Anomalous Monism
Most contemporary non-reductive physicalists subscribe to a position called anomalous monism (or something very similar to it). Unlike epiphenomenalism, which renders mental properties causally redundant, anomalous monists believe that mental properties make a causal difference to the world.
The position was originally put forward by Donald Davidson in his 1970 paper Mental Events, and holds that in addition to this principle of causal interaction between the mental and the physical, two other principles can also be granted. One is the "nomological character of causality" - the idea that when two events are in causal relation to one another, there must be a strict law connecting them (i.e. the occurrence of the cause must by law guarantee the occurrence of the effect). Davidson's third principle is the "anomalism of the mental", which states that the kind of strict deterministic law described above cannot apply to the mental events, which can be neither predicted nor explained in a decisively law-like manner. The mental is rather governed by ‘guidelines’ of normativity. These three principles, Davidson claims, are incompatible. We can accept any two but not the third, lest we be led into contradiction.
There is only one way to resolve this Mexican stand-off, according to Davidson, and that is to stake an identity claim between mental and physical tokens based on the notion of supervenience (see below). If mental events are identical to physical events, then they can enter into strict law-governed causal relationships, upholding the first and second principles. But since mental properties are supervenient on, and so not reducible to, physical properties, they can retain their anomological status, preserving the third principle.
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