Is Knowledge Closed Under Known Logical Implication?

Closure is the epistemic principle that for any propositions p and q, if someone knows that p, and that p entails q, then that individual is in an epistemic position to know q – that individual has sufficient evidence to know q. In other words, were that individual to believe q on the basis of her knowledge that p and that p entails q, she would thereby know q. The basic idea is that we can add to what we know by deducing things from what we already know. Such a principle seems obviously true and denying it appears to be an extravagant cost to any epistemological theory. Nonetheless, closure has been under attack in recent literature in epistemology. Closure has principally been denied as a way of avoiding skepticism. The skeptic cannot get to the conclusion that I do not know that I have hands, even if I do not know that I am not radically deceived by an evil demon, if closure is false (if it is false that we have knowledge of known entailments of what we know).

Fred Dretske gives a case like the following as a purported counterexample to closure. Joe goes to a cage marked ‘zebras’ at the zoo. Joe looks at the striped animals in the cage and believes they are zebras. In fact, it seems that Joe knows that they are zebras. For an animal to be a zebra, it cannot be a cleverly disguised mule. In knowing what a zebra is, Joe knows this fact as well. Does Joe know that the animals before him are not cleverly disguised mules? Dretske says no, and he claims that this is the intuitive answer. In supporting this claim, Dretske states that all of Joe’s evidence in favor of the animals being zebras has been neutralized and does not count in favor of the animals not being cleverly disguised mules, since Joe would have the same perceptual evidence in that case as well. In this case, (a) Joe knows there are zebras before him, and (b) Joe does not know that these are not cleverly disguised mules, even though (c) Joe knows that if an animal is a zebra then it is not a cleverly disguised mule. If (a) through (c) hold, then this is a counterexample to the closure principle. [for the contemporary relevance of this example see this].

However, (a) through (c) do not hold in the above example, this is not a counterexample to closure. Several objections can be brought to Dretske’s account. First, Joe’s evidence that there is a zebra before him is also evidence that there is not a cleverly disguised mule in front of him. Dretske seems to assume that Joe’s evidence for the former claim is solely his perceptual experience, but this is a mistake. Joe will not be justified in believing that there is a zebra before him without further background evidence regarding the nature of zoos, zookeepers, and the difficulty in so disguising a mule (that zoos are not in the buisness of trickery, that zoos often have actual zebras, etc.). Without such background evidence that is assumed in the case, it is false that Joe is justified in believing that there is a zebra in front of him. This can be seen by imaging a case where Joe has the same perceptual experience as of a zebra, but is in a context completely unfamiliar to him. In such a case it is far from clear that Joe knows that there is a zebra before him.

However, when coupled with the relevant background evidence, Joe’s perceptual evidence is also evidence that there is not a cleverly painted mule before him. Considerations concerning the nature of zoos, zookeepers, and the difficulty in disguising a mule, make the belief that there is a zebra before him the belief that Joe epistemically ought to have. What more is needed for knowledge? Joe’s belief is justified, true, and not subject to any Gettier considerations. Therefore, if this evidence is good enough for Joe to know that there is a mule before him, it is also good enough to know that there is not a cleverly disguised mule before him. Thus, closure is preserved.

Dretske might reply that a child lacks this background evidence, yet still knows that the animal before her is a zebra. If one lacked this background evidence, then one would not be justified in believing (and thus would not know) that the animals were not cleverly disguised mules, but in such a case the individual would also not know that there were zebras before her. It is by no means clear that a child in such an epistemic situation actually knows that there is a zebra before her. Even if the child can distinguish between ‘is a zebra’ and ‘merely looks like a zebra’ without the relevant background evidence, she has insufficient reason to affirm that the animal is a zebra. In addition, even if it is granted that the child knows that the animal is a zebra, it is doubtful that she can draw the inference regarding cleverly disguised mules, and if she cannot do so, then the case does not bear on the closure principle.

Second, the intuitions that Dretske appeals to do not support a denial of closure. Though people typically do claim to have seen zebras in the zoo, and though when confronted with the possibility that what they saw was actually cleverly disguised mules these people may deny that they know that what they saw was not cleverly disguised mules; when this denial is made, the former self-attribution of knowledge is often withdrawn, at least temporarily. This response is the response dictated by the closure principle. It seems that at no one time do individuals claim knowledge of a proposition and simultaneously deny knowledge of one of that propositions known entailments. As such, intuitions are on the side of closure, not against it. We have not been presented a counterexample to the closure principle.