There are fundamental irreducible mental properties.
There are fundamental irreducible physical properties.
Necessarily, anything that has a physical property is a physical thing.
Recombination: Fundamental irreducible properties are all mutually compatible.
So, it is possible for something to have both a mental and a physical property. (1, 2 and 4)
So, it is possible for a physical thing to have a mental property. (3 and 5)
This seems to imply at least the logical possibility of property dualism. But not quite: for property dualism, you need that a purely physical thing has a mental property. And I think there are no purely physical things, because each substance has an immaterial form.
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As I mentioned in the other thread, I don't know what it means for something to be a "physical property". I know what it means when you say that the thing itself is purely physical, but what does it mean to say that a plant's "life" or its "wellbeing" are "purely physical properties"? What does it mean to say that a cheetah's knowledge of how ("know-how") to attack a gazelle and knowledge of where to bite ("know-where") and knowledge of when to strike ("know-when") are all "purely mental properties"... or are they? I just don't understand these distinctions at all.
It seems to me that we are better to say that living creatures of certain capacities and properties that are characteristic of being alive, animate beings have capacities and properties characteristic of being animate (on top of the "living" ones), and rational, language-using creatures like us have rational capacities and properties (on top of the animate and living ones). At no point does the property itself need to be defined as being "physical" or "non-physical", since properties are not substances or objects at all.
A sufficient condition for being a physical property is that the property reduces to one of the fundamental properties of physics. And that's all I need for the argument.
I don't think the properties of even the simplest of living things "reduce to one of the fundamental properties of physics", or even any collection of them. So, why should the capacities of self-moving, language-using creatures like us reduce to such? If that means our capacities (or those of a flower, for that matter) are "non-physical", then so be it; but I don't think the property dualists will be happy.
The core problem stems from Descartes' mistaken impression that living bodies are like machines, and that there is a dividing line between the machine of the body and the powers of the "mind". Living organisms are not machines.
I guess the conceptual crowbar needs to turn on the fulcrum of something like "health" or "wellbeing". We know that those are meaningful terms describing real properties that bacteria, plants, animals, and humans all have. But they are manifestly not describable using nothing but the language of physics (spin, charge, position, trajectory, etc.). Are these then what the property dualists mean by "non-physical properties"?
If so, then sciences like biology are primarily concerned with non-physical properties, long before we ever get to anything about consciousness, voluntary action, perception, etc. If not, then the definition given ("reducible to fundamental properties of physics") is rebuked.
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