##  [Proof-Theoretic Semantics](/proof-theoretic-semantics-0) 

 Definition

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

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

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

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

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

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

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

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

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.