Context

Sections 7-10 of Part 3 have been devoted to the examination of belief: its nature, its causes, and its influence on the mind. The author returns to the subject of reasoning, which was examined in part in sections 4-6. The topic in those sections was how the mind makes particular causal inferences, in which a present impression of an object (the cause) is the occasion of a new idea of an object (the effect).

Background

The author notes in the Abstract that Leibniz complained that logicians have been “too concise when they treat of probabilities, and those other measures of evidence on which life an action depend.” The balance of Part 3 is an attempt “to supply this defect.”

The Treatise

1. To this point, the author has produced a “system” to explain belief, including the belief in the existence of an object, which comes about through causal reasoning. In this and the next two sections, he undertakes to mobilize his system to explain other kinds of reasoning. The resulting “consequences” of the system serve to increase the credibility of the system itself, bestowing upon it “its full force and evidence.”

2. Before proceeding with the application of his system to other forms of reasoning, the author apparently modifies a key distinction that was made in the first section of Part 3: the distinction between knowledge and probability. In Section 1, the author had deemed the “objects of knowledge and certainty” to be relations that can be established through the mere comparison of ideas. In Section 2, the author considers relations which cannot be established merely by comparing ideas. Although he does not call our reasoning about these latter relations “probability” in the body of the text, the title of the section begins, “Of probability.” In the present section, the author states that earlier he had followed the “method of expression” of other philosophers by denominating as knowledge “that evidence, which arises from the comparison of ideas,” and as probability “all our arguments from causes or effects.” But he notes that this division conflicts with “common discourse,” in which “we readily affirm, that many arguments from causation exceed probability, and may be receiv’d as a superior kind of evidence.” Someone who states that it is probable that the sun will rise tomorrow, or that all humans must die, would sound ridiculous. On the other hand, the only basis for our beliefs of the truth of these propositions is experience. The author thus decides that it would be “more convenient” to follow the common usage and distinguish between three “degrees of evidence.”

• Knowledge: “the assurance arising from the comparison of ideas,”
• Proofs: “those arguments, which are deriv’d from the relation of cause and effect, and which are entirely free from doubt and uncertainty,”
• Probability: “that evidence, which is still attended with uncertainty.”

3. Probability is now described another way, as “reasoning from conjecture,” and it is divided into to types, according to whether the reasoning is based on:

• Chance,
• Causes.

4. The author re-states his account of the idea of cause and effect. First, it is derived from experience, which presents us with objects which are constantly conjoined with each other. The result is “a habit of surveying them in that relation.” The habit is so strong that surveying the objects in some other relation would require “sensible violence.” Chance is said to be “nothing real in itself, and, properly speaking is merely the negation of a cause.” As it negates a cause, chance leaves us with no basis for associating ideas of objects. Hence, we are left in a state of perfect indifference with respect to the existence or non-existence of any “contingent” object for which there is no idea of its cause. In opposition to the idea of a cause, which “in a manner forces us to survey such certain objects, in such certain relations,” chance destroys this determination of the mind and “instantly” reinstates “its native situation of indifference.”

5. The “perfect” or “entire” indifference of chance implies that the only way one possible outcome could be “superior” to another is if the first outcome is “compos’d of a superior number of equal chances.” To adapt an example given in paragraphs 6 and 9, suppose there is a six-sided die, with one kind of marking (say, a square) on four sides and another kind of marking (say, a triangle) on the two remaining sides. Suppose that the chances are described in terms of which side comes up. Then each chance is equal to the other. But if we describe the chance in terms of a square coming up, then we may call it superior to the chance of a triangle coming up, because the former chance is composed of four chances and the latter is composed of two. If there were any other way in which one chance could be superior to another, it must be regarded as a cause, in which case it would not be a “chance,” by the definition given above. The author notes that the following “truth” is shared with “every one, that forms calculations concerning chances.” That truth is that, “A perfect and total indifference is essential to chance, and one total indifference can never in itself be either superior or inferior to another.”

6. Although chance and cause are “directly contrary,” causes play a role in conceiving the “superiority of chances” described in the last paragraph. We must have some constraints to be able to combine or aggregate the initial chances so that one combination is superior to the other. In a throw of dice, we must allow “that there are some causes to make the dice fall, and preserve their form in their fall, and lie upon one of their sides” in order to be able to calculate the odds according to the “laws of hazard.” If there were total indifference with respect to the outcome of the throw of the dice, no circumstance could “give one the advantage over another,” as we can imagine the most extravagant possibilities, such as the shape of a die changing so that two sides come up simultaneously. So we are indifferent with respect to which side of a die comes up but not to the causes which govern, in a general way, how it can come up. “The mind is here limited by the causes to such a precise number and quality of the events; and at the same time is undetermin’d in its choice of any particular event.”

