Conjunction and Probability

What do we do in ordinary life when we have to calculate the probability of finding a sequence repeated, where the events that are conjoined are not known to be connected in some other way?

Suppose we throw a die and get a six. The chances that six will turn up in a single throw are one in six, or 1/6. Assuming that we know of nothing in the way the die is thrown that will secure one side rather than some other, the chances that a second throw will have the same result is 1/6 x 1/6 or 1/36. The chances of getting three sixes in a row is (1/6) * (1/6) * (1/6), or 1/216. The probability of getting six sixes in a row would be one in almost 50,000, and if we carried the repetitions to ten or more, the figures would become astronomical.

If we found that anyone kept getting sixes regularly, we would begin to get suspicious. We would accept their results perhaps through two or three repetitions, or perhaps a few more. But there would come a time when the hypothesis of mere luck would strain our belief, compared to the theory of some sort of control, that we would only a dupe would continue believing it.

We can't say what the chances are of getting any one result, because we can't set a limit to the number of possible alternatives (Strictly speaking, we can't set such a limit in the other case either. We can't rule out beforehand the possibility that the die would stand on one of its corners and spin there permanently.). But we can say something if one of these results goes on repeating itself to the exclusion of all other results.

If there is nothing in one event that would compel or require another event to follow, then it's more likely that the first event would be followed by by something other than some other third event than by the second event itself. And if, in spite of this, only the second event continues to appear with the first event, it's just as naive to assume they are connected by anything but chance, as it would be to assume the same thing about winning throws of the die.

But that's what the denial of intrinsic relations commits us to. It says that if water continues to put out fires, it's just luck.

But if you believe in merely chance conjunctions, then among the possibilites covered by that view is not only repetition of the first event with varying consequents, but also its repetition with the same result.

It's always the improbable that happens. Watever side of a die comes up, the chances agsint it before it happend were five to one. Yet for all that, it happened. The simultaneous occurrence of the events composing the present universe is only one out of an infinite number of possibilities, and hence was all but infinitely improbably a moment ago. Yet here it is.

In the same way, the repetition of two events toegther a dozen times, or a hundred, or a thousand, is unlikely beforehand, but is still one of the possibilities conceivable under the rule of chance. Therefore, it's absurd to offer it as evidence against the rule of chance.

I don't think there is a reply to this, on it's own grounds. I think there are other ways in which the theory that all succession is chance can be refuted, such as by demonstrating necessity in inference. Out of an infinite number of chance combinations, the known history of the world may present one, and it might be argued that any extension of that history, with any multiplication of its uniformities, could only produce another.

But the argument commits the fallacy of inexhaustive division. The question at issue is how the successions we find are to be interpreted, and the original argument proceeded by offering the alternative of chance or some form of necessity, and then eliminating chance. The alternatives considered in the reply are not these. These are combinations that would be possible only by chance.

You've been so eager to show that your hypothesis of chance will cover the facts that you've forgotten that to the hypothesis under which all your combinations fall, there is a further alternative which may cover the facts equally well. And when we turn from chance to intrinsic connection, it not only covers the facts much better.

For on the chance hypothesis, every successive repetition of a conjunction given in the past is the occurrence of the progressively more improbable, while with intrinsic connection, it's only a confirmation, more impressive at each recurrence, of what the hypothesis predicted. So far, neither theory can exclude its rival. But the chance hypothesis, while consistent with the known arrangement, would accord equally with any other arrangement. The hypothesis of connection accords selectively with the arrangement actually found.

32.17

Causation, Physical Nature, and Regular Sequence

So not only in inference but also in many other mental processes, the causal relation involves necessity.

But what about causation in physical nature? This is the field in wich most cases of causation are considered to occur.

The only relation between physical cause and effect is mere regular conjunction. You can't show that the relation of a hammer to a nail it hits is the same as the relation between statements in the process of inference and other mental processes.

The question

"Why is any physical event followed by any other?

has not been answered. But failure in specific cases does not show that causality is merely regular sequence. Regular sequence is not enough. There must be some intrinsic connection between cause and effect. And this intrinsic connection includes logical necessity.

Mental processes supply genuine cases of causality, and any situation in which the causal relation involves necessity is enough to invalidate an explanation which denies that causality involves necessity.

But we are dealing with causality in physical nature. And even there, the theory is disproved by several criticisms.

32.16

Necessity, Tautology, and the Empirical

The consequent that seems to be entailed by love, or low self-esteem, or the incomplete work of the painter, is not a predicate distinct from the subject and necessitated by it, but part of the subject itself. To feel gratitude when praised by others, for example, is part of the meaning of low self-esteem. The judgment is analytic.

But an analytic judgment is one in which the predicate forms part of the subject concept, part of the subject as thought of.

"A circle is round." 

is analytic, if to think of a circle is to think of a round figure.

