What Is a Necessary Condition? The Distinction Most Arguments Get Wrong

A necessary condition is one without which the thing cannot occur, so its absence guarantees the absence of what it conditions, and its presence guarantees nothing. A sufficient condition is one whose presence guarantees the thing, and whose absence guarantees nothing, since something else might produce the same result. Being over eighteen is necessary but not sufficient for holding a driving licence. Being decapitated is sufficient but not necessary for dying. Some conditions are both, which is what a definition aims for, and a proposed definition fails as soon as a case satisfies it without being the thing, or is the thing without satisfying it. The logical relationship is simply implication read in two directions, so saying that A is necessary for B is the same as saying that B implies A, and that A is sufficient for B is the same as saying A implies B, which is why the two are so easily swapped by anyone not attending carefully.

The two errors have names and appear constantly in ordinary argument:

• •Affirming the consequent, which reasons from a sufficient condition backwards, so noting that illness would explain an absence and concluding from the absence that somebody is ill
• •Denying the antecedent, which reasons that because a sufficient condition is absent, the result must be absent, ignoring the other routes to the same result
• •Treating a necessary condition as a cause, which is why oxygen is not blamed for fires and why identifying something present in every case of a problem does not identify what produced it
• •Treating a sufficient condition as the only route, which underlies a great deal of single-cause explanation of complex outcomes
• •Policy arguments where a measure is shown to be necessary and is then treated as adequate, or shown to be inadequate alone and dismissed although it is required
• •Medical and legal reasoning, where a test result may be necessary for a diagnosis without establishing it, and where a legal element may be required without being decisive

The classical project of defining a concept by giving conditions that are jointly necessary and sufficient has a poor record, and the failures are instructive. Attempts to define knowledge as justified true belief were refuted by cases satisfying all three conditions without being knowledge. Attempts to define a game were the occasion for Wittgenstein's argument that many concepts have no such definition and are held together by overlapping resemblances rather than by a shared essence, in the way that members of a family resemble each other without any single feature running through all of them. Attempts to define art, life, causation and personhood have each generated large literatures without settled answers. The pattern suggests that many ordinary concepts are not the kind of thing that admits necessary and sufficient conditions, which does not make them useless, and which does mean that demanding a precise definition before discussing something is frequently a way of preventing the discussion.

The distinction earns its keep when applied deliberately. In diagnosing a failure, separating what was required from what was decisive prevents both the error of blaming a background condition and the error of ignoring one that could have been removed. In evaluating a proposal, asking whether it is necessary, sufficient, both or neither clarifies disputes quickly, since much disagreement consists of one party arguing a measure is not sufficient and the other arguing it is necessary, which are compatible positions. In causal reasoning about accidents, the standard analysis distinguishes conditions present in every instance from the factor that made the difference in this one, which is what a counterfactual test does. And in constructing arguments, checking whether each premise is doing necessary work or sufficient work reveals which ones could be dropped and which cannot, which is the most direct route to seeing what an argument actually requires.

A necessary condition cannot be missing and guarantees nothing by its presence, while a sufficient one guarantees the result and may not be required. Confusing them produces affirming the consequent and denying the antecedent, and treating a background requirement as a cause. Definitions aim for both at once and mostly fail, which is why many ordinary concepts hold together by family resemblance instead.