Paradox

Overview

A paradox refers to a statement that, despite valid reasoning from true premises, leads to a self-contradictory or logically unacceptable conclusion.

Antinomy

An antinomy is a paradox which reaches a self-contradictory result with valid reasoning.

Russell's Paradox

Let $R=\{x\mid x\notin x\}$. Then $R\in R⟺R\notin R$.

Grelling-Nelson Paradox

An autological adjective is one that describes themselves, e.g. short and multisyllabic. A heterological adjective is one that does not describe itself, e.g. hyphenated.

The Grelling-Nelson paradox, also known as the paradox of non-self-instantiation, is the contradiction that forms when considering whether the word "heterological" is either heterological or autological.

Veridical

A veridical paradox is a statement that produces a counterintuitive, but demonstrably true, result. It is not an actual paradox in the literal sense of the word.

Berkson's Paradox

Berkson's paradox refers to the phenomenon in which two independent events $A$ and $B$ become negatively correlated when conditioning on $C=A\cup B$. That is, assuming $0<P(C)<1$,

It usually occurs because of a sampling bias in the study design.

Bertrand's Box Paradox

Bertrand's box paradox assumes existence of three boxes:

1. a box containing two gold coins,
2. a box containing two silver coins, and
3. a box containing one gold coin and one silver coin.

If a coin withdrawn randomly from one of the three boxes is gold, what is the probability the remaining coin is also gold? Counterintuitively, the answer is $2/3$.

Boy or Girl Paradox

The boy or girl paradox involves the answers to the following two questions:

1. Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
2. Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

The answers are $1/2$ and $1/3$ respectively. If $p$ denotes the probability of a characteristic that can be used to identify a specific child, we see that we can interpolate between the two answers via formula

Hilbert's Paradox of the Grand Hotel

Consider a hypothetical hotel with rooms numbered $1$, $2$, $3$, and so on with no upper limit. That is, there is a countably infinite number of rooms in this hotel. Furthermore, it's assumed every room is occupied.

Hilbert's paradox of the grand hotel shows that any finite or countably infinite number of additional guests can still be accommodated for.

Simpson's Paradox

Simpson's paradox refers to the phenomenon in which a trend appears in several groups of data but disappears or reverses when the data is aggregated.

For events $A$, $B$, and $C$, an instance of Simpson's paradox occurs if

but $P(A|B)\ge P(A|B^C)$. In this situation, $C$ is called the confounding variable.