This is another attempt at an argument against inferentialism about logical constants.
Given a world w, let w* be a world just like w except that it has added to it an extra spatiotemporally disconnected island universe containing exactly one hydrogen atom with a precisely specified wavefunction ψ0. Suppose that in in the actual world there is no such isolated hydrogen atom. Now, given a nice first-order language L describing our world, let L* be a language whose constants are the same as the constants of L with an asterisk added to every logical constant, name and predicate. Given a sentence ϕ of L, let ϕ* be the corresponding sentence of L*—i.e., the sentence with all of L’s logical constants asterisked.
Let the rules of inference of L* be the same as those of L with asterisks added as needed.
Let the semantics of L* be as follows:
Every predicate P* in L* means the same thing as P in L.
Every name a* in L* means the same thing as a in L.
Any sentence ϕ* in L* without quantifiers means the same thing as ϕ in L.
But if ϕ* has a quantifier, then ϕ* means that ϕ would be true if there were an extra spatiotemporally disconnected island universe containing exactly one hydrogen atom with wavefunction ψ0.
Thus, ϕ* is true in world w if and only if ϕ is true in w*.
Observe that because L contains only names for things that exist in the actual world, and hence not for the extra hydrogen atom or its components, an atomic sentence P(a1,...,an) in L is true if and only if the corresponding sentence P*(a1*,...,an*) is true in L*.
Logical inferentialism tells us that the logical constants of L* mean the same thing as those of L, modulo asterisks. After all, modulo asterisks, we have the same inferences, the same meanings of names, and the same meanings of predicates. But this is false: for if ∃* in L* were an existential quantifier, then it would be true that there exists an isolated hydrogen atom with wavefunction ψ0. But there is none such.
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