Logic

The Cave

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Book VII of Plato’s Republic begins with this conversation between Socrates and Glaucon:

And now, I said, let me show in a figure how far our nature is enlightened or unenlightened: –Behold! human beings living in a underground den, which has a mouth open towards the light and reaching all along the den; here they have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning round their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised way; and you will see, if you look, a low wall built along the way, like the screen which marionette players have in front of them, over which they show the puppets.

I see.
And do you see, I said, men passing along the wall carrying all sorts of vessels, and statues and figures of animals made of wood and stone and various materials, which appear over the wall? Some of them are talking, others silent.

You have shown me a strange image, and they are strange prisoners.
Like ourselves, I replied; and they see only their own shadows, or the shadows of one another, which the fire throws on the opposite wall of the cave?

True, he said; how could they see anything but the shadows if they were never allowed to move their heads?

And of the objects which are being carried in like manner they would only see the shadows?

Yes, he said.
And if they were able to converse with one another, would they not suppose that they were naming what was actually before them?

Very true.
And suppose further that the prison had an echo which came from the other side, would they not be sure to fancy when one of the passers-by spoke that the voice which they heard came from the passing shadow?

No question, he replied.
To them, I said, the truth would be literally nothing but the shadows of the images.

That is certain.

The human situation with respect to truth is somewhat like Plato’s cave dwellers. I argued in the previous post that all human knowledge is imperfect and vague. A more precise argument to the same effect would be the following.

There are two basic ways that we can learn the meaning of words. In one way, by having the meaning of the word explained to us. This requires the knowledge of other words. In another way, by learning the contexts in which the word is used. This does not necessarily require the previous knowledge of other words. Thus for example we learn the meaning of the word “red” by seeing it used in reference to red things, without a necessity of explaining the idea of red using other words. Third, we might learn the meaning of some words by a combination of the above methods.

Defining a word by other words cannot lead to an absolutely precise meaning unless the other words that are used have an absolutely precise meaning, so such a definition cannot be a source of absolute precision. And unless one points to all possible cases in which a word might be used, which is a physical impossibility, experiencing the contexts in which a word is used will also never lead to an absolutely precise meaning. It is consequently unlikely that you can arrive at an absolutely precise meaning by any combination of these methods, and consequently unlikely that you can arrive at such absolute precision in any way.

Language and thought are very closely connected. There are perhaps not many real cases of feral children, but the experience we do have of such situations suggests that someone who does not learn a language is basically unable to think. This is perhaps not true absolutely, but it is surely true that such a child cannot think clearly. Clear thinking requires language. This at least suggests that perfectly clear thinking would require perfectly clear language. But this is impossible, by the above argument. Consequently perfectly clear thinking is impossible for human beings.

This argument does not refute itself. It it of course impossible to have a proof that there are no proofs. Likewise, if absolute subjective certainty is impossible, as I have suggested, then it is impossible to have such certainty about this very fact. There is no inconsistency here. Likewise, if all thoughts and all arguments are vague, then these very thoughts and this very argument is vague. There is no inconsistency here.

The Paradox of the Heap

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The paradox of the heap argues in this way:

A large pile of sand is composed of grains of sand. But taking away a grain of sand from a pile of sand cannot make a pile of sand stop being a pile of sand. Therefore if you continually take away grains of sand from the pile until only one grain of sand remains, that grain must still be a pile of sand.

A similar argument can be made with any vague word that can differ by an apparently continuous number of degrees. Thus for example it is applied to whether a man has a beard (he should not be able to change from having a beard to not having a beard by the removal of a single hair), to colors (an imperceptible variation of color should not be able to change a thing from being red to not being red), and so on.

The conclusion, that a single grain of sand is a pile of sand, or that a shaven man has a beard, or that the color blue is red, is obviously false. In order to block the deduction, it seems necessary to say that it fails at a particular point. But this means that at some point, a pile of sand will indeed stop being a pile of sand when you take away a single grain. But this seems absurd.

Suppose you don’t know the meaning of “red,” and someone attempts to explain. They presumably do so by pointing to examples of red things. But this does not provide you with a rigid definition of redness that you could use to determine whether some arbitrary color is an example of red or not. Rather, the probability that you will call something red will vary continuously as the color of things becomes more remote from the examples from which you learned the name, being very high for the canonical examples and becoming very low as you approach other colors such as blue.

This explains why setting a boundary where an imperceptible change of color would change something from being red to being not red seems inappropriate. Red doesn’t have a rigid definition in the first place, and assigning such a boundary would mean assigning such a definition. But this would be modifying the meaning of the word. Consequently, if the meaning is accepted in an unmodified form, the deduction cannot logically be blocked, just as in the previous post, if the meaning of “true” is accepted in an unmodified form, one cannot block the deduction that all statements are both true and false.

Someone might conclude from this that I am accepting the conclusions of the paradoxical arguments, and therefore that I am saying that all statements are both true and false, and that a single grain of sand is a pile, and so on.

I am not. Concluding that this is my position is simply making the exact same mistake that is made in the original paradoxes. And that mistake is to assume a perfection in human language which does not exist. “True,” “pile,” and so on, are words that possess meaning in an imperfect way. Ultimately all human words are imperfect in this way, because all human language is vague. The fact that logic cannot block the paradoxical conclusions without modifying the meanings of our words happens not because those conclusions are true, but because the meanings are imperfect, while logic presupposes a perfection of meaning which is simply not there.

In a number of other places I have talked about how various motivations can lead us astray. But there are some areas where the very desire for truth can lead us away from truth, and the discussion of such logical paradoxes, and of the vagueness of human thought and language, is one of those areas. In particular, the desire for truth can lead us to wish to believe that truth is more attainable than it actually is. In this case it would happen by wishing to believe that human language is more perfect than it is, as for example that “red” really does have a meaning that would cause something in an a definitive way to stop being red at some point with an imperceptible change, or in the case of the Liar, to assert that the word “true” really does have something like a level subscript attached to its meaning, or that it has some other definition which can block the paradoxical deductions.

These things are not true. Nor are the paradoxical conclusions.

The Liar

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The paradox of the Liar is a logical problem that results from a sentence that implies that the very sentence itself is false, or at least that it is not true. Consider the following statement:

(1) Statement (1) is not true.

Is statement (1) true or not? We might reason about it as follows.

(2) If statement (1) is true, then statement (1) is not true, since this is what it says.

