otherwise minimally different5 " (Stalnaker, 1968, pg 102); Lewis fills out this distinction formally using the concept of spheres of accessibility (Lewis, 1973, pg 1-43). The idea here is simple: it will not do to appeal to a world where, in addition to the kangaroos not having tails, there is no gravity. Obviously, without gravity to pull them over, tail-less kangaroos would not topple over. But this is not one of the closest possible worlds6 which is minimally different from the actual world - it is hard to believe that to have tail-less kangaroos, the world must be one without gravity. The possible world used to evaluate this counterfactual must be one where no other changes to the actual world are made other than those absolutely necessary to guarantee that kangaroos have no tails.
CAUSAL PROPOSITIONS AND COUNTERFACTUAL LOGIC
The logic of counterfactuals, as constructed by Lewis, is extensional7. This is to say, the logic can address only those properties which reside in the objects of the propositions themselves (Burks, 1951, pg 363-365). Hume argued, however, that the causal relation between causes and their effects was not an extensional property of the objects:
A CAUSE is an object precedent and contiguous to another, and so united with it, that the idea of the one determines the mind to form the idea of the other... (Hume, Section VII, Part II)The causality, Hume argued, is actually in our minds, and not to be found in any object of nature.
We recognize causality, Hume argued (Hume, Section VII, Part II) through a habit of the mind wherein, because in our experience the effect has always followed its cause, we anticipate the effect upon seeing its cause. But in point of fact, the causal relation is not contained within either the cause or the effect - the relation is within our minds and nowhere in nature. The constant connection we attribute to a cause and its effect is limited strictly to those causes and effects of our limited experience. At the very least, to know of a truly causal relation between a given cause and its purported effect, we would have to have experience of every such cause, and in this experience, we would have to find the effect each and every time the cause was found.
This is the problem of induction which Hume so forcefully demonstrated in the Treatise. Even so, the claim of natural laws is that the causal relationship between cause and effect is a property of the cause and the effect, and therefore, an extensional one. While we can never know this to be the case, this is what we assert in statements of natural law. Given that this is the case, we must treat the causal relation as actually residing within the cause and effect, and for the purpose of constructing a logic to model natural laws, treat the causal relation as an extensional property.
CAUSAL LAWS
There is something more to law statements than this 'pseudo-extensional' causal relation between two given objects or classes of objects.
Law statements, unlike non-law statements, seem to warrant inference to statements of the form, "If a, which is not S, were S, a would be P" and "For every x, if x were s, x would be P". (Chisholm, 1955, pg 97)Here, "S" is to be understood as a property which "a" does not have, but which we understand other objects to have, and which, because they have this property, have another property "P". This distinction becomes clear when one considers the following two statements: "All charged particles in electric fields experience an acceleration," and "If I were to spill my beer on your trouser leg, you would hit me." Both seem to involve some sort of causal relation, but the former is easily recognized to be governed by a causal law, while the latter is not.
Unfortunately, the philosophical difference between law and non-law statements is not well understood. Chisholm offers three suggestions on how this difference is to be recognized (Chisholm, 1955, pg 99-100). First, law statements contain no non-universal (i.e. specifically named or identified objects), non-logical terms (no one in particular is credited with this suggestion). Second, non-law statements have the epistemic quality that they are known to have been observed while law statements do not suffer from this restriction (Strawson, 1966, pg 109). Third, law statements form and partake of a cohesive web of relations, and the amenability of previously accepted laws drives the selection of statements as new laws (Braithwaite, 1953, page 303). Chisholm himself suggests that all these are valid to some extent, but also demonstrates that any one of them, by itself, is insufficient. Taken together, it is not clear that these three suggestions account fully for the distinction though.
LEWIS AND THE LAW
Given that natural laws admit the kind of causal relation developed above, and that counterfactual implication is that logical form that ought to be employed to express such statements in formal logic, the question is: how does Lewis and his system address such statements? If one allows the assertion that all laws of nature are identical with causal laws8, then what Lewis says of the former will also be true of the latter.
Lewis appears to accepts something like this9, and in fact, his statements about the laws of nature reflect a union of the three suggested distinctions between law and non-law statements as given by Chisholm above. But Lewis' evaluation of counterfactuals entails an appeal to possible worlds where the matters-of-fact differ from the actual world. Further, the natural laws of these possible worlds (which, may in part, explain these different matters-of-fact) may also differ. So in addition to the "cohesive web of relations" each possible world's natural laws engender, they must also be amenable to matters-of-fact of that possible world. For this reason, Lewis regards laws of nature as contingent (Lewis, 1973, pg 74).
Because we live in only one world, the actual world, we might think of the natural laws of our actual world as necessary. But a logic which allows for possible worlds which can be wildly different from the actual world drives home the point that natural laws are clearly contingent. This leads Lewis to say:
I doubt that laws of nature have as much of a special status as has been thought. Such special status as they do have, they need not have by fiat. (Lewis, 1973, pg 73)Lewis then has cause to question which is more important, laws or facts. His answer is:
Laws are very important, but great masses of particular fact count for something too; and a localized violation is not the most serious difference of law. (Lewis, 1973, pg 75)MIRACLES
The "localized violation" he is allowing is a miraculous and inexplicable violation of a law.
