Non-contradiction

The principle that nothing can both be and not be, in the same respect and at the same time.

Theme: Logic, language and interpretation
Tradition: Greek
Term fixed: c. 330 BC

In plain terms

Nothing can be the case and not be the case at once, in the same respect. It looks too obvious to state.

It is worth stating because of what depends on it. In classical logic, a single contradiction lets you derive absolutely anything - if P and not-P are both true, then any claim whatever follows. That is not a curiosity; it means an inconsistent system is not partly reliable, it is entirely useless, because it endorses every statement including the negation of every statement.

Which is why an inconsistency in a body of rules, a database or a legal code is not a small blemish to be tidied up when convenient. It is a hole through which anything can be justified, and the practical world discovers this regularly.

As a philosopher would put it

Aristotle states it as the firmest principle and denies that it admits of demonstration, since any proof would have to assume it. His method is elenctic: he argues that a denier who says anything determinate has already relied on it, since to mean one thing rather than another is to exclude.

In classical logic the principle is tied to explosion, ex contradictione quodlibet: from a contradiction, every proposition follows. This is what makes consistency non-negotiable in formal systems, and it is exactly what paraconsistent logics reject - they retain contradiction while blocking the derivation, so an inconsistent theory can still be informative.

Dialetheism goes further: Priest argues that some contradictions are true, the liar sentence being the standard candidate, and that the cost of denying this is worse than the cost of accepting it. Hegel's dialectic is often cited here and is a different claim - that reality develops through contradictions - rather than a denial of the logical principle, and running the two together is a common error.

Where you meet it

A hospital medication record holds an allergy list. If one entry records a penicillin allergy and another, entered elsewhere in the same system, records no known allergies, the record now asserts both.

What makes this dangerous is not the missing information but the presence of both. A clinician querying the record can be shown either, depending on which module they open and which entry sorts first. Automated decision support, which fires alerts by checking a prescription against the allergy list, may find the negative entry and stay silent. Reconciliation of conflicting entries across systems is a recognised category of patient-safety work for this reason.

The general lesson generalises well beyond medicine. Any body of rules that contains a contradiction will, on classical inference, license the conclusion the reader wants - and the failure is silent, because nothing in the system looks broken. It answers every question. It just answers them both ways.

What is actually disputed

Whether any contradictions are true.

The orthodox position is that none are, and that apparent counterexamples show a fault in the description rather than in the world. Dialetheists disagree, pointing to the liar paradox, certain legal situations where a person is both entitled and not entitled under conflicting statutes, and the boundaries of vague predicates. Their claim is not that reasoning collapses but that with paraconsistent logic it does not have to.

Aristotle's defence is also disputed as a defence. Showing that a denier presupposes the principle in order to say anything meaningful demonstrates that it cannot be coherently denied in speech; whether that establishes that it is true of the world, or only that our language is built to exclude, is a question his argument arguably cannot reach - and it is the reason he was careful to call it a defence rather than a proof.

Commonly confused with

the excluded middle Non-contradiction says not both. The excluded middle says at least one. They are different principles and intuitionists reject the second while keeping the first.
inconsistency in the loose sense Behaving inconsistently is a matter of not matching a pattern. Logical contradiction is a specific relation between a proposition and its negation.
Hegelian contradiction Hegel's contradictions are tensions that drive development, and he is not proposing that P and not-P are both true in the logician's sense.

Defined against

Paradox A paradox is a case where the principle appears to fail, which is why the two entries have to be read together

Philosophers

Aristotle The statement, and the argument that it cannot be proved
Heraclitus The apparent denier Aristotle has in view
William of Ockham The medieval treatment of consequence and inconsistency
Georg Wilhelm Friedrich Hegel Contradiction as the motor of development, which is a different claim
Nagarjuna The four-cornered analysis, which refuses both alternatives without asserting both

Works

Metaphysics Book Gamma, where the principle is defended
Fragments The unity of opposites, and what it does and does not deny
Summa Logicae The medieval logic of consequence
The Phenomenology of Spirit Dialectical contradiction as productive
Mulamadhyamakakarika The tetralemma, and its refusal of all four corners

Schools

Peripatetic school (Aristotelianism) Where the principle is stated and defended
Heraclitean School The apparent opposition, and how much of it is real
Megarian School Where the logic of contradiction and conditionals is worked out
Hegelianism Where contradiction becomes developmental
Madhyamaka Buddhism Where a four-valued structure replaces the two-valued one

Related concepts

Excluded middle c. 330 BC · Validity c. 330 BC · Catuṣkoṭi c. 150 · Paradox c. 450 BC · Coherence 1807

Aristotle calls it the firmest of all principles and argues that it cannot be demonstrated, only defended against a denier - which is a different and stranger kind of argument.