Proof-theoretic semantics is an approach to the
semantics of logic that attempts to locate the meaning of
propositions and
logical connectives not in terms of
interpretation
Interpretation may refer to:
Culture
* Aesthetic interpretation, an explanation of the meaning of a work of art
* Allegorical interpretation, an approach that assumes a text should not be interpreted literally
* Dramatic Interpretation, an event ...
s, 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 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 rule 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 axiom ...
s within
natural deduction. 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 In mathematics, an analytic proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not predominantly make use of algebraic or geometrical methods. The term was first used by Bernard Bolzano, who first ...
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.
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 other ...
*
Truth-conditional semantics
References
Proof-Theoretic Semantics at the
Stanford Encyclopedia of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...
Logical Consequence, Deductive-Theoretic Conceptions at the
Internet Encyclopedia of Philosophy
The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia, dealing with philosophy, philosophical topics, and philosophers. The IEP combines open access publication with peer reviewed publication of original pape ...
.
*
Nissim Francez, "On a Distinction of Two Facets of Meaning and its Role in Proof-theoretic Semantics", ''
Logica Universalis
''Logica Universalis'' is a peer-reviewed
Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work (peers). It functions as a form of self-regulation by qualified members of a profess ...
'' 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