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