Deduction

Reasoning in which the conclusion cannot be false if the premises are true.

Theme: Knowledge, truth and reason
Tradition: Greek
Term fixed: c. 330 BC

In plain terms

A deduction has a guarantee attached. If the premises are true, the conclusion is true - not probably, not usually, but necessarily, because the conclusion was already contained in what you started with.

That guarantee is what makes deduction the only kind of reasoning that cannot go wrong, and also what makes it feel like it adds nothing. Everything the conclusion says was there in the premises before you started. All the work is in noticing.

Except that noticing is not trivial. Mathematics is entirely deductive and is not obvious. A proof can take three hundred years to find, and once found it tells you something you did not know - which is a puzzle about how a form of reasoning that adds no content can be so difficult and so informative.

As a philosopher would put it

Validity is a property of form: an argument is valid when no interpretation makes the premises true and the conclusion false. Soundness adds true premises. Aristotle's syllogistic covers a fragment of what can be expressed; Frege's quantificational logic handles relations and multiple generality, which is what mathematics needs and what the syllogism could not reach.

The limits are known precisely, which is unusual. Gödel's completeness theorem shows first-order logic proves everything valid in it; his incompleteness theorems show that any consistent formal system strong enough for arithmetic contains truths it cannot prove. Deduction is therefore complete relative to a logic and incomplete relative to mathematics.

The paradox of analysis is the standing philosophical puzzle: if a valid argument's conclusion is contained in its premises, deduction is uninformative, and yet it plainly informs. Answers appeal to the difference between containment and recognition, or to bounded rationality. And the question of what justifies deduction is worse - Carroll's tortoise shows that no additional premise can compel the step from premises to conclusion, since accepting it would itself require another step.

Where you meet it

The four colour theorem states that any map can be coloured with four colours so that no two adjacent regions share one. It was conjectured in 1852 and resisted proof for a hundred and twenty-four years.

In 1976 Kenneth Appel and Wolfgang Haken proved it by reducing the problem to 1,936 configurations and checking each by computer - roughly twelve hundred hours of machine time. No human has ever verified the case analysis, and no human could.

The mathematical community's reaction was not doubt about the result but discomfort about the kind of thing it was. A proof is traditionally something a mathematician can survey and be convinced by; this one required trusting a program and the hardware it ran on. A formally verified version was completed in 2005 using the Coq proof assistant, which addressed the correctness worry by adding another layer of machine checking. The philosophical question it raised - whether a deduction nobody can follow is a proof - has not gone away.

What is actually disputed

Whether deduction is informative, and what justifies it.

The paradox of analysis has no agreed solution. Saying that the conclusion is contained in the premises but not obviously so relocates the puzzle into the word obviously, and any account of what makes recognition hard has to explain why the difficulty is not merely psychological if the informativeness is not.

Carroll's regress shows that deduction cannot be justified by adding premises. Some conclude that inference rules are not premises but practices, which is Wittgenstein's direction; others that logic is constitutive of thought and cannot be justified from outside because there is no outside. And a minority reject classical logic itself - intuitionists deny the excluded middle, paraconsistent logicians allow contradictions without explosion - which makes even the guarantee a matter of which logic you are in.

Commonly confused with

reaching a conclusion by reasoning Most reasoning is not deductive. The detective who deduces from clues is almost always inferring to the best explanation.
proof in the everyday sense Everyday proof means enough evidence. Deductive proof means the conclusion cannot be otherwise, which is a far stronger and much rarer thing.

Defined against

Induction The two are defined against each other by whether the conclusion is guaranteed, and every account of reasoning has to say how they relate

Philosophers

Aristotle The syllogistic, and the first formal account of validity
William of Ockham The medieval development of the logic of terms
Gottfried Wilhelm Leibniz The dream of a calculus that would settle disputes by computation
Bertrand Russell Logic as the foundation of mathematics
Rudolf Carnap Logical syntax as the framework for all rigorous discourse

Works

Metaphysics The classical logic
Summa Logicae The medieval system at its most developed
Principia Mathematica Mathematics derived from logic, at enormous length
The Foundations of Arithmetic Frege's programme, stated philosophically
The Logical Syntax of Language Deduction as the whole of rigour
Tractatus Logico-Philosophicus Logic as showing the structure of what can be said

Schools

Peripatetic school (Aristotelianism) Where the subject begins
Ockhamism The medieval logic of terms
Logical Atomism Where logic becomes the skeleton of the world
Vienna Circle Where it becomes the only source of necessity
Cambridge Analysis Where the formal tools are turned on philosophy itself

Related concepts

Inference c. 330 BC · Necessity c. 330 BC · Reason c. 500 BC · Induction c. 330 BC · Truth c. 330 BC

Aristotle's syllogistic was the whole of logic for two thousand years; Frege's Begriffsschrift in 1879 replaced it with something powerful enough for mathematics, which is when the subject changed shape.