E.g. in this video at 9:00, I have the same question as a user there:

At 9:00, I don't see how the example you gave violate the closure principle. If he were in fact looking at a disguised mule, the belief P = "I am looking at a Zebra" would not meet the sensitivity test in the first place. So in this example Q = "I am not looking at a disguised mule" failed the sensitivity test because P also failed the sensitivity test.

So at least I'm not alone!

Or take the following passages in the IEP article on "Epistemic Closure Principles"

Let’s illustrate this with an example similar to Nozick’s own (1981, 207). Let p be the belief that one is sitting in a chair in Jerusalem. Let q be the belief that one’s brain is not floating in a tank on Alpha Centauri, being artificially stimulated so as to make one believe one is sitting in a chair in Jerusalem. Suppose one has a true belief that p. In the “closest” counterfactual situations (to employ the terminology of one account of truth-conditions for subjunctives) in which p is false (say, one is standing in Jerusalem, or one is sitting in Tel Aviv), one will not believe p. In close counterfactual situations in which one is sitting in Jerusalem, one does believe that p. One’s belief of p tracks the truth of p and thus counts as knowledge.

Suppose, on the other hand, that one has a true belief that q. If one’s belief that q were false, however (and one really was in this predicament on Alpha Centauri), one would still believe (falsely) that one was not in Alpha Centauri (q). One’s belief that q, while actually true, does not track the truth of q (being held when q is true but not when q is false). Hence, the belief that q does not count as knowledge.

Here I have the same problem: you'll believe p even if q is false, but if q is false then p will also be false and so p will not adhere to the sensitivity requirement.

I really don't get the point of those examples. They seem to show sensitivity in a certain context where it indeed works, and then when it comes to showing the contradiction forget about the context.

Since those examples are common, it must be my fault. What am I overlooking here?

  • Isn't the point that closure under entailment fails? If q is false then p is false, that is p entails q. But it does not mean that p is not sensitive, it means that knowledge is not preserved by entailment. IEP emphasizes this in the next paragraph:"We may suppose that one can correctly deduce q from p. Even so, since one’s belief that p tracks the truth of p and counts as knowledge and one’s belief that q does not do so, knowledge fails to be closed under known entailment."
    – Conifold
    Jan 16 at 21:56
  • @Conifold yes, that's the point. I just don't understand why p is sensitive in the given examples (not in general). Regarding the 2nd example: if the BiaV scenario is possible, then we see how ¬ p can be true, but the person will believe p. Why doesn't this contradict the requirement "the person would not have believed that p if p had been false"?
    – viuser
    Jan 16 at 22:06
  • This is where "closest counterfactual situations" come in. In PW semantics, so-called accessibility relations are defined on possible worlds, and only counterfactuals in accessible worlds 'count', see SEP and @Dennis's post here. Sensitivity is analogous to continuity for functions, only values in a 'neighborhood' matter in the definition. The BiV world is inaccessible from the chair-in-Jerusalem world, it is too 'far' from it, so counterfactuals in it do not count.
    – Conifold
    Jan 16 at 23:44
  • 1
    Hope my answer for another post helps clarify a bit for you... Jan 17 at 2:28
  • 1
    Here's an answer from me on a related question.
    – Bumble
    Jan 17 at 6:40


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