Please read this all and l believe it will be really enjoyable to get through it.

This is from Wittgenstein's lectures on the foundation of mathematics.


Explanation
What would go wrong, if anything, if we didn't recognize the law of contradiction-or any other proposition in Russell's logic? We treated the question of double negation as parallel to that: If some people used double negation to mean affirmation, and others used double negation to mean negation, should we say then that they were using negation-or double negation-with "different meanings"? We discussed whether a particular meaning of negation made a certain usage correct, or whether that meaning consists in using negation in that way. This is a difficulty which arises again and again in philosophy: we use "meaning" in different ways. On the one hand we take as the criterion for meaning, something which passes in our mind when we say it, or something to which we point to explain it. On the other hand, we take as the criterion the use we make of the word or sentence as time goes on.First of all, to put the matter badly and in a way which must be corrected later, it is clear that we judge what a person means in these two ways. One can say that we judge what a person means by a word from the way he uses it. And the way he uses it is something which goes on in time. On the other hand, we also say that the meaning of a word is defined by the thing it stands
for; it is something in our minds or at which we can point.

The connexion between these two criteria is that the picture in our minds is connected, in an overwhelming number of cases-for the overwhelming majority of human beings-with a particular use. For instance: you say to someone "This is red" (pointing); then you tell him "Fetch me a red bookm-and he will behave in a particular way. This is an immensely important fact about us human beings. And it goes together with all sorts of other facts of equal importance, like the fact that in all the languages we know, the meanings of words don't change with the days of the week. Another such fact is that pointing is used and understood in a particular way-that people react to it in a particular way.If you have learned a technique of language, and I point to this coat and say to you, the tailors now call this colour 'Boo' ", then you will buy me a coat of this colour, fetch one, etc.The point is that one only has to point to something and say,"This is so-and-so", and everyone who has been through a certain preliminary training will react in the same way. We could imagine this not to happen.

If I just say, "This is called 'Boo' " you might not know what I mean; but in fact you would all of you automatically follow certain rules. Ought we to say that you would follow the right rules?-that you would know themeaning of "boo"? No, clearly not. For which meaning? Are there not 10,000 meanings which "boo" might now have?-It sounds as if your learning how to use it were different from your knowing its meaning. But the point is that we all make the SAME use of it. To know its meaning is to use it in the SAME way as other people do. "In the right way" means nothing. You might say, "Isn't there something else, too? Something besides the agreement? Isn't there a more natural and a less natural way of behaving? Or even a right and a wrong meaning?"-Suppose the word "colour" used as it is now in English. "Boo" is a new word. But then we are told, "This colour is called 'boo' and then everyone uses it for a shape. Could I then say, "That's not the straight way of using it"? I should certainly say they behaved unnaturally. This hangs together with the question of how to continue the series of cardinal numbers. Is there a criterion for the continuation-for a right and a wrong way-except that we do in fact continue them in that way, apart from a few cranks who can be neglected? We do indeed give a general rule for continuing the series; but this general rule might be reinterpreted by a second rule, and this second rule by a third rule, and so on.


One might say, "But are you saying, Wittgenstein, that all this is arbitrary?"-I don't know. Certainly as children we are punished if we don't do it in the right way. Suppose someone said, "Surely the use I make of the rule for continuing the series depends on the interpretation 1 make of the rule or the meaning I give it." But is one's criterion for meaning a certain thing by the rule the using of the rule in a certain way, or is it a picture or another rule or something of the sort? In that case, it is still a symbol-which can be reinterpreted in any way whatsoever. This has often been said before. And it has often been put in the form of an assertion that the truths of logic are determined by a consensus of opinions. Is this what I am saying? No. There is no opinion at all; it is not a question of opinion. 'They are determined by a consensus of act a consensus of doing the same thing, reacting in the same way. There is a consensus but it is not a consensus of opinion. We all act the same way, walk the same way, count the same way. In counting we do not express opinions at all. There is no opinion that 25 follows 24-nor intuition. We express opinions by means of counting. People say, "If negation means one thing, then double negation equals affirmation; but if it means another thing, double negation equals negation." But I want to say its use is its meaning. 'There are various criteria for negation.-Think of the ways in which a child is taught negation: it may be explained by a sort of ostensive definition. You take something away from him and say "No". A child is trained in a certain technique of applying negation long before the question of double negation arises. If a child is taught the use of negation apart from all this, and then goes on to use double negation as equivalent to negation, would you say he is necessarily using negation now to mean something different? If you say, "It must have a different meaning now-this says nothing, unless you mean that a different picture will be associated with it. Let us go back to the law of contradiction. We saw last time that there is a great temptation to regard the truth of the law of contradiction as something which follows from the meaning of negation and of logical product and so on. Here the same point arises again.

