Deductive closure principle

1. Closure principles

According to the ‘deductive closure principle’, if S knows that P, and S correctly deduces Q from P, then S knows that Q (call this principle ‘Closure’). The set X of propositions which one knows is not closed (in the mathematical sense) under logical implication simpliciter. This is because there are countless propositions which are logical consequences of what one knows which one does not recognize as such. Since one does not believe these logically implied propositions, one does not know them, and so they are not members of X. A similar point holds for the closure principle for known logical implication: it seems possible for one to know that P, know that P implies Q, and yet fail to deduce Q. According to Closure, those logical consequences which one deduces from what one knows are members of X. Such deduced consequences meet two of the conditions for knowledge: they are true (since they follow from known, and therefore true, propositions), and they are believed. If these deduced consequences are to be known, as Closure says, then they must satisfy whatever further necessary conditions for knowledge there might be.

Suppose that having justification for believing that P is a necessary condition for knowing that P (see Knowledge, concept of §2; Justification, epistemic). Then it is plausible to suppose that Closure will hold only if closure for justification holds: if S has justification for believing that P, and S correctly deduces Q from P, then S has justification for believing that Q. This principle is used in constructing Gettier examples in which one starts with a justified belief of a false proposition (such as that Jones has ten coins in his pocket and will win the race) and deduces a proposition (say, that someone who has ten coins in his pocket will win the race) which just happens to be true. If closure for justification holds, then one will have a justified, true belief that does not amount to knowledge (see Gettier problems).

S may have justification for believing that P without actually believing that P (for example, S may fail to realize that S’s evidence justifies a belief about the identity of the killer). Thus it is plausible to hold that justification is closed under logical implication simpliciter: if S has justification for believing propositions which logically imply Q, then S has justification for believing that Q. This principle is employed in generating the lottery paradox. One holds a ticket in a very large, fair lottery which one justifiably believes will have just one winner. Assume that the very high probability that one’s ticket will not win justifies one in believing that one’s ticket will not win. Since similar reasoning will apply to every other ticket, one has justification for believing that the first will not win, one has justification for believing that the second will not win, and so on. Given that these propositions logically imply that no ticket will win, the closure principle now under consideration implies that, paradoxically, one has justification for believing that no ticket will win (see Paradoxes, epistemic §1).