The idea that mental states are non-reducible properties of brain states is the central tenant of a theory of mind called property dualism. However, before we can assess the theory we must be aware that the question assumes the existence of mental states and as such we cannot answer this question from some perspectives (e.g. eliminative materialism)
Property dualism proclaims the existence of a single, physical substance (unlike Cartesian dualism), but argues that this single substance has two potential properties: physical and mental states that are not reducible. It is not just that we might talk of mental and physical states in different ways, but that the difference is in ontology as well as language. This is equivalent to historical notions that living things contained some 'vital force'. Essentially mental states are an extra property of matter in the brain.
Property dualists argue that consciousness is an emergent phenomenon of the physical processes of the brain and thus it is important to note that this leads to a belief in asymmetrical causation. Mental properties are caused by physical properties, but have no effect themselves on the physical properties, making the relationship one way.
This can be seen in the illustration below where the same object has both physical and mental properties that exist inextricably alongside each other. The object's physical properties can cause a change in its physical properties, but not the other way around:
The theory of property dualism arose as a defence of consciousness that responded to the key problems of Cartesian dualism (i.e. the problems of location and interaction) caused by the notion of two different substances. Property dualism also allows for the primacy of the physical over the mental. This is critical because it allows for matter to cause mental states, which in turn allows for the rejection of the notion of two substances.
By supporting the notion that our experience of mental states cannot be reduced or eliminated the theory is intuitively compelling. This is a benefit over materialist theories of mind that fail to capture or explain our intuitive experience of consciousness.
Fundamentally, property dualism is an advancement of substance dualism, and over this theory it has several advantages. Firstly, by invoking only a single substance it avoids to the problems of interaction and location associated with the non-spatial Cartesian mental substance. Secondly, it is not rooted in religious beliefs and is thus more scientifically based than Descartes' theory. Thirdly property dualism is compatible with Descartes' arguments that the mind has properties that are distinct from the body, thus taking the benefits whilst leaving the drawbacks. Finally, property dualism is compatible with advances in brain science in the same way that materialist theories are, thus seemingly creating a 'best of both worlds' scenario.
Another advantage of property dualism is that it allows for first-person perspective. Materialist theories do not normally allow for the same degree of privileged access to one's own internal states that Descartes proposed. However, the Cartesian model is solipsistic, whilst property dualism, being materialist, allows for scientific investigation and public third party access to the causes of mental states.
However, there are also a number of objections to property dualism. The first applies Occam's razor ("Entities should not be multiplied more than necessary"). Surely it would be simpler to suggest that we are simply talking about one thing, as in reductive or eliminative materialist accounts of mind? The notion of two properties seems overly complex.
A further point...