Hume will begin by examining probability.
§277 Conjecture Juncture
We may give probability another name: reasoning from conjecture. There are two kinds of this reasoning:
1) conjectural knowledge based on chance, and
2) conjectural knowledge based on causal relations.
§278 Causal Necessity and Chance Contingency
We often experience objects that are constantly conjoined to each other. This produces the habit of thinking them together. In this way experience teaches us the idea of cause and effect.
[We keep our house in pristine condition. Our neighbor let's his degrade. We are embarrassed to live so close.
A terrible storm arrives one day. Tornadoes brew above. One touches down. It destroys our house. But our neighbor's is left intact. Now he is embarrassed to live near us. But we are the ones who took care of our home. This is not the way things should be.]
Causal relations are necessary. Those who take care of their home maintain its condition. Those who do not, allow it to deteriorate. But consider when something happens by chance like the tornado event. The result seems to lack just cause. It is an outcome seemingly without a cause at all. In fact, considering we were the ones who kept our house maintained, its destruction seems to have come about despite causal influences. So chance is the negation of cause. It is "nothing real in itself." (125ab)
Causation causes us to believe in the reality of some relation between objects. But the tornado was not fair. There was no logic to its path. Hence when we next see a tornado, we do not infer that it will go one way or another. So causation leads us to believe in the reality of some effect, whereas chance leads us to doubt there is any such a predetermined outcome. In chance relations, the corresponding object is considered contingent and not necessary. So we must decide for ourselves whether it exists or not.
A cause traces the way to our thought, and in a manner forces us to survey such certain objects, in such certain relations. Chance can only destroy this determination of the thought, and leave the mind in its native situation of indifference; in which, upon the absence of a cause, it is instantly re-instated. (125b emphasis mine)
§279 Chance Indifference
The tornado was indifferent to what lie in its path. It could strike the least deserving while narrowly avoiding those who have earned such punishment.
Now, winning the lottery is chance also. We do not think that there is really any legal way to cause oneself to win.
However, some might say that winning the lottery is slightly more likely than getting hit by a tornado. But then we are confusing probabilities with chance.
Our neighbor's home stands beside ours. The tornado comes toward both, but shifts towards our house. Because the tornado is entirely unpredictable, we are not inclined to say that there was a greater or lesser probability that the tornado would come our way. For one thing, there is no way to know which way it will go. But also recall Hume's use of the meaning for probability.
We see patterns in the world around us. We let-go-of something in our hands. It drops. Every time we let go, the object falls. The more times such a connection repeats, the stronger our mind comes to associate our impressions and ideas. This makes the evoked idea more vibrant and more forceful upon our mind. So we see someone holding an egg. She lets-go. Instinctively we reach out to catch it before it messes our floor. We did not need to reason in this case. Our mind instantly evoked the idea of the egg falling when we saw her hands release it.
But in the case of the tornado, we see it going one way. Suddenly it goes another. There is only one thing that is consistent about the tornado's movement: it is constantly unpredictable. So when the tornado comes to our house, it is useless to wonder what the statistical probabilities are of it hitting our house rather than our neighbor's.
Similarly for the lottery. We might have a one in a million chance of winning, because there are a million tickets in the pool. But when the official reaches into the pile, where his hand finally goes is not so easy to quantify. Like the tornado, his hand might be near ours and our neighbor's ticket. And our neighbor could be selected, even though he again least deserves such good fortune.
Now there are some people who chase storms. After seeing so many tornadoes, they begin to find some regularities in their behavior. When they saw the tornado approach your house, they foresaw it shift your way. For, they could tell from its behavior that it might do so. In this case, the storm-chaser might say that there is a greater chance of the tornado striking our home. But he could only do so on account of certain sense-impressions evoking habitually associated ideas of where other tornadoes turned. So then we are dealing with probability, and not chance.
For if we affirm that one chance can, after any other manner, be superior to another, we must at the same time affirm, that there is something, which gives it the superiority, and determines the event rather to that side than the other: That is, in other words, we must allow of a cause, and destroy the supposition of chance; which we had before established. A perfect and total indifference is essential to chance, and one total indifference can never in itself be either superior or inferior to another. (125c.d)
§280 Hazard of the Dice
Hume now addresses the irreducibility of chance.
Shake the dice. Spill. Anything can happen. This is pure chance.
Any of the six numbers can show. There is no greater likelihood for any number.
