Context

The author has in the last section discussed the probability of chances, i.e., the degree of belief one has in the face of ignorance of causes. The intensity of the belief that a “chance” event will occur is said to be a function of the superiority in numbers of our ideas of that outcome versus those of opposing outcomes. The subject of the present section is belief in a causal relation in the face of the occurrence of events contrary to that relation.

Background

For the most part, the treatment of the author is original. One notable place where something like the present topic is Newton’s fourth “Rule of Reasoning in Philosophy,” which states:

In experimental philosophy we are to look upon propositions collected by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.
This rule we must follow to ensure that the argument of induction may not be evaded by hypotheses.

The Treatise

1. The only use of the just-completed treatment of the probability of chances is that it aids in the examination of the probability of causes, which is “what we must chiefly examine.” The reason is that what the vulgar call chance, the philosophers commonly attribute to hidden causes. [The account of the probability of chances does not explain what happens “by chance,” but rather uncovers the mechanism whereby we reason about probabilities in general, and therefore about the probability that a given effect will follow from a given cause.]

2. There are several kinds of probabilities of causes are “deriv’d from the same origin, viz. the association of ideas to a present impression.” Each new resembling conjunction of impressions (or of their objects) strengthens the habit that produces the association between them, until the association reaches “perfection.” Suppose an object of type A is observed to be followed by one of type B. “The first instance has little or no force: The second makes some addition to it: The third becomes still more sensible; and ’tis by these slow steps, that our judgment arrives at a full assurance.” This assurance is called “proof” [as had been described in the previous section, paragraph 2]. At each step before it reaches this point, the association produces a judgment that is only a “presumption or probability.” The gradation is often hard to detect, though it is most easily observed in the earlier stages, when the assurance is not very strong.

3. The author remarks that although this process of the accumulation of conforming instances is “the first in order, and naturally takes place before any entire proof can exist,” no mature person can take notice of it. It is a fact that most commonly, people of the most advanc’d knowledge will form imperfect habits of expectation when they have only an imperfect experience of many particular events. But this tendency is countered by the fact that the mind has made an observation concerning the causal connection, which gives a new force to our reasoning concerning the given causal relations. This allows them ind to build an argument for a causal connection on the basis of one single experiment, when duly prepar’d and examin’d.” [For a fuller discussion of this mode of inference, see Section 8, paragraph 14.] What we have found once to follow from any object, we will conclude will for ever follow from it. If we have any doubt about the argument, is not because we have not considered enough cases, as one will do, but because there are contrary cases. Our reaction to contrary cases gives rise to a a second species of probability, where there is a contrariety in our experience and observation. [The first species is that described in the previous paragraph, based on an accumulation of conforming instances.]

4. It would be a good thing if there were uniformity in observed nature, since the only uncertainty in our judgments about causes and effects would be due to imperfections in our judgment. But in fact, ’tis frequently found, that one observation is contrary to another, and that causes and effects follow not in the same order, of which we have had experience. Because of the uncertainty induced by this contrariety, we are forced to take it into account in our reasoning. The first step toward understanding this process is to investigate the nature and causes of the contrariety.

5. The author begins the investigation by contrasting the views of the vulgar, who base their judgments on mere appearances, and the philosophers, who look into matters more deeply. The former take it that the observed contrariety is due to the failure of the cause to bring about its usual effect, despite there being nothing to hinder its so doing. The latter recognize that there are many causes that are too far away or too small to be observed, in which case they may be what interferes with the normal operation of a cause. And they become certain that this is the explanation when they discover such hindrances when they carefully examine the causal processes. An example is the stoppage of a watch. A peasant can only say that the watch does not go right, but the philosopher will attribute it to the presence of a small foreign particle that prevents the mechanism from moving. From the observation of several parallel instances, philosophers form a maxim, that the connexion betwixt all causes and effects is equally necessary, and that its seeming uncertainty in some instances proceeds from the secret observation of contrary causes.

6. The differences between the vulgar and philosophical explanations of contrariety notwithstanding, their inferences from it are always of the same kind, and founded on the same principles. There are two kinds of inference. The first, explained in this paragraph, is simply that the observation of contrary effects produces an imperfect habit and transition from the present impression to the related idea. Only a perfectly uniform experience is sufficient to produce a perfect habit. Common experience of our actions and reasoning shows that a constant perseverance in any course of life produces a strong inclination and tendency to continue in the future. There are as well weaker habits produced by the inferior degrees of steadiness and uniformity in our conduct.

