Proof-theoretic semantics
Encyclopedia
Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connective
s not in terms of interpretation
s, as in Tarskian approaches to semantics, but in the role that the proposition or logical connective plays within the system of inference.
Gerhard Gentzen
is the founder of proof-theoretic semantics, providing the formal basis for it in his account of cut-elimination for the sequent calculus
, and some provocative philosophical remarks about locating the meaning of logical connectives in their introduction rules within natural deduction
. It is not a great exaggeration that the history of proof-theoretic semantics since then has been devoted to exploring the consequences of these ideas.
Dag Prawitz
extended Gentzen's notion of analytic proof
to natural deduction
, and suggested that the value of a proof in natural deduction may be understood as its normal form. This idea lies at the basis of the Curry–Howard isomorphism, and of intuitionistic type theory
. His inversion principle lies at the heart of most modern accounts of proof-theoretic semantics.
Michael Dummett
introduced the very fundamental idea of logical harmony
, building on a suggestion of Nuel Belnap
. In brief, a language, which is understood to be associated with certain patterns of inference, has logical harmony if it is always possible to recover analytic proofs from arbitrary demonstrations, as can be shown for the sequent calculus by means of cut-elimination theorems and for natural deduction by means of normalisation theorems. A language that lacks logical harmony will suffer from the existence of incoherent forms of inference: it will likely be inconsistent.
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
s not in terms of interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
s, as in Tarskian approaches to semantics, but in the role that the proposition or logical connective plays within the system of inference.
Gerhard Gentzen
Gerhard Gentzen
Gerhard Karl Erich Gentzen was a German mathematician and logician. He had his major contributions in the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus...
is the founder of proof-theoretic semantics, providing the formal basis for it in his account of cut-elimination for the sequent calculus
Sequent calculus
In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi, systems LK and LJ, were introduced by Gerhard Gentzen in 1934 as a tool for studying natural deduction in...
, and some provocative philosophical remarks about locating the meaning of logical connectives in their introduction rules within natural deduction
Natural deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning...
. It is not a great exaggeration that the history of proof-theoretic semantics since then has been devoted to exploring the consequences of these ideas.
Dag Prawitz
Dag Prawitz
Dag Prawitz is a Swedish philosopher and logician. He is best known for his work on proof theory and the foundations of natural deduction....
extended Gentzen's notion of analytic proof
Analytic proof
In mathematical analysis, an analytical proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not make use of results from geometry...
to natural deduction
Natural deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning...
, and suggested that the value of a proof in natural deduction may be understood as its normal form. This idea lies at the basis of the Curry–Howard isomorphism, and of intuitionistic type theory
Intuitionistic type theory
Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism. Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher,...
. His inversion principle lies at the heart of most modern accounts of proof-theoretic semantics.
Michael Dummett
Michael Dummett
Sir Michael Anthony Eardley Dummett FBA D.Litt is a British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford...
introduced the very fundamental idea of logical harmony
Logical harmony
Logical harmony, a name coined by Sir Michael Dummett, is a supposed constraint on the rules of inference that can be used in a given logical system....
, building on a suggestion of Nuel Belnap
Nuel Belnap
Nuel D. Belnap, Jr. is an American logician and philosopher who has made many important contributions to the philosophy of logic, temporal logic, and structural proof theory. He has taught at the University of Pittsburgh since 1961; before that he was at Yale University. His best known work is...
. In brief, a language, which is understood to be associated with certain patterns of inference, has logical harmony if it is always possible to recover analytic proofs from arbitrary demonstrations, as can be shown for the sequent calculus by means of cut-elimination theorems and for natural deduction by means of normalisation theorems. A language that lacks logical harmony will suffer from the existence of incoherent forms of inference: it will likely be inconsistent.