But what if we make a safer bet? We wager on rolling any of the first four numbers. Each number had an equal chance of showing. But now we have increased our chances by combining the likelihoods of four numbers. This is much more promising than rolling just one specific number. Hume calls this the "combination of chances." By making such calculations, we can say that one chance is greater than another.
And yet, luck is against us. We roll a five. Obviously we still did not eliminate chance. So long as hazard remains, we cannot know chance's outcomes, even if the likelihoods are precisely calculated.
A dye that has four sides marked with a certain number of spots, and only two with another, affords us an obvious and easy instance of this superiority. The mind is here limited by the causes to such a precise number and quality of the events; and at the same time is undetermined in its choice of any particular event. (126c)
§281 Chance Unknowing
So far we have three steps to our argumentation:
1) Causes are necessarily determined, but chance outcomes cannot be determined. Hence "chance is merely the negation of cause."
2) No chance can be greater or lesser than another. A chance is both a negation of a cause, and an indifference to outcome despite a totality of possibilities. Thus, "one negation of a cause and one total indifference can never be superior or inferior to another." (126c)
3) Reasoning requires that there be necessary connections between ideas. Such necessary connections are obtained when one idea is continually connected with another. We call such a constant connection a causal relation. Hence "there must always be a mixture of causes among the chances, in order to be the foundation of any reasoning." (126c)
Let's return to the tornado and the expert storm observer. He is not able to predict the directions of tornadoes. But he has seen many of them. He has found that they are impossible to predict. And yet, they also are not completely random in their movements. For example, he notices that when there is a dip in the ground, the tornado might suddenly shift direction. The tornado, he observed, encountered such a dip in front our neighbor's home. So as he saw the tornado approaching the depression in the ground, he reasoned there was a chance that it would shift towards our house. But he was only able to reason this way because there were causal influences mixed-in with the chances. He could thereby assess that there was a "superior" or "greater chance" that the tornado would shift directions.
Another way that we were able to better know outcomes is by a "combination of chances." We previously considered betting on more than one side of the dice showing-up. We bet on any of the first four numbers. The die has six sides. The chance of any one number showing 1/6. We bet on four. So we combine four chances: 1/6 + 1/6 + 1/6 + 1/6. This equal 4/6 or 2/3. So in every three rolls, we should probably win twice.
Hume will now examine these cases when causes are mixed with chances and when chances are combined. He wants to know how they influence our mind's judgment and opinion.
We know that we cannot say with any certainty at all how the dice will land, even if we combine many chances. Nor can we know exactly where a tornado will go, even though we observe certain influencing factors. Now recall how we are defining demonstration and probability. In a demonstration, a counter-argument would be impossible. And we take a probability not to be a statistical probability, but rather a history of constant connection between ideas. So technically it is probable that the sun will arise tomorrow, because that is always what we have experienced. But it is also conceivable that some day the earth's rotation will slow down until it is like the moon, where one side always faces away from its center of orbit. So can be that one day the sun does not rise. So we cannot demonstrate that the sun will rise tomorrow based on past evidence. We perhaps would have to use knowledge of astrophysics to demonstrate that no other outcome is conceivable.
But consider our dice wager. In the past, we have seen all numbers come-up in a random way. So we cannot reason that we will win when we bet on four numbers. For, past experience indicates that any of six numbers can show. Hence we are not reasoning by probability. And certainly we can conceive of the dice landing a five or six. So clearly there is no demonstrable knowledge involved.
So regarding cases where there superior chances and combinations of chances, we may return to our previous discoveries to explain how they influence our reasoning. They cannot cause us to reason anything.
'Tis indeed evident that we can never by the comparison of mere ideas make any discovery, which can be of consequence in this affairs and that 'tis impossible to prove with certainty, that any event must fall on that side where there is a superior number of chances. (126d)
§282 Wagers of Likelihood
But some might still object. They argue: we can "reason" that it is more statistically probable and hence more likely that we will win the dice game if we bet on rolling the first four numbers. So it is not certain the outcome, but it is certain one outcome is more likely than another.
Try as we might, day-in to night, we cannot conceive a circular square. Such a logical impossibility demonstrates that circles are curvilinear and not rectilinear. And we have never seen such a shape in our lives, or heard from anybody who has. Every time we have seen circles, they were perfectly curved. This constant connection gives us the impression there is a necessary connection. And our demonstration proves it. So we are certain that there are no square circles.
