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Prior's Tonk
Logical harmony, a name coined by Michael Dummett, is a supposed constraint on the rules of inference that can be used in a given logical system. Overview The logician Gerhard Gentzen proposed that the meanings of logical connectives could be given by the rules for introducing them into discourse. For example, if one believes that ''the sky is blue'' and one also believes that ''grass is green'', then one can introduce the connective ''and'' as follows: ''The sky is blue AND grass is green.'' Gentzen's idea was that having rules like this is what gives meaning to one's words, or at least to certain words. The idea has also been associated with the Wittgensteinian notion that in many cases we can say, '' meaning is use''. Most contemporary logicians prefer to think that the introduction rules and the elimination rules for an expression are equally important. In this case, ''and'' is characterized by the following rules: An apparent problem with this was pointed out by Arthur Pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 was, until 1992, Wykeham Professor of Logic at the University of Oxford. He wrote on the history of analytic philosophy, notably as an interpreter of Frege, and made original contributions particularly in the philosophies of mathematics, logic, language and metaphysics. He was known for his work on truth and meaning and their implications to debates between realism and anti-realism, a term he helped to popularize. In mathematical logic, he developed an intermediate logic, a logical system intermediate between classical logic and intuitionistic logic that had already been studied by Kurt Gödel: the Gödel–Dummett logic. In voting theory, he devised the Quota Borda system of proportional voting, based on the Borda count, and conj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuel Belnap
Nuel Dinsmore Belnap Jr. (; May 1, 1930 – June 12, 2024) was 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 retirement in 2011. Early life and education Belnap was born on May 1, 1930. He attended New Trier High School in Winnetka, Illinois, and earned a Bachelor of Arts degree from the University of Illinois. He recalled Max Fisch assigned Whitehead readings. Belnap worked as a programmer on the IBM 701 for the National Security Agency through the United States Air Force for two years before attending graduate school at Yale University. He enjoyed metaphysics, and his professors included Paul Weiss, Arthur Pap, Henry Margenau, Frederic Fitch, and Rulon Wells. On a Fulbright Fellowship in 1958 he went to Louvain to study with Canon Robert Feys. Belnap domiciled in Brussels with wife and 2-year-old. Feys directed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greg Restall
Greg Restall (born 11 January 1969) is an Australian philosopher and Professor of Philosophy at the University of St Andrews. He is a fellow of the Australian Academy of the Humanities. Restall is known for his research on logic and theories of meaning. After working at the University of Melbourne for years he was appointed the Shelby Cullom David Professor of Philosophy at the University of St Andrews. Books * ''An Introduction to Substructural Logics'', Routledge, 2000 * ''Logic'', Routledge, 2006 * ''Logical Pluralism'', with Jc Beall, Oxford University Press, 2006 * ''Logical Methods'', with Shawn Standefer, MIT Press, 2023 See also * Substructural logic *Validity (logic) *Logical harmony Logical harmony, a name coined by Michael Dummett, is a supposed constraint on the rules of inference that can be used in a given logical system. Overview The logician Gerhard Gentzen proposed that the meanings of logical connectives could be giv ... * Relevance logic References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Wittgenstein taught at the University of Cambridge. Despite his position, only one book of his philosophy was published during his entire life: the 75-page ''Logisch-Philosophische Abhandlung'' (''Logical-Philosophical Treatise'', 1921), which appeared, together with an English translation, in 1922 under the Latin title ''Tractatus Logico-Philosophicus''. His only other published works were an article, "Some Remarks on Logical Form" (1929); a book review; and a children's dictionary. #Works, His voluminous manuscripts were edited and published posthumously. The first and best-known of this posthumous series is the 1953 book ''Philosophical Investigations''. A 1999 survey among American university and college teachers ranked the ''Investigations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semantic Theory Of Truth
A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences. Origin The semantic conception of truth, which is related in different ways to both the correspondence and deflationary conceptions, is due to work by Polish logician Alfred Tarski. Tarski, in "On the Concept of Truth in Formal Languages" (1935), attempted to formulate a new theory of truth in order to resolve the liar paradox. In the course of this he made several metamathematical discoveries, most notably Tarski's undefinability theorem using the same formal technique Kurt Gödel used in his incompleteness theorems. Roughly, this states that a truth-predicate satisfying Convention T for the sentences of a given language cannot be defined ''within'' that language. Tarski's theory of truth To formulate linguistic theories without semantic paradoxes such as the liar paradox, it is generally necessary to distinguish the language that one is tal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Tarski
Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician and mathematician. A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, type theory, and analytic philosophy. Educated in Poland at the University of Warsaw, and a member of the Lwów–Warsaw school, Lwów–Warsaw school of logic and the Warsaw school of mathematics, he immigrated to the United States in 1939 where he became a naturalized citizen in 1945. Tarski taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death in 1983.#FefA, Feferman A. His biographers Anita Burdman Feferman and Solomon Feferman state that, "Along with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 that capture the pre-theoretic notions of truth, validity, and logical consequence. While logical syntax concerns the formal rules for constructing well-formed expressions, logical semantics establishes frameworks for determining when these expressions are true and what follows from them. The development of formal semantics has led to several influential approaches, including model-theoretic semantics (pioneered by Alfred Tarski), proof-theoretic semantics (associated with Gerhard Gentzen and Michael Dummett), possible worlds semantics (developed by Saul Kripke and others for modal logic and related systems), algebraic semantics (connecting logic to abstract algebra), and game semantics (interpreting logical validity through game ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type System
In computer programming, a type system is a logical system comprising a set of rules that assigns a property called a ''type'' (for example, integer, floating point, string) to every '' term'' (a word, phrase, or other set of symbols). Usually the terms are various language constructs of a computer program, such as variables, expressions, functions, or modules. A type system dictates the operations that can be performed on a term. For variables, the type system determines the allowed values of that term. Type systems formalize and enforce the otherwise implicit categories the programmer uses for algebraic data types, data structures, or other data types, such as "string", "array of float", "function returning boolean". Type systems are often specified as part of programming languages and built into interpreters and compilers, although the type system of a language can be extended by optional tools that perform added checks using the language's original type synta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof System
In mathematical logic, a proof calculus or a proof system is built to prove statements. Overview A proof system includes the components: * Formal language: The set ''L'' of formulas admitted by the system, for example, propositional logic or first-order logic. * Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems. * Axioms: Formulas in ''L'' assumed to be valid. All theorems are derived from axioms. A formal proof of a well-formed formula in a proof system is a set of axioms and rules of inference of proof system that infers that the well-formed formula is a theorem of proof system. Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determined and can be used for radically different logics. For example, a paradigmatic case is the sequent calculus, which can be used to express the consequence relations of both intuitionistic logic and relevance logic. Thus, lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur Prior
Arthur Norman Prior (4 December 1914 – 6 October 1969), usually cited as A. N. Prior, was a New Zealand–born logician and philosopher. Prior (1957) founded tense logic, now also known as temporal logic, and made important contributions to intensional logic, particularly in Prior (1971). Biography Prior was born in Masterton, New Zealand, on 4 December 1914, the only child of Australian-born parents: Norman Henry Prior (1882–1967) and his wife born Elizabeth Munton Rothesay Teague (1889–1914). His mother died less than three weeks after his birth and he was cared for by his father's sister. His father, a medical practitioner in general practice, after war service at Gallipoli and in Francewhere he was awarded the Military Crossremarried in 1920. There were three more children: Elaine, the epidemiologist Ian Prior, and Owen. Arthur Prior grew up in a prominent Methodist household. His two Wesleyan grandfathers, the Reverends Samuel Fowler Prior and Hugh Henwood Teag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rules Of Inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the logical structure of valid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. ''Modus ponens'', an influential rule of inference, connects two premises of the form "if P then Q" and "P" to the conclusion "Q", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet." There are many other rules of inference for different patterns of valid arguments, such as '' modus tollens'', disjunctive syllogism, constructive dilemma, and existential generalization. Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and rules of replacement, which state that two expressions are equivalent and can be freely swapped. Rules of inference contrast with formal fallaciesinvalid argument forms involving lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elimination 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 axioms as much as possible to express the logical laws of deductive reasoning. History Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, and Russell (see, e.g., Hilbert system). Such axiomatizations were most famously used by Russell and Whitehead in their mathematical treatise ''Principia Mathematica''. Spurred on by a series of seminars in Poland in 1926 by Łukasiewicz that advocated a more natural treatment of logic, Jaśkowski made the earliest attempts at defining a more natural deduction, first in 1929 using a diagrammatic notation, and later updating his proposal in a sequence of papers in 1934 and 1935. His proposals led to diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
