4

Consider the following Skeptical Argument:

P1. You do not know that you are not a brain in a vat.
P2. You know that if you have hands, then you are not a brain in a vat
P3. If you know that A, and you know that if A then B, then you know that B.
C. You do not know that you have hands.

Nozick says that the principles are wrong and rejects premise 3, that "knowledge is closed under known implication". "I don’t know that I’m not a brain in a vat because if I were a brain in a vat, then I would still believe, and still be sure, that I’m not a brain in a vat. So Nozick says, I do know that I have hands, though I don’t know that I’m not a brain in a vat". So according to Nozick:

[4.] You know that you have hands.
[5.] You know that if you have hands, then you are not a brain in a vat.
[6.] You do not know that you are not a brain in a vat.

Isn't P3 above just modus ponens? So how can Nozick reject P3? What's the meaning of 4-6, how can they all be true? Applying modus ponens to 5, yields:

[7.] you are not a brain in a vat.

7 differs from 6, because 6 contains 'You do not know that'. But so what?

Source: 10 minutes 53 seconds juncture; Lecture 7, Video 5 (transcription nonexistent);
MITx: 24.00x Introduction to Philosophy; by MIT Associate Prof Caspar Hare PhD (Princeton) See after 1 minutes 2 seconds in Lecture 7, Video 4.

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P3 is not modus ponens because the implication is under the knowledge operator, K(p->q) is not the same as Kp->Kq, where Kx stands for "S knows that x". It is widely accepted by epistemologists however as the Epistemic Closure Principle: if S knows that p and knows that p implies q, then S knows that q. Nozick denies it because he has a different take on what constitutes knowledge.

The traditional definition is:"S knows that p if and only if S believes p, p is true, and S is justified in believing p". The last condition is a problem for a "brain in a vat" hooked up to a Matrix feeding it non-stop hallucinations. But Nozick favors "externalist" approach, where S is allowed to be entirely unaware of the factors that justify her belief and make it knowledge. Specifically, according to him these justifying factors are the following counterfactuals:

(a) Variation condition: were p not true S would not believe it.

(b) Adherence condition: were p still true in somewhat different circumstances, S would still believe it, and would not believe not p.

If these are met the belief "tracks the truth", as Nozick puts it, and is knowledge.

In your example S can obviously know that if she has hands then she is not a brain in a vat (this is a definitional tautology). And S can know that she has hands, e.g. because S's brain receives nerve signals from them. And if they didn't she wouldn't, so the variation condition is met, although S is unaware of her nervous system's workings. But this does not mean that S knows that she is not a brain in a vat. In fact, S can never know that she is not a brain in a vat because the variation condition can never be met: even if S was a brain in a vat, she would still believe that she isn't.

See more in Stanford and Internet Encyclopedias of Philosophy.

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