Now, if we bet on any of the six sides of the die showing, we can be certain that we will win. Yet, only fools would bet against us, and they tend not to have money to wager. So we create a likelihood that our opponent will win. We say we will win if one through four lands. This might even be about the same as the casino's odds of winning. So gamblers take our offer. We are certain we have a greater statistical probability, but what is that reasoning based on?
We conclude that the likelihood of rolling one-through-four are superior or higher than rolling five or six. We came to this by first taking the equal chance of rolling any one number, 1/6, then adding it four times. In other words, there is a superior number of equal chances for us to win. Then we conclude that there is a superior likelihood.
Now let's say we had a very unlucky day. We bet on one-through-four, and after 12 losing rolls, we were out of money. This is not likely. But also, it is unlikely that this has never happened. The chances of this anomaly might be something like 1 in 531441. But certainly dice have been cast many more times than that. So it is perfectly conceivable that it could happen to us. Thus when we are "certain" that something is probable, that does not mean we conceive of any necessary associations between ideas. We might instead say that we tend to believe-in one outcome more than another. But now we know that we do not assent to some possibility because we may rationally demonstrate it or because we have frequently experienced it in the past. So Hume asks, what causes us to believe certain likelihoods more than others?
§283 Gambling is Our Mind's Most Curious Operation
Hume experiments. Someone raises a dice-box. He inserts a die. His wager: one-through-four. To dramatize, we paint a skull over the loosing sides 5 and 6, and we paint golden coins overtop sides 1 through 4. As he shakes the box, he feels that the odds are on his side. But just as he tips the can to cast, he hesitates. Why? He could not get the skull-image out of his mind. So we lower the stakes and up the odds. He will win on one-through-five. Now the dreadful scull image is less vivid, so he shakes the box with more belief that he will win.
Hume will analyze this simple example to explain "the most curious operations of the understanding" which led to the man's increase in belief.
§284 The Properties of Our Destiny
So Hume will use the dice example to illustrate the ways that superior chances influence our mind's operation. To do so, he distinguishes three categories of the dice's properties which help explain this influence.
1) the physical causes:
Certain causes such as gravity, the die's density, and its cubical shape help determine how the die will fall and land.
2) the quantity of possibilities:
There a six possible sides that may turn-up.
3) the appearances of the outcomes:
Two sides have a skull figure, the other four have gold coins.
These three sets of features make up all the die's properties that are relevant to our inquiry. So we will work carefully through each to determine their role in the strength of our beliefs.
§285 The Physics of Chance
Consider how we experience longer days every summer [that we do not spend near the equator.] So whenever we imagine a summer day, we conceive it to be a long one. And when it is summer and we awake to the sunrise, we will schedule our day's activities based on the longer sunlight. For, our minds so readily conceive the summer day as long. It would require a measure of "violence" to force it to imagine a short summer day.
Hume considers our sense impression of the summer dawn to be the cause for our conclusion that the day is long. We likewise have had many experiences of dice falling, and also of anything at all falling. When someone's hand loses its grasp of a fresh egg, we instantly reach out to catch it before it messes our floor. We did not reason rationally from the cause to the effect. For, the sense-impression of the person's hand losing hold of the egg automatically called to mind the idea of it falling. We have seen many things fall this way in the past, so we develop the habit of associating our impressions and images this way.
Now the die has physical properties that associate it with other dice we have seen thrown. And we also have been in many circumstances where we experienced dice be cast. So we cannot but expect the die to fall to the table and land with one side up.
We have very frequently seen it do so. And we have never seen it do otherwise.
Now imagine we are hammering shingles to a roof. The hammer slips from our hands. We watch it fall to the ground. Did the wooden handle hit the ground first? No, its heavy head did. Many experiences like this one cause us to associate the heavier side of a falling object with the part that first touches ground.
And, we might notice that there the numbers for the die's sides are marked with dots. These dots are carved into the surface of the die. So there are more holes on the die's sides '5' and '6' than there are for the smaller numbered sides. And also, our gold paint is heavier than the black paint we used to make the skull image. So when we throw the dice, we might tend more to anticipate the heavier 1-through-4 sides showing. So we see how such physical properties factor-in when we calculate chances. But in our example we will suppose that the imbalances have be rectified. So the die is evenly weighted on each side.
§286 Chance Forces
So we suppose that we are casting the die onto a hard surface. If it were pudding, the die will often land unevenly and show more than one possible outcome at once. So it is necessary that our hard table-top will only show one number. But, chance alone decides which number shows.
