What Is Logical Consequence? The Relation That Makes Arguments Work

Direct Answer

Logical consequence is the relation that holds between a set of premises and a conclusion when the conclusion follows of necessity: it is impossible for all the premises to be true and the conclusion false. The relation is usually written Γ ⊨ φ — "φ is a logical consequence of Γ" — and it is what makes a deductive argument valid. An argument is valid exactly when its conclusion is a logical consequence of its premises.

The notion has three defining features. It is necessary: the connection holds in all circumstances, not merely in fact. It is formal: it depends on the logical form of the sentences, not on their subject matter. And it is truth-preserving: it transmits truth from premises to conclusion. Modern logic makes this precise with the model-theoretic definition: φ is a logical consequence of Γ when every interpretation (model) that makes all members of Γ true also makes φ true.

Historical Context

Aristotle's syllogistic was the first systematic theory of logical consequence: the validity of a syllogism consists in the conclusion following necessarily from the premises, a fact he tested by considering counterexamples — arguments of the same form with true premises and a false conclusion. The medieval logicians refined this into the notion of consequentia and the rule that an inference is valid when the truth of the premises is incompatible with the falsity of the conclusion.

The modern era transformed the concept in two stages. Gottlob Frege's predicate logic (1879) supplied a precise language in which logical form could be displayed, making it possible to say exactly what a counterexample would look like. The decisive step came from Alfred Tarski, whose 1936 paper "On the Concept of Logical Consequence" defined consequence in terms of models: φ follows from Γ if every model of Γ is a model of φ. Tarski's definition, refined by the development of model theory in the 1950s and 1960s, remains the standard account, though philosophers of logic continue to debate whether the formal definition captures the intuitive, modal notion of necessity it was meant to explicate.

Philosophical Significance

Logical consequence is the concept that gives logic its authority. When we say a conclusion is a logical consequence of premises, we claim that no facts about the world could break the connection — the relation holds in virtue of logical form alone. This is why logic is topic-neutral and why deductive arguments are "truth-preserving" in the strongest sense.

The philosophical debates about logical consequence are deep. First, there is the modal question: is the model-theoretic account (truth in all models) the same as the intuitive notion (impossibility of true premises with false conclusion)? Tarski himself worried that formalization cannot capture all cases, since "model" must be interpreted relative to a fixed domain. Second, there is the question of logical constants: which words — "and," "or," "all," "some" — determine logical form, and on what grounds? Different answers yield different logics and hence different consequence relations. Third, consequence is paired with provability: Godel's completeness theorem showed that for first-order logic, the semantic relation Γ ⊨ φ coincides with the syntactic relation Γ ⊢ φ — what is true in all models is exactly what is provable — a result that unifies the meaning and the practice of logic.

Examples

A logical consequence in syllogistic form:

• All humans are mortal; Socrates is human; therefore Socrates is mortal.
• Model-theoretically: in every model where "human" picks out a subset of "mortal" and Socrates is in "human," Socrates is in "mortal."

An example that is not a consequence:

• All dogs are animals; all cats are animals; therefore all dogs are cats.
• Countermodel: a model with dogs and cats as disjoint subclasses of animals makes the premises true and the conclusion false.

Tautological consequence:

• P and Q entails P — in every row of the truth table where P∧Q is true, P is true. Here logical consequence coincides with the truth-table test of validity.

Related Concepts

Logical consequence is the semantic core of validity, the relation that defines a good deductive argument, and the subject matter of the philosophy of logic. It is studied in propositional logic and predicate logic, and it connects to truth — since consequence is defined in terms of preservation of truth — and to deductive reasoning. Its formal theory was created by Tarski on foundations laid by Frege.

Further Learning

Tarski's "On the Concept of Logical Consequence" (1936) is the classic source, reprinted in Logic, Semantics, Metamathematics. The Stanford Encyclopedia of Philosophy's entry "Logical Consequence" surveys the modern debate between model-theoretic, proof-theoretic, and modal accounts in authoritative detail. For the historical arc from Aristotle to Tarski, William and Martha Kneale's The Development of Logic and John Etchemendy's The Concept of Logical Consequence (a sharp philosophical critique of the model-theoretic account) are the essential companions.