Proof-theoretic semantics is an approach to the

semantics of logic In logic, the semantics of logic or formal semantics is the study of the semantics, or interpretations, of formal and (idealizations of) natural languages usually trying to capture the pre-theoretic notion of entailment. Overview The truth cond ...

that attempts to locate the meaning of

proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

s and

logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

s not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the proposition or logical connective plays within the system of inference.

Overview

Gerhard Gentzen Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died ...

is the founder of proof-theoretic semantics, providing the formal basis for it in his account of

cut-elimination The cut-elimination theorem (or Gentzen's ''Hauptsatz'') is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark 1934 paper "Investigations in Logical Deduction" for ...

for the

sequent calculus In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology i ...

, and some provocative philosophical remarks about locating the meaning of logical connectives in their introduction rules within

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. This contrasts with Hilbert-style systems, which instead use ax ...

. The history of proof-theoretic semantics since then has been devoted to exploring the consequences of these ideas.

Dag Prawitz Dag Prawitz (born 1936, Stockholm) is a Swedish philosopher and logician. He is best known for his work on proof theory and the foundations of natural deduction. Prawitz is a member of the Norwegian Academy of Science and Letters, of the Roya ...

extended Gentzen's notion of analytic proof to

, 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 (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician an ...

. His inversion principle lies at the heart of most modern accounts of proof-theoretic semantics.

Michael Dummett Sir Michael Anthony Eardley Dummett (27 June 1925 – 27 December 2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He w ...

introduced the very fundamental idea of logical harmony, building on a suggestion of

Nuel Belnap Nuel Dinsmore Belnap Jr. (; born 1930) is an American logician and philosopher who has made contributions to the philosophy of logic, temporal logic, and structural proof theory. He taught at the University of Pittsburgh from 1963 until his reti ...

. 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.

References

Proof-Theoretic Semantics
at the Stanford Encyclopedia of Philosophy
Logical Consequence, Deductive-Theoretic Conceptions
at the Internet Encyclopedia of Philosophy. * Nissim Francez, "On a Distinction of Two Facets of Meaning and its Role in Proof-theoretic Semantics", '' Logica Universalis'' 9, 2015. * Thomas Piecha, Peter Schroeder-Heister (eds)
"Advances in Proof-Theoretic Semantics"
Trends in Logic 43, Springer, 2016.

External links

Arché Bibliography on Proof-Theoretic Semantics.
{{logic-stub Mathematical logic Philosophical logic Proof theory Semantics

Overview

See also

References

External links