(3) But this is absurd, since statement (1) would then be both true and not true.

(4) Therefore (1) is not true.

(5) But this is just what (1) says. So (1) is true.

And so on. There does not appear any way to avoid the conclusion that (1) is both true and not true, which is a contradiction. Nor is it helpful to say that it is neither true nor not true, since the same contradiction will follow: if something fails to be not true, it is surely true.

Any statement whatever will follow from a contradiction, so if one accepts this contradiction, one will be forced to accept that every statement is both true and false.

A. N. Prior discusses the idea of an analytically valid inference:

It is sometimes alleged that there are inferences whose validity arises solely from the meanings of certain expressions occurring in them. The precise technicalities employed are not important, but let us say that such inferences, if any such there be, are analytically valid.

One sort of inference which is sometimes said to be in this sense analytically valid is the passage from a conjunction to either of its conjuncts, e.g., the inference ‘Grass is green and the sky is blue, therefore grass is green’. The validity of this inference is said to arise solely from the meaning of the word ‘and’. For if we are asked what is the meaning of the word ‘and’, at least in the purely conjunctive sense (as opposed to, e.g., its colloquial use to mean ‘and then’), the answer is said to be completely given by saying that (i) from any pair of statements P and Q we can infer the statement formed by joining P to Q by ‘and’ (which statement we hereafter describe as ‘the statement P-and-Q’), that (ii) from any conjunctive statement P-and-Q we can infer P, and (iii) from P-and-Q we can always infer Q. Anyone who has learnt to perform these inferences knows the meaning of ‘and’, for there is simply nothing more to knowing the meaning of ‘and’ than being able to perform these inferences.

A doubt might be raised as to whether it is really the case that, for any pair of statements P and Q, there is always a statement R such that given P and given Q we can infer R, and given R we can infer P and can also infer Q. But on the view we are considering such a doubt is quite misplaced, once we have introduced a word, say the word ‘and’, precisely in order to form a statement R with these properties from any pair of statements P and Q. The doubt reflects the old superstitious view that an expression must have some independently determined meaning before we can discover whether inferences involving it are valid or invalid. With analytically valid inferences this just isn’t so.

I hope the conception of an analytically valid inference is now at least as clear to my readers as it is to myself. If not, further illumination is obtainable from Professor Popper’s paper on’ Logic without Assumptions’ in Proceedings of the Aristotelian Society for 1946-7, and from Professor Kneale’s contribution to Contemporary British Philosophy, Volume III. I have also been much helped in my understanding of the notion by some lectures of Mr. Strawson’s and some notes of Mr. Hare’s.

He proceeds to draw some conclusions from this:

I want now to draw attention to a point not generally noticed, namely that in this sense of ‘analytically valid’ any statement whatever may be inferred, in an analytically valid way, from any other. ‘2 and 2 are 5’, for instance, from ‘2 and 2 are 4 ‘. It is done in two steps, thus:

2 and 2 are 4.

Therefore, 2 and 2 are 4 tonk 2 and 2 are 5.

Therefore, 2 and 2 are 5.

There may well be readers who have not previously encountered this conjunction ‘tonk’, it being a comparatively recent addition to the language; but it is the simplest matter in the world to explain what it means. Its meaning is completely given by the rules that (i) from any statement P we can infer any statement formed by joining P to any statement Q by ‘tonk’ (which compound statement we hereafter describe as’ the statement P-tonk-Q ‘), and that (ii) from any ‘contonktive’ statement P-tonk-Q we can infer the contained statement Q.

A doubt might be raised as to whether it is really the case that, for any pair of statements P and Q, there is always a statement R such that given P we can infer R, and given R we can infer Q. But this doubt is of course quite misplaced, now that we have introduced the word ‘tonk’ precisely in order to form a statement R with these properties from any pair of statements P and Q.

As a matter of simple history, there have been logicians of some eminence who have seriously doubted whether sentences of the form ‘P and Q’ express single propositions (and so, e.g., have negations). Aristotle himself, in De Soph. Elench. 176 a 1 ff., denies that ‘Are Callias and Themistocles musical?’ is a single question; and J. S. Mill says of ‘Caesar is dead and Brutus is alive’ that ‘we might as well call a street a complex house, as these two propositions a complex proposition’ (System of Logic I, iv. 3). So it is not to be wondered at if the form ‘P tonk Q’ is greeted at first with similar scepticism. But more enlightened views will surely prevail at last, especially when men consider the extreme convenience of the new form, which promises to banish falsche Spitzfindigkeit from Logic for ever.

His point is quite clear. Given the way the word “tonk” is defined, one cannot avoid drawing all possible conclusions. But this means the word “tonk”, defined in this way, is quite unacceptable in the first place.

If we define the word “true” by saying that “P is true” is a statement such that it necessarily follows from P, and such that P necessarily follows from “P is true,” and we consider this an acceptable definition, then the rules of logic will force us to accept all possible conclusions.

Like the definition of the word “tonk”, therefore, this definition of the word “true” is unacceptable, and in the same sense, namely that if the definition is accepted, all possible conclusions follow.

This explains why all solutions to the Liar paradox seem to fail, in the sense that in the end either they admit a contradiction, or they insist that we change the meaning of our language, as for example by talking about levels of truth and so on. For despite the consequences, the word “true” does basically have the meaning stated. The only real difference in comparison with the word “tonk” is that the latter word would never be used in any real language, because the consequences are obvious. In the case of “true,” the consequences are subtle and only follow in special circumstances, namely the kind that are found in the case of the Liar paradox, and so the word could be incorporated into human language, and basically with this meaning, before the implications were noticed.

Note that this is quite different from saying that the word “true” has an inconsistent meaning. The problem is even deeper than that. We could define the word “zrackled” to mean “white and not white in the same respect,” and the meaning would be inconsistent. The only consequence would be that nothing is zrackled, and no contradiction would follow. But if we said that “true” has an inconsistent meaning, and consequently that nothing is true, it would follow from the meaning stated that “nothing is true” is true, and consequently that “nothing is true” is not true. The problem is that we are attempting to define the word, at least in part, by certain rules of usage, and those rules themselves force a contradiction, and ultimately force one to draw all possible conclusions, as with the word “tonk.”

