Friday, 8 May 2015

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) xF (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!

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