Let's start with a quote by Donald Davidson:
The notion of supervenience, as I have used it, is best thought of as a relation between a predicate and a set of predicates in a language: a predicate p is supervenient on a set of predicates S if for every pair of objects such that p is true of one and not of the other there is a predicate of S that is true of one and not of the other (Davidson, 1985, p. 242).
Previous theories had argued that claims concerning the identity of particular mental and physical events depended upon the discovery of lawlike relations between mental and physical properties. These theories thus held that empirical evidence supporting such laws was required for particular identity claims. According to Anomalous Monism, however, it is precisely because there can be no such strict laws that causally interacting mental events must be identical to some physical event. The token-identity thesis thus requires no empirical evidence and depends on there being no lawlike relations. It in effect justifies the token-identity of mental and physical events through arguing for the impossibility of type-identities between mental and physical properties or kinds (Davidson 1970, 209, 212-13; see Johnston 1985).
To draw his ideas out further, x=y IFF z (z causes x and implies y or z causes y and implies x) and z (x causes z and implies y or y causes z and implies x).
An explanation of an agent's action can be considered adequate only if it shows the action in question to be reasonable against the background of an agent's beliefs and desires. This latter
condition together with the truth condition, which states that the propositional attitudes a rationalization attributes to an agent must be true, form the necessary conditions for the justification model of explanation.
Davidson considers the above conditions necessary but not sufficient. The deficiency of the justification model is explained by drawing attention to the distinction between having a
reason for an action and having the reason why one performs an action. For a reason to be the reason why one performs an action the reason must cause the action.
For example, one has a reason to turn on the television, say, to watch one's favorite TV show. But this need not be the reason why one turns on the television. This is because the above reason did not cause one to turn on the television.
As Davidson puts it:
[S]omething essential has certainly been left out, for a person can have a reason for an action, and perform the action, and yet this reason not be the reason why he did it.
The reason for one to turn on the television could simply be because one is lonely and desires company. Thus, one reason (namely, to keep one company) was the cause of the action while the other reason (namely, to watch one's favorite show) wasn't.
Of course, we can include this idea too in justification; but then the notion of justification becomes as dark as the notion of reason until we can account for the force of that "because."
The mere possibility that a person acted on the basis of one reason rather than another presents an insurmountable obstacle. The anti-causalist has no way of accounting for the force of the "because" in the rationalization.
The only solution, according to Davidson, is to view the efficacious reasons (the ones that account for the correct rationalization) as causes of action. This leaves us, according to Davidson, with only one alternative to justificationalism, namely, the view that reason explanation is a species of causal explanation.
According to Anomalous Monism, however, it is precisely because there can be no such strict laws that causally interacting mental events must be identical to some physical event.
X=X but is not dependent on brain-state.