C.S. Lewis on Bulverism

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C.S. Lewis begins his essay on Bulverism:

It is a disastrous discovery, as Emerson says somewhere, that we exist. I mean, it is disastrous when instead of merely attending to a rose we are forced to think of ourselves looking at the rose, with a certain type of mind and a certain type of eyes. It is disastrous because, if you are not very careful, the color of the rose gets attributed to our optic nerves and its scent to our noses, and in the end there is no rose left. The professional philosophers have been bothered about this universal black-out for over two hundred years, and the world has not much listened to them. But the same disaster is now occurring on a level we can all understand.

We have recently “discovered that we exist” in two new senses. The Freudians have discovered that we exist as bundles of complexes. The Marxians have discovered that we exist as members of some economic class. In the old days it was supposed that if a thing seemed obviously true to a hundred men, then it was probably true in fact. Nowadays the Freudian will tell you to go and analyze the hundred: you will find that they all think Elizabeth [I] a great queen because they all have a mother-complex. Their thoughts are psychologically tainted at the source. And the Marxist will tell you to go and examine the economic interests of the hundred; you will find that they all think freedom a good thing because they are all members of the bourgeoisie whose prosperity is increased by a policy of laissez-faire. Their thoughts are “ideologically tainted” at the source.

The person probably most known for making the first point, about sensation, is John Locke. He says in his Essay Concerning Human Understanding:

9. Primary qualities of bodies. Qualities thus considered in bodies are,

First, such as are utterly inseparable from the body, in what state soever it be; and such as in all the alterations and changes it suffers, all the force can be used upon it, it constantly keeps; and such as sense constantly finds in every particle of matter which has bulk enough to be perceived; and the mind finds inseparable from every particle of matter, though less than to make itself singly be perceived by our senses: v.g. Take a grain of wheat, divide it into two parts; each part has still solidity, extension, figure, and mobility: divide it again, and it retains still the same qualities; and so divide it on, till the parts become insensible; they must retain still each of them all those qualities. For division (which is all that a mill, or pestle, or any other body, does upon another, in reducing it to insensible parts) can never take away either solidity, extension, figure, or mobility from any body, but only makes two or more distinct separate masses of matter, of that which was but one before; all which distinct masses, reckoned as so many distinct bodies, after division, make a certain number. These I call original or primary qualities of body, which I think we may observe to produce simple ideas in us, viz. solidity, extension, figure, motion or rest, and number.

10. Secondary qualities of bodies. Secondly, such qualities which in truth are nothing in the objects themselves but powers to produce various sensations in us by their primary qualities, i.e. by the bulk, figure, texture, and motion of their insensible parts, as colours, sounds, tastes, &c. These I call secondary qualities. To these might be added a third sort, which are allowed to be barely powers; though they are as much real qualities in the subject as those which I, to comply with the common way of speaking, call qualities, but for distinction, secondary qualities. For the power in fire to produce a new colour, or consistency, in wax or clay — by its primary qualities, is as much a quality in fire, as the power it has to produce in me a new idea or sensation of warmth or burning, which I felt not before — by the same primary qualities, viz. the bulk, texture, and motion of its insensible parts.

It is possible that this is actually mostly true, and mostly consistent with the philosophy of Aristotle, even though Locke would likely wish that the latter were not the case. Aristotle says in On the Soul:

If it is true that the movement, both the acting and the being acted upon, is to be found in that which is acted upon, both the sound and the hearing so far as it is actual must be found in that which has the faculty of hearing; for it is in the passive factor that the actuality of the active or motive factor is realized; that is why that which causes movement may be at rest. Now the actuality of that which can sound is just sound or sounding, and the actuality of that which can hear is hearing or hearkening; ‘sound’ and ‘hearing’ are both ambiguous. The same account applies to the other senses and their objects. For as the-acting-and-being-acted-upon is to be found in the passive, not in the active factor, so also the actuality of the sensible object and that of the sensitive subject are both realized in the latter. But while in some cases each aspect of the total actuality has a distinct name, e.g. sounding and hearkening, in some one or other is nameless, e.g. the actuality of sight is called seeing, but the actuality of colour has no name: the actuality of the faculty of taste is called tasting, but the actuality of flavour has no name. Since the actualities of the sensible object and of the sensitive faculty are one actuality in spite of the difference between their modes of being, actual hearing and actual sounding appear and disappear from existence at one and the same moment, and so actual savour and actual tasting, &c., while as potentialities one of them may exist without the other. The earlier students of nature were mistaken in their view that without sight there was no white or black, without taste no savour. This statement of theirs is partly true, partly false: ‘sense’ and ‘the sensible object’ are ambiguous terms, i.e. may denote either potentialities or actualities: the statement is true of the latter, false of the former. This ambiguity they wholly failed to notice.

Saying the sound and color are a “potentiality” when they are not being sensed, and actualized when they are being sensed, suggests very much the same thing as Locke when he says that sensible qualities are “such qualities which in truth are nothing in the objects themselves but powers to produce various sensations in us.”

However, by using the terms of “primary” and “secondary,” and saying that sensible qualities are “nothing in the objects themselves but…”, Locke suggests that there is something especially unreal about qualities such as color and odor, as Lewis mentions in his opening paragraph. This at least is a mistake on Locke’s part, since he failed to notice that there is no huge distinction between the aspects of a body that he calls “primary” and the ones that he calls “secondary.” Just as the color of an object appears differently in different lighting and so on, so a body looks larger or smaller depending on where it is situated relative to me. Likewise an elliptical object appears to have a different shape depending on my point of view. In other words, if there is reason to think that color is “nothing but the ability to look colored,” then there is an equal reason to think that shape is “nothing but the power to appear shaped.”

And I would say that both of these are true in a certain way, and false in a certain way, just as Aristotle did about the existence of white and black. They are false, if they are taken to mean that “grass is green,” is a false statement, or that is a subjective one. It is a true statement, and an objective fact about grass. They are true, if they mean that what we know about greenness is basically what we know from sensing it, and that being green means having the possibility of being sensed in this way. And the like is equally true of shape, size, quantity, hardness, and the other aspects that Locke calls primary.

C.S. Lewis is using this as a comparison with a critical analysis of people’s thought processes. Just as analyzing the causality involved in sensation can lead someone to say that sensible things don’t have an objective existence, so analyzing people’s thought processes could lead someone to believe that nothing is true. Thus he continues:

Now this is obviously great fun; but it has not always been noticed that there is a bill to pay for it. There are two questions that people who say this kind of thing ought to be asked. The first is, are all thoughts thus tainted at the source, or only some? The second is, does the taint invalidate the tainted thought – in the sense of making it untrue – or not?

