@kennethamy,
kennethamy wrote:
Leibniz had two principles:
1. The indiscernability of identicals.
2. The identity of indiscernibles.
The first says that if (putatively) two things have the very same properties, then they are not two things, but really one (and the same thing). The second is the converse of the first. It says that no two things can have the very same properties. It is generally agreed that the first is true, and indeed, it is the definition of identity logicians accept (Leibniz's Law of Identity). The second is obviously controversial. Leibniz held it was true just as he held that the first was true, but although the first is intuitively true, Leibniz had to give (dubious) metaphysical reasons for the second. Of course, the second might be true (for all we know). But if it is true, it is clearly not a necessary (or intuitive) truth as is the first. And the issue is not whether the second is true, but whether it must be true as the first is not only true, but it must be true.
Both are necessarily true, ie. [](x=y <-> ∀F(Fx ↔ Fy)), is a theorem.
Stanford: The Identity of Indiscernibles
The Identity of Indiscernibles (hereafter called the Principle) is usually formulated as follows: if, for every property F, object x has F if and only if object y has F, then x is identical to y. Or in the notation of symbolic logic:
∀F(Fx ↔ Fy) → x=y.
This formulation of the Principle is equivalent to the Dissimilarity of the Diverse as McTaggart called it, namely: if x and y are distinct then there is at least one property that x has and y does not, or vice versa.
The converse of the Principle, x=y → ∀F(Fx ↔ Fy), is called the Indiscernibility of Identicals. Sometimes the conjunction of both principles, rather than the Principle by itself, is known as Leibniz's Law.
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(x=y) ↔ ∀F(Fx ↔ Fy), is a theorem of Principia Mathematica ..Russell and Whitehead.