Things that happen by causal relations are necessary. But if they happen by chance, they are not necessary. We saw that chance is the negation of causes. This leaves us unable to know what will happen. Our mind is "in a perfect indifference among those events, which are supposed contingent." (128d)
Past experience has taught us that if we roll the dice, one side will show. But which one we do not know. For each possibility is equally probable.
So we cast the die. We have a sense impression of it in the air. This causes our mind to associate it with the outcomes we have previously experienced. But we cannot imagine all six sides coming up at once, because only one comes-up. But we are also not inclined to imagine any one showing instead of any other. For, they are all equally likely. Our sense-impression does not
direct us with its entire force to any particular side; for in that case this side would be considered as certain and inevitable; but it directs us to the whole six sides after such a manner as to divide its force equally among them. (129b emphasis mine)
So we imagine each possibility individually. But none has greater vivacity than the others. So none has greater force. Hence we do not believe any one outcome more than another.
We conclude in general, that some one of them must result from the throw: We run all of them over in our minds: The determination of the thought is common to all; but no more of its force falls to the share of any one, than what is suitable to its proportion with the rest. It is 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. (129bc emphasis mine)
§287 The Intensity of Chance
So we have examined the first two qualities:
a) the causes, and
b) our indifference to the many possible outcomes.
We found that both factors give us an "impulse" to consider all the possibilities. And because all have equal chances, the force of that impulse was divided six ways for the six sides.
We now will analyze the third factor: the figures inscribed on the sides.
In our modified example, we have skulls on two sides, and gold coins on the other four.
But each one of the six sides is equally vivacious in our imagination. So the force of our impulse to believe any one of them has been divided six ways. However, four of those six possibilities share one inscribed figure, and the two other possibilities share a very different image. So even though each of the six possibilities influences our minds equally, four of them combine forces and have greater influence than the other two.
'Tis evident that where several sides have the same figure inscrib'd 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. (129d)
...
'tis evident, that the impulses belonging to all these sides must re-unite in that one figure, and become stronger and more forcible by the union. (130a emphasis mine)
So there are more coins than skulls. And the images unite forces. This makes the idea for the coin-sides more vivacious. Hence the impulses to believe in a gold coin showing is greater than our impulses to believe a skull turning up.
The vivacity of the idea is always proportionable to the degrees of the impulse or tendency to the transition; and belief is the same with the vivacity of the idea, according to the precedent doctrine. (130b emphasis mine)
Our aim was to explain why we believe in one outcome more than another. Some argue that this is because we rationally determine that one is more likely. And then we believe more in the likelier outcome. Hume then showed that we could not possibly come to such a conclusion rationally. By means of his dice illustration, he gave an account based on our tendencies. We believe one outcome more than another, because it is more vivacious in our minds. An idea has more vivacity when we have a stronger tendency to call it to mind. When we believe in one outcome more than another, it is because we unite many equal chances together. Each one of these possibilities has equal force. But when combined they have increased force. The power of this force causes us to have a stronger tendency to call to mind that outcome, which makes it a more vivid idea. We know that we believe more in vivacious ideas. So we believe greater likelihoods because they are made up of many smaller equal but combined tendencies that together impress upon our imaginations with greater force.
[So the greater force causes the ideas to have more intensity. And, we obtain that greater force from a stronger tendency to call something to mind. But this tendency is not something that extends in space. It is an impulsion. It is like Leibniz' conatus. It tends without extending. It tends inwardly. So it is an intensity.
Now also consider that these associative tendencies place our mind in a mental relationship with its object. So it is like Husserl's intentional consciousness. Our minds have a stronger relationship-with or gravitation-towards the idea for the more likely outcome. So we might also say that in this case there is more intentionality in the phenomenological sense. We are more motivated to intend the more probable result.]
From the original text:
Sect. xi. Of the Probability of Chances.
But in order to bestow on this system its full force and evidence, we must carry our eye from it a moment to consider its consequences, and explain from the same principles some other species of reasoning, which are derived from the same origin.