Small wonder if the closest antecedent-worlds to A [a possible world under consideration] are worlds where the particular facts before 't' [a particular time] are preserved at the cost of a small miracle... (Lewis, 1973, pg 75)The demand here is to consider two possible worlds:
In attempting to justify this claim, Lewis asserts:
The violated deterministic law has presumably not been replaced by a contrary law. Indeed, a version of the violated law, complicated and weakened by a clause to permit the one exception, may still be simple and strong enough to survive as a law. (Lewis, 1973, pg 75)It is not clear how sense can even be made of this statement. How such a miracle can be incorporated into a law without violating the first of Chisholm's distinctions between law and non-law statements (the restriction against non-universal non-logical terms), which Lewis appears to accept, is not clear. In fact, on the face of it, this seems quite impossible and contradictory. Perhaps there is some clever or non-obvious way in which this apparent contradiction can be removed.
LEWIS' MOTIVATION
Such statements almost seem to be an attempt to call attention to the fact that the "minimal difference" criterion might require one to regard a world where there is a single difference in natural law as closer to the actual world than a world where the matters-of-fact are grossly different. For a world wherein gravity varied not with the inverse square of the distance between two massive objects (as in the actual world), but instead varied with the inverse of the cube of the distance, might look very similar to the actual world with regard to the matters-of-fact. If there were such a possible world, the "minimal difference" criterion might require us to consider this world closer to the actual world than some world in which the law of gravity is the same as it is here, but the matters-of-fact are wildly different from the actual world.
On the other hand, to allow a one time exception of an "inverse-cube" gravitational "law" between two objects simply cannot be regarded as an "allowable exception" to the "law" - such a one time exception destroys the very notion of natural law. The occurrence of such an event is the exact sort of thing which is called upon to prove a seeming relationship is not a law. The observation of such an event requires one to disregard the relationship and conclude that there is some hitherto unrecognized law in force.
CONCLUSION
Lewis has successfully developed a formal system of counterfactual logic which allows for the evaluation of causal dependence and implication. As naturally law appears to rely upon such dependence, this system also allows for formal evaluation of such laws. Lewis' statements about miracles are incompatible with what appear to be the necessary conditions (although possibly not sufficient) for a statement to be regarded as a natural law. If ignored, they in no way cripple the system, and the basic tenets of what it is for a statement to be a expression of natural law are maintained. This leaves Lewis' formal system of counterfactual logic as a method for both the rendering and evaluation of natural laws.
1...Stalnaker's earlier work...
However, It is well known that Lewis takes exception with certain aspects of Stalnaker's theory, e.g. the principle of Counterfactual Excluded Middle (CEM) which results from Stalnaker's insistence that there is only one closest possible world. Stalnaker's CEM claims it is a tautology that any given antecedent counterfactually implies any given consequent or it counterfactually implies the consequent's negation. Lewis' denial of there being only one closest possible world entails the denial of CEM. (Bonevac, 1987, pg 282)
2...the effect temporally follows the cause...
Lewis calls this a "brute force" solution (Lewis, 1993, pg 556) to the "problem of epiphenomena," on the grounds that certain modern quantum mechanical theories purport backward or simultaneous causation. But this is not a valid objection because these theories assign a "relevant temporal direction" for the causation. It might, however, be appropriate to weaken this third trait in the following way: "...has the characteristic that the effect follows the cause in the relevant sense..."
3...The relation is asymmetrical...
As an example of this, Nagel offers: "...the passage of the spark through the mixture of gasses is the cause of their transformation into water, but the formation of water is not the case of the passage of the spark." (Nagel, 1961, pg 74)
4...the concept of possible worlds...
Some have argued that these possible worlds are epistemologically questionable entities, and Lewis claims that one must take a realistic view of them:
I can only ask him to admit that he knows what sort of thing our actual world is, and then explain that other worlds are more things of that sort, differing not in kind but only one world among others. (Lewis, 1973, pg 85)However, it is not clear that this is the case, i.e., one must be a realist with regard to possible worlds, as there is nothing in Lewis' formal counterfactual analysis of causal dependence that requires it.
5...otherwise minimally different...
Another common complaint is that possible worlds are unacceptable for use in analysis due to the inherent vagueness of the "minimally different" criterion. Lewis responds to this somewhat in the following quote, but full justice can not be done to this objection here. Suffice it to say that the present author agrees with Lewis when he says, "It is vague - very vague - in a well understood way." (Lewis, 1973, pg 91)
6...one of the closest possible worlds...
As mentioned previously, Lewis and Stalnaker disagree on the allowable number of closest possible worlds - with regard to any possible counterfactual antecedent, the former allows an infinite number of closest possible worlds while the latter requires that there be only one closest possible world.
7...is extensional...
This is, in fact, true of most formal logic systems, though there have been modern systems which are purposefully intensional.
8...laws of nature are identical with causal laws...
This will be left here as an unjustified assumption - to argue that there are no laws of nature which are not causal, and that there are no causal laws which would not be recognized as laws of nature is beyond the scope of this paper. Hopefully no one's common sense will be offended by the assumption.
9...Lewis appears to accepts something like this...
Lewis makes use a theory of lawhood developed by F. P. Ramsey (Ramsey, 1999).
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