OK, I've read this paper now, and it's opened up a new line of attack. It is strong stuff.
I agree. It is strong stuff. Um... No, I meant to say that I need strong stuff. The last time I've felt like this was after I graded exam papers of group of students who were on a quick conversion track from almost-bachelor's-degree to master's-degree and who hadn't taken lectures before coming to the exam.
It is not clear whether the author of this paper is as dependent on S5 as Marten thinks he is.
I don't know about other zombie-provers, but he certainly isn't dependent on S5...
[Edited to add: I should have proof-read this one more time before posting, I'm using 1., 2., and 3. in two different meanings in different places, hopefully it doesn't confuse too much.]
Like I wrote, there is much that I could say about the paper. A lot of it would be unprintable in polite circles. I might start by stating that there is no chance on earth that a paper like that would be accepted in any serious conference on mathematical logic. His proof leaves too much undefined. I have once reviewed a paper that contained a side result that was left on the air in a similar way. Even though the paper was otherwise relatively good, all three referees (I don't know who the other two were) quoted that theorem as one of the reasons why it should be rejected. [Side note: I wrote this paragraph before I realized what is really happening in the proof. If I was writing a review for, I would be very tempted to include nasty comments in it, something that I have never done.]
I haven't actually checked what I intended to check about the semantics, yet. That is because I came up with a simpler proof for refuting computationalism using similar methods as what the author uses. And the proof is so simple that it, in my opinion, demonstrates that the formalization used in the paper is fatally flawed.
For starters, the author gives three natural language propositions that come from Searle:
- Brains cause conscious mental phenomena.
- There is some sort of conceptual or logical connection between conscious mental phenomena and external behavior.
- The capacity of the brain to cause consciousness is conceptually distinct from its capacity to cause motor behavior. A system could have consciousness without behavior and behavior without consciousness.
I'm not a philoshopher but from what I understand, the idea is that computationalism implies proposition two. He gives later an altered version of the statement that is essentially equivalent but speaks of computations, instead. But since his proof is about this formulation, I use it here. The third proposition corresponds with the claim that mind is not physical.
He then gives a modal logic formalization for these propositions. It goes like this:
[latex](1) $\exists x . (B(x) \land M(x))$ \\
(2) $\Box\forall x . (B(x) \land M(x) \rightarrow E(x)) \land
\Box\forall x . (B(x) \land E(x) \rightarrow M(x)) $ \\
(3) $\Diamond(B(a) \land M(a) \land \neg E(a)) \land \Diamond( B(a)
\land \neg M(a) \land E(a))$[/latex]
I could go on quite long about the reasons why (1) isn't a good formalization but that would be a lenghty tangent to non-monotonic logics and it is not relevant to the main content of this post, especially since Bringsjord doesn't use it for anything anywhere.
When I read these the first time I wondered why he fixed the system to 'a' in (3) instead of using quantifiers and variables as in (2). Because now he only says that there exists
one specific brain whose external behavior may be separated from the consciousness. I'm not a native English speaker but how I understand 3. is that it speaks of the brains of all fully functioning human beings. He should have written something like this (I'm not certain whether this is the best form):
[latex]$\forall x . (B(x) \rightarrow \Diamond ((M(x) \land \neg E(X)) \lor
(\neg M(x) \land E(X))))$[/latex]
[Side note: Yes, some variants of quantified modal logic forbid modal operators inside quantiers but Bringsjord himself uses them so later in the paper so it shouldn't have been an issue. And if your logic prevents you from expressing a vital proposition accurately, you should be using some other logic anyway.]
It took me three readings of his proof before I realized that there is a very good reason for this specific formulation. Stay tuned, film at 11.
Now, this set of modal formulas is inconsistent under the modal logic M (or T as it is called in the paper). The proof is in the paper, it is nothing really important. However, it is important to note that inconsistency comes from (2) and (3) together, and (1) has no function in it.
Next, Bringsjord sets to resolve this inconsistency by proving that (2) has to be false. He does this by using the thought experiment where a persons neurons are replaced one by one by their functional equivalents. He states that Searle has identified that there are at least three possible results from it:
- V1: Everything goes smoothly and nothing strange manifests itself.
- V2: The person loses his conscious experience but the body keeps on working as usual.
- V3: The person maintains his concsiousness but the body stops working.
