1. A p-zombie has all the physical characteristics of a human being.
2. A p-zombie is not conscious"
Let's put that one into modern syntax and look it more closely. I'll use the notation where "A(x)" denotes that x is an attribute of a human being, "P(x)" denotes that x is physical, "Z(x)" that x is a zombie, "H(x, y)" denotes "x has y", and finally, the constant "c" denotes the specific property of consciousness.
The first proposition becomes:
[latex]$P_1$: $\forall p\forall a ( A(a) \land P(a) \land Z(p) \rightarrow H(p, a))$[/latex]
Or in plain English: "If 'a' is a physical attribute and 'p' is a zombie, then 'p' has the attribute 'a'."
The second proposition becomes:
[latex]$P_2:$ $\forall p (Z(p) \rightarrow \neg H(p, c))$[/latex]
In plain English: "If 'p' is a zombie, then 'p' does not have consciousness."
I hope that you agree with this formulation.
Then there was also the zeroth premise, that consciousness exists as an attribute. I am willing to grant this to you for the sake of this argument so let's formalize it also:
[latex]$P_0$: $A(c)$[/latex]
The question then is that is nonphysicality of consciousness a logical consequence of the above set of three formulas.
In predicate calculus a formula f is a logical consequence of a set S of formulas if and only if f is true in all models of S. That is, if all premises are true, then f has to be true also. There is a nice theorem that states that f is a consequence of S if and only if the set of formulas that we get by adding the negation of f to S is inconsistent.
So, we add the negation of our desired conclusion to the set of propositions and check what happens. In effect, we are trying to prove that the following set of formulas is inconsistant:
[latex]$P_0$: $A(c)$ \\
$P_1$: $\forall p\forall a ( A(a) \land P(a) \land Z(p) \rightarrow H(p, a))$\\
$P_2:$ $\forall p (Z(p) \rightarrow \neg H(p, c))$ \\
$C$: $\neg \neg P(c)$[/latex]
(Note that the double negation in C is in purpose, the desired conclusion is that c is not physical, so by negating that we get the second negation symbol).
What happens is that we are not able to derive a contradiction from the set. In fact, we can easily construct a model for it. The model is the following structure:
[latex] $U = \{ c \}$ \\
$A = \{ c \}$ \\
$P = \{ c \}$ \\
$Z = \emptyset$ \\
$H = \emptyset$
[/latex]
Let's go through the set of sentences to verify that all of them are satisfied. Since c is in interpretation of A, P
0 is satisfied. Since the interpretation of Z is empty, the left sides of the implications P
1 and P
2 are trivally false, so the implications themselves are satisfied. Finally, since c is in the interpretation of P, neg neg P(c) is also true.
So the above structure is a counter model for the claim that neg P(c) was a logical consequence of the three premises.
What does this mean in plain English? It means that from the set of premises we can't conclude that consciousness is necessarily not physical.
Why was my conclusion different from yours? It is because your syntax hid one crucial assumption, and it did it so well that you probably didn't notice it yourself: That p-zombies exist. It is there: "
a p-zombie has...
a p-zombie is not". If we add the premise that there exists at least one p-zombie, then neg P(c) becomes a consequence. But without it, it doesn't work.
In effect, your proof says: "If p-zombies exist, then consciousness is not physical". I have nothing to say against that. I fully agree.
Of course, I don't believe that p-zombies exist, but that is a completely different matter.