Proof-theoretic semantics

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 interpretations, 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 Consequence, Deductive-Theoretic Conceptions] , at the Internet Encyclopedia of Philosophy.

ee also

* Inferential role semantics
* Truth-conditional semantics

External links

* [ Arché Bibliography on Proof-Theoretic Semantics.]

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Proof theory — is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively defined data structures such as plain lists, boxed… …   Wikipedia

  • Semantics — is the study of meaning in communication. The word derives from Greek σημαντικός ( semantikos ), significant , [cite web|url= bin/ptext?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3D%2393797|title=Semantikos, Henry… …   Wikipedia

  • Formal semantics — See also Formal semantics of programming languages. Formal semantics is the study of the semantics, or interpretations, of formal languages. A formal language can be defined apart from any interpretation of it. This is done by designating a set… …   Wikipedia

  • Truth-value semantics — In formal semantics, truth value semantics is an alternative to Tarskian semantics. It has been primarily championed by Ruth Barcan Marcus, H. Leblanc, and M. Dunn and N. Belnap. It is also called the substitution interpretation (of the… …   Wikipedia

  • Inferential role semantics — (also: conceptual role semantics, functional role semantics, procedural semantics) is an approach to the theory of meaning that identifies the meaning of an expression with its relationship to other expressions, typically its inferential… …   Wikipedia

  • Truth-conditional semantics — is an approach to semantics of natural language that sees the meaning of a sentence being the same as, or reducible to, the truth conditions of that sentence. This approach to semantics is principally associated with Donald Davidson, and carries… …   Wikipedia

  • Analytic proof — In structural proof theory, an analytical proof is a proof whose structure is simple in a special way. The term does not admit an uncontroversial definition, but for several proof calculi there is an accepted notion of analytic proof. For example …   Wikipedia

  • Denotational semantics — In computer science, denotational semantics (initially known as mathematical semantics or Scott–Strachey semantics) is an approach to formalizing the meanings of programming languages by constructing mathematical objects (called denotations)… …   Wikipedia

  • Kripke semantics — (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal… …   Wikipedia

  • Meaning (philosophy of language) — The nature of meaning, its definition, elements, and types, was discussed by philosophers Aristotle, Augustine, and Aquinas. According to them meaning is a relationship between two sorts of things: signs and the kinds of things they mean (intend …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.