The Law of Identity by Bertrand Russell says
Whatever is, is.
It's hard to find any kind of justification on this statement. Why should this be true and what are the arguments for any opposite hypotheses that this might not be a true law?
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[formalization of identity] in the language L of classical first-order logic (FOL) by selecting a two place predicate of L, rewriting it as ‘=’, and adopting the universal closures of the following two postulates:
Ref: x = x
LL: x = y → [φ(x) → φ(y)],
where the formula φ(x) is like the formula φ(y) except for having occurrences of x at some or all of the places φ(y) has occurrences of y. Ref is the principle of the reflexivity of identity and LL (Leibniz’ Law) is the principle of the indiscernibility of identicals.
See all the entry for an introductory discussion of the many interesting philosophiocal issues related to identity :
Identity is often said to be a relation each thing bears to itself and to no other thing. This characterization is clearly circular ("no other thing") and paradoxical too, unless the notion of "each thing" is qualified. More satisfactory (though partial) characterizations are available and the idea that such a relation of absolute identity exists is commonplace. Some, however, deny that a relation of absolute identity exists. [...] The concept of identity, simple and settled though it may seem (as characterized by the standard account), gives rise to a great deal of philosophical perplexity.
Due to the small number of words in the definition, this may be better thought of as a "definition for 'is'" than a law of identity. Russel had to worry about particularly frustrating types of things that "are," especially those which try to describe what they are (as we see in this sentence).
There are contrary positions, such as "all that 'is' is an illusion," and you can generate an entire world view from that. One effect of Russel's definition is that he is not interested in an illusionary "whatever is." If another person talks about an "is" that is an illusion, Russel's words indicate that he does not intend to use language in the same way as that person, so there should be no surprise if combining that person's words with Russel's words might lead to a contradiction.
This is not a formal statement of the law of identity as it is used in Principia Mathematica, it plays the same role as Euclid's "point is that which has no part". The purpose of it, as is Euclid's, is to give an informal idea of where the claim is coming from, a=a in this case. At the time of the writing Russell was part of a logicism program, the purpose of which was to reduce all of mathematics to laws of thought, logic. Just as Euclid points in the direction of a basic physical experience as the source with his non-definition of a point, Russell points in the direction of an instantly recognizable law of thought, "whatever is, is [itself]".
As for the contrary arguments, the law of identity is the most agreed upon logical law. Heraclitus and Hegel are sometimes interpreted as rejecting it, and Heraclitus's argument is famous:"we do not walk into the same river twice", a=a attempts to do just that. According to Heraclitus and Hegel, everything is becoming, everything is in flux, so much so that any self-identity is negated, nothing is [itself]. Most, like Aristotle, see this as a major exaggeration, see Is there a theory of time consistent with Heraclitus?
Wittgenstein in the Tractatus criticized not so much the law of identity itself as its expression in Russell's logic:"To say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing". According to him in "proper" logic such a law should be redundant. See discussion of Russell's response in How does Russell's argument for identity refute that of Wittgenstein's?
The Leibniz-Russell definition of identity: x=y =def ∀F(Fx <-> Fy), is invalid for non-referring names and non-referring descriptions.
i.e. x=x <-> ∀F(Fx <-> Fx).
Even though 'the present King of France' is indiscernible from itself, it is not self-identical.
∀F(F(the present King of France) <-> F(the present King of France)), is true but, (the present King of France)=(the present King of France), is false.
The same can be said about non-referring names such as: Vulcan, Pegasus..
It seems to me that we need to re-define identity to account for these apparent exceptions.
x=y =def Exist(x) & Exists(y) & ∀F(Fx <-> Fy).
What do you think about that?