General idea: the essential nature of a mental state such as pain (i.e. the characteristics that all pains share by virtue of which they are pains) is to occupy a certain functional role, in particular a causal role.
More informally: mental states are defined by the "causal work" they do.
What's a functional state? Usually this is taken to be a state defined by its relations to inputs, outputs, and other (also functionally defined) states.
This sounds circular, because every functional state is defined in terms of other functional states. However, we can define all of them at once.
Versions of functionalism:
machine functionalism: minds are Turing machines; mental states are Turing machine states. (Or, better: the brain is a physical computing machine, and the mind is an abstract computing machine (i.e. a program). So mind is to brain as software is to hardware. However, this doesn't require thinking that the architecture of the human mind is the same as that of the Turing machine!
causal-theoretical functionalism: mental states are defined in terms of a psychological theory.
which psychological theory? Two versions:
(a) common-sense psychological theory. (This can be regarded as defining common-sense psychological terms like belief, desire, etc.)
(b) scientific psychological theory.
Over behaviorism: according to functionalism, psychological states are internal states that cause behavior. For (logical and metaphysical) behaviorism, by contrast, mental states are behaviors, not internal causes of behavior. (So in particular, functionalism can allow for the possibility that a surgical patient given curare and an amnestic drug feels pain even though they never exhibit any behavior associated with that pain.)
Over the identity theory: functionalism allows for the multiple realizability of mental states. The same functional state can be realized by various physical states.
The beginning point for discussion of the identity theory is the existence of systematic and extensive relationships between mental phenomena and neural phenomena.
There is a lot we don't know about these relationships, but there are also many things we do know.
Problems for Functionalism
Background point: recall the distinction between intentional and phenomenal states.
Intentional mental states are representational.
Phenomenal mental states are characterized by feelings, or "what it's like," or qualia.
Some states are intentional but not necessarily phenomenal: "dispositional" belief, desire
Some states are phenomenal but not intentional: "raw feels," perhaps itchiness or sadness
Some states are both: "occurrent" belief, sense perception
1. Problems of absent and inverted qualia. (Kim 162ff)
absent qualia: it seems theoretically possible to build something that has the same functional architecture as you but which has no qualia. If so, then a functional account of qualia can't be correct.
inverted qualia: it seems theoretically possible to build a functional duplicate of you, but which has different qualia than you do. If so, then, again, a functional account of qualia can't be correct.
Possible responses: (a) incoherent; there can't really be functional duplicates with different qualia; (b) perhaps functionalism is right about intentional phenomena, while the old-fangled identity theory is right about qualia
(Does this show -- if correct -- that a functional account of pain can't be correct? Depends on whether 'pain' refers to qualia, the feeling of being in pain, or rather to something that gives rise to qualia.)
2. Chinese Room argument.
This is John Searle's argument designed to show that functionalism can't account for intentionality. You could think of it as an "absent intentionality" argument.
3. Problem of inflexibility. (see Kim p 156)
Kim points out that causal-theoretical functionalism seems to face the problem that if the theory is false, then no terms for mental states refer to anything. [This doesn't seem all that serious to me, and David Lewis anticipated it in the articles Kim cites. His suggestion: terms for mental states refer to the real-world state that comes the closest to satisfying the theory.]
4. Problem of holism.
The problem with defining all mental states at once is that the definition of each depends on the definitions of all the others. (see Kim, p 138)