Peter D. Klein

Peter D. Klein

Peter David Klein is a professor of philosophy and chair of the department at Rutgers University, New Jersey. Peter Klein received a BA at Earlham College, and a PhD from Yale University...

, in the second edition of The Cambridge Dictionary of Philosophy

The Cambridge Dictionary of Philosophy

The Cambridge Dictionary of Philosophy is a dictionary of philosophical terms published by Cambridge University Press and edited by Robert Audi. The dictionary, having been in publication since 1995, is now in its second edition. There are 28 members on the Board of Editorial Advisors and 440...

, defines closure

Closure (mathematics)

In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

 as follows:

A set of objects, O, is said to exhibit closure or to be closed under a given operation

Closure operator

In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....

, 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. [links not in original]

In propositional logic

Propositional calculus

In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...

, the set of all propositions exhibits deductive closure: if set O is the set of propositions, and operation R is logical entailment (“”), then provided that proposition p is a member of O and p is R-related to q (i.e., p  q), q is also a member of O. In the philosophical branch of epistemology, many philosophers have and continue to debate whether particular subsets of propositions—especially ones ascribing knowledge or justification of a belief to a subject—are closed under deduction.

Epistemic closure

It is not the case that knowledge is closed under deduction; that is, if person S knows p, and p entails q, then S knows q (sometimes called the straight principle). A subject may not actually believe q, for example, regardless of whether he or she is justified or warranted. Thus, one might instead say that knowledge is closed under known deduction: if, while knowing p, S believes q because S knows that p entails q, then S knows q. An even stronger formulation would be as such: If, while knowing various propositions, S believes p because S knows that they entail p, then S knows p. While the principle of epistemic closure is generally regarded as intuitive, philosophers such as Robert Nozick

Robert Nozick

Robert Nozick was an American political philosopher, most prominent in the 1970s and 1980s. He was a professor at Harvard University. He is best known for his book Anarchy, State, and Utopia , a right-libertarian answer to John Rawls's A Theory of Justice...

 and Fred Dretske

Fred Dretske

Frederick Irwin Dretske is a philosopher noted for his contributions to epistemology and the philosophy of mind. His more recent work centers on conscious experience and self-knowledge. Additionally, he was awarded the Jean Nicod Prize in 1994...

 have argued against it.

Justificatory closure

In the seminal 1963 paper, “Is Justified True Belief Knowledge?”, Edmund Gettier gave an assumption (later called the “principle of deducibility for justification” by Irving Thalberg Jr.

Irving Thalberg Jr.

Irving Thalberg Jr. was the son of 1930s Hollywood producer Irving Thalberg and Academy award-winning actress Norma Shearer. He taught philosophy at the University of Illinois at Chicago....

) that would serve as a basis for the rest of his piece: “for any proposition P, if S is justified in believing P and P entails Q, and S deduces Q from P and accepts Q as a result of this deduction, then S is justified in believing Q.” This was seized upon by Thalberg, who rejected the principle in order to demonstrate that one of Gettier's examples fails to support Gettier's main thesis that justified true belief is not knowledge (in the following quotation, (1) refers to “Jones will get the job”, (2) refers to “Jones has ten coins”, and (3) is the logical conjunction

Logical conjunction

In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

of (1) and (2)):

Why doesn't Gettier's principle (PDJ) hold in the evidential situation he has described? You multiply your risks of being wrong when you believe a conjunction. [… T]he most elementary theory of probability indicates that Smith's prospects of being right on both (1) and (2), namely, of being right on (3), are bound to be less favorable than his prospects of being right on either (1) or (2). In fact, Smith's chances of being right on (3) might not come up to the minimum standard of justification which (1) and (2) barely satisfy, and Smith would be unjustified in accepting (3). [Thalberg p. 798]