Application of Leibniz's law to infer that things are different

Question

In terms of practical reasoning, what are the constraints/conditions where one can infer that (putatively) two things are not identical on the basis of them seeming to have different properties?

At first read this might seem odd, but consider the story of the blind men and the elephant: One day, an elephant is brought to town. The six blind men of the town, never having encountered one before, go to it. The first, at its trunk, declares "It is like a snake!", the second at the tusk declares "It is like a spear!", the third, at its ear declares "It is like a fan!", the fourth, at its leg declares "It is like a column!", the fifth, at its side declares "It is like a wall!" and the fifth, touching just the tip of its tail declares "It is like a mouse!".

One place where this comes up is in Descartes' Meditation -- he infers that mind and body are distinct, but he could just be a blind man who touched the elephant in two different places and declared that it must be two things. It seems like one needs additional assumptions/constraints to make this leap.

More Discussion

I prefer to use natural(ish) language, so let "all properties same" indicate the formal statement "for a given x,y, for all properties p, x has p iff y has p" (or however you'd like to more formally state it). Leibniz's law is usually stated "if 'all properties same for x,y' then 'x,y are identical'". Implication from the properties from the identity. However, the converse implication is (seems?) obviously true: "if 'x,y are identical' then 'all properties are the same'". Thus the relationship is one of bidirectional implication. Therefore you should be able to infer "different properties therefore not identical", but it's not clear to me that you can make this inference in general; hence the need for additional constraints.

Answer 1

The Identity of Indiscernibles:

is a principle of analytic ontology first explicitly formulated by Wilhelm Gottfried Leibniz in his Discourse on Metaphysics.

The priciple has been an is debated: see Arguments for and against the Principle.

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.

The fact that we (or someone) cannot "apply" in practice the principle, does not per se contradict his ontological status.

See Indiscernibility of identicals:

one famous application of the indiscernibility of identicals was by René Descartes in his Meditations on First Philosophy.[...] This argument [...] allegedly derives a conclusion about what is true from a premise about what people know. What people know or believe about an entity, they [the critics of Descartes] argue, is not really a characteristic of that entity.

Answer 2

Well, a trunk and a leg are indeed different, so there really aren't any issues in trying to employ Leibniz's law. However, I think the issue arises when the men claim that they've found ontologically independent things; they think they're entities in themselves that are not part of a whole. Thus I'm not sure if this is a problem of Leibniz's law but of mereology. Correct me if I'm misunderstanding something.

Answer 3

(Fx <-> Fy) -> x=y, is invalid for non-referring names or non-referring descriptions.

F(the present King of France) <-> F(the present King of France), for all F.. is a tautology and ~((the present King of France)=(the present King of France)), is true.

F(Vulcan) <-> F(Vulcan), for all F..is a tautology and ~(Vulcan=Vulcan), is true.

The Leibnitz-Russell definition of identity x=y <-> (for all F)(Fx <-> Fy) will not do.

I suggest: x=y <-> [(x exists) & (y exists) & (for all F)(Fx <-> Fy)].