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
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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 in a proof system ( or )
Philosophical underpinnings
principle asserts that the meaning of a linguistic expression is determined by how it is used in a language
holds that the meaning of an expression is determined by the inferences it licenses and the circumstances under which it may be asserted
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 (∧ introduction, → introduction)
specify what can be inferred from a formula containing a logical connective (∧ elimination, → elimination)
Introduction and elimination rules for a connective should be in , 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
provides an alternative to model-theoretic validity
Allows for the study of logical systems that may not have a clear model-theoretic interpretation ()
Key Principles and Contributors
Prawitz's inversion principle
Proposed by , 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
, 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