Leibniz's Law in Suppes's 'Introduction to Logic'

Question

I've been reading Patrick Suppes's book 'Introduction to Logic'. In chapter 5 of the book he introduces a reader to the Leibniz's law or the principle of the indentity of indiscernibles. He provides a definition and a small proof for this principle. The definion looks incomplete although the proof seems to be valid. Here's a paragraph from the book:

If every property of x is also a property of $y$, then x = y; for x has the property of being identical with x, and hence if every property of x is a property of y, then y has the property of being identical with x, so that y = x, and hence x = y.

If we assume that $y$ has all properties that $x$ has and an additional one which $x$ doesn't possess. Using the definition above we can infer that $x = y$ but $y \ne x$. This contradicts the reflective property of equivalence relation.

In my opinion the valid definition should look like this:

'If every property of $x$ is also a property of $y$ AND if every property of $y$ is also a property of $x$ then $x = y$.'

Am I missing something?

Answer 1

See Identity of Indiscernibles:

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 [not first-order]:

$∀F \ (Fx ↔ Fy) → x=y$.

Your suggested formulation:

If every property of $x$ is also a property of $y$ AND if every property of $y$ is also a property of $x$,

is correct. It is formalized with the bi-conditional: $\leftrightarrow$.

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Suppes' formulation amounts to:

$∀F \ (Fx \to Fy) → x=y$.

Suppes' heuristic reasoning is the following:

Granted the previous stated laws: reflexivity: $x=x$ (called law of identity), simmetry: $x=y \to y=x$ and transitivity:

assume that every property of $x$ is also a property of $y$. Then $x$ has the property of being identical with $x$ (reflexivity), and hence if every property of $x$ is a property of $y$, then $y$ has the property of being identical with $x$, so that $y = x$.

But then, by symmetry: $x = y$.

Answer 2

Yes, if we assume that $y$ has every property possessed by $x$ and also a property that $x$ does not have that leads to a contradiction. It does not follow that we need to change Leibniz - all that follows is that our assumption was false.

Answer 3

There's another property that's a converse to the identity of indiscernibles called the indiscernibility of identicals: $$\forall x,y.(x = y)\land P(x) \implies P(y)$$ This and reflexivity is often taken as defining equality. On the linked Wikipedia page, it gives a derivation of symmetry and transitivity from these. This is often presented as a rule of inference for equality rather than an axiom which avoids needing second-order logic.

This principle states that if $x = y$, then any property of $x$ is also a property of $y$ (and by the derived symmetry, vice versa). As David C. Ullrich states, you've simply assumed something contradictory (that $y$ has an additional property that $x$ does not) and derived a contradiction from it.