The laws of identity states that everything is identical with itself, i .e. a thing is what it is. In other words, a thing is not other than itself.
Symbolically we may say that for anything x, x is x is a ways true, For example, accepting this law, we assert “table is a table”, “Chair is a chair”, Man is a man” and so on. It does not assert anything about the nature of x, it does not tell us whether x in question is white or heavy or soft. In does not tell us about any particular character of the world. It gives us a very useful instruction concerning the use of concepts occurring in an argument. It tells us that in any good argument or in any process of good reasoning, every concept occurring in it must be used in the same sense throughout the argument.
In other words, the meaning of concepts occurring in an argument should remain constant throughout the argument. This law carnal so be taken as an instruction for assigning truth-values to proposition variables in classical logic.
In this sense, it states that distinct occurrences of the same propositional variable ways receive the same truth value throughout the argument. In propositional sense it states that the very proposition implies itself.
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Symbolically the law states that for any proposition P, (P) P) holds. (Read ‘P) P’ as ‘if P then P’.) This means that if a proposition is used in an argument to state something then whenever this proposition occurs in the argument it is used to state that thing. It states that the same proposition should not state different things in the same argument.
One cannot accept and reject a given proposition at the same breath in any given argument. If we accept and rejected proposition P at the same time in an argument then the very use of P becomes pointless. Hence, for any discourse or argument to be possible, we have to accept the law of identity.
The fundamental nature of the law of identity can be understood in the following way. For example, if we deny this law i.e. if we deny x is x then it implies that there is a denial. Hence by use of the law of identity we have ‘denial is a denial’. It is of the form‘d is d’, where d stands for ‘denial’.
Thus the law of identity is back. The very denial of this law implies its presence. Therefore, without presupposing this law we can not even state or assert anything. For any discourse or argument to be possible we have to accept the law of identity. In this sense, it is a fundamental or basic principle of logic.