The Probability of Chances

Chance is the Negation of Cause

To show the full force and evidence of this system, we must:

• view its consequences, and
• explain from the same principles some other species of reasoning derived from the same origin.

The following philosophers are obliged to comprehend all our arguments from causes or effects under the general term of probability:

• Those who have:
  • divided human reason into knowledge and probability, and
  • defined the reason to be that evidence which arises from the comparison of ideas.

Everyone is free to use his words in what sense he pleases.

However, many arguments from causation:

• exceed probability, and
• may be received as a superior kind of evidence.

It would be ridiculous to say that it is only probable:

• that the sun will rise tomorrow, or
• that all men must die.

We have no assurance of these facts other than what experience affords us.

Let us categorize human reason into three kinds in order to preserve the meaning words and mark the several degrees of evidence.

From knowledge

Knowledge is the assurance arising from the comparison of ideas.

From proofs

Proofs are those arguments:

• derived from the relation of cause and effect, and
• entirely free from doubt and uncertainty.

From probabilities

Probability means that evidence which is still attended with uncertainty.

Probability is reasoning from conjecture.

It may be divided into 2 kinds:

Probability founded on chance Probability arising from causes.

The idea of cause and effect is derived from experience.

Experience presents us with certain objects constantly conjoined with each other.

It produces a habit of surveying them in that relation.

We cannot survey them in any other relation, without a sensible violence.

On the other hand, chance is not real in itself.

It is merely the negation of a cause.

Its influence on the mind is contrary to the influence of causation.

Chance is essential to the mind.

It leaves the imagination perfectly indifferent to consider the existence or non-existence of that object it regards as contingent.

A cause traces the way to our thought. It 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 it is instantly re-instated on the absence of a cause. A perfect and total indifference is essential to chance.

We can only say that Chance A is superior to Chance B if Chance A has had more chance in the past. By doing so, we: allow of a cause, and destroy our indifference to chance. One total indifference can never in itself be superior or inferior to another. This truth is acknowledged by every system that forms calculations on chances. Chance and causation are directly contrary.

It is remarkable that it is impossible for us to conceive this combination of chances needed 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 imagined idea is on a footing of equality. There cannot be any circumstance to give one the advantage above another. We cannot calculate the laws of hazard unless we allow that there are some causes to make the dice: fall and preserve their form in their fall, and lie on some one of their sides. It is easy to arrive at a notion of a superior combination of chances, supposing: these causes to operate, and all the other causes are indifferent and to be determined by chance. This superiority can be easily seen in a die that has four sides with one dot and two sides with two dots.

The mind here is:

• limited by the causes to two outcomes, and
• not determined in its choice of any outcome.

Chance is merely the negation of a cause.

It produces a total indifference in the mind. A negation of a cause and a total indifference can never be superior or inferior to each other. There must always be a mixture of causes among the chances to be the foundation of any reasoning.

What effect does a superior combination of chances have on the mind?

How does it influence our judgment?

We may repeat all the same arguments we used to examine that belief, which arises from causes. We may prove that a superior number of chances produces our assent not by demonstration nor probability. By the comparison of mere ideas, we can never make any discovery which can be of consequence in these affairs. 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 any certainty in this case, is to overthrow what we have established about the opposition of chances and their perfect equality and indifference. In an opposition of chances, it is impossible to determine on which side the event will fall.

Yet we can say that it is more likely and probable that it will be on the side where there are more chances. What is meant by likelihood and probability here? The likelihood and probability of chances is a superior number of equal chances. Consequently, when we say it is likely that the outcome will fall on the superior side, we only affirm that: the superior side is where there are more chances, and the inferior side is where there are fewer chances. These are identical propositions, and of no consequence.

The question is: how a superior number of equal chances:

• operates on the mind
• produces belief or assent?

Since by arguments, it is not derived from demonstration nor probability.

To clear up this difficulty, let us suppose that a person throws the die that has one dot on four sides and two dots on two sides.

He concludes that one outcome will be more probable than the other. He prefers the one dot which is inscribed on more sides. He believes that this will be the most likely outcome even if, with hesitation and doubt, it is proportional to the number of contrary chances.

His belief gets more stability and assurance as these chances change. This belief arises from an operation of the mind on the simple and limited object of the die. The nature of the die can be easily explained. We only have one die to contemplate, to comprehend one of the most curious operations of the understanding.

This die has three circumstances worthy of our attention.

Physical causes, such as gravity, solidity, a cubical figure, etc. determine it to:

• fall,
• preserve its form in its fall, and
• turn up one of its sides.

Its number of sides which are indifferent.

A dot or two dots inscribed on each side.

These three items make up the whole nature of the die

These are are the only circumstances regarded by the mind in its judgment on the outcome of such a throw.

Therefore, let us consider the influence of these circumstances on the mind.

Influence 1: The mind habitually passes from any cause to its effect.

On the appearance of the one, it is almost impossible for the mind not to create an idea of the other. Their constant conjunction in past instances has produced such a habit in the mind.

It always:

• conjoins them in its thought, and
• infers the existence of the one from that of its usual attendant.

When it considers the die as no longer in the box, it cannot, without violence, regard it as suspended in the air.

It naturally:

• places it on the table, and
• views it as turning up one of its sides.

This is the effect of the intermingled causes needed to form any calculation concerning chances.

Influence 2: There is nothing to fix the particular side of the die when it lands.

This is determined entirely by chance.

The very nature and essence of chance is a negation of causes, and in leaving the mind in a perfect indifference to those outcomes which are supposed contingent. The mind is determined by the cause to consider the die as falling and turning up one of its sides.

The chances:

• present all these outcomes as equal, and
• make us consider every outcome, as similarly probable.

The cause is the throwing of the die.

The effect is the turning up one of the six sides.

The mind passes from the cause to the effect

It feels an impossibility of:

• stopping short in the way, and
• forming any other idea.

This principle does not lead us to think of all the outcomes because:

• all these six sides are incompatible, and
• the die cannot show more than one side at a time.

This principle does not make us think with its entire force to any particular side:

For in that case, this side would be considered as certain and inevitable.

Instead, it makes us think of the six sides, dividing its force equally among them.

We conclude 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. In this way, the original impulse, and consequently the vivacity of thought, arising from the causes, is split by the intermingled chances. We have already seen the influence of the two first qualities of the die:

the physical causes of the die, and the number and indifference of the sides. We have learned how they:

give an impulse to the mind, and divide that impulse into as many parts as there are the number of sides. We must now consider the effects of the third quality: the inscriptions on each side.

A dot is inscribed on several sides of the die.

These sides:

• have the same influence on the mind, and
• unite on one idea of a dot all those divided impulses, that were dispersed on the sides that had that dot.

What side will be the outcome?

All the sides have a perfectly equal chance.

Will the outcome be one dot?

At this question, our impulses on these sides:

• re-unite at the idea of one dot, and
• become stronger and more forcible by the union on the dot and the sides.

4 sides presently have one dot inscribed on them.

2 sides have two dots inscribed on them.

Therefore, the impulses of the one dot are superior to those of the two dots.

But:

• the events are contrary, and
• it is impossible for both the one dot and two dots to be the outcome at the same time.

Thus:

• the impulses likewise become contrary, and
• the inferior destroys the superior, in terms of strength.

The vivacity of the idea is always proportional to the impulse or tendency to the transition.

Belief is the same with the vivacity of the idea, according to the precedent doctrine.