7. Although the author acknowledges that this first species of probability is in play on some occasions, he is persuaded that this is not the most common kind, but that another principle governs our reasoning in the face of contrariety. He notes that this kind of probability is produced immediately and without deliberation. While there are few cases in which a perfect habit is produced by observation of perfect uniformity, there are even fewer cases in which an imperfect habit is produced without deliberation. When we do deliberate, we commonly take knowingly into consideration the contrariety of past events; we compare the different sides of the contrariety, and carefully weigh the experiments, which we have on each side. Thus, the habit we form upon observing contrariety does not produce the probability directly, but must do so only in an indirect or oblique way. [This is the second kind of inference noted in the preceding paragraph,] which the author will now explain.

8. The inference to a probability in the face of contrary causes is based on past experience. If a contrary effect of a cause has been observed in the past, we regard it at least as possible in the future. And as past experience regulates our judgment concerning the possibility of these effects, so it does that concerning their probability. Specifically, the possible effect that has occurred most frequently is deemed most probable. The author now asks two questions. 1) why is past experience the basis of our probable judgments about the future, and 2) how a single judgment of probability can be extracted from a contrariety of past events.

9. The answer to the first question is that we move from past to future out of habit. We suppose that the future resembles the past not because of reasoning, but because we are accustomed to its having done so before. This habit or determination to transfer the past to the future is full and perfect; and consequently the first impulse of the imagination in this species of reasoning in endow’d with the same qualities. [That is, in judgments of probability, the first movement of the imagination, to suppose that future events will resemble past ones, is full and perfect.]

10. The full and perfect first impulse of the imagination cannot operate successfully when experiences has presented cases of contrariety, where the future does not resemble the past. In that case, there is no steady object presented, but instead there is a number of disagreeing images in a certain order and proportion. This breaks up into pieces, so to speak, the first impulse. Each one of the images then takes up an equal share of the force and vivacity that the first impulse conveys. Any of these past events may again happen; and we judge, that when they do happen, they will be mix’d in the same proportion as in the past. [To take a simple case, suppose that cause c is observed to have effect e₁ once and e₂ twice. Each e_i gets 1/3 of the force and vivacity that it would have had if the effect had been uniform: all e₁ or all e₂. The proportion of vivacity would be 1:2, and e₂ would be thought twice as probable as e₁.]

11. The foregoing explanation accounts for probabilities in terms of the proportions of contrary events in a great number of instances. This requires that the images that are the basis of probability judgments remain in their original or first form: a plurality that preserves the original proportions. The author illustrates this form with the example of twenty ships that have departed from a harbor. Experience has revealed a proportion of nineteen ships returning and one being lost at sea. This past experience is transferred to the future, resulting in an image of nineteen ships returning and one not. This is not the only form of probabilistic judgment, however. We also judge concerning the probability of single events which appear uncertain. [We might ask whether the first ship to sail will return.] To do this, we must unite the concurring individual images into a single image. These agreeing images unite together, and render the idea more strong and lively, not only than a mere fiction of the imagination, but also than any idea, which is supported by a lesser number of experiments. [Thus, the idea of the first ship’s returning will be 19 times more vivid than that of its perishing at sea.] Each new experiment is a new stroke of the pencil, which bestows an additional vivacity on the colours, without either multiplying or enlarging the figure. This process of enlivening has been described in the previous section, and the author professes that he cannot make it any more clear than he has done already. In fact, everything that has been said about the probability of chances applies to the probability of causes. The reason is that the uncertainty about the outcome in cases of contrary experience is treated as if it were a matter of chance. And for this reason every thing that has been said on the one subject is applicable to both.

12. The overall consequence of contrary observations is an imperfect belief. The belief may be imperfect in one of two ways. The habit we have of expecting the future to resemble the past may be weakened, as described in paragraph 3. Or, the habit may remain in a state of perfection, in the sense that it makes us conclude in general, that instances of which we have no experience, must necessarily resemble those of which we have. In that case, the habit is divided and then rejoined in the manner described in the previous paragraph.

13. The author now turns to a further justification of his account of the second kind of probability, which involves reasoning with knowledge and reflection from a contrariety of past experiments. He recognizes that the considerations are subtle, but he claims that subtlety does not detract from the justness of any reasoning, any more than the fineness of matter in air, fire, etc. removes any of its qualities.

14. The first preliminary point is that the probability of causes is distinguished from certainty [or proof, as it is called in Section 11] because of the possibility of a contrary case, no matter how high the probability is. In the case of the kind of probability under consideration, there has been a contrary experience in the past, and this is proof of the possibility of a contrary experience in the future.

15. The second preliminary point is that each of the possibilities or parts that contribute to the probability is of the same nature, and differ in number only, but not in kind. In the previous section, it was claimed that each of the individual chances that are compounded in the probability of chances are entirely equal. The only difference between the probability of any contingent event is the relative number of chances. The case is the same with the probability of causes. Each of the contrary observations in the past is assigned equal weight when we transfer our thought from the past to the future. Only if the collective weight of one contrary is greater than that of another is it deemed more probable. So, the all possibilities have equal weight in both the probability of chances and that of causes.

