The Nature of Logical Consequence

What is it for a conclusion genuinely to follow from premises? This topic examines competing analyses of the consequence relation that sits at the heart of logic.

Definition

Logical consequence is the relation holding between a set of premises and a conclusion such that, necessarily and in virtue of form, the conclusion is true whenever all the premises are true.

Scope

This topic addresses the conceptual analysis of logical consequence: the model-theoretic account on which validity is preservation of truth across all interpretations; the proof-theoretic account on which it is derivability; and the philosophical question of whether either captures the apparently necessary, formal, and a priori character of 'following from'. It also covers the interplay of soundness and completeness theorems, which connect the two analyses for classical first-order logic.

Core questions

• Is consequence best understood as truth-preservation across models or as derivability in a proof system?
• What grounds the necessity and formality that intuitively belong to logical consequence?
• Do soundness and completeness results show that the two analyses pick out the same relation?
• Can a purely extensional definition capture an essentially modal notion?

Key concepts

• truth-preservation
• necessity and formality
• model-theoretic consequence
• proof-theoretic consequence
• soundness and completeness
• logical form

Key theories

Tarski's model-theoretic analysis

A sentence is a logical consequence of a set of sentences iff every model of the set is a model of the sentence; consequence is reduced to truth in all reinterpretations of the non-logical constants.

The modality objection

Etchemendy contends that the model-theoretic account, by quantifying over actual interpretations, cannot capture the genuine necessity of consequence and yields the right verdicts only contingently, depending on how rich the domain happens to be.

History

Tarski's 1936 paper introduced the model-theoretic definition that became orthodox after the mid-century development of model theory. Etchemendy's 1990 critique prompted a sustained reassessment of whether the formal definition tracks the intuitive notion, and subsequent work (e.g., Shapiro) examined how modality and the choice of logical constants enter the analysis.

Debates

Extensional adequacy vs. conceptual analysis

Whether the model-theoretic definition merely happens to deliver the correct extension of valid arguments, or genuinely analyzes what consequence is, given that it appears to omit the relation's modal force.

Key figures

• Alfred Tarski
• John Etchemendy
• Stewart Shapiro
• Gottlob Frege

Seminal works

• tarski1936
• etchemendy1990

Frequently asked questions

Are the model-theoretic and proof-theoretic accounts equivalent?

For classical first-order logic, the soundness and completeness theorems show that the two accounts coincide extensionally: a conclusion is derivable from premises exactly when it is true in every model of them. Whether they are conceptually the same relation, however, remains philosophically contested.