Bringsjord formalises these propositions (well, the latter two, he is not interested in the first case) by simply using propositional letters V2 and V3. He doesn't stop to wonder if the thought experience and its result even can be expressed using his logic and he certainly doesn't express it. [Side note: I would go for option V4: the operation is so involved that he dies under the knife losing both consciousness and the body functions.]
Next, comes his argument A1 that is the actual proof that in his opinion refutes computationalism (well, he doesn't spell it out completely, I add the missing steps. Possible mistakes in missing steps are mine). His idea is to prove that (3) is true, and then from that conclude that (2) has to be false.
[latex]P1. $\Diamond V2$ [V2 can be conceived] \\
P2. $\Diamond V3$ [V3 can be conceived] \\
P3. $\Diamond V2 \rightarrow \Diamond (B(a) \land E(a) \land \neg
M(a))$ ["clearly"] \\
P4. $\Diamond V3 \rightarrow \Diamond (B(a) \land M(a) \land \neg
E(a))$ ["just as clearly"] \\
5. $\Diamond (B(a) \land E(a) \land \neg M(a))$ [1, 3] \\
6. $\Diamond (B(a) \land M(a) \land \neg E(a))$ [2, 4] \\
7. $\Diamond (B(a) \land E(a) \land \neg M(a)) \land \Diamond (B(a)
\land M(a) \land \neg E(a))$ [5, 6]
[/latex]
Bringsjord's view is that everything that can be conceived is logically possible. That is his justification for including premises P1 and P2.
His justification for premises P3 and P4 are really "clearly" and "just as clearly" in the paper. This is the first case of
Proof by Blatant Assertion that I see in the wild. [There is one "unclear" part here: is it allowed to use the same constant 'a' in both P3 and P4? I skip this problem because I didn't bother to find out what exact first-order modal semantics Bringsjord is using.]
And here we see the real reason why the (3) is defined for one specific brain instead of all brains:
Because otherwise Bringsjord couldn't have concluded it from 5. and 6.. He wants to prove that 3. is true, so he formalizes it into a weaker form that he then proves.
The great irony of this cheap trick is that it was completely unneccessary. His real aim is to disprove 2. Proving (3) was just a tool to that aim.
There is a shorter, more direct proof for this result that uses less blatant assertions and that doesn't need (3) at all:
[latex]
P1. $\Box\forall x . (B(x) \land M(x) \rightarrow E(x)) \land
\Box\forall x . (B(x) \land E(x) \rightarrow M(x)) \rightarrow \neg
\Diamond V2$ ["clearly"] \\
P2. $\Diamond V2$ \\
3. $\neg \Box\forall x . (B(x) \land M(x) \rightarrow E(x)) \land
\Box\forall x . (B(x) \land E(x) \rightarrow M(x))$ [P1, P2]
[/latex]
This "clearly" is just as justified as his two instances were. He said that "it is possible that a person loses his mind while body keeps working" implies one half of statement (3). So it should be equally clear that the formula that states that minds and external behavior are equivalent implies that it is not possible for a person to lose only one of them. And we get the conclusion that (2) is false.
So, I needed only 3 steps and one blatant assertion while he needed 7 steps and two blatant assertions. I didn't even have to assume that the one person whose brain we are messing with just happens to be the only guy in the whole universe whose brain is guaranteed to produce funny effects when messed with.
But there is still more. Note that the above proof doesn't use any modal logic machinery whatsoever. We don't use any fancy inference rules involving boxes and diamonds at all. In fact, we could have saved us some trouble in writing if we had used the principle of substitution to abstract the modalities from it, and simply used C to denote (2) and T to denote diamond V2. The the proof would be:
[latex]
P1. $C \rightarrow \neg T$ \\
P2. $T$ \\
3. $\neg C$ [P1, P2]
[/latex]
See,
we can reduce Bringsjord's argument to one textbook application of modus tollens!. And his own proof doesn't use any modal logic constructs, either. You can do similar substitutions there. All the boxes and diamonds in the definitions are pure smoke screen, Bringsjord could as well have worked with propositional logic from start! Even his cheating in formalizing 3. was unnecessary. I don't even go to his quite peculiar usage of "physically possible" later in the paper.
In conclusion: Bringsjord's key argument A1 can be represented in propositional logic. But we already agreed that you need modalities to discuss zombies. So his formalization is inadequate to the extreme.