14.2 Proof-theoretic semantics

Proof-theoretic semantics assigns meaning to logical connectives based on their roles in formal proofs. It's a fresh take on meaning, focusing on how we use expressions in proofs rather than just their truth conditions.

This approach is built on some big ideas in philosophy. It's all about how we use language, make inferences, and verify statements. These concepts form the backbone of proof-theoretic semantics, tying meaning to how we reason and prove things.

Foundations of Proof-Theoretic Semantics

Meaning derived from proofs

Top images from around the web for Meaning derived from proofs

• 

  Propositional Logic Proof using I.P. or C.P or rules of inference - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  logic - Natural deduction proof of $\forall x (\exists y (P(x) \vee Q(y))) \vdash \exists y ... View original

  Is this image relevant?

• 

  logic - Given P ∨ ¬ P prove (P → Q) → ((¬ P → Q) → Q) by natural deduction - Philosophy Stack ... View original

  Is this image relevant?

• 

  Propositional Logic Proof using I.P. or C.P or rules of inference - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  logic - Natural deduction proof of $\forall x (\exists y (P(x) \vee Q(y))) \vdash \exists y ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Meaning derived from proofs

• 

  Propositional Logic Proof using I.P. or C.P or rules of inference - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  logic - Natural deduction proof of $\forall x (\exists y (P(x) \vee Q(y))) \vdash \exists y ... View original

  Is this image relevant?

• 

  logic - Given P ∨ ¬ P prove (P → Q) → ((¬ P → Q) → Q) by natural deduction - Philosophy Stack ... View original

  Is this image relevant?

• 

  Propositional Logic Proof using I.P. or C.P or rules of inference - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  logic - Natural deduction proof of $\forall x (\exists y (P(x) \vee Q(y))) \vdash \exists y ... View original

  Is this image relevant?

1 of 3

• Proof-theoretic semantics assigns meaning to logical connectives based on the rules that introduce or eliminate them in formal proofs
• Contrasts with truth-conditional semantics which defines meaning in terms of truth conditions
• Focuses on the role expressions play in the practice of proving statements
• Meaning of a logical constant is given by its introduction rules in a proof system (natural deduction or sequent calculus)

Philosophical underpinnings

• Meaning-as-use principle asserts that the meaning of a linguistic expression is determined by how it is used in a language
• Inferentialism holds that the meaning of an expression is determined by the inferences it licenses and the circumstances under which it may be asserted
• Verificationism maintains that a statement is meaningful only if there is a method to verify its truth
• These philosophical views provide the foundation for proof-theoretic semantics by grounding meaning in the inferential rules and proof conditions associated with expressions

Inference Rules and Harmony

Introduction and elimination rules

• Introduction rules specify the conditions under which a logical connective can be introduced into a proof ($\wedge$ introduction, $\rightarrow$ introduction)
• Elimination rules specify what can be inferred from a formula containing a logical connective ($\wedge$ elimination, $\rightarrow$ elimination)
• Introduction and elimination rules for a connective should be in harmony, justifying each other
• Harmony ensures that the rules are not too strong or too weak, and that they coherently specify the meaning of the connective

Proof-theoretic validity

• A formula is considered valid if there exists a proof of it using the given inference rules
• Validity is defined in terms of the existence of a proof, rather than truth conditions
• Proof-theoretic validity provides an alternative to model-theoretic validity
• Allows for the study of logical systems that may not have a clear model-theoretic interpretation (intuitionistic logic)

Key Principles and Contributors

Prawitz's inversion principle

• Proposed by Dag Prawitz, a key figure in the development of proof-theoretic semantics
• States that the elimination rules for a logical connective should be inverses of its introduction rules
• Ensures that the rules are in harmony and that the meaning of the connective is fully determined by its introduction rules
• Provides a criterion for assessing the coherence and justification of inference rules in a proof system

Dummett's justification procedures

• Michael Dummett, another prominent advocate of proof-theoretic semantics, emphasized the role of justification procedures
• A justification procedure is a method for proving or refuting a statement based on the inference rules of the language
• The meaning of a statement is given by the conditions under which it can be proven or refuted using these procedures
• Dummett argued that this approach provides a more constructive and intuitively appealing account of meaning compared to truth-conditional semantics