The law of identity states that each thing is identical to itself and can be represented as A=A. Snow, for example, has a set of characteristics that make snow what it is. The law says that snow, by having these characteristics and these alone, has one identity and not two. It can, of course, have various characteristics. But then these characteristics make up its identity. To exist is to exist as something with an identity that enables us to differentiate it from others.
When we apply this law to propositions and their meaning in the context of arguments, we see that fallacy of equivocation occurs when one term fails to have one identity in the course of argument. So here we see that the word ‘miracles’ has two meanings which makes the argument fallacious due to the ambiguity of equivocation:
If you believe in the miracles of science then you should believe in the miracles of the Bible.
You believe in the miracles of science.
So you should believe in the miracles of the Bible.
So the intelligibility of our discourse and thinking requires identity. And if we disagree with the law of identity we run into contradictions and thus logical and physical impossibilities. If a shape is BOTH (1) a triangle or a figure with three sides whose angles add up to 180 degrees (the traits taken to make a triangle what it is and not something else) AND (2) a square with four sides, then we have a contradiction because the law of identity was violated: one and the same shape would have two identities – three sided AND four sided – at the same time and in the same respect. Since this is contradictory, we don’t think there exist entities that have three and four angles. So the law of non-contradiction, which states that we cannot have A and not-A at the same time and in the same respect, is grounded in the law identity.
So obviously a lot rides on things having an identity or being what they are and not something else. The reality of such identities allows us to intelligibly think, communicate, and investigate what is possible or actual. Thus we see how it is indeed our most fundamental law of logic.
Of course, some may wish to express objections to the law of identity. For example, consider this passage from Friedrich Nietzsche:
“Logic is bound up with the proviso: granted that identical cases exist. As a matter of fact, before one can think and conclude in a logical fashion, this condition must first be assumed. That is to say, the will to logical truth cannot be consummated before a fundamental falsification of all phenomena has been assumed. From which it follows that an instinct rules here, which is capable of employing both means: first, falsification; and secondly, the carrying out of its own point of view: logic does not spring from a will to truth.” (Will to Power, section 516)
Here we see Nietzsche agreeing that logic is bound up with identity and yet going to assert that logic does not get us to the truth. He also writes:
“Supposing there were no such thing as A identical with itself, as every logical (and mathematical) proposition presupposes, and that A is in itself an appearance, then logic would have a mere world of appearance as its first condition. As a matter of fact, we believe in that proposition, under the influence of an endless empiricism which seems to confirm it every minute. The “thing”—that is the real substratum of A; our belief in things is the first condition of our faith in logic. The A in logic is, like the atom, a reconstruction of the thing…. By not understanding this, and by making logic into a criterion of real being, we are already on the road to the classification of all those hypostases, substance, attribute, object, subject, action, etc., as realities—that is to say, the conception of a metaphysical world or a “real world” (—this is, however, once more the world of appearance…).” (Will to Power, section 516)
Here he offers a genetic account of the law of identity: we experience, over and over, so-called identical things through our senses. But we mistakenly think these experiences are reality when, in fact, they are just appearances. And this mistake, in turn, leads us to develop all kinds of conceptual and linguistic distinctions that have no connection to truth at all!
But if the law is false then Nietzsche’s claims would be unintelligible. After all he must employ concepts and words that are what they are to make his case against logic (and not against something other than logic). Thus his critique would be presupposing as true the very law of identity he seeks to expose as a fiction. And if the law is true his claims about the law being a fiction are obviously false.
For some thoughts on philosophy and contradiction, go here.