16. The third preliminary point concerns the imperfect belief that is the effect of probable reasoning. The author claims that the belief is the result of compounding possibilities, which themselves may be either chances or observations. He argues for this conclusion on the basis of a general principle which holds in both the mind and in nature. The principle is that an effect is compounded if its cause has parts and the degree of the effect varies proportionally to the number of its parts. In that case, each of the parts gives rise to a partial effect, which is a component of the final effect. Consider for example the weight of a body consisting of four parts of equal weight, say one pound each. If the subtraction of any one of the parts results in a weight of three pounds or the addition of a one-pound part results in a weight of five pounds, etc., the weight to the compound is the result of the weight of its parts. The absence or presence of a part of the cause is attended with that of a proportionable part of the effect. This kind of variation is enough to prove that each of the parts of the cause is responsible for a corresponding part of the effect. In the example, the first body is the cause of one pound of the total weight, etc. It has been shown that the degree of belief [the force and vivacity of the idea] of a future event varies proportionally with the possibilities which are entertained by the mind, belief is an effect compounded of those possibilities.

17. The three preliminary points are now joined together in a single conclusion. The three points are summarized as follows. First, every probability admits of an opposite possibility [which may be greater or lesser, depending on the number of chances or past experiences]. Second, the possibility is composed of parts of the same nature as those comprising the probability, and so influence the mind in the same way. Third, the belief in the probability is a compounded effect, consisting of the effects of each of the parts of the probability. The conclusion is that each part of the opposite possibility influences the mind in the same way as does each part of the probability, since they are of the same nature. The contrary belief, attending the possibility, implies a view of a certain object, as well as the probability does an opposite view. In this particular both these degrees of belief are alike. The only way that one view can prevail over another is by containing a superior number of parts, which produces a stronger and livelier view of the effect. Each part presents a particular view; and all these views uniting together produce one general view, which is fuller and more distinct by the greater number of causes or principles, from which it is deriv’d.

18. Because the causes of belief are the same in kind, the only difference in their effect can be the superiority in number. In the case of both the probable object [the probable effect of a given cause] and its possible contrary, there is presented a full and entire view, leaving out no part of the object, there is no way to distinguish the two views except in terms of their force and vivacity. The probable prevails because of the superiority of its view of its object.

19. The author now presents almost the same argument in a different light. He begins by observing that, All our reasonings concerning the probability of causes are founded on transferring of past to future. This transferring gives us a view of the object in question. The past experience may be single or a multitude of similar instances. It may also be consistent or contrary. Now take the case where there are a multitude of past observations, some of which are contrary to others. In the transference from past to future, there remain views of the object. We must only explain what takes place when the views concur and when they are opposed. There are two hypotheses that could explain concurrence. The first is that each of the similar past experiences is transferred to a separate view of the object. The second is that they increase the force and vivacity of a single view. The author maintains that experience shows the first hypothesis to be erroneous. Probable reasoning produces only a single belief, and if it were to produce many, the mind would be distracted or even unable to comprehend them all distinctly, due to its limited capacities. So the only remaining hypothesis is that these similar views run into each other, and unite their forces; so as to produce a stronger and clearer view, than what arises from any one alone. This explains how the views concur in a belief of probability. When the views are opposed, one is incompatible with the other, and no object can embody both of them. As a result, the influence of the two views becomes mutually destructive, and the mind is determin’d to the superior only with that force, which remains after subtracting the inferior.

20. At this point, the author has concluded his subtle defense of his account of probable belief. He reflects that the common reader will find his reasoning difficult, inasmuch as he is not accustomed to the deep reflections of the philosopher. As a result, such a reader is likely to reject it as chimerical, as it departs from the common receiv’d notions, and . . . the easiest and most obvious principles of philosophy. The author admits that it takes some effort to understand his reasoning. However, he thinks that it takes little effort to realize the deficiencies of common explanations of probability and the small extent to which philosophy can explain it. He conjectures that if the reader were to accept two principles, he would be ready to entertain even the most extraordinary philosophical system, because these principles overthrow the common principles. The principles are: that there is nothing in any object, consider’d in itself, which can afford us a reason for drawing a conclusion beyond it; and that even after the observation of the frequent or constant conjunction of objects, we have no reason to draw any inference concerning any object beyond those of which we have had experience. [These principles had been reached in Section 2, paragraph 5 and Section 6, paragraphs 1 and 2, respectively.] The author states that these principles have been found to be sufficiently convincing with regard to our most certain reasoning about causation, and he thinks that they are made even more evident in the case of less-certain probable reasoning in the face of contrary causes.

21. The first reason supporting the principles is that the fact that we have contrary views of the future state of an object makes us uncertain about what that state will be. The basis of the uncertainty is that the causes of that state are hidden. So there is no way that we can have a basis for drawing a conclusion about the object beyond its past and present states merely by considering the object in itself or on the basis of frequent conjunction. [The case of constant conjunction would apply only to our most certain reasonings, not the conjectural reasoning of probabilities.]