"A circle is that plane figure which encloses the largest area with the smallest perimeter."

is not analytic, because even though it's as necessary as the previous statement, the predicate does not have to be a part of the buejct concept. I can think of a circle without thinking of this property of it. To say that a judgment is still analytic when the predicate is part of the subject, not as thought of, but as existent in rerum natura, would make the drawing of the ddistinction impossible in most of our judgments.

The feeling of gratitude is not part of what we mean by low self-esteem. I can think about low self-esteem without any reference to gratitude, and yet see when it is pointed out to me that there is more than an accidental connection between them.

To call such statements necessary is to take them as prior, and to take them as prior is to say that they are not dependent on experience, and to say that all statements about special types of emotion and desire are not dependent on experience is absurd.

But the mark of prior connections is not our ability to conceive them before they are presented in fact, but simply their necessity. And necessary connections are just as truly presented through the given as any other connections.

That whatever is read is extended is learned through experience, although this does not involve saying that like the statement

"Red coals burn."

it's a mere empirical connection.

But necessity holds only among formal characteristics, such as those studied in arithmetic and geometry, not among such qualitative characters as love, hate, and desire.

That's mere superstition. The statements

"What is read is extended."

and

"Pleasure is a good."

are just as prior as

"2 + 2 = 4"

I'm not suggesting that in the instances mentioned about mental causality the factors connected and the connections between them can be isolated as they can in these simpler instances. The causal relation is complex, and the relation of necessity is merely one of its strands.

And when we speak of low self-esteem producing gratitude, those two terms are so vague and complex, that if they were fully analyzed, we would see that we had included something in each which was not intimately connected with the other, and omitted from each something that was connected in that way.

But that only proves that in these cases no necessary nexus has been analyzed out yet. The failure to discern connections has been used too frequently to decide against the conviction that we do have insight into why the result occurs in these cases.

32.15

Necessity and Inference

Inference is of course just one of many mental processes. And the influence of necessity shown in inference can be traced to other mental processes.

Not that it's presence in other processes is equally plain. And I don't expect this anyway. Inference is concerned with necessity in a unique way.

But necessity, no matter what our first impressions are, is a matter of degree. Between a complete demonstration and a mere accidental conjunction it might be present in many degrees. Rigorous inference approximates a complete demonstration, and assocation by mere contiguity is just an accident.

And between demonstration and accidental conjunction there are many processes in which there is more contingency than in inference, but more necessity than in association.

A painter is painting a landscape that is half-completed, and he finds himself moved to put a tree in the foreground. Is that kind of development normally intelligible? Most painters would not say so.

And it's not an example of pure necessity, but clearly falls somewhere in between.

Let's say someone is afraid of dogs. Through psycho-analysis they discover that at an early age they were badly injured by the attack of a dog. It may come to them as a quasi-intuitive insight that it was this that caused the fear. And let's assume they're right. I can't claim how the fear arose is unintelligible to them.

"Why are modest men grateful? Because they think lightly of their own deserts. This implies a syllogism in Barbara.

All who think lightly of their own deserts are grateful,

and

Modest men think lightly of their own deserts." (Joseph, Logic, 306)

Does the major premise

All who think lightly of their own deserts are grateful.

express a causal connection or a logical connection?

It expresses both.

If someone we know has low self-esteem becomes grateful for someone else's esteem, is that as surprising, unpredictable, and unintelligible, as if they had started speaking in a Sumerian language or become violent?

I'm not saying that between thinking lightly of oneself and being grateful there is the same simple abstract connection that one finds in geometry. I can't isolate in human nature the precise reciprocating conditions of gratitude, or formulate the rule in anything better than a statement of tendency.

But tendency is after all not utter darkness. We do have some insight into why the person of low self-esteem should be grateful for the esteem of others.

We can see and to some extent really understand why an insult tends to anger, why love leads to grief if the object of one's love dies or is untrustworthy, why a success should give pleasure, why the anticipation of physical pain should arouse fear. It does seem more reasonable on other than inductive grounds to suppose that if Ann loves Bob that will tend to make her sorry when Bob dies than to suppose that it will make her intensely glad. . . . At any rate anybody who denies altogether the insight for which I am contending will have to hold that it is just as reasonable to think of love as causing intense joy at the death of the person loved, except that this does not happen in fact.

If the regularity view were true, all practical life would be nonsense. All practical life assumes we can do things and are moved by motives and desires.

For example, to say that some actions are motivated by power is more than saying that some actions generally follow a desire for power.

It is saying:

"Some actions are driven by the desire for power."

does not merely follow, but is itself determined by the desire for power.

Causation is not merely regular sequence, but intrinsic connection as well.

(Ewing, Idealism, 176-178, 161-162)

32.14

Denying Necessity Renders Argument Irrational

Unless necessity is a part of inference, no argument will prove anything.

