If we define an “A” as a “B” (eg, a “clade” as a “natural group”), and then assume as a premise that only A:s are B:s, then we arrive to the logical conclusion that **all** and **only** A:s and B:s are A:s and B:s (eg, that only clades are natural groups).

This conclusion is totally logical given the definition and the premise, but is none the less false according to Russell’s paradox. The reasoning is as follows. If we call any definable collection (like A:s and B:s) a set, and if we let *R* be the set of all sets that are not members of themselves, then, if *R* is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves (ie, Russell’s paradox). R is thus impossible. It means that the conclusion is false specifically by generally contradicting its definition. The conclusion is simply incompatible with its definition.

This is the fundamental problem for typologists (traditionally called (class) realists), like like cladists, particle physicists and race biologists, but which nominalists like traditional (Popperian) scientists avoid. The fundamental question is whether universals are real or abstract, which (class) realists (thus erroneously) answer with “real”, but which traditional (Popperian) scientists answer with “abstract”, The answer does not mean that traditional (Popperian) scientists are right, but just that only they aren’t wrong. Russell’s paradox means that being right is an impossibility. The best we can do is not being wrong.

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