What is the closure principle for knowledge? Is it refuted by counterexamples?

Submitted for PH144 (Mind and Reality) Take Home Exam

The ‘closure principle for knowledge’ is expressed as follows: if S knows p, and S knows that p entails q, then S knows q (where S is a person, and p and q are declarative sentences).

Initially, this seems trivially true, stemming from two key points. The first is the definition of entailment: if ‘p entails q’, then there is no logically possible world in which p, but not q. Thus, the inference that ‘if p, and p entails q, then q’ is readily accepted. Consequently, truth itself is closed under entailment. The second relates to the nature of knowledge. It seems plausible that knowledge is a subset of belief: if S knows p, then S must believe p. Moreover, one cannot know a falsehood. Hence, true belief is a necessary condition for knowledge (though it is not by itself sufficient). Returning to the closure principle, we see that ‘S knows p’ necessarily means that p is true; ‘S knows that p entails q’ necessarily means that p does, in fact, entail q. By the first point, q is necessarily true. So, when S thereby comes to hold the true belief q, it seems undeniable that S knows q (not solely because it is a true belief, but because its truth was logically necessary, and this was understood by S – luck played no part). To take an example, if I know that I live on the University of Warwick campus, and I know that this entails that I live in Coventry, then by the closure principle, I know that I live in Coventry.

But the closure principle can be abused in sceptical arguments in such a way that makes some philosophers – including Dretske, Nozick, and Vogel – argue that it is not as sound as it seems in the abstract. The most robust counterexample, from Vogel (1990:16) is as follows: having parked one’s car in Avenue A, one is later inclined to say one knows one’s car is in Avenue A. But one does not know that one’s car hasn’t been stolen – statistically, there is no reason to think that it hasn’t. But the proposition “My car is in Avenue A” (p) entails the proposition “My car hasn’t been stolen” (q). Here, one knows p; one knows that p entails q; but one doesn’t know q – in violation of the closure principle.

The issue arises because in day-to-day life, we say we ‘know’ things which are not certain, but merely (highly) probable. But when we say that ‘truth is closed under entailment’, we refer only to certainties. If one is committed to a definition of knowledge which requires such certainty, then the scope of one’s knowledge will be very small indeed (I can’t know that there is such a country as Russia; there is a small probability that everyone is lying or mistaken about it). In this case, the closure principle is not refuted (or refutable) – one just can’t claim to know that one’s car is in Avenue A. More plausibly, one could dispense with absolute certainty (‘truth’) being a necessary condition of knowledge. Thus, we maintain the intuitive notion that one does know where one’s car is in Vogel’s example; but in this case, the closure principle is refuted, and no longer holds.

Result

Mark: 68% (High 2.1)

Feedback:

You appear to argue that we should respond to the closure principle by rejecting the idea that knowledge requires certainty.

There are some excellent things about this answer. I was struck by the clarity and sophistication of your writing. You explain the closure principle elegantly (albeit with one slip), as well as Vogel's counterexample. It was great to give your own example of the closure principle in operation - this made the principle vivid and intuitive. Your analysis is interesting and shows excellent independence of thought. There is scope to improve it, however.

Some more specific comments:

• The minor slip in your explanation of closure. A couple of times you described the Q of the principle - i.e. the entailed proposition from P - as being necessarily true. I know what you meant, but there is a slip here that you need to watch: it doesn't follow from the fact that Q necessarily follows from P and "P entails Q" that Q is a necessary truth. (That I have two hands entails that I have at least one hand, but it isn't a necessary truth that I have at least one hand.)

• Your discussion at the end seemed to conflate the truth condition on knowledge (i.e. S knows that P only if P) with the condition that knowledge requires certainty. However, these aren't the same thing. The truth condition simply says that, if I know P, then P is true. It doesn't imply anything about the strength of my grounds for believing P - e.g. that these must amount to "certain" grounds. I think you were onto something in this part of your essay, but you didn't succeed in formulating it as clearly as you might have done.