14.2 Proof-theoretic semantics
3 min read•august 7, 2024
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
Key Terms to Review (16)
Dag prawitz: Dag Prawitz is a significant figure in proof theory, known for developing the proof-theoretic semantics approach which emphasizes the role of proofs in understanding the meaning of logical expressions. His work laid the foundation for how the structure of proofs can represent logical consequence, showcasing that the meaning of statements is inherently linked to their derivation within a formal system. This perspective marks a shift from traditional truth-conditional semantics to a more proof-oriented approach in logic.
Dummett's justification procedures: Dummett's justification procedures are methods that focus on the justification of assertions and the way in which they can be validated through proof. These procedures emphasize a proof-theoretic perspective, arguing that the meaning of a statement is closely tied to the methods of its verification, rather than simply its truth conditions. This idea leads to an understanding of meaning as inherently linked to the processes of establishing claims through logical inference and proof.
Elimination Rules: Elimination rules are logical rules used in proof systems that allow for the removal of a specific logical operator from a formula, enabling the conclusion to be derived from premises containing that operator. These rules are crucial in both natural deduction and sequent calculus, as they help define how one can infer new statements from existing ones by breaking down complex expressions into simpler components. The concept also ties into proof-theoretic semantics, emphasizing how the meaning of logical constructs can be understood through their usage in proofs.
Gerhard Gentzen: Gerhard Gentzen was a German mathematician and logician known for his groundbreaking contributions to proof theory, particularly in developing natural deduction and sequent calculus. His work laid the foundation for many modern concepts in logic, impacting various aspects of mathematical logic, including soundness, completeness, and proof systems.
Harmony: In the context of proof-theoretic semantics, harmony refers to the idea that the rules for deriving judgments should align with the intuitive meaning of those judgments, creating a balance between proof and meaning. This concept ensures that the way we construct proofs is coherent and resonates with the semantics of the logical system, allowing for a clear understanding of how statements are validated and what they truly signify.
Inferentialism: Inferentialism is a philosophical theory that emphasizes the role of inference in understanding meaning and knowledge, positing that the significance of a statement arises from its inferential connections to other statements rather than from its correspondence to an external reality. This approach highlights how assertions gain meaning through the inferences they allow, focusing on the practical use of language in reasoning processes.
Introduction Rules: Introduction rules are specific guidelines in proof systems that dictate how to introduce logical connectives or quantifiers in a formal proof. These rules are crucial because they define how certain propositions can be derived based on the logical structure of the proof, whether in natural deduction or sequent calculus. They establish the conditions under which new formulas can be added to the proof, serving as a foundation for constructing valid arguments.
Intuitionistic logic: Intuitionistic logic is a form of logic that emphasizes the constructive nature of mathematical proofs, where a statement is only considered true if there is a method to construct an example demonstrating its truth. This approach leads to different interpretations of logical connectives and quantifiers compared to classical logic, making it essential for understanding various proof systems, the foundations of logic, and connections between different logical frameworks.
Meaning-as-use: Meaning-as-use is a philosophical perspective that emphasizes the role of language in practical contexts, suggesting that the meaning of a term is derived from how it is used in communication rather than from abstract definitions. This concept focuses on the pragmatic aspects of language, highlighting how context and usage shape understanding and interpretation.
Michael Dummett: Michael Dummett was a prominent British philosopher known for his influential work in logic, philosophy of language, and proof theory. His ideas helped shape the field of proof-theoretic semantics, which emphasizes the role of proof in determining meaning and truth in language, connecting linguistic practices to logical deduction.
Natural Deduction: Natural deduction is a proof system used in logic that allows one to derive conclusions from premises using a set of inference rules in a structured and intuitive way. It emphasizes the natural reasoning process, enabling proofs to be constructed through the application of these rules without the need for additional axioms or complex structures, making it particularly useful in various fields of mathematics and philosophy.
Prawitz's Inversion Principle: Prawitz's Inversion Principle is a concept in proof theory that asserts a duality between proofs and their corresponding derivations, emphasizing that the structure of a proof can be understood in terms of the rules applied to derive it. This principle connects the syntactic representation of proofs to their semantic interpretations, highlighting how the way a conclusion is derived can influence its meaning. The principle serves as a foundation for understanding proof-theoretic semantics, allowing for a more comprehensive exploration of the relationship between proofs and logical consequence.
Proof-theoretic semantics: Proof-theoretic semantics is an approach in the philosophy of logic that focuses on the meanings of logical expressions based on their proof conditions rather than their truth conditions. This perspective emphasizes the role of formal proofs and how they establish the meaning of statements through the rules and processes used to derive them. It connects closely to various branches of logic, particularly intuitionistic logic, which relies heavily on constructive proofs, and it provides a framework for understanding the nature of logical consequence.
Proof-theoretic validity: Proof-theoretic validity refers to the property of a statement or formula being provable within a given formal system, indicating that the statement holds true based on the rules and axioms of that system. This concept is crucial in proof-theoretic semantics, where the focus is on understanding meaning through the structure of proofs rather than just truth conditions. By examining the relationship between proofs and the validity of statements, proof-theoretic validity helps illuminate how logical systems can be interpreted and understood.
Sequent Calculus: Sequent calculus is a formal proof system designed for deriving sequents, which express the validity of implications between formulas in logic. It serves as a key framework for studying proof theory, enabling structured reasoning about logical statements and their relationships, particularly in first-order logic and intuitionistic logic.
Verificationism: Verificationism is a philosophical doctrine asserting that a statement or proposition is meaningful only if it can be definitively verified or is analytically true. This view emphasizes the connection between language and empirical verification, focusing on the idea that statements must be testable through observation or experience to hold significance.