If they say that all thoughts are thus tainted, then, of course, we must remind them that Freudianism and Marxism are as much systems of thought as Christian theology or philosophical idealism. The Freudian and Marxian are in the same boat with all the rest of us, and cannot criticize us from outside. They have sawn off the branch they were sitting on. If, on the other hand, they say that the taint need not invalidate their thinking, then neither need it invalidate ours. In which case they have saved their own branch, but also saved ours along with it.

The only line they can really take is to say that some thoughts are tainted and others are not – which has the advantage (if Freudians and Marxians regard it as an advantage) of being what every sane man has always believed. But if that is so, we must then ask how you find out which are tainted and which are not. It is no earthly use saying that those are tainted which agree with the secret wishes of the thinker. Some of the things I should like to believe must in fact be true; it is impossible to arrange a universe which contradicts everyone’s wishes, in every respect, at every moment. Suppose I think, after doing my accounts, that I have a large balance at the bank. And suppose you want to find out whether this belief of mine is “wishful thinking.” You can never come to any conclusion by examining my psychological condition. Your only chance of finding out is to sit down and work through the sum yourself. When you have checked my figures, then, and then only, will you know whether I have that balance or not. If you find my arithmetic correct, then no amount of vapouring about my psychological condition can be anything but a waste of time. If you find my arithmetic wrong, then it may be relevant to explain psychologically how I came to be so bad at my arithmetic, and the doctrine of the concealed wish will become relevant – but only after you have yourself done the sum and discovered me to be wrong on purely arithmetical grounds. It is the same with all thinking and all systems of thought. If you try to find out which are tainted by speculating about the wishes of the thinkers, you are merely making a fool of yourself. You must find out on purely logical grounds which of them do, in fact, break down as arguments. Afterwards, if you like, go on and discover the psychological causes of the error.

If being “tainted” means having causes that are not completely correlated with truth, then the true answers to Lewis’s questions are that either all, or nearly all, thoughts are tainted, and no, this does not necessarily mean that they are false. Virtually all thoughts are tainted in this sense because human beings do not usually have only a single motive for things that they choose to do, and this includes the choice to believe certain things. This need not “invalidate” the thoughts because obviously it does not guarantee that the thoughts are false.

Lewis is correct that pointing out that someone’s opinions match his desires does not prove that his opinions are false. However, we saw earlier that a person’s claim is evidence for what is claimed. This will be affected, however, by a person’s motives. If a person is motivated mostly by reasons which make his claim more likely to be true, then his claim is stronger evidence. And likewise, if he is motivated mostly by things which do not make his claim more likely to be true, then his claim is weaker evidence. Consequently, if I point out that such motives are a strong factor in a person’s belief, this is not a waste of time, nor is it irrelevant. It is quite relevant, because it weakens the evidence present in his claim, and consequently it becomes more likely that the thing is actually false.

Lewis continues:

In other words, you must show that a man is wrong before you start explaining why he is wrong. The modern method is to assume without discussion that he is wrong and then distract his attention from this (the only real issue) by busily explaining how he became to be so silly. In the course of the last fifteen years I have found this vice so common that I have had to invent a name for it. I call it “Bulverism.” Some day I am going the write the biography of its imaginary inventor, Ezekiel Bulver, whose destiny was determined at the age of five when he heard his mother say to his father – who had been maintaining that two sides of a triangle were together greater than the third – “Oh, you say that because you are a man.” “At that moment,” E. Bulver assures us, “there flashed across my opening mind the great truth that refutation is no necessary part of argument. Assume your opponent is wrong, and then explain his error, and the world will be at your feet. Attempt to prove that he is wrong or (worse still) try to find out whether he is wrong or right, and the national dynamism of our age will thrust you to the wall.” That is how Bulver became one of the makers of the Twentieth Century.

I find the fruits of his discovery almost everywhere. Thus I see my religion dismissed on the grounds that “the comfortable parson had every reason for assuring the nineteenth century worker that poverty would be rewarded in another world.” Well, no doubt he had. On the assumption that Christianity is an error, I can see clearly enough that some people would still have a motive for inculcating it. I see it so easily that I can, of course, play the game the other way round, by saying that “the modern man has every reason for trying to convince himself that there are no eternal sanctions behind the morality he is rejecting.” For Bulverism is a truly democratic game in the sense that all can play it all day long, and that it give no unfair advantage to the small and offensive minority who reason. But of course it gets us not one inch nearer to deciding whether, as a matter of fact, the Christian religion is true or false. That question remains to be discussed on quite different grounds – a matter of philosophical and historical argument. However it were decided, the improper motives of some people, both for believing it and for disbelieving it, would remain just as they are.

I see Bulverism at work in every political argument. The capitalists must be bad economists because we know why they want capitalism, and equally Communists must be bad economists because we know why they want Communism. Thus, the Bulverists on both sides. In reality, of course, either the doctrines of the capitalists are false, or the doctrines of the Communists, or both; but you can only find out the rights and wrongs by reasoning – never by being rude about your opponent’s psychology.

Until Bulverism is crushed, reason can play no effective part in human affairs. Each side snatches it early as a weapon against the other; but between the two reason itself is discredited. And why should reason not be discredited? It would be easy, in answer, to point to the present state of the world, but the real answer is even more immediate. The forces discrediting reason, themselves depend of reasoning. You must reason even to Bulverize. You are trying to prove that all proofs are invalid. If you fail, you fail. If you succeed, then you fail even more – for the proof that all proofs are invalid must be invalid itself.

Lewis is himself engaging in something very like Bulverism here, or at least he is attempting to get others to do so. In the first place, the very invention of the name “Bulverism” is such an attempt. It has a ridiculous sound, and the only contribution it makes to the discussion is to make it appear that someone is doing something ridiculous. This is part of Lewis’s plan to “crush” Bulverism, namely by making people dismiss it out of hand, just as he says that they dismiss his positions out of hand.

Of course, as I already stated, Bulverism as defined by Lewis is indeed a bad thing. But this is not because a person’s motives are irrelevant, but because a person may be right despite his motives. And even if a person’s motives may weaken the evidence present in his claim, it would indeed be silly to think that the full response to an argument could be that the person does not have good motives relative to truth. If you are going to refute an argument, you should do that by discussing the matter of the argument. Lewis is quite right on this point.

