Consider the principle of deductive closure:

PDC. K(p) ∧ K(p → q) ⊢ K(q)

Informally (and roughly) this means that we know the logical consequences of the things we know. Despite being a fundamental epistemic principle, ensuring that our deductions are knowledge yielding, it has been challenge by many.

One representative example is the challenge raised by the contextualists who argue that PDC is valid only if the same context is kept passing from the premises to the conclusions.

I am looking at other kinds of challenges, different from the contextualist approach.


Two usually discussed challenges are by Nozick and Dretske.

Nozick's analysis of knowledge includes the following condition for S to know that p:

(sensitivity) Were p false, S would not believe p.

Truth conditions for subjunctive conditionals are given in the usual way: Sensitivity requires that in the close worlds to the actual world in which not-p holds, S does not believe p.

Sensitivity is motivated for example, because it explains failure to know in Gettier-type cases and also helps us to deal with the notorious skeptical paradox. (See the links.) Sensitivity enables us to give counter examples to closure.

Take the following propositions:

p=Lucky has hands. q=Lucky is not a brain in a vat.

Now assume that Lucky (who has hands) is actually in a world where he is not in a brain-in-a-vat scenario. Now, sensitivity is satisfied for p (and in fact, so are all the other Nozick's conditions for knowledge), and so Lucky knows that he has hands. (Sensitivity is satisfied, because in all close worlds where he does not have hands, e.g. when he loses them in an accident or something like that, he does not believe he has hands.)

However Lucky does not know that he is not a brain in a vat, because for that proposition sensitivity is not satisfied: Take a close world where q is false, so in that world he is a brain in a vat. Lucky would still, in that world, believe that he is not a brain in a vat. So "were q false, Lucky would not believe q" is not satisfied.

(One could also claim that both of these results are even intuitive.)

Lucky knows that p.
Lucky knows (assuming he has thought about the matter) that p --> q.
But Lucky does not know, that q.

There are also more mundane examples of p and q where closure fails, because sensitivity is not closed under entailment. Nozick's analysis of knowledge is obviously in some way externalistic. Nozick's analysis of knowledge has been criticized by showing that it leads to some other counterintuitive claims. It's up to the critic to provide a better more plausible analysis, that can also handle Gettier cases and the skeptical paradox.

Another issue related to closure is transmission. In failure of transmission the justification, for believing that p, fails to "transmit" to some entailed consequence q. Some failures of transmission are uncontroversial.

For more on Nozick, Dretske and transmission see: http://www.iep.utm.edu/epis-clo/ http://www.iep.utm.edu/transmis/

There is also a good chapter titled "Skepticism and Closure" in A Companion to Epistemology (Dancy, Sosa, Steup).

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