Conversation

I will now use an awful expression. I wanted to talk of a stationary meaning, such as a picture that one has in one's mind, and a dynamic meaning. I was going to say, "No dynamic meaning follows from a stationary meaning." But that is very badly put and had better be forgotten immediately. Another way of putting it is to warn you: Don't think any use collides with a picture, except in a psychological way.Don't imagine a sort of logical collision. But that is also very badly expressed. For one then wants to ask where I got the idea of logical collision from. And one would be perfectly justified in asking. One is tempted to say, "A contradiction not only doesn't work-it can't work." One wants to say, "Can't you see? I can't sit and not sit at the same time." One even uses the phrase "at the same timew-as when one says, "I can't talk and eat at the same time." The temptation is to think that if a man is told to sit and not to sit, he is asked to do something which he quite obviously can't do. Hence we get the idea of the proposition as well as the sentence. The idea is that when I give you an order, there are the words-then something else, the sense of the words-then your action. And so with "Sit and don't sit", it is supposed that besides the words and what he does, there is also the sense of the contradiction-that something which he can't obey. One is inclined to say that the contradiction leaves you no room for action, thinking that one has now explained why the contradiction doesn't work. Suppose that we give the rule that "Do so-and-so and don't do it" always means "Do it". The negation doesn't add anything. So if I say "Sit down and don't sit down", he is to sit down. If I then say, "Here you are, the contradiction has a good sense", you are inclined to think I am cheating you. This is an immensely important point. Am I cheating you? Why does it seem so?

Turing: I should say that we were discussing the law of contradiction in connexion with language as ordinarily used, not in connexion with language modified in some arbitrary way which you like to propose.'

Malcolm: The feeling one has was that we were talking of 'p. -p'as it is now used-to express a contradiction; and you have merely suggested a use in which it would no longer express a contradiction.

Wittgenstein: Yes; you speak of the sentence as expressing a contradiction-as if the contradiction were something other than the sentence and expressed by it.-But doesn't the explanation of this feeling that I have cheated lie perhaps in the fact that I have made a wrong continuation? Now what is it that I have continued wrongly?

Turing: Could one take as an analogy a person having blocks of wood having two squares on them, like dominoes. If I say to you "White-green", you then have to paint one of the squares on the domino which I give you white and the other green. If the point of this procedure is to be able to distinguish the two squares, you will probably hesitate when I say "White-white". -Your suggestion comes to saying that when I say "White-white" you are to paint one of the squares white and the other grey.

Wittgenstein: Yes, exactly. And where does the cheating come in? What is the wrong continuation I have suggested? Why is this continuation in your analogy a wrong continuation? Might it not be the ordinary jargon among painters? The point is: Is it or is it not a case of one continuation being natural for us? Or ought one to say that there is something more to it than that? Ought one to give a reason why one continuation is natural for us? Ought one to say this, for example: "If we learn to use orders of the form 'p', 'q', 'p and q', 'p and not-q'etc, then so long as we give the phrase 'p and not-p'the sense which is
determined by the previous rules of training, it is clear that this cannot be a sensible order and cannot be obeyed. If the rules for obeying these orders-for logical product and negation-are laid down, then if we stick to these rules and don't in some arbitrary way deviate from them, then of course 'p and not-p'can't make sense and we can't obey it." Isn't that the sort of thing you would consider not cheating?

Turing: I should say that it is another kind of cheating. I should say that if one teaches people to carry out orders of the form 'p and not-q'then the most natural thing to do when ordered and not-p'is to be dissatisfied with anything which is done.

Wittgenstein: I entirely agree. But there is Just one point: does "natural" mean "mathematically natural"?

Turing: No.

