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Property Dualism/Consciousness/Chalmers: from the lack of a logical supervenience of consciousness from the physical follows that conscious experiences imply the properties of an individual that are not implied by the physical properties of that individual. It is not about a separate "substance".
Consciousness is a feature, a property of the world, beyond the physical facts.
Property Dualism: there is a weaker kind of property dualism, according to which biological fitness is not implied by physical facts. Such a property dualism would be compatible with materialism.
This variety is not to be confused with our present one.
Consciousness/Chalmers: consciousness can result, according to our variety, from property dualism of physical properties without being implied by them.
This view is completely naturalistic and compatible with our scientific knowledge.
Definition Naturalistic Dualism/Chalmers: I call my variety of dualism naturalistic dualism, according to which properties of the phenomenal consciousness supervene on physical facts in a still to be determined manner, although not logically.
What could happen in the future would be what happened with Maxwell's equations: known laws could be extended as far as their scope of application is concerned. There could be a psychophysical theory, as there is an electromagnetic theory.
This dualism is naturalistic because it proceeds from the validity of all known laws. It is closer to materialism than many other forms of dualism by negating any transcendental elements.
My dualism, by the way, can still turn out as a kind of monism if it should be shown that the phenomenal and the physical are two aspects of an overlapping kind, as it turned out to be the case with matter and energy. I have a certain sympathy for this view. But this could not be a materialistic ((s) eliminative) monism.
The Conscious Mind Oxford New York 1996
Constructing the World Oxford 2014