Regarding Lewis’s last point in this paragraph, it is unlikely that many people actually had the intention to “prove that all proofs are invalid.” People really do have various motives for their beliefs, but this does not prove that their beliefs are false, nor does it show that their arguments do not work.

Lewis concludes the essay:

The alternative then is either sheer self-contradicting idiocy or else some tenacious belief in our power of reasoning, held in the teeth of all the evidence that Bulverists can bring for a “taint” in this or that human reasoner. I am ready to admit, if you like, that this tenacious belief has something transcendental or mystical about it. What then? Would you rather be a lunatic than a mystic?

So we see there is justification for holding on to our belief in Reason. But can this be done without theism? Does “I know” involve that God exists? Everything I know is an inference from sensation (except the present moment). All our knowledge of the universe beyond our immediate experiences depends on inferences from these experiences. If our inferences do not give a genuine insight into reality, then we can know nothing. A theory cannot be accepted if it does not allow our thinking to be a genuine insight, nor if the fact of our knowledge is not explicable in terms of that theory.

But our thoughts can only be accepted as a genuine insight under certain conditions. All beliefs have causes but a distinction must be drawn between (1) ordinary causes and (2) a special kind of cause called “a reason.” Causes are mindless events which can produce other results than belief. Reasons arise from axioms and inferences and affect only beliefs. Bulverism tries to show that the other man has causes and not reasons and that we have reasons and not causes. A belief which can be accounted for entirely in terms of causes is worthless. This principle must not be abandoned when we consider the beliefs which are the basis of others. Our knowledge depends on our certainty about axioms and inferences. If these are the results of causes, then there is no possibility of knowledge. Either we can know nothing or thought has reasons only, and no causes.

“Either we can know nothing or thought has reasons only, and no causes,” indicates that Lewis is confused.

Thoughts surely do have causes, and the efficient cause is generally the human will. Reasons and motives are both “why” people choose to believe things, and consequently are the formal aspect of that efficient cause, or in other words are final causes. It might seem a little strange to say that the fact that two and two make four is the final cause of my belief that if I have two shoes and acquire two more, I will have four shoes. But it only seems strange because we do not notice that final causes themselves come in varieties. And in any case, this is partly a question of framing. If I say, “I think I would have four shoes because to think otherwise would call into question that two and two make four,” then it is clear enough that my belief is for the sake of the truth that two and two make four.

Lewis fails to note the distinction between various types of causes, and so supposes that if thoughts have causes in an ordinary sense, they cannot have reasons, but this does not follow. Reasons are simply one of the causes that thoughts have, but they can have other causes at the same time.

Someone might say that this blog engages in a good deal of what C.S. Lewis calls Bulverism, and this concern is one of the “tainted thoughts” which led me to compose this post. Thus I summarize my response to this concern:

More or less all claims are affected by motives other than truth. But this is not irrelevant, because some claims are more affected by such motives than others, and to the extent that this is the case, this weakens the evidence contained in those claims. Thus I would not consider it irrelevant even if I point out someone’s motives without proving directly that he is mistaken.

Nonetheless, in practice I do not usually do this. Generally if I think that someone is wrong, I will argue this directly, and only reveal someone’s motives in addition, which even Lewis concedes can be relevant in that situation.

There is one possible additional objection, which Lewis does not formulate explicitly, but which he indicates when he says, “The modern method is to assume without discussion that he is wrong and then distract his attention from this (the only real issue) by busily explaining how he became so silly.” The objection would be that whether he is right or wrong is “the only real issue.” So even if you first argue that someone is wrong, proceeding to discuss his motives is inappropriate, since it is no part of the issue, and it seems merely insulting.

I disagree that whether someone is right or wrong is the only real issue. As I have stated before, I do not think I really understand someone until I understand why he says what he does. And real people do not only have reasons for their beliefs, but motives as well. So in order to understand someone as well as possible, it is necessary to know not only his reasons, but also any other motives that he has.

But since people rarely notice their own motives, this suggests that I am saying that frequently people do not even understand themselves. And this is quite true, and in this way St. Paul says that some have “turned to meaningless talk, desiring to be teachers of the law, without understanding either what they are saying or the things about which they make assertions.”

In

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In Book IV of his Physics, Aristotle discusses the meanings of the word “in”:

(1) As the finger is ‘in’ the hand and generally the part ‘in’ the whole.

(2) As the whole is ‘in’ the parts: for there is no whole over and above the parts.

(3) As man is ‘in’ animal and generally species ‘in’ genus.

(4) As the genus is ‘in’ the species and generally the part of the specific form ‘in’ the definition of the specific form.

(5) As health is ‘in’ the hot and the cold and generally the form ‘in’ the matter.

(6) As the affairs of Greece centre ‘in’ the king, and generally events centre ‘in’ their primary motive agent.

(7) As the existence of a thing centres ‘in its good and generally ‘in’ its end, i.e. in ‘that for the sake of which’ it exists.

(8) In the strictest sense of all, as a thing is ‘in’ a vessel, and generally ‘in’ place.

Since the meaning and usage of prepositions tends to vary somewhat according to language, not all of these meanings are very customary in English. Still, none of the meanings are unintelligible to us.

The eighth sense is said to be the “strictest of all,” in the sense that it is the basic physical reality to which we are comparing the other meanings, just as before in time is the first sense of “before” according to the order of time. In this basic sense of “in”, water might be in a glass, or a man in a house.

But just as we saw that “before by nature” is first by nature, here too one sense of “in” is first by nature, and this is Aristotle’s fifth sense, namely as form is in matter. We can see this by noting that every sense of “in” listed by Aristotle, and in my opinion every other reasonable sense, can be analyzed in terms of an analogy with the presence of form in matter.

What do we mean in the strictest sense, when we say that water is in a glass, or a man in a house? This surely implies that the glass contains the water, and the house the man, namely that the container physically surrounds the thing contained. But what is so interesting about this? Wherever I am, there are things all around me, but I don’t say that I am in those things. I might say that I am in the ring composed of this wall, this chair, this table, the computer on which I am working, and so on, but this would be an ad hoc claim made in order to apparently establish that the statement that one thing is in another simply means that some things are surrounding something else. Also, in the case of the glass, we do not have a problem saying that the water is in the glass, even in the strictest sense, even though the water is not completely surrounded. Likewise the man is in the house even if the windows and the doors are open.