7. The author now summarizes his description of chance in “three steps:”

• Chance is nothing more than the negation of cause, and it produces a total indifference in the mind.
• No single chance or negation of cause can be superior or inferior to any other.
• In order to reason about chances, there must be some mixture of causes among the chances.

8. It could be objected that there is in fact certainty with respect to chances: though not with respect to which particular event of a kind (or “side”) will occur, but “that ’tis more likely and probable, ’twill be on that side where there is a superior number of chances, than where there is an inferior.” Thus, we may be certain about the probabilities of chance events. The author responds by questioning what is meant by ‘likelihood’ and ‘probability.’ These terms merely indicate the superiority of chances. Thus, the certainty we have here is a trivial identity: “where there is a superior number of chances there is actually a superior, and where there is an inferior there is an inferior.” So there is no argument from probability to produce the belief in one of the chances in the superior combination. But if not, what is the source of the belief?

9. In order to answer this question, the author invokes the die example discussed above. He thinks that it is “plain” that the belief of the outcome would be that one of the four sides with a square will come up. This belief is not entire, but is “still with hesitation and doubt, in proportion to the number of chances, that are contrary.” If the superior side is augmented (say by replacing one triangle with a square), “his belief acquires new degrees of stability and assurance.” We should be able to understand why this occurs, as the example focuses on “the simple and limited object before us.” All we have is “one single dye to contemplate, in order to comprehend one of the most curious operations of the understanding.”

10. The author now recounts three features of the example.

• There are causes, such as gravity, solidity and figure, which determine the fall of the die.
• There are six sides of the die, and the mind is indifferent with respect to which side will come up.
• Each side is marked with one of the two figures.

11. The first observation is that in causal reasoning, the mind is influenced by habit based on constant conjunction, so that one object is presented, the existence of another is inferred. “Upon the appearance of the one, ’tis almost impossible for it not to form an idea of the other.” Our habit is such that when we see the die begin to fall, we expect it to land on the table and not to be suspended in mid-air. This is one of the causes that are “mixed” into our calculation of chances.

12. The second observation is that which side comes up is entirely a matter of chance, due to the fact that this event is supposed to be contingent, and we have no idea of a cause that would determine how it will turn out. So we consider all sides to be equally likely to come up. Given our beliefs about causes, we form a “principle” that one of the six sides will come up: the mind “feels a kind of impossibility both of stopping short or forming any other idea” than that of one of the sides coming up. Because it is a causal inference, the idea of one side’s coming up has a certain vivacity or “force.” The principle does not allow all six sides to come up, which is impossible, and it does not “direct us with its entire force to any particular side,” since this would make that side’s coming up “certain and inevitable.” The only alternative is that the principle that one side will come up “directs us to all six sides after such a manner as to divide its force equally among them.” As there are six sides, the force is divided six ways. “’Tis after this manner the original impulse, and consequently the vivacity of thought, arising from the causes, is divided and split in pieces by the intermingled chances.”

13. Of the three factors noted in paragraph 12, the first two (causes and indifference) have been accounted for: “they give an impulse to the thought, and divide that impulse into as many parts as there are unites in the die.” What is left to explain is the figures that mark the die. Where different sides have the same figure marked on them, they “must concur in their influence on the mind, and must unite upon one image or idea of a figure all those divided impulses, that were dispers’d over the several sides, upon which that figure is inscrib’d.” The question now is not which side will come up (all have an equal likelihood), but which side with a given figure will come up. Because the same figure is on each of several sides, “the impulses belonging to all of these sides must re-unite in that one figure, and become stronger and more forcible by the union.” Thus, there will be a more vivid idea of the figure because of the repeated thought of it. Given that four sides have squares and two sides have triangles, “the impulses of the former are, therefore, superior to those of the latter.” But this is not the end of the story, for now we have ideas of two figures, though with different degrees of vivacity. Since the two figures are incompatible, given that one side will come up, there can be a belief of only one of them. The stronger force will prevail, but it will be diminished by the amount of force the weaker one has. The result will be that the belief is not as strong as it would have been were there no competing possibilities. “The vivacity of the idea is always proportionable to the degrees of the impulse or tendency to the transition; and the belief is the same with the vivacity of the idea.”

The Enquiry

The probability of chances is discussed in the first three paragraphs 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 states that the belief in the more probable event is due to the concurrence of a greater number of sides than the alternative possibility. This takes place “by an inexplicable contrivance of nature.” He ventures as a mere supposition that “belief is nothing but a firmer and stronger conception of an object than what attends the mere fictions of imagination,” and he uses it to give “in some measure,” an account of how belief occurs. “The concurrence of these several views or glimpses imprints the idea more strongly on the imagination; gives it superior force and vigour; renders its influence on the passions and affections more sensible; and in a word, begets that reliance or security, which constitutes the nature of belief and opinion.” Also in this section there is a footnote distinguishing proofs, “such arguments from experience as leave no room for doubt or opposition,” from probabilities.