Those philosophers, who have divided human reason into knowledge and probability, and have defined the first to be that evidence, which arises from the comparison of ideas, are obliged to comprehend all our arguments from causes or effects under the general term of probability. But though every one be free to use his terms in what sense he pleases; and accordingly in the precedent part of this discourse, I have followed this method of expression; it is however certain, that in common discourse we readily affirm, that many arguments from causation exceed probability, and may be received as a superior kind of evidence. One would appear ridiculous, who would say, that it is only probable the sun will rise to-morrow, or that all men must dye; though it is plain we have no further assurance of these facts, than what experience affords us. For this reason, it would perhaps be more convenient, in order at once to preserve the common signification of words, and mark the several degrees of evidence, to distinguish human reason into three kinds, viz. THAT FROM KNOWLEDGE, FROM PROOFS, AND FROM PROBABILITIES. By knowledge, I mean the assurance arising from the comparison of ideas. By proofs, those arguments, which are derived from the relation of cause and effect, and which are entirely free from doubt and uncertainty. By probability, that evidence, which is still attended with uncertainty. It is this last species of reasoning, I proceed to examine.
Probability or reasoning from conjecture may be divided into two kinds, viz. that which is founded on chance, and that which arises from causes. We shall consider each of these in order.
The idea of cause and effect is derived from experience, which presenting us with certain objects constantly conjoined with each other, produces such a habit of surveying them in that relation, that we cannot without a sensible violence survey them iii any other. On the other hand, as chance is nothing real in itself, and, properly speaking, is merely the negation of a cause, its influence on the mind is contrary to that of causation; and it is essential to it, to leave the imagination perfectly indifferent, either to consider the existence or non-existence of that object, which is regarded as contingent. A cause traces the way to our thought, and in a manner forces us to survey such certain objects, in such certain relations. Chance can only destroy this determination of the thought, and leave the mind in its native situation of indifference; in which, upon the absence of a cause, it is instantly re-instated.
Since therefore an entire indifference is essential to chance, no one chance can possibly be superior to another, otherwise than as it is composed of a superior number of equal chances. For if we affirm that one chance can, after any other manner, be superior to another, we must at the same time affirm, that there is something, which gives it the superiority, and determines the event rather to that side than the other: That is, in other words, we must allow of a cause, and destroy the supposition of chance; which we had before established. A perfect and total indifference is essential to chance, and one total indifference can never in itself be either superior or inferior to another. This truth is not peculiar to my system, but is acknowledged by every one, that forms calculations concerning chances.
And here it is remarkable, that though chance and causation be directly contrary, yet it is impossible for us to conceive this combination of chances, which is requisite to render one hazard superior to another, without supposing a mixture of causes among the chances, and a conjunction of necessity in some particulars, with a total indifference in others. Where nothing limits the chances, every notion, that the most extravagant fancy can form, is upon a footing of equality; nor can there be any circumstance to give one the advantage above another. Thus unless we allow, that there are some causes to make the dice fall, and preserve their form in their fall, and lie upon some one of their sides, we can form no calculation concerning the laws of hazard. But supposing these causes to operate, and supposing likewise all the rest to be indifferent and to be determined by chance, it is easy to arrive at a notion of a superior combination of chances. A dye that has four sides marked with a certain number of spots, and only two with another, affords us an obvious and easy instance of this superiority. The mind is here limited by the causes to such a precise number and quality of the events; and at the same time is undetermined in its choice of any particular event.
Proceeding then in that reasoning, wherein we have advanced three steps; that chance is merely the negation of a cause, and produces a total indifference in the mind; that one negation of a cause and one total indifference can never be superior or inferior to another; and that there must always be a mixture of causes among the chances, in order to be the foundation of any reasoning: We are next to consider what effect a superior combination of chances can have upon the mind, and after what manner it influences our judgment and opinion. Here we may repeat all the same arguments we employed in examining that belief, which arises from causes; and may prove, after the same manner, that a superior number of chances produces our assent neither by demonstration nor probability. It is indeed evident that we can never by the comparison of mere ideas make any discovery, which can be of consequence in this affairs and that it is impossible to prove with certainty, that any event must fall on that side where there is a superior number of chances. To, suppose in this case any certainty, were to overthrow what we have established concerning the opposition of chances, and their perfect equality and indifference.
Should it be said, that though in an opposition of chances it is impossible to determine with certainty, on which side the event will fall, yet we can pronounce with certainty, that it is more likely and probable, it will be on that side where there is a superior number of chances, than where there is an inferior: should this be said, I would ask, what is here meant by likelihood and probability? The likelihood and probability of chances is a superior number of equal chances; and consequently when we say it is likely the event win fall on the side, which is superior, rather than on the inferior, we do no more than affirm, that where there is a superior number of chances there is actually a superior, and where there is an inferior there is an inferior; which are identical propositions, and of no consequence. The question is, by what means a superior number of equal chances operates upon the mind, and produces belief or assent; since it appears, that it is neither by arguments derived from demonstration, nor from probability.