Despite the physical incompleteness of the containment in these situations, however, I would still agree that in this basic physical sense, if one thing is in another, it is contained by that other. But something else is going on even here, before we move to any other sense of “in”. This is the implication that the thing which is in something, fills it or partly fills it, so that the thing would be empty if there were not something in it. A glass without water or something similar is an empty glass, and likewise a house without people is an empty house, especially if no one lives there even at other times, and most especially if there is nothing else, such as furniture, in the house.

This is basically the idea that the thing within forms the container, which is otherwise empty and formless. Thus Genesis speaks of the unformed world as a “formless void,” namely as an empty, formless thing. The order of the days of Genesis signifies the order of matter and form, and the last three days consist in filling the empty world with moving things, filling the sea with fish, the air with birds, and the dry land with animals and men. These things are seen as forming the earth, which would otherwise be unformed, as for example is Mars in its current state.

Each of the cases mentioned by Aristotle can be analyzed in a similar way, being understood by analogy with the presence of form in matter.

The part is in the whole because, as was said in the consideration of whole and part, the part is an aspect of the existence of the whole; something which the whole exists as. In this way it is like form, just as redness is a form of the apple, and likewise an aspect of its existence. The whole is in the parts because the whole is made of the parts as its material cause.

The species is in the genus because the genus is like matter which is formed in a specific way to get the species. The genus is in the species because it is a part of the definition of the species, and thus is in the species as a part in the whole.

The effect is in the efficient or final cause because the cause is like something generic, i.e. material, which is made specific, i.e. formed, by the particular effect which it produces.

One thing is in another, therefore, as form is in matter.

Vagueness

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Vagueness comes in various kinds. In the first place, everything that human beings think or say is vague in principle, because of the weakness of human understanding. Our understanding is to the fullness of reality “as the eyes of bats are to the blaze of day,” and this weakness of our understanding is found within everything we think or say. Discussing why there is something rather than nothing, we saw that there must be a being which is necessary in itself. But even after making this argument, this does not become evident to us in itself, and this is because we do not know the nature of being.

Considering whether names said of God belong primarily to him or to us, St. Thomas says,

I answer that, In names predicated of many in an analogical sense, all are predicated because they have reference to some one thing; and this one thing must be placed in the definition of them all. And since that expressed by the name is the definition, as the Philosopher says (Metaph. iv), such a name must be applied primarily to that which is put in the definition of such other things, and secondarily to these others according as they approach more or less to that first. Thus, for instance, “healthy” applied to animals comes into the definition of “healthy” applied to medicine, which is called healthy as being the cause of health in the animal; and also into the definition of “healthy” which is applied to urine, which is called healthy in so far as it is the sign of the animal’s health. Thus all names applied metaphorically to God, are applied to creatures primarily rather than to God, because when said of God they mean only similitudes to such creatures. For as “smiling” applied to a field means only that the field in the beauty of its flowering is like the beauty of the human smile by proportionate likeness, so the name of “lion” applied to God means only that God manifests strength in His works, as a lion in his. Thus it is clear that applied to God the signification of names can be defined only from what is said of creatures. But to other names not applied to God in a metaphorical sense, the same rule would apply if they were spoken of God as the cause only, as some have supposed. For when it is said, “God is good,” it would then only mean “God is the cause of the creature’s goodness”; thus the term good applied to God would included in its meaning the creature’s goodness. Hence “good” would apply primarily to creatures rather than to God. But as was shown above, these names are applied to God not as the cause only, but also essentially. For the words, “God is good,” or “wise,” signify not only that He is the cause of wisdom or goodness, but that these exist in Him in a more excellent way. Hence as regards what the name signifies, these names are applied primarily to God rather than to creatures, because these perfections flow from God to creatures; but as regards the imposition of the names, they are primarily applied by us to creatures which we know first. Hence they have a mode of signification which belongs to creatures, as said above.

St. Thomas does not say it explicitly, but the principle he presents here, “this one thing must be placed in the definition of them all,” implies that according to his argument, God is contained in the definitions of creatures. And in this way the being which is necessary in itself must be in the definition of every being. Consequently our failure to understand the nature of being in itself implies a failure to fully understand the nature of any being.

Second, someone can say something vague for the sake of accuracy, namely because he knows that his knowledge is vague, and he wishes to express his knowledge as it is, rather than claiming to know more than he does. Freeman Dyson and others say that “it is better to be wrong than vague,” but they are, strictly speaking, wrong about this. It is better to be vague and right, rather than clear and distinct, but wrong.

Third, someone can say something vague because he does not primarily care about whether or not it is true, but about something else. In this case it may be left vague because there is no need for the effort that it would take to make it clear, since the person’s purposes can be achieved without that effort, or it may be left vague because those purposes are achieved even better when it remains vague. Thus there is a story about Hegel, which I was unable to track down while writing this post, which says that he explained to someone that the Phenomenology of Spirit had to be extremely difficult to understand, in order to ensure that Hegel would become famous. I do not find the story particularly credible, but I do find the motive credible.

I said “strictly speaking” above in discussing people who say that it is better to be wrong than vague, because in many cases such people are actually opposing the third kind of vagueness, and are simply intending to say that it is better to care about the truth but to make a mistake, than not to care about the truth at all, and therefore not to bother to say something which could turn out to be wrong.

Extraordinary Claims and Extraordinary Evidence

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Marcello Truzzi states in an article On the Extraordinary: An Attempt At Clarification, “An extraordinary claim requires extraordinary proof.” This was later restated by Carl Sagan as, “Extraordinary claims require extraordinary evidence.” This is frequently used to argue against things such as “gods, ghosts, the paranormal, and UFOs.”

However, this kind of argument, at least as it is usually made, neglects to take into account the fact that claims themselves are already evidence.

Here is one example: while writing this article, I used an online random number generator to pick a random integer between one and a billion inclusive. The number was 422,819,208.

Suppose we evaluate my claim with the standard that extraordinary claims require extraordinary evidence, and neglect to consider the evidence contained within the claim itself. In this case, given that I did in fact pick a number in the manner stated, the probability that the number would be 422,819,208 is one in a billion. So readers should respond, “Either he didn’t pick the number in the manner stated, or the number was not 422,819,208. The probability that both of those were true is one in a billion. I simply don’t believe him.”

