closure. A set of objects O, is said to exhibit closure or to be closed under a given operation, R, provided that for every object, x, if x is a member of O and x is R-related to any object, y, then y is a member of O. For example, the set of propositions is closed under deduction, for if p is a proposition and p entails q, i.e., q is deducible from p, then q is a proposition (simply because only propositions can be entailed by propositions). In addition, many subsets of the set of propositions are also closed under deduction. For example, the set of true propositions is closed under deduction or entailment. Others are not. Under most accounts of belief, we may fail to believe what is entailed by what we do, in fact, believe. Thus, if knowledge is some form of true, justified belief, knowledge is not closed under deduction, for we may fail to believe a proposition entailed by a known proposition. Nevertheless, there is a related issue that has been the subject of much debate, namely: Is the set of justified propositions closed under deduction? Aside from the obvious importance of the answer to that question in developing an account of justification, there are two important issues in epistemology that also depend on the answer.
Subtleties aside, the so-called Gettier problem depends in large part upon an affirmative answer to that question. For, assuming that a proposition can be justified and false, it is possible to construct cases in which a proposition, say p, is justified, false, but believed. Now, consider a true proposition, q, which is believed and entailed by p. If justification is closed under deduction, then q is justified, true, and believed. But if the only basis for believing q is p, it is clear that q is not known. Thus, true, justified belief is not sufficient for knowledge. What response is appropriate to this problem has been a central issue in epistemology since E. Gettier’s publication of ‘Is Justified True Belief Knowledge?’ (Analysis, 1963).
Whether justification is closed under deduction is also crucial when evaluating a common, traditional argument for skepticism. Consider any person, S, and let p be any proposition ordinarily thought to be knowable, e.g., that there is a table before S. The argument for skepticism goes like this: (1) If p is justified for S, then, since p entails q, where q is ‘there is no evil genius making S falsely believe that p’, q is justified for S. (2) S is not justified in believing q. Therefore, S is not justified in believing p. The first premise depends upon justification being closed under deduction. See also EPISTEMIC LOGIC, EPISTEMOLOGY, JUSTIFICATION , SKEPTICIS. P.D.K.