We come to a little more complex statement. I am showing you all this because these chapters 9, 10, 11, 12 are great examples of the presentation of data which are known and presentation of that which is not known. They are presented in such a way that the conclusion seems to be necessarily true or most probably true because the structure of the argument is of this Then he might scratch his head, it’s a new question. So he might say ultimately that it is necessarily so. If you put gold on black, may be some other colour might come up, or vice versa. Therefore, he will tell me, it is necessarily so that what is black is black, what is gold is gold. Having said this Tushar may derive a conclusion that logically it is impossible for black to be gold and for gold to be black. But you must see that he has actually said on the basis of experience. He has tried to make gold- black or black-gold but he failed to do so, therefore he concluded that it is logically impossible. What is black is black and what is gold is gold and the two are opposite of each other. Now he has become a good logician in the mean time. He will answer my question that gold and black are contradictory of each other and two contradictions cannot be true of an object. An object cannot be such that it is what it is and it is opposite of it. If somebody says, I am riding a horse and I am not riding a horse at the same time, in the same place, in the same state of consciousness. I may be riding in my dream while I am flying on my bed that is possible. In the same state of consciousness I am riding and not riding is not possible. The two are contradictory of each other; therefore logically it is impossible that the same object can be having two contradictory predicates.
This is what is normally called a basic principle of logic. Let me state it in a classical form, logic is a science which tells you that when certain data are presented to you and you derive from those data a certain imaginative conclusion and if your conclusion is true then what has happened in your thinking by which your conclusion has been found to be true?
I will show him a diamond and ask him a question is it also a pearl? And by this time he has learnt logic, so he says diamond is a diamond and pearl is pearl. What is diamond cannot be a pearl and pearl cannot be a diamond. The two are contradictory of each other or contrary of each other, therefore either the object is a diamond or it cannot be a pearl, necessarily it will not be a pearl. He has applied, what is called the law of contradiction and this is the basic law of logic.
The entire logic says nothing but this basically that when certain data are given to you and you are required to derive some other data which are not known to you. When you can go from this to that provided you follow and apply the law of contradiction. When you apply this law of contradiction then you can be sure that your conclusion is not a pure fiction, is not fancy, is not fantasy. Even if it is imagination it will be necessarily true. This law of contradiction is also called, the law of identity. It’s very easy to see why the law of contradiction is also called law of identity. Law of identity says that a black cat is necessarily black; a black cat is necessarily black. Black is identical with black, therefore this statement must be true. There is no contradiction involved in this sentence. Therefore the sentence must be true, why, because, I am repeating an identity. A black cat is black; therefore, this statement is true, since it involves no contradiction. You don’t need to look at the cat again and verify whether it is true or not. A black cat is black. If I tell you just like that you will not need to go to black cat and find out that this statement is true or not. It is necessarily true, why, because no contradiction arises or there is no possibility of raising any contradiction.
Whenever you state an identity, the possibility of contradiction ceases, therefore the law of identity is just the other way of stating the law of contradictions because in identities there are no possibilities of contradictions. Therefore, the other side of the law of contradiction is the law of identity. What is free from contradiction is necessarily true and what states identity is necessarily true, so both the stations are basically identical.
There is also third law of logic, which is another form of stating the law of identity or the law of contradiction. And what is that, it is called the law of excluded middle. When there are two contradictions regarding any object, one must be true, another must be false. The middle course is excluded. If there are two contradictions about any object then you are sure that one of them must be true another one must be false, therefore, it is called the law of excluded middle. Law of contradiction supports the law of excluded middle and vice versa and law of contradiction is basically the law of identity. You can apply all the three laws whenever you like or you can say, I am applying the law of excluded middle, it is all the same thing.
Logic has only one law fundamentally, when you can show that the data which are before you and the data which are not known to you are basically identical in character; then you are certain that your conclusion is valid. When the known data are shown to be identical with the data which are not known but by some method, if you can show that they are identical then you can be sure that your conclusion must be true. This is called the rigour of logic and the rigour of argument. Your argument must be such that from the known data the conclusion that you are deriving is fundamentally identical. That is why you must have seen in mathematics, you have got identities. Most important part of mathematics consists of identities. The full chapter of identities is only this; complete proof in mathematics is based upon identity. You are sure that this is exactly identical with this then your proof is conclusive. That is why mathematics is also called logic and logic is called mathematics. Mathematics also is the science of identities. Logic tells you that all reasonings are valid when you can show identities. So far everything seems to be quite simple.