Paradox

An argument that appears sound and leads somewhere unacceptable - which means something in it must go, and it is never obvious what.

Theme: Logic, language and interpretation
Tradition: Cross-tradition
Term fixed: c. 450 BC

In plain terms

A paradox is not a puzzle with a hidden trick. It is an argument where every step looks acceptable, the premises look true, and the conclusion cannot be.

That combination is informative rather than annoying. Since the conclusion is unacceptable, at least one thing you believed is wrong, and the paradox has narrowed down where. Finding out which is often where the real advance is.

The most productive ones in history have been the ones nobody could dismiss. Zeno's arguments about motion forced clarity about infinity that took two thousand years to reach. The liar sentence forced a distinction between a language and the language it talks about. And Russell's letter to Frege forced set theory to be rebuilt, by pointing out one sentence.

As a philosopher would put it

Paradoxes are usually sorted by what they demand be given up. Veridical paradoxes have conclusions that are surprising and true. Falsidical ones have a flaw to be located. Antinomies are the serious kind: the reasoning appears impeccable and the conclusion cannot be accepted, so a principle has to go.

The liar - this sentence is false - forced Tarski's hierarchy of object language and metalanguage, and remains a live driver of work on truth. Russell's paradox - the set of all sets that do not contain themselves - destroyed Frege's foundations of arithmetic and produced type theory and axiomatic set theory in response. The sorites drives the whole literature on vagueness. Newcomb's problem divides decision theory into two camps that have not converged in fifty years.

What the important ones share is that they are not solved by finding an error. They are resolved by adopting a new framework in which the argument cannot be stated - which is a different kind of answer, and the reason paradoxes have done more work in logic than almost anything else.

Where you meet it

In June 1902 Gottlob Frege received a letter from Bertrand Russell. The second volume of his Grundgesetze der Arithmetik was already at the printer.

Russell's letter was short. Consider the set of all sets that are not members of themselves. Is it a member of itself? If it is, it is not; if it is not, it is. Frege's Basic Law V, which allowed a set to be formed from any well-defined condition, generated the set and therefore the contradiction.

Frege added an appendix. It opens by saying that hardly anything more unwelcome can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished, and states plainly that Russell's letter had done exactly that. He attempted a repair, which did not work, and the logicist programme in his form did not recover. What replaced it - type theory, then axiomatic set theory - is the foundation mathematics has used ever since, and it exists because of one paragraph.

What is actually disputed

Whether paradoxes have solutions or only settlements.

One view is that each contains a specific error, and that the work is to find it. Another is that the important ones do not: the premises are individually plausible and jointly inconsistent, and what looks like a solution is always a choice about which plausible thing to abandon. On that view Tarski did not solve the liar but built a framework where it cannot be formed, which is not the same.

The dialetheist position takes the third exit and accepts that some paradoxical conclusions are simply true, contradictions and all, on the grounds that every other option has costs and this one at least takes the argument at its word. Whether that is courage or capitulation is the sort of thing philosophers do not settle, and the sorites, the liar and Newcomb's problem all remain open by the ordinary standard of open, which is that competent people still disagree.

Commonly confused with

irony or a witty remark Ordinary usage calls a surprising juxtaposition a paradox. The logical sense requires an actual argument.
a hard puzzle A puzzle has a solution you have not found. A genuine antinomy has premises you cannot all keep.

Defined against

Non-contradiction A paradox is a demonstration that a set of plausible commitments cannot all be held, which is why the principle is what it threatens

Philosophers

Zeno of Elea The first constructed paradoxes, and still among the best
Bertrand Russell The letter that broke Frege's system
Ludwig Wittgenstein Paradox as a symptom of a confusion in the picture
Immanuel Kant The antinomies, where both sides of a question are proved
Nagarjuna Argument by driving every alternative into unacceptability

Works

On Nature The setting of Zeno's arguments
The Foundations of Arithmetic The programme Russell's letter ended
Principia Mathematica The reconstruction that followed
Critique of Pure Reason The antinomies of pure reason
Mulamadhyamakakarika Paradox used as method rather than met as obstacle

Schools

Eleatic School Where paradox is first used as a weapon
Logical Atomism Where set-theoretic paradox forces reconstruction
Vienna Circle Where the liar drives the theory of truth
Cambridge Analysis Where paradoxes become the standard test of a formal system
Madhyamaka Buddhism Where the method is deliberate

Related concepts

Non-contradiction c. 330 BC · Excluded middle c. 330 BC · Validity c. 330 BC · Catuṣkoṭi c. 150 · Infinity c. 570 BC

Zeno's paradoxes are the first constructed to defend a thesis by making its denial absurd; the term itself, para-doxa, means contrary to expectation.