In order to clear up this difficulty, we shall suppose a person to take a dye, formed after such a manner as that four of its sides are marked with one figure, or one number of spots, and two with another; and to put this dye into the box with an intention of throwing it: It is plain, he must conclude the one figure to be more probable than the other, and give the preference to that which is inscribed on the greatest number of sides. He in a manner believes, that this will lie uppermost; though still with hesitation and doubt, in proportion to the number of chances, which are contrary: And according as these contrary chances diminish, and the superiority encreases on the other side, his belief acquires new degrees of stability and assurance. This belief arises from an operation of the mind upon the simple and limited object before us; and therefore its nature will be the more easily discovered and explained. We have nothing but one single dye to contemplate, in order to comprehend one of the most curious operations of the understanding.
This dye, formed as above, contains three circumstances worthy of our attention. First, Certain causes, such as gravity, solidity, a cubical figure, &c. which determine it to fall, to preserve its form in its fall, and to turn up one of its sides. Secondly, A certain number of sides, which are supposed indifferent. Thirdly, A certain figure inscribed on each side. These three particulars form the whole nature of the dye, so far as relates to our present purpose; and consequently are the only circumstances regarded by the mind in its forming a judgment concerning the result of such a throw. Let us, therefore, consider gradually and carefully what must be the influence of these circumstances on the thought and imagination.
First, We have already observed, that the mind is determined by custom to pass from any cause to its effect, and that upon the appearance of the one, it is almost impossible for it not to form an idea of the other. Their constant conjunction in past instances has produced such a habit in the mind, that it always conjoins them in its thought, and infers the existence of the one from that of its usual attendant. When it considers the dye as no longer supported by the box, it can not without violence regard it as suspended in the air; but naturally places it on the table, and views it as turning up one of its sides. This is the effect of the intermingled causes, which are requisite to our forming any calculation concerning chances.
Secondly, It is supposed, that though the dye be necessarily determined to fall, and turn up one of its sides, yet there is nothing to fix the particular side, but that this is determined entirely by chance. The very nature and essence of chance is a negation of causes, and the leaving the mind in a perfect indifference among those events, which are supposed contingent. When therefore the thought is determined by the causes to consider the dye as falling and turning up one of its sides, the chances present all these sides as equal, and make us consider every one of them, one after another, as alike probable and possible. The imagination passes from the cause, viz. the throwing of the dye, to the effect, viz. the turning up one of the six sides; and feels a kind of impossibility both of stopping short in the way, and of forming any other idea. But as all these six sides are incompatible, and the dye cannot turn up above one at once, this principle directs us not to consider all of them at once as lying uppermost; which we look upon as impossible: Neither does it direct us with its entire force to any particular side; for in that case this side would be considered as certain and inevitable; but it directs us to the whole six sides after such a manner as to divide its force equally among them. We conclude in general, that some one of them must result from the throw: We run all of them over in our minds: The determination of the thought is common to all; but no more of its force falls to the share of any one, than what is suitable to its proportion with the rest. It is 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.
We have already seen the influence of the two first qualities of the dye, viz. the causes, and the number and indifference of the sides, and have learned how they give an impulse to the thought, and divide that impulse into as many parts as there are unites in the number of sides. We must now consider the effects of the third particular, viz. the figures inscribed on each side. It is evident that where several sides have the same figure inscribe 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 dispersed over the several sides, upon which that figure is inscribed. Were the question only what side will be turned up, these are all perfectly equal, and no one coued ever have any advantage above another. But as the question is concerning the figure, and as the same figure is presented by more than one side: it is evident, that the impulses belonging to all these sides must re-unite in that one figure, and become stronger and more forcible by the union. Four sides are supposed in the present case to have the same figure inscribed on them, and two to have another figure. The impulses of the former are, therefore, superior to those of the latter. But as the events are contrary, and it is impossible both these figures can be turned up; the impulses likewise become contrary, and the inferior destroys the superior, as far as its strength goes. The vivacity of the idea is always proportionable to the degrees of the impulse or tendency to the transition; and belief is the same with the vivacity of the idea, according to the precedent doctrine.
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