Validity

The property of an argument whose conclusion cannot be false if its premises are true - a guarantee about form, and about nothing else.

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

In plain terms

An argument is valid when the conclusion follows. Not when it is true - when it follows.

Those come apart in both directions, and keeping them apart is the first thing anyone learns in logic and the thing most often forgotten afterwards. A valid argument from false premises gives you a false conclusion with a perfect guarantee attached. An invalid argument can have true premises and a true conclusion and still be worthless, because the truth of the conclusion had nothing to do with the premises.

So validity is a property of the shape of an argument. It tells you that if you put truth in, truth comes out. It is completely silent about whether you put truth in, and that is where almost all real reasoning goes wrong.

As a philosopher would put it

The modern definition is model-theoretic: an argument is valid when there is no interpretation of its non-logical vocabulary on which the premises are true and the conclusion false. Aristotle's version is syntactic and restricted to the syllogistic forms; the quantificational logic that replaced it is powerful enough for mathematics.

Validity is a property of forms, not of contents, which is what makes logic general and also what makes it easy to misapply. Soundness adds true premises, and soundness is what anyone actually wants.

The standing puzzles are structural. Classical validity licenses explosion: from a contradiction everything follows, which paraconsistent logics reject. It also makes any argument with a necessarily true conclusion valid regardless of premises, which sits badly with the idea that the premises should be relevant - the objection relevance logics were built to answer. And Carroll's tortoise shows that validity cannot be justified by adding a premise saying the argument is valid, since that premise would need another.

Where you meet it

On 23 September 1999 the Mars Climate Orbiter fired its engine to enter orbit and was never heard from again. It had come in about a hundred and seventy kilometres lower than intended and either burned up or passed on into solar orbit.

The navigation software at the Jet Propulsion Laboratory expected thruster impulse data in newton-seconds. The ground software supplied by the contractor produced it in pound-force seconds. The two differ by a factor of about 4.45.

Every calculation performed on that number was correct. The trajectory model was valid, the software worked as specified, and the conclusions followed from the inputs with complete reliability. What the mission review board found was not a reasoning error but an input error, propagated flawlessly through a system that had no way to notice - which is validity doing exactly what it promises and precisely nothing more.

What is actually disputed

Whether there is one correct logic.

Classical validity is the default and is not universally accepted. Intuitionists reject the excluded middle and therefore reject proofs by contradiction, which changes what follows from what. Relevance logicians reject the classical result that anything follows from a contradiction. Paraconsistent logicians go further and allow some contradictions without collapse. Each is a different account of what validity is, not a different opinion about the same one.

Logical pluralists hold that several are correct for different purposes; monists hold that one of them describes consequence and the others are useful formalisms. And underneath sits Carroll's problem, which suggests that whatever validity is, it cannot be something we accept on the strength of an argument, since accepting an argument is the thing being explained.

Commonly confused with

Truth c. 330 BC An argument is valid; a statement is true. A valid argument can have a false conclusion, and most bad reasoning in public life is perfectly valid.
validity in measurement In statistics, a valid instrument measures what it claims to. Same word, unrelated concept.

Defined against

Truth The pair is the first distinction in logic and the one most often collapsed: an argument is valid or not, a statement is true or not, and neither implies the other

Philosophers

Aristotle The first systematic account of forms that preserve truth
William of Ockham The medieval logic of consequence
Gottfried Wilhelm Leibniz The project of reducing reasoning to calculation
Bertrand Russell The logic that could carry mathematics
Rudolf Carnap Validity as a matter of the syntax of a chosen language

Works

Metaphysics The syllogistic
Summa Logicae Medieval consequence in its developed form
Principia Mathematica Formal validity applied at scale
The Logical Syntax of Language Validity relative to a language framework
Tractatus Logico-Philosophicus Logical form as what shows itself

Schools

Peripatetic school (Aristotelianism) The original account
Ockhamism Medieval consequence
Megarian School The ancient logic of conditionals, and the disputes about them
Logical Atomism Where logical form becomes central
Vienna Circle Where logic is the only source of necessity

Related concepts

Deduction c. 330 BC · Non-contradiction c. 330 BC · Inference c. 330 BC · Necessity c. 330 BC · Truth c. 330 BC

Aristotle's syllogistic is the first systematic account of which argument forms preserve truth; the modern model-theoretic definition, in terms of interpretations, is Tarski's from 1936.