Paradoxes

A paradox is something against or beyond common opinion. In modern languages, it is generally known as an absurd claim with sound arguments sustaining it.

Epistemic paradoxes

The lottery (H. Kyburg, 1961).

A lottery with a thousand tickets is there. Each ticket is highly unlikely to win. Therefore no ticket will win. But we know that some ticket is going to win.

The preface (D.C. Makinson, 1965).

In the preface, an author is justified in believing the content of his book. He explains in the preface that, due to human nature, errors may be there. Then he is justified to believe both that in the book everything is true and that something is false.

The knowability (F. Fitch, 1963).

All truths are knowable. Some truths are not known. The latter claim is a truth, so it must be knowable. Therefore there is a truth that is known to be true and known not to be true.

The surprise test (D.J. O'Connor, 1948).

A surprise test is announced by the teacher sometimes next week. It will not be on Friday, the last day available, because at that point it would not be a surprise anymore. Once it is ascertained that it will not be on Friday, it will not be on Thursday either, because it is the last day available at this point. And so on. So there cannot be a surprise test next week.

The crows.

Crows are black, as proven by induction. But induction also proves that non-crows are non-black. So every experience of non-black non-crows strengthen the evidence that crows are black.

Semantic paradoxes

The liar.

Said a Cretian: "All the Cretians are liars, I tell you the truth". How could he say the truth?

The bald.

Which is the single hair fall that make the not bald into bald? None, therefore no one is bald.

The barber.

There is a barber who shaves everyone in town but those who do not shave themselves. He cannot shave.

Logic paradoxes

The largest ordinal (Burali-Forti, 1897 and Russell, 1903).

The set of all ordinals is well-ordered and so has an ordinal number (Ω) which is the largest ordinal. But every ordinal α has a larger ordinal α + 1. So Ω + 1 is larger than Ω, and this is contradictory.

Set and property (Russell, 1903).

The set S of all sets x such that x is not a member of x. Is S a member of itself? S is a member of itself iff S is not a member of itself.

Theological paradoxes

Evil.

God is not omnipotent, because he does not stop evil. But then He is not God

God is cruel, because he does not know about human evil. But then He is not all-good.

God is not omniscient, because he does not stop evil. But then He is not God.