OK Paul,
I have found an article : "Zombies versus Materialists: The Battle for Conceivability".
http://www.brown.edu/Departments/Philosophy/zombie.html
Nice article. I intended to reply to this already in the morning but decided that I should referesh my memory about the formal definitions of the original model logics (K and its derivations) before doing so.
Here it is with my annotations:
And I add a couple of comments to it.
If Zombies are possible, materialism is false. This was the structure of the argument I supplied and our resident logician has confirmed that this is the case.
Not exactly. I stated that if the
do exist, then materialism is false. If it is "possible" that they exist, then the conclusion depends on what we mean with "possible".
I agreed with this, and also stated that I guessed that the argument would need the strongest scheme of modal logic (S5) to push it through. If this article is correct, then so was my guess. We need S5.
This brings us to the first serious issue of the p-zombie proof. Is it reasonable to use S5? In the paper the author states that for the sake of the argument he is willing to concede this to the zombist. And he then goes on to show that even that doesn't result in falsification of materialism. I'm not as willing.
S5 operates under the assumption that "If X is possible, then it is necessarily possible". Let's see what that means formally.
The semantics of modal logics is given with respect to
frames. A frame contains a collection of worlds W (states) and an accessibility relation R between them. We get a
model by assigning a truth valuation for each world. For example, if we had two worlds s
1 and s
2 and a proposition P, then one valuation would be that P is false in s
1 and true in s
2.
The definition of modal logics usually don't use the terms "possibly" and "necessary" but instead "diamond" and "box". This is because then you can use the same definitions if you want to attach different meanings for the modalities. For example, "diamond" might be "believes" and "box" be "knows".
A formula "diamond P" is true in a state s of W if there exists at least one world that is accessible from s where P is true. For example, suppose that R is defined in the previous example so that s
2 is reachable from s
1 and that is the only accessibility. Then, "diamond P" is true in s
1 but false in s
2 even though P itself is true in it. Conversely, "box P" is true if P is true in all successors of the state. In this example it is true for both states. So, essentially the modal operators chase through the different worlds following the arcs of the accessibility relation.
When it comes to validity, there are three different validities involved. First, a formula may be
valid in a model if it is true in all worlds of it. The "box P" is such a formula in our example. Next, a formula is
valid in a frame if it is valid in all models that can be created from it. Our "box P" is not valid in the given frame since it is not true in any model where P is false in s
2. Finally, a formula is
valid in a set of frames if it is valid in every frame of the set.
Essentially, each different modal logic can be characterized as a set of frames. The idea is that the logic has a characteristic formula, and the logic then corresponds to the set of frames where that formula is valid.
In the specific case of S5 the characteristic formula is "diamond P implies box diamond P". This corresponds with the set of frames where R is an equivalence-relation. Informally, this means that we can divide the worlds into one or more partitions and within each partition all worlds are accessible from each other (and worlds are accessible from themselves, too). The R of the prior example doesn't fulfill this condition so it is not a S5 frame.
Now, after this long-winded introduction I finally can come to the actual problem:
Can we model our intuition of what "possible" means in the Real World using S5 frames?.
I'm not claiming that it absolutely can't be done. But I see enough difficulties in it that I can't take it as given. What are the possible worlds?
A natural candidate would be the idea that every time when there is some choice involved there is a world where each different was choosen. But this doesn't work, because then the accessibility relation is not an equivalence relation and we have to be able to move back and forth between the worlds. If I kill someone, he stays dead no matter what I do. I can't enter the world where I didn't kill him.
If we use "possible" in S5 sense, then we lose its natural language sense. Right now it is possible for me to stand up, walk fifteen steps, and grab a cup of coffee [ETA: eleven steps actually, and I ended microwaving it because the timer had went off] . But in S5 I'm not able to use the word like this. Because I'm often in a situation where I can't do that.
So, before I accept even that it is possible to even formulate the p-zombie proof in S5 [that the article shows is futile in end], I would need to see how the S5 frame is constructed and what is the intuitive meaning of "possible" under that formalization. It won't be the everyday sense in any case.
It all comes down to the quirk in S5 which the zombist relies on to force the proof.
Is there any source where the
complete S5 p-zombie proof is presented?
Even without seeing the complete proof I'm quite willing to bet that it is the very same quirk that enabled the zombist to make it in the first place (and this supposing that we could use S5).