If, following Place and Smart's topic-neutral analysis, mental concepts describe functional-causal roles rather than intrinsic qualia, then 'pain' and 'C-fiber firing' can rigidly co-designate the same underlying physical state without modal residue.
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A leftover gap or difference between two things when you consider all possible scenarios—the idea that something could be true in one possible world but not another, leaving room for them to come apart.
Place and Smart(as referenced in the philosophy of mind)
Two philosophers (U.T. Place and J.J.C. Smart) who argued in the 1950s-60s that mental states like thoughts and feelings are actually just physical brain processes, not something separate or mysterious.
Rigidly co-designate(in philosophy of language and metaphysics)
Two names or terms that refer to exactly the same thing in every possible situation—like how 'water' and 'H₂O' always refer to the same substance, no matter what.
topic-neutral analysis(Philosophy of mind, identity theory of mind)
An analysis of sensation reports that does not presuppose either a physical or a mental vocabulary, remaining neutral between physicalist and non-physicalist interpretations, in the way that arithmetic is neutral between physical and mental domains