Donald Davidson wanted to resolve what he saw as a conflict in all materialist philosophies of mind, and in particular with identity theory, in the idea that mental phenomena have a causal role, and yet physical science has no need to refer to the mental. The problem stems from three plausible principles:
1) The Principle of Mental Causation: At least some mental events interact causally (directly or otherwise) with physical events
(1) and (2) seem to imply that some mental events can be predicted and explained on the basis of laws. (3) explicitly denies this. Davidson accepts all three and tries to show they are compatible. To this end, he presents a version of the identity theory which shows how the three principles can be reconciled, anomalous monism. It is 'monism' because it entails that there is only one kind of thing, the physical. However, it's not a type-identity theory. Anomalous monism postulates token event identity without psychophysical laws.
From the three principles above, Davidson categorises three possible positions in the philosophy of mind:
Nomological monism - there are correlating laws between the
mental and the physical, but they both come under one category (either
entirely mental or entirely physical)
Davidson's new position allows him to avoid the reductionist problems associated with token-token identity theory whilst sticking to a strictly materialist conception of the mind. It allows him his next move, which is to insist that causality and identity are relations between events themselves, and that the study of the mind is the study of mental events, themselves purely physical but not explicable in terms of the physical. This allows him to avoid a further problem of identity theory that is intentionality, for all mental events cannot be understood purely in terms of their physical nature since one also needs to understand how these events stand in relation to each other and their surrounding circumstances.
Davidson holds that events can be described in different ways, and that events can only be related to one another given a specific description. Causality between mental events therefore need not be explicable in physical terms, and in fact Davidson denies that this is possible. So for example, if I have just read a book by Graham Greene I might reach up to the bookshelf to find another Greene novel; those two events are causally related and can explained by a strict law relating mental events, but there is no physical law that can link the brain states at the time of each event, nor of the physical events themselves.
This denial of psycho-physical laws stretches so far as for Davidson to assert that general principles of rational discourse associated with the physical sciences, e.g. coherence and consistency, need not apply to discourse of mental events, such that although correlations between the mental and the physical can be shown, they needn't be able to form strict laws of the kinds we associate with the physical sciences.
Davidson is thus saying that the mental supervenes on the physical, in that the mental is dependent upon the physical without, in linguistic terms, being strictly identical with the physical. That is to say that there cannot be two identical physical events with two different associated mental events, nor that a mental event cannot occur without a correlative physical event, but that an event might occur in the brain that cannot be explained satisfactorily in physical terms and that requires explanation of the mental event. That one set of events or facts, A, supervenes on another set, B, doesn't entail that A-events can be reduced to B-events, either by law or by definition. All it means is that the latter kind of facts fix or determine facts of the former kind.