Wittgenstein: Exactly. "Natural" there is not a mathematical term. It is not mathematically determined what is the natural thing to do. We most naturally compare a contradiction to something which jams. I would say that anything which we give and conceive to be an explanation of why a contradiction does not work is always just another way of saying that we do not want it to work. If you have a tube and a cock which shuts or opens it, your experience may have led you to think that always when the handle is parallel to the tube, the tube is open, and when it is at right angles to it, the tube is closed. But at home I have a cock which works the other way about. And in order to get used to it, I had to think of the handle as lying along the tube and blocking it, so that the tube was closed when the handle was parallel to it. I had to invent a new imagery. Similarly, one needs to change one's imagery in the case of contradictions. One can change one's imagery in such a way that 'p and not-p 'sounds entirely natural, as when we say, "The negative doesn't add anything".

This is most important. We shall constantly get into positions where it is necessary to have a new imagery which will make an absurd thing sound entirely natural. I want to talk about the sense in which we should say that the law of contradiction: - (p. - p) is a true proposition. Should we say that if '-(p. -p)' is a true proposition, it is true in a different sense of the word from the sense in which it is a true proposition that the earth goes round the sun? In logic one deals with tautologies-propositions like '- (p. -p)'. But one might just as well deal with contradictions instead. So that Principia mathematica would not be a collection of tautologies but a collection of contradictions. Should one then say that the contradictions were true? Or would one then say that "true" is being used in a different sense?

Turing: One would certainly say that it was being used in a different way.

Wittgenstein: It is used in a different way because you now say it of things of which you would not say it before. One could put the point this way. One often hears statements about "true" and "false" for example, that there are true mathematical statements which can't be proved in Principia Mathematica, etc. In such cases the thing is to avoid the words "true" and "false" altogether, and to get clear that to say that p is true is simply to assert p; and to say that p is false is simply to deny p or to assert -p. It is not a question of whether p is "true in a different sense". It is a question of whether we assert p. If a man says "It is fine" and I say "It is not fine", I am correcting him and asserting the opposite; and we can then argue about whether it is fine or not, and we may be able to settle the question. But if I am trained in logic, I am trained to assert certain things and not to assert others. This is an entirely different case from being trained to assert that Smith looks sad. I am not trained to assert that he looks sad or that he doesn't look sad. But I am actually trained to assert mathematical propositions that 3 X 6= 18, and not 19-and logical propositions. ''Trained to assert under what conditions? Well, for instance, when I have to pass an exam.-And if, for example, we did logic by means of contradictions, we should be trained to assert contradictions in examinations. It is important in this connexion that there is an inflexion of asserting. We make assertions with a peculiar inflexion of the voice; and there are gestures with this. This is one thing which is very characteristic of assertion. It is also important that assertions in our language have a peculiar jingle; we make them with sentences of a certain form. For instance, "'Twas brillig" is an assertion, although "brillig" is not a normal word. Now suppose that we were trained to use contradictions instead of tautologies in logic. There are circumstances in which we should call it the same logic as our present logic. What are these circumstances? What would be our criterion for saying that this other logic is all absurd, or for saying that it is essentially the same as our present logic?

Malcolm: Wouldn't we say it was the same as our present logic if we used "-" in a different way?

Wittgenstein: Yes, But using "-" in a different way does not here refer to the way in which it is used in the proofs. [In the proofs it] might be just the sameIn ironical statements, a sentence is very often used to mean just the opposite of what it normally means. For instance, one says "He is very kind", meaning that he is not kind. And in these cases the criterion for what is meant is the occasion on which it is used. One might make a deduction and say "He is very kind, therefore we will give him a birthday present "ironically, meaning "He is not kind, therefore we will not give him a birthday present." Thus we could have proofs in our supposed new logic just like the ones in Principia Mathematics, and the assertion sign would appear before contradictions. By the way, this is the way in which a proposition can assert of itself that it is not provable. Besides putting the assertion sign before contradictions I could put it before propositions like 'p--> q'. In the one case I-- p. -p would mean p. -p is refutable and in the other 'I- p--> q' would mean p--> q is not provable.Thus we see that Principia might not only be a collection of tautologies or a collection of contradictions; it might even be a collection of propositions which are neither contradictions nor tautologies.