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 meaning and interpretation of formal languages, formal systems, and (idealizations of) natural languages. This field seeks to provide precise mathematical models tha ...
that attempts to locate the meaning of
proposition
A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...
s and
logical connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the ...
s not in terms of
interpretations, as in
Tarskian approaches to semantics, but in the role that the proposition or logical connective plays within a
system of inference.
Overview
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
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 tautolog ...
, 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 ...
. 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, and for his contributions to proof-theoretic semantics.
Prawitz is a member of the ...
extended Gentzen's notion of
analytic proof to
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 ...
, 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
Language is a structured system of communication that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed language, signed forms, and may also be conveyed through writing syste ...
, 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.
See also
*
Inferential role semantics
Inferential role semantics (also conceptual role semantics, functional role semantics, procedural semantics, semantic inferentialism) is an approach to the theory of meaning that identifies the meaning of an expression with its relationship to othe ...
*
Truth-conditional semantics
References
Proof-Theoretic Semantics at the
Stanford Encyclopedia of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...
Logical Consequence, Deductive-Theoretic Conceptions at the
Internet Encyclopedia of Philosophy
The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia with around 900 articles about philosophy, philosophers, and related topics. The IEP publishes only peer review, peer-reviewed and blind-refereed original p ...
.
*
Nissim Francez, "On a Distinction of Two Facets of Meaning and its Role in Proof-theoretic Semantics", ''
Logica Universalis'' 9, 2015. {{doi, 10.1007/s11787-015-0118-8
* 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.Proof-Theoretic Semantics Network
Mathematical logic
Philosophical logic
Proof theory
Semantics