An approach that assigns meaning to logical connectives and formulas by their roles in inference—rules of introduction and elimination—rather than by model-theoretic truth conditions.

Principle

Meaning is given through canonical inference patterns: connectives are characterized by the inferential moves that introduce and discharge them in proofs, making proof transformations central to semantics.

Demonstration

In natural deduction, the conjunction connective is given meaning by its introduction rule (from A and B infer A∧B) and by its elimination rules (from A∧B infer A or B); together these rules determine how the connective functions semantically in proof contexts.

Misapplication

Treating arbitrary proof systems as semantic without ensuring harmony or normalization can yield inconsistent meanings; taking syntactic inference rules as meaning without checking stability under proof transformations is a misuse.

Consequence

Connectives and formulas acquire meanings tied to inferential use; this yields a close connection between proof theory and semantics, supports normative accounts of assertion and inference, and can guide constructive interpretations of logical constants.

Reversal

Model-theoretic semantics inverts the focus by defining meaning through truth in structures and then deriving proof rules, rather than starting from inferential roles.

Boundary

Applies primarily to systems where proof rules are well-behaved (canonical, harmonious, and supporting normalization); it is limited in addressing purely extensional, model-based properties or logics lacking suitable proof systems.

Semantic Tension

Tension between inferential (use-based) and model-theoretic (truth-based) meanings: some phenomena are naturally captured by proofs (constructive content) while others demand extensional models (truth conditions across structures).

Synthesis

Proof-Theoretic Semantics treats logical meaning as arising from canonical proof operations: by explicating introduction and elimination behavior and their harmony one obtains a semantics that links normative inference patterns to the interpretation of logical constants.