22. The second reason, which the author takes to be just as obvious, is that there is no way for the understanding by itself to transfer past experience to a belief about the future. Belief cannot arise merely by repetition of ideas. It can only arise from concurring experiences being combined, melted together, so to speak, into one single idea or image, which is intense and lively in proportion to the number of experiments from which it is deriv’d, and their superiority above their antagonists. Past experience does not present us with a single object, but rather with many objects. Yet belief is fixed on a single object, and this requires an act of the fancy or imagination. Thus, the imagination plays a role in probable reasoning as well as in our most certain causal reasoning [the latter being explained in Section 6, paragraph 12].

23. The author concludes the discussion of the probability of contrary effects with two reflections. The first concerns the way in which the mind forms multiple views of the object that are central to probable reasoning. In his account, the author found these views to originate in experience. Contrary views are presented, and belief results from the uniting of a preponderance of concurring views into a single lively view. The question arises as to whether such a view could be generated by the imagination, by repeating the same view again and again. The author claims that it cannot, or at least cannot to the same degree as does experience. The repetition might be involuntary, as with custom and education. This at least takes a long time and requires very frequent repetition. If it is voluntary, it will never produce any greater degree of belief than would a single observation. Although voluntary actions share the characteristic of being undertaken by design, they have nothing else to unite them: each act of the mind, being separate and independent, has a separate influence, and joins not its force with that of its fellows. Moreover, the voluntary acts of repetition are not all produced by the same object, so there is nothing to unite them. The author notes that this aspect of the mind will be treated later [in Part IV, Section 2, where the author argues that the mind consists of perceptions that are not united in identical object.]

24. The second reflection concerns a potential counter-example to the author’s account. We in fact distinguish small differences in large probabilities, such as between those resulting from the observation of 1,000 concurring objects and 1,001 contrary objects. Yet the mind cannot discern the differences in the views of the two, and in particular not by running through all the cases. The author notes that this phenomenon holds in parallel for the affections. Suppose I desire 1,001 pounds more than 1,000 pounds. In such a case, the principle enunciated in paragraph 16 applies. A change in the parts brings about a proportional change in the effects, so each of the parts brings about its own effect, which is compounded. Thus, the desire for 1,001 pounds is the compound of 1,001 desires for one pound [which is the unit that distinguishes the two desires] or even more than that [if, say, the desires were for pennies]. Yet nothing can be more certain, than that so small a difference woul’d not be discernible in the passions, nor coul’d render them distinguishable from each other. The only explanation for the difference is the effects of custom and general rules. We become accustomed to detecting distinguishable differences, as with the desire for three pounds is palpably greater than that for two. Inequalities in larger numbers resemble felt inequalities. A general rule formed on this basis gives us a stronger desire for more money than for less. General rules will be discussed more fully beginning at paragraph 6 of the next section.

25. Having explained probabilities based on imperfect experience [paragraph 6] and contrary causes [paragraphs 7 through 24], the author turns to a third species of probable reasoning, which arises from analogy. This species differs from the first two in some material circumstances. The author begins by recounting his explanation of beliefs about causes. The two elements are the invariable conjunction, one after the other, of objects of the same kind and the resemblance of a present object to the first of the conjoined objects. These two circumstances enliven the idea or view of an object of the same kind as the following conjoined objects, resulting in belief. Weakening the constancy of the union or the resemblance weakens the transition from the impression of the present object to the idea of what will follow it. In the case of the probability of chances or of causes, it is the constancy that is weakened. In the case of analogy, it is the resemblance, and only the resemblance, that is diminished. Resemblance comes in degrees, and the lesser the degree of resemblance, the lesser the force of the idea, and the greater that degree, the greater the force. On the other hand, as long as there is some resemblance, there may be enough for some probability.

The Enquiry

The probability of causes is discussed in the fourth paragraph of Section 6 of the Enquiry. The account is quite similar to that in the Treatise, and it is also more clearly and simply presented. Hume distinguishes between causes that are followed by certain events without exception, such as the production of motion by the universal law of gravitation, and those which admit of exceptions, such as a person‘s drowsiness upon receiving a dose of opium. Philosophers attribute the exceptions to the operation of secret causes, but our ordinary reasoning works differently. We transfer our past experience to the future. If we have experienced a perfectly regular sequence of causes and effects, we expect the next effect to be of the same nature of all the previous ones. If there is irregularity, on the other hand, we may believe that an effect will occur if it conforms to most of our experience, but we must give some weight to other experienced effects. The weight given to any given future effect is proportional to its frequency of occurrence in our past experience. Hume claims that other existing explanations of probability are defective in their treatments of it.