There is obviously a problem here, since in fact I did pick the number in the way stated, and that was actually the number. And the problem is precisely leaving out of consideration the evidence contained within the claim itself. Given that I make a claim that I picked a random number between one and a billion, the probability that I would claim 422,819,208 in particular is approximately one in a billion. So when you see me claim that I picked that number, you are seeing evidence (namely the fact that I am making the claim) which is very unlikely in itself. The fact that I made that claim is much more likely, however, if I actually picked that number, rather than some other number. Thus the very fact that I made the claim is strong evidence that I did pick the number 422,819,208 rather than some other number.

In this sense, extraordinary claims are already extraordinary evidence, and thus do not require some special justification.

However, we can consider another case, a hypothetical one. Suppose that in the above paragraphs, instead of the number 422,819,208, I had used the number 500,000,000, claiming that this was in fact the number that I got from the random number generator.

In that case you might have found the argument much less credible. Why?

Assuming that I did in fact pick the number randomly, the probability of picking 422,819,208 is one in a billion. And again, assuming that I did in fact pick the number randomly, the probability of picking 500,000,000 is one in a billion. So no difference here.

But both of those assume that I did pick the number randomly. And if I did not, the probabilities would not be the same. Instead, the fact that simpler things are more probable would come into play. At least with the language and notation that we are actually using, the number 500,000,000 is much simpler than the number 422,819,208. Consequently, assuming that I picked a number non-randomly and then told you about it,  is significantly more probable than one in a billion that I would pick the number 500,000,000, and thus less probable than one in a billion that I would pick 422,819,208 (this is why I said above that the probability of the claim was only approximately one in a billion; because in fact it is even less than that.)

For that reason, if I had actually claimed to have picked 500,000,000, you might well have concluded that the most reasonable explanation of the facts was that I did not actually use the random number generator, or that it had malfunctioned, rather than that the number was actually picked randomly.

This is relevant to the kinds of things where the postulate that “extraordinary claims require extraordinary evidence” is normally used. Consider the claim, “I was down in the graveyard at midnight last night and saw a ghost there.”

How often have you personally seen a ghost? Probably never, and even if you have, surely not many times. And if so, seeing a ghost is not exactly an everyday occurrence. Considered in itself, therefore, this is an improbable occurrence, and if we evaluated the claim without considering the evidence included within the claim itself, we would simply assume the account is mistaken.

However, part of the reason that we know that seeing ghosts is not a common event is that people do not often make such claims. Apparently 18% of Americans say that they have seen a ghost at one time or another. But this still means that 82% of Americans have never seen one, and even most of the 18% presumably do not mean to say that it has happened often. So this would still leave seeing ghosts as a pretty rare event. Consider how it would be if 99.5% of people said they had seen ghosts, but you personally had never seen one. Instead of thinking that seeing ghosts is rare, you would likely think that you were just unlucky (or lucky, as the case may be.)

Instead of this situation, however, seeing ghosts is rare, and claiming to see ghosts is also rare. This implies that the claim to have seen a ghost is already extraordinary evidence that a person in fact saw a ghost, just as my claiming to have picked 422,819,208 was extraordinary evidence that I actually picked that number.

Nonetheless, there is a difference between the case of the ghost and the case of the number between one and a billion. We already know that there are exactly one billion numbers between one and a billion inclusive. So given that I pick a number within this range, the probability of each number must be on average one in a billion. If it is more probable that I would pick certain numbers, such as 500,000,000, it must be less probable that I would pick others, such as 422,819,208. We don’t have an equivalent situation with the case of the ghost, because we don’t know in advance how often people actually see ghosts. Even if we can find an exact measure of how often people claim to see ghosts, that will not tell us how often people lie or are mistaken about it. Thus although we can say that claiming to see a ghost is good evidence of someone actually having seen a ghost, we don’t know in advance whether or not the evidence is good enough. It is “extraordinary evidence,” but is it extraordinary enough? Or in other words, is claiming to have seen a ghost more like claiming to have picked 422,819,208, or is it more like claiming to have picked 500,000,000?

That remains undetermined, at least by the considerations which we have given here. But unless you have good reasons to suspect that seeing ghosts is significantly more rare than claiming to see a ghost, it is misguided to dismiss such claims as requiring some special evidence apart from the claim itself.

Claims and Evidence

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Earlier I have mentioned the fact that when someone holds a position, this very fact is evidence for his position. Here I will consider this in more detail.

The reason to think that the claim is evidence for the position is that it seems more likely that someone would hold a position if the position is true than if it is false. It is evident that this must hold in general, or it would be impossible to learn a language, since people would be equally likely to say “the sky is blue” even if it was not blue, and therefore it would be impossible for children to learn that this sentence says that the sky is blue rather than that the sky is green.

However, someone might object that it is not true in general, and that in some cases claims either have no evidential effect, or that they are evidence that the claim is false.

What would be necessary for this to be true? Let’s take a case where the claim might have no evidential effect at all. Suppose someone says that exactly one year from today, you will eat strawberries for dessert. We might suppose this has no effect: the person has no way of knowing what you will be eating, and therefore he seems equally likely to make the claim, whether you will be eating strawberries or not.

But unless we unreasonably think that it is absolutely certain that prophetic knowledge of the future does not exist, then there is some probability that the statement is prophetic. This will make him somewhat more likely to make the statement if you will in fact be eating strawberries, unless there is a completely equal chance of his statement being anti-prophetic, that is, being made because you will not be eating strawberries. But this would equally require that he know the future, and consequently this amounts to saying that he is equally likely in general to assert or deny the eating of strawberries, even when he knows the truth. But we already admitted that this is not the case: someone who knows the truth is, in general, more likely to assert the truth than to deny it. Thus it is unreasonable to deny that such a statement is in fact evidence that you will eat strawberries for dessert a year from now.

In order for a claim to be evidence that the thing is false, we would have to have something similar: a case where someone who knows the truth is more likely to deny it than to assert it. This would not clearly be the case even, e.g. if we knew that someone was inventing an alibi. It may be that people who invent alibis include more truths than falsehoods in them, taken as a whole. But it could be the case in very concrete circumstances, and taking these circumstances into account. For example, if someone writes a novel “based on a true story,” the fact that the protagonist is called “Peter Smith,” may be evidence that in real life the person’s name was not Peter Smith.

