WOW! Two pickled herrings later and common sense has broken out in this thread! I basically agree with everything you say here, you simply don't take account of the wider context in which this logical argument arose. When Robin originally tried to formalise it, he only took account of one portion of a larger argument. However, the portion you have very nicely turned into predicate calculus is just fine.
Nice that you agree. I want to empasize that this means that your earlier argument that sentences P
0 - P
2 entail that consciousness is not physical was premature and you need add something more before you get the result.
In effect, this logic proves that IF p-zombies can exist, consciousness can't be physical.
Now we move into dangerous waters. The argument above was that if they
do exist, then consciousness isn't physical. Here we have
can exist and we have to consider this more clearly. What do we mean here with "can exist"? This is not a standard concept of predicate logic so we have to pin down its exact semantics before we know if the quoted sentence follows logically from P
0 - P
2.
I can immediately think of three possible interpretations for it:
- That there exists a model for P0 - P2 where the interpretation of the predicate symbol Z is not empty. That is, that if we augment the set of premises with the sentence "exists p . Z(p)" we do not get an inconsistency.
- That we move from predicate calculus to some modal logic where "possible" is a modality and we express "can exist" as "possible exists p . Z(p)".
- That we do otherwise as in the second option but use only predicate calculus syntax to define a new predicate symbol to denote the "can exist" concept.
I would immediately rule out the third one since that approach is suitable only for really simple problems with small domains and it will quickly break down when complexity of the model increases. [Side note: I have used this approach successfully for solving simple planning puzzles where you can define the moves that can be made at any state with a couple of simple rules.]
Of the other two, the first one is the one that is to me the more natural interpretation of "can exist". This is because my background is in computational logic where we are interested in solving real world problems with mathematical logic and it is more useful in practice to say that "possibly A" means that "there is a model where A is true". Those who are more into philosophical logic may disagree and they have the right to do so.
However, I'm not particularly keen on trying to model the system using some modal logic. This is mainly because it has been a long time since I've worked with them and even then I have mostly used LTL and CTL (linear time logic and computation tree logic) that are not suitable for this case, so I would have to brush up my memory before doing the modeling. But I'm willing to examine the formalization if someone else provides it.
This leaves us with the first, and to me the natural, interpretation. However, here we can easily construct a structure that is a model for P
0 -- P
2 and where exists p . Z(p) is true:
[latex]$U = \{ c, p \}$ \\
$A = \{ c \}$ \\
$P = \emptyset$ \\
$Z = \{ p \}$ \\
$H = \emptyset$ [/latex]
Note that in this structure consciousness is not physical.
This means that the "logically consistent" interpretation for "can exist" is not strong enough to conclude the non-physicality of consciousness. Under this formalization, p-zombies "can exist" but neg P(c) was not entailed. The same holds also for the other direction, from this we can't conclude that consciousness is physical, either.
So, to salvage the proof you have to model it using some modal logic. I have to warn you that if you choose to undertake that project, you will have hard time in justifying your choice of the logic as well as your formalization. Predicate logic is nice and clean (for some values of 'nice' and 'clean', at least) and, most importantly, there is only one predicate logic (well, only one logic that is
the predicate logic). But as soon you open the doors for modal logic, you face a choice of several dozens of candidate logics, each with its own axioms.
Our current conceptual scheme makes them very easy to imagine.
There are many things that that are very easy to imagine but that don't exists. For example, right now I'm imagining a walrus that is as big as the block of flats opposite my window. But unfortunately such walruses don't exist. And can't exist, if our current theories of biology and physics are even reasonably close to being correct.
That I'm able to conceive something doesn't say anything about the existence or even the possibility of existence of that thing.
We have an apparently neccesary connection which we are conceptually incapable of explaining. Nagel's position is that this suggests there is something wrong with our concepts. He suggests we look for a new conceptual scheme, under which the neccessary connection between minds and brains becomes obvious and transparent. At this point, p-zombies will become conceptually impossible as well as actually impossible. Do you follow?
I think that I get the basic idea. I haven't had time to read that Nagel's paper so I'm not certain if I get the details of his argument.
How I see the matter is this: Suppose that we somehow could construct a formal structure S that modeled our universe and its history perfectly (this is of course impossible to do in reality, the simplest argument against it is that we would have to construct another much larger universe to store it). Suppose further that in addition of containing information of what really is and was, it contains all information of what might have happened: every time there is some random event, the structure branches and both choices are included. And this structure covers the whole history of universe, from the beginning to the end. So, essentially S is an accurate model of our noumenal reality. And as it contains also all possibilities, saying that something is possible means that there some branch in S where the thing is true.
Now, with respect to p-zombie and consciousness question, there are four possibilities:
- Consciousness is non-physical and there are no p-zombies in S.
- Consciousness is physical and there are no p-zombies in S.
- Consciousness is non-physical and there is at least one p-zombie in S.
- Consciousness is physical and there is at least one p-zombie in S.
Our noumenal reality corresponds to one of these choices. Note that if it happens to be the last, it means that the laws of logic don't hold in the universe and all attempts to use reason to deduce anything about it are in the end futile.
Supposing that the universe is not contradictory, we live in one of the three first choices. Nagel's problem occurs only in the first one. In the second one a p-zombie is as impossible as a non-prime number 7, and if we are in the third one, Nagel is barking the wrong tree when he tries to eliminate the zombies. So, you need the new conceptualization only if you think that we are in the first situation.
My personal opinion is that we will never be able to distinguish which of the above choices is really true. I believe that the second one is the most probable one but if I ever claim that I can prove it, you can (and should) take away my licence to practice logic.