Quick Thoughts on the Identity of Indiscernibles

The law of the Identity of Indiscernibles is said to be the converse of Leibniz’s law. This is the Indentity of Indiscernibles:

If a and b have all their properties in common, then they are one and the same thing.

In symbols:

(x) (y) (F) ((F) x ≡ F (y) ⊃. x = y)

The basic question is:

Imagine two steel balls which have all their properties in common. Could they still be two?

Intuitively, most people (I think) would say 'yes'. It doesn't seem inconceivable prior to modal philosophising.

For a start, wouldn’t the balls still be spatially or temporally separate? If that is the case, then surely they wouldn't have all their properties in common. (That's if you accept spatial and temporal properties, which many philosophers do.)

Now we go deeper into this thought experiment.

One could say (I suppose) that if the balls were suddenly frozen in space, then their positions would be different. Hence they'd have different spatial properties. However, what if the balls will never be frozen in space and never have been frozen in space? (Is that a hypothetical scenario about a hypothetical scenario?)

As they are now, and in five minutes, etc., the balls are constantly on the move relative to one another. Thus they have all their spatial (as well as temporal) properties in common. And because they're both in an empty world, there can be no relational properties (care-of other objects, conditions, events, etc.). Any relational properties a has relative to b, b has relative to a. Thus they have all their properties in common.

This, then, appears to break Leibniz’s law in that the balls are indiscernible; though not identical!

Leibniz’s Law & Intensional Contexts

What is Leibniz’s law? This:

If a is the same as b, then everything true of a is also true of b.

Or in symbols: (x) (y) (F) (x = y ⊃. F (x) ≡ F (y))

There's at least one way in which it can be taken to be false. Take Roger Scruton’s example:

“Suppose John is thinking tenderly of Mary, and Mary is the person who, unknown to John, ate his beloved cat. Is John thinking tenderly of the person who ate his cat?” 

This is primarily a question of John’s beliefs about Mary (de dicto): not about Mary herself (de re). It's indeed the case the cat-eater is the same person as the person thought tenderly of by John. However, John doesn't know that Mary ate his cat. This is a fact about John; not a fact about Mary. (Can there be psychological facts?)... Unless what is thought by other people about Mary also constitute facts about Mary. In that case, there would be a multitude of facts about Mary that she and others (as individuals) couldn't know about Mary (which isn't in itself implausible).

Does this story break Leibniz’s law? No.

What we would require to save the day is a theory of contexts: 'intensional' contexts. Do we need such intensional (or belief contexts) at all? Quine said no – at least not in science, logic and perhaps in philosophy too. That is, what John is thinking of doesn't belong to this particular extension (or reference) – that is, to Mary. Thus it plays no part in science or logic. (Though it would, perhaps, play a part in the psychological descriptions of John.) Then again, it could also be said that Mary (as a single person or human being) couldn't really play a part in science or logic either.