In this case, of course, there is not even a claim that Peter Smith was the person’s name in the first place. So we actually have still not established the existence of such a claim. And if such a case is found, it will be the circumstances, rather than the general fact of the claim, which are evidence against it. Considered in itself, the fact that someone makes a claim or holds a position, is evidence for that claim or position.

Division Into Two

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I pointed out in the last post that Parmenides is mistaken in maintaining the absolute unity of all being. But the refutation was simply from experience. One can still ask about the real reason for this. Why is being not absolutely one in the way he supposed?

Distinction consists in the fact that one thing is not another. But why is it not the other? We can find two kinds of distinction in things, material and formal.

Material distinction consists in the fact that one thing is not another, even though the things are of the same kind. Thus one man is not another man. Formal distinction consists in the fact that one thing is not another thing because they are different in kind, as for example a dog is distinct from a man, or as blue is distinct from green.

It is quite difficult to understand the existence of material distinction, and I will not try to explain it at this time. But formal distinction always happens because of some opposition between the forms in question. And opposition results from things that are opposite to one another, while opposites come in pairs. Consequently formal distinction always results first into a division into two, although the things which are divided into two may possibly be divided again.

We can illustrate this with the way that St. Thomas divides a text into parts. For example, in his commentary on the Gospel of St. John, discussing the wedding at Cana, he says:

Above, the Evangelist showed the dignity of the incarnate Word and gave various evidence for it. Now he begins to relate the effects and actions by which the divinity of the incarnate Word was made known to the world. First, he tells the things Christ did, while living in the world, that show his divinity. Secondly, he tells how Christ showed his divinity while dying; and this from chapter twelve on. As to the first he does two things. First, he shows the divinity of Christ in relation to the power he had over nature. Secondly, in relation to the effects of grace; and this from chapter three on. Christ’s power over nature is pointed out to us by the fact that he changed a nature. And this change was accomplished by Christ as a sign: first, to his disciples, to strengthen them; secondly, to the people, to lead them to believe (2:12). This transformation of a nature, in order to strengthen the disciples, was accomplished at a marriage, when he turned water into wine. First, the marriage is described. Secondly, those present. Thirdly, the miracle performed by Christ. In describing the marriage, the time is first mentioned. Hence he says, “On the third day there was a wedding,” i.e., after the calling of the disciples mentioned earlier. For, after being made known by the testimony of John, Christ also wanted to make himself known. Secondly, the place is mentioned; hence he says, at Cana in Galilee. Galilee is a province, and Cana a small village located in that province.

Every division here is into two except when he talks about the description of the marriage, saying, “First, the marriage is described. Secondly, those present. Thirdly, the miracle performed by Christ.” But it is easy to see that he divides into three here in order to omit a distinction that would not be very helpful, namely the division between describing the background to the miracle and describing the miracle itself. The background is then divided once again into the marriage and into those present.

Thus, in theory every formal division is into two. But in practice in can happen that it is sometimes useful to divide into three, and in rare cases larger numbers. This happens first of all when some divisions are obvious and can be skipped over, as is the case here with St. Thomas. Second, the division into beginning, middle, and end is usually best left as a division into three, even though in principle the beginning can be divided against the rest. Finally, cases which consist of a list are best left as such, as when I mention seven interesting things that happened to me yesterday. Basically such cases are cases of material distinction, not formal; here is one interesting thing, here is another, and here is still another.

For additional illustration, we may divide the above paragraph:

  1. Statement of the theoretical principle: every formal division is into two.
  2. Discussion of practical exceptions.
    1. General statement regarding exceptions: sometimes it is useful to divide into larger numbers.
    2. Consideration of various cases.
      1. Consideration of cases which in fact contain formal distinction.
        1. The general case in which some divisions are omitted.
        2. The special case of beginning, middle, and end.
      2. Consideration of cases in which material distinction is involved instead.
        1. Description of such cases: situations where we basically have a list.
        2. Explanation of such cases: the fact that they consist in material distinction.

Someone may argue that such an explanation of a text is artificial, and that the author was not thinking of such a breakdown of his text, and consequently that it cannot be a true explanation. But the reality is that it does not matter whether he was thinking of it or not. If his text is in fact coherent, it will have such an explanation, and one that is basically most correct, in comparison to others which are less correct or incorrect.

This is true not only of texts, but of any whole which is coherently divided into parts based on formal distinctions.

Defining Order

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Earlier I discussed Aristotle’s senses of before and after. Using yesterday’s discussion of one and many, we can now find a more exact definition of the same terms.

We can define “second” as the formal part of something two, namely the part by which the two is two.

Thus “first” is the part of something two which is not the second. This way of defining first and second may seem backwards, but it is analogous with how we defined the unity of a thing by negating division.

Something first, as such, implies the existence of something second, and likewise something second implies the existence of something first. However, the existence of two implies the existence of one (as a part), while the existence of one does not imply the existence of two. This corresponds to the difference in the definitions of first and second given above. The second is the part by which two is two, while the first is a part, but not the part by which two is two. If first and second are considered only with respect to what is formal in them, then, as “not that by which two is two”, and “by which two is two”, then according to this consideration the first does not imply a second, but the second implies a first.

From this we can see that being before by nature, or as Aristotle says, “what does not reciprocate according to consequence of being,” if not the first thing to which the words before and after are applied, is nonetheless first in the nature of things to possess the before and after.

We can also see that this sense of before and after must be found in some way in all other senses, for every case of before and after will involve something first and something second.

This can be illustrated with the order of time, the first thing to which the words before and after are actually applied. At first it might seem that such a before and after are completely separate from the idea of reciprocation according to consequence of being, since one day can exist without another, nor is it evident that the existence of one day implies that another day existed or that another will exist.

If we consider our actual experience of the past and present (since we have no experience of the future), however, we find something different. Our experience of the present includes our memory of the past, and in this respect implies the past existence of the past. But our experience of the past, namely not the present experience of remembering the past, but the remembered experience of the past that was once present, does not include anything of the present. In this way the present implies the past, but the past does not imply the present, and thus according to these considerations the past is before the present even according to consequence of being.

This is not to deny that according to other considerations the present might be before the past. Rather, these considerations show why the past is considered to be before the present, namely because the present seems to build on the past, as though the past were one block of wood, with the present being a second block of wood stacked on top of the first block. We will find that something similar is the case in every way in which we can say that one thing is before or after another thing.