Now I have already pointed out that the Socrates argument is from universals to particulars and yours is not. Please let your logic lecturer and LW know about this.
The structure of an argument is relevant only if you use syllogisms. And there hasn't been any good reason to use syllogisms after Hilbert presented his axioms for predicate logic. And now that there are symbolic proof procedures that work by breaking down the structure of an arbitrary formula (such as Smullyan-style semantic tableaux), there is no real reason to use axiom proofs, either, except for trivial toy problems.
What is relevant is that there is a set of formulas S. They may be completely arbitrarty sentences having completely arbitrary structures as long as they are well-formed formulas. That set entails a set of concequences Cn(S). And by definition, Cn(S) contains all those formulas that are true in all models of S.
In case of the flawed zombie formalization the set P was:
[latex]$P_1$: $\forall x . ( T(X) \rightarrow S(X) )$ \\
$P_2$ $\neg S(c)$[/latex]
In the case of Socrates the set S is:
[latex]$S_1$: man$(s)$ \\
$S_2$: $\forall x . ( $man$(x) \rightarrow $mortal$(x))$[/latex]
Note that symbolic logic doesn't care what you call those formulas: you may call them definitions, premises, assumptions, or even blue-globs-of-skrbl. What matters is that there exists a set of well-formed formulas.
In the case of set P, we can prove (using any sufficiently powerful sound proof system) that neg T(c) is in Cn(P). And in the case of set S, we can prove (using any sufficiently powerful sound proof system) that mortal(s) is in Cn(S). And in case of both sets, we can prove that they are consistent.
On the other hand, we have also sets of M and M' of formulas:
[latex]$M$: $\forall x . T(C)$ \\
$M'$: $\forall x . \neg $mortal$(x)$[/latex]
When we put P and M together we get an inconsistent set (and we can prove it inconsistent using any sufficiently powerful sound proof system). When we put S and M' together we get an inconsistent set.
The situation is the same.
Another difference is that the conclusion of the Socrates argument follows from the argument – a feature that used to be considered quite important.
The conclusion that follows from your argument is “human minds are not physical”.
Careful here. You always have to be extremely careful when you try to apply logical arguments to the Real World
TM.
This is because to apply logic, you have to formalize your problem. [Note: I use formalize in the sense of: "express it using a formal language", not in the more restricted "write it down as a syllogisms"]. When you prove things, you prove that they hold in the formal system. This is an extremely crucial point.
Your results apply to the Real World only if you have done the formalization correctly. So what you get out is: "If I didn't make any errors in formalization, then this result holds." This is the best thing that you can ever hope to achieve with logic.
In practice, it is not possible to model anything completely. [And I'm not certain that it could be done in theory] You have to abstract things and hope that you don't throw out anything relevant.
The p-zombie arguments are abstracted to the extreme. The paper that Geoff posted reduced the whole question of minds into only four statements (well, they have five but they ignore one completely and could have left it out). By sheer accidence this is exactly the same number of formulas that I needed to model the game of Sudoku. So they reduce the whole complexity of brains and minds down to a level of a simple puzzle game. I see no reason to suppose that the formalization corresponds to reality.
So, the conclusion [of the M-logic based p-zombie argument from the paper that Geoff linked to] when translated back to natural language is: "If you can accurately model the problem of human consciousness using only four statements, two of which are not actually formalized at all and presented only in natural language, then consciousness is not physical".
Now this brings us to a third major difference and the one which you, LW and your logic lecturer appear to have completely missed.
If you add the negation of the conclusion to the premise of the Socrates argument you do not get an inconsistency.
I certainly missed that one.
Let's see, the conclusion was (using the previous syntax) mortal(s). So let's add neg mortal(s) to S to see what we get:
[latex]$S_1$: man$(s)$ \\
$S_2$: $\forall x . ( $man$(x) \rightarrow $mortal$(x))$ \\
$C: $\neg$ mortal$(s)$ \\
$N_1$: man$(s) \rightarrow $ mortal$(s)$ [instantiating $S_2$] \\
$N_2$: mortal$(s)$ [MP $S_1$, $N_1$]
[/latex]
Do you have an explanation why having both $C$ and $N_2$ among the conclusions is not an inconsistency? [Espcially given that one perfectly good way to define inconsistency is "Set P is inconsistent iff for some formula f, Cn(P) contains both f and neg f".]
See above for my comment on accuracy of formalizations: garbage in, garbage out.
