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Thomas Polger (2004) handles multiple realizability by developing a "non-reductive mind-brain identity theory".
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But the arguments of Shapiro and other recent critics (see section 2 above) challenge the existence of multiple realizability only by assuming flat realization, and no recent critic has defended flat realization as the correct account involved in the scientific cases at issue.
If successful, this challenge undercuts the multiple realizability argument by denying that the same mental kinds obtain across species, to be realized by different physical mechanisms.
So this type of multiple realizability is not by itself a barrier to reducibility.
Headed in the opposite direction, arguments concerning new concepts such as multiple realizability and supervenience by Putnam, Kim, Fodor and others led to higher-level functionalism, a distinction between type-type and token-token reductions and the examination of its implications.
Despite its initial plausibility, the identity theory faces a strong challenge in the form of the thesis of multiple realizability, first formulated by Hilary Putnam.
One approach challenges the way that mental kinds are individuated by multiple realizability proponents.
This sort of psychophysical property identity would seem to be blocked by the multiple realizability argument sketched earlier.
Associated with functionalism was the doctrine of multiple realizability: mental properties can, in practice as well as principle, be realized by a variety of media which do not share anything in common physically other than a capacity to support the distinctive pattern (Fodor 1974).
In the philosophy of mind, the multiple realizability thesis contends that a single mental kind (property, state, event) can be realized by many distinct physical kinds.
This point was often expressed by saying that functional states exhibit "multiple realizability".
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com