Proof-theoretic semantics is an approach to the semantics of logic Formal semantics is the study of the semantics, or interpretations, of formal and also natural languages. A formal language can be defined apart from any interpretation of it. This is done by designating a set of symbols and a set of formation rules (also called a formal grammar) which determine which strings of symbols are well-formed formulas that attempts to locate the meaning of propositions and logical connectives In logic, a logical connective is a symbol or word used to connect two or more sentences (of either a formal or a natural language) 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 not in terms of interpretations, as in Tarskian Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death approaches to semantics, but in the role that the proposition or logical connective plays within the system of inference.
Gerhard Gentzen Gerhard Karl Erich Gentzen was a German mathematician and logician. He had his major contributions in the foundation of mathematics, proof theory, especially on natural deduction and sequent calculus. He died in 1945 after the Second World War, because he was deprived of food after being arrested in Prague is the founder of proof-theoretic semantics, providing the formal basis for it in his account of cut-elimination for the sequent calculus In proof theory and mathematical logic, sequent calculus is a widely known 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 first-order logic. Gentzen's so-called ", 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 the axiomatic systems which instead use axioms as much as possible to express the logical laws of deductive 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 extended Gentzen's notion of 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. The term was first used by Bernard Bolzano, who first provided a non-analytic proof of his intermediate value theorem and then, several years later provided proof of 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 the axiomatic systems which instead use axioms as much as possible to express the logical laws of deductive 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, 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, in 1972. Martin-Löf has modified his proposal a. His inversion principle lies at the heart of most modern accounts of proof-theoretic semantics.
Michael Dummett Sir Michael Anthony Eardley Dummett FBA D.Litt is a leading British philosopher. He has both written on the history of analytic philosophy, and made original contributions to the subject, particularly in the areas of philosophy of mathematics, philosophy of logic, philosophy of language and metaphysics. He also devised the Quota Borda system of introduced the very fundamental idea of logical harmony, building on a suggestion of 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 his collaboration with Alan Ross Anderson on relevance. 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
- Logical Consequence, Deductive-Theoretic Conceptions, at the Internet Encyclopedia of Philosophy The Internet Encyclopedia of Philosophy is a free online encyclopedia on philosophical topics and philosophers founded by James Fieser in 1995. The current general editors are James Fieser and Bradley Dowden. The staff also includes numerous area editors as well as volunteers.
See also
- Inferential role semantics
- Truth-conditional semantics Truth-conditional semantics is an approach to semantics of natural language that sees the meaning of assertions as being the same as, or reducible to, their truth conditions. This approach to semantics is principally associated with Donald Davidson, and attempts to carry out for the semantics of natural language what Tarski's semantic theory of
External links
- Arché Bibliography on Proof-Theoretic Semantics.
Categories: Philosophical logic | Semantics Categories: Linguistics | Philosophy of language | Semiotics | Theoretical computer science | Philosophical logic | Proof theory Categories: Mathematical logic | Proofs | Theories of deduction | Metalogic
