Knowledge paradox

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The knowledge paradox (known in English-language literature as the "knower paradox") is a paradox to which, according to Kaplan and Montague, the paradox of the unexpected execution can be reduced, and has been discussed in two variants.

version 1

The variant originally formulated by Kaplan and Montague consists of the following sentence p:

"It is known that this sentence is wrong."

Using the knowledge operator K and the negation ~, p can be written as follows:

K (~ p)

The sentence is paradoxical as it can apparently both be proven and disproved. Because firstly (according to the rules of epistemic logic ) a sentence that is known is also true. From p it follows that ~ p. Which would disprove p.

But if p can be refuted, this is - according to the rules of classical logic - a proof of ~ p. Since, according to the rules of epistemic logic, everything that can be proven is also known, it follows that K (~ p). Which would prove p.

Variant 2

Strangely enough, a paradox also results from the following sentence q:

It is not known that this sentence is true.

It can be written as

~ K (q).

Again q can be both proven and disproved:

If we start this time with the assumption that ~ q, then it clearly follows that K (q), and - since everything that is known must also be true - that q.

As in variant 1, however, this is a proof of the inconsistency of ~ q and must (in the usual two-valued logic) apply as a proof of q. Now that we have proved q, q is known, symbolically: K (q). But this also proves the negation of q - which must count as a refutation of q.

Self-referentiality

Due to the self-referentiality of p and q, there is a close relationship to the liar paradox .

Individual evidence

  1. ^ D. Kaplan and R. Montague, A Paradox Regained, Notre Dame Journal of Formal Logic Volume 1, Number 3 (1960), 79-90.
  2. ^ R. Sorensen, "Epistemic Paradoxes", The Stanford Encyclopedia of Philosophy (Fall 2013 Edition), Edward N. Zalta (ed.)

literature