Principle of Minute Differences: Proof or Counter-Example

[Oinkyboinky]

In The Problem of the Many, Peter Unger claims that he sees no reason to reject the principle of minute differences:

(PMD): If x is a paradigm F and y is an entity that differs from x in any respect relevant to being an F only very minutely, then y is an F.

There seem to be plenty of instances where this principle is obeyed.

For example, if a cloud-candidate differs only by a few water droplets from a paradigm cloud, then it seems the cloud-candidate is also a cloud.

Of course, though, many instances of the principle do not show that it is true in general.

What are some potential proofs of the PMD or any counter-examples where it does not hold?

[-1-]

I read this just the other day. I forgot the name of the author and of the argument.

I present a counter-example to your claim.

A could with a few droplets fewer than the previous state of the same cloud.

If you still take some drops away, it's still a cloud.

If you keep taking drops still away, eventually you are down to one or two droplets, which a cloud don't make.

So there is a qualitative transformation resulting from quantitative transformations.

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Marx expressed the same thing in reverse: "Quantitative changes always precede qualitative changes."

[Oinkyboinky]

I read this just the other day. I forgot the name of the author and of the argument.

I present a counter-example to your claim.

A could with a few droplets fewer than the previous state of the same cloud.

If you still take some drops away, it's still a cloud.

If you keep taking drops still away, eventually you are down to one or two droplets, which a cloud don't make.

So there is a qualitative transformation resulting from quantitative transformations.

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Marx expressed the same thing in reverse: "Quantitative changes always precede qualitative changes."

Ok, but this isn't a counter-example, since you did not specify an object that is a paradigm F, and a different object that differs minutely from the first, and show that this second object is a non-F.

Rather, what you gave is a reductio argument against the principle (in the form of a sorites paradox). While this is a valid way to show the PMD is false, I'm looking for a particular counter-example.

In addition, one might respond to your argument as follows:

At some point in the sorites series, the typical cloud changes from typical to atypical, so the principle is not violated when the atypical cloud changes into a non-cloud (since we did not go from a typical cloud to a non-cloud abruptly).