When I reach a conclusion from evidence, I commonly assume I have been moved by the uniqueness of this evidence, that the relevance of the evidence and the cogency of the argument had something to do with my concluding as I did.

But if they have nothing to do with my concluding things, then no conclusions are ever arrived at because the evidence requires them.

And the assumption that I have been moved by reasons is an illusion.

I've been moved only by causes, and between those causes and their effects there is no rational connection.

So the fact that a conclusion is reached at the end of an argument, no matter how necessary each statement may seem because of the statements before it has nothing to do with its validity, since logical factors had no part in the actual causation of the of the conclusion.

But how can you even argue for that view consistently? If you do argue for it, you assume that your own belief and that of those you're trying to convince might be affected by your argument, and that the more reasonable or logically compelling you can make your case, the better your prospects of persuading reasonable minds.

If you were to act consistently with your theory, you would abandon reasoning and never seek to reason with others.

No one is really moved by reasoning to accept a conclusion, are they?

32.13

Necessities Within Content

You're confusing two different kinds of relation. Logical relations are timeless, connecting essences or characters. Causal relations are temporal, and connect events. Hence in reasoning, logic is not the cause of the outcome. Logic is not a series of causes and effects, whether physiological or psychological.

Logic is simply the fact of the consistency of one statement with another. It's the knowledge of these consistencies and inconsistencies which is to some slight extent the cause. Causality links events and the events in the case being considered are first my acceptance of the conclusion, then my apprehension that the premises entail the conclusion, and my acceptance of that conclusion.

It is between these events, or psychical facts, that the causality holds, and it's as meaningless to say that one such event entails another as it is to say the premises cause the conclusion.

An attractive view, for the moment. You're trying to draw a line horizontally through a train of events, and say that above that line are the characters or universals of logic, and below that line are the existences or occurrences of causality.

But that kind of line can't be drawn. Even if it could, causality would have no necessity or meaning. The characters dealt with by logic do enter into the causal process. An event without characters is no event at all.

If what happens is nothing, then nothing happens.

When a hammer drives a nail, the cause in part is the motion of a certain mass at a certain velocity. These characters are an integral part of the cause. It's because of their presence that this effect rather than some other is produced, and no statement of what caused the effect could disregard them.

In the same way, the psychical event called the occurrence of a judgment in a mind is not a naked act or event. It's always a presentation of this rather than that.

And the character of this content enters vitally into the causation.

Now if the contents or characters of the events enter into the process, we can't say that the relations between these characters are irrelevant to it. Maybe no one would deny that there is a fact such as association by similarity. And however difficult it may be to construe the mechanism at work, it would be futile to deny that the similarity of content sometimes has something with the appearance of an associate. That relations within the content play a part in causation is even clearer when the relation is logical necessity.

In explicit inference we have a process in which we can directly see not only that one event succeeds another, but in large measure why one event succeeds another. It would be absurd if after the presence in our minds of the judgments, 'the squire was the abbe's first penitent' and 'that first penitent was a murderer', and the conclusion, 'the squire is a murderer', that we said we didn't have the slightest notion of why this conclusion emerged rather than a judgment about Florida grapefruit.

The fact that this was the logical conclusion to draw is relevant to its appearance as an event. And this does not commit us to say that necessary relations, connecting the content now before us with something else that is not before us, have something to do with the development of this content in one direction rather than another, that necessities within the subject-matter lay under some degree of compulsion the temporal path of thought. You may not accept the metaphysical theory I offered earlier to explain this compulsion, but the compulsion is an undeniable fact.

32.12

Logical Necessity in Inferential Causality

Consider any instance of reasoning, for example the case of the abbe and the squire. "Ladies", says the abbe, "do you know that my first penitent was a murderer?" "Ladies,", says the squire, entering shortly afterward, "do you know that I was the abbe's first penitent?"

Of course a conclusion was produced in the ladies' minds, and the question is about the nature of the causation that produced it. Now the ladies' entertainment of the premises had something causally to do with the emergence in their minds of the conclusion. The question is whether, when we say that the one contributed causally to the other, our only proper meaning is that whenever the first appears the second one does, or whether we must also say that the special logical relation in which the content of the premises stands to the content of the conclusion had something to do with the appearance of that conclusion. It did have something to do with it.

According to the regularity theory, our only reason for expecting the judgment that arose rather than some other judgment, such as that Florida raises grapefruit, is that thoughts of the first kind have regularly been followed by thoughts of the second. The question why what followed did follow is one which we cannot answer. The connection between the events is no more intelligible than the connection of lightning and thunder for a savage.

If the ladies were asked how they came to have the belief, they would say that it was because this belief was implied by what they were thinking in the previous moment, and this is the natural and correct answer. Other causes could have contributed to the result. We are not saying that causality reduces to logical necessity. But when one passes in reasoning from ground to consequent the fact that the ground entails the consequent is one of the conditions determining the appearance of this consequent rather than something else in the thinker's mind.

32.11