The knower paradox is a

paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...

belonging to the family of the paradoxes of

self-reference Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding. In philoso ...

(like the

liar paradox In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth ...

). Informally, it consists in considering a sentence saying of itself that it is not known, and apparently deriving the contradiction that such sentence is both not known and known.

History

A version of the paradox occurs already in chapter 9 of

Thomas Bradwardine Thomas Bradwardine (c. 1300 – 26 August 1349) was an English cleric, scholar, mathematician, physicist, courtier and, very briefly, Archbishop of Canterbury. As a celebrated scholastic philosopher and doctor of theology, he is often call ...

’s ''Insolubilia''. In the wake of the modern discussion of the paradoxes of self-reference, the paradox has been rediscovered (and dubbed with its current name) by the US logicians and philosophers David Kaplan and

Richard Montague Richard Merritt Montague (September 20, 1930 – March 7, 1971) was an American mathematician and philosopher who made contributions to mathematical logic and the philosophy of language. He is known for proposing Montague grammar to formalize th ...

, and is now considered an important paradox in the area. The paradox bears connections with other

epistemic Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Episte ...

paradoxes such as the hangman paradox and the paradox of knowability.

Formulation

The notion of

knowledge Knowledge can be defined as awareness of facts or as practical skills, and may also refer to familiarity with objects or situations. Knowledge of facts, also called propositional knowledge, is often defined as true belief that is distinc ...

seems to be governed by the principle that knowledge is factive: :(KF): If the sentence ' ''P'' ' is known, then ''P'' (where we use single quotes to refer to the linguistic expression inside the quotes and where 'is known' is short for 'is known by someone at some time'). It also seems to be governed by the principle that

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...

yields knowledge: :(PK): If the sentence ' ''P'' ' has been proved, then ' ''P'' ' is known Consider however the sentence: :(K): (K) is not known Assume for

reductio ad absurdum In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...

that (K) is known. Then, by (KF), (K) is not known, and so, by ''reductio ad absurdum'', (K) is not known. Now, this conclusion, which is the sentence (K) itself, depends on no undischarged assumptions, and so has just been proved. Therefore, by (PK), we can further conclude that (K) is known. Putting the two conclusions together, we have the contradiction that (K) is both not known and known.

Solutions

Since, given the

diagonal lemma In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories of the natural numbers—specificall ...

, every sufficiently strong theory will have to accept something like (K), absurdity can only be avoided either by rejecting one of the two principles of knowledge (KF) and (PK) or by rejecting

classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...

(which validates the reasoning from (KF) and (PK) to absurdity). The first kind of strategy subdivides in several alternatives. One approach takes its inspiration from the hierarchy of truth predicates familiar from

Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...

's work on the Liar paradox and constructs a similar hierarchy of knowledge predicates. Another approach upholds a single knowledge predicate but takes the paradox to call into doubt either the unrestricted validity of (PK) or at least knowledge of (KF). The second kind of strategy also subdivides in several alternatives. One approach rejects the

law of excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradic ...

and consequently ''reductio ad absurdum''. Another approach upholds ''reductio ad absurdum'' and thus accepts the conclusion that (K) is both not known and known, thereby rejecting the

law of non-contradiction In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the sa ...

.Priest, G. (1991), 'Intensional Paradoxes', ''Notre Dame Journal of Formal Logic'' 32, pp. 193–211.

References

External links

* *{{cite SEP , url-id=epistemic-paradoxes , title=Epistemic Paradoxes , last=Sorensen , first=Roy} Mathematical paradoxes Self-referential paradoxes