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List Of Hilbert Systems
This article contains a list of sample Hilbert-style deductive systems for propositional logics. Classical propositional calculus systems Classical propositional calculus is the standard propositional logic. Its intended semantics is bivalent and its main property is that it is strongly complete, otherwise said that whenever a formula semantically follows from a set of premises, it also follows from that set syntactically. Many different equivalent complete axiom systems have been formulated. They differ in the choice of basic connectives used, which in all cases have to be functionally complete (i.e. able to express by composition all ''n''-ary truth tables), and in the exact complete choice of axioms over the chosen basis of connectives. Implication and negation The formulations here use implication and negation \ as functionally complete set of basic connectives. Every logic system requires at least one non-nullary rule of inference. Classical propositional calculus typica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert System
:''In mathematical physics, ''Hilbert system'' is an infrequently used term for a physical system described by a C*-algebra.'' In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob FregeMáté & Ruzsa 1997:129 and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well. Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference. Hilbert systems can be characterised by the choice of a large number of schemes of logical axioms and a small set of rules of inference. Systems of natural deduction take the opposite tack, including many deduction rules but very few or no axiom schemes. The most commonly studied Hilbert systems have either just one rule of inference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, artificial intelligence, cognitive science, computer science and various areas of analytic philosophy, especially philosophy of mathematics, philosophy of language, epistemology, and metaphysics.Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003. He was one of the early 20th century's most prominent logicians, and a founder of analytic philosophy, along with his predecessor Gottlob Frege, his friend and colleague G. E. Moore and his student and protégé Ludwig Wittgenstein. Russell with Moore led the British "revolt against idealism". Together with his former teacher A. N. Whitehead, Russell wrote ''Principia Mathematica'', a milestone in the development of classical logic, and a major attempt to reduce the whole ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intermediate Logics
In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate logics (the logics are intermediate between intuitionistic logic and classical logic). Definition A superintuitionistic logic is a set ''L'' of propositional formulas in a countable set of variables ''p''''i'' satisfying the following properties: :1. all axioms of intuitionistic logic belong to ''L''; :2. if ''F'' and ''G'' are formulas such that ''F'' and ''F'' → ''G'' both belong to ''L'', then ''G'' also belongs to ''L'' (closure under modus ponens); :3. if ''F''(''p''1, ''p''2, ..., ''p''''n'') is a formula of ''L'', and ''G''1, ''G''2, ..., ''G''''n'' are any formulas, then ''F''(''G''1, ''G''2, ..., ''G''''n'') belongs to ''L'' (closure under substitution). Such a logic is intermediate if furthermore :4. ''L'' is not the set of all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peirce's Law
In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written in a form that involves only one sort of connective, namely implication. In propositional calculus, Peirce's law says that ((''P''→''Q'')→''P'')→''P''. Written out, this means that ''P'' must be true if there is a proposition ''Q'' such that the truth of ''P'' follows from the truth of "if ''P'' then ''Q''". In particular, when ''Q'' is taken to be a false formula, the law says that if ''P'' must be true whenever it implies falsity, then ''P'' is true. In this way Peirce's law implies the law of excluded middle. Peirce's law does not hold in intuitionistic logic or intermediate logics and cannot be deduced from the deduction theorem alone. Under the Curry–Howard isomorphism, Peirce's law is the type of continuation operators, e.g. call/ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excluded Middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law (or principle) of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'': "no third ossibilityis given". It is a tautology. The principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable now, but provable in the futu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intuitionistic Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. Other properties related to completeness The property converse to completeness is called soundness: a system is sound with respect to a property (mostly semantical validity) if each of its theorems has that property. Forms of completeness Expressive completeness A formal language is expressively complete if it can express the subject matter for which it is intended. Functional completeness A set of logical connectives associated with a formal system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implicational Propositional Calculus
In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus which uses only one connective, called implication or conditional. In formulas, this binary operation is indicated by "implies", "if ..., then ...", "→", "\rightarrow ", etc.. Functional (in)completeness Implication alone is not functionally complete as a logical operator because one cannot form all other two-valued truth functions from it. For example, the two-place truth function that always returns '' false'' is not definable from → and arbitrary sentence variables: any formula constructed from → and propositional variables must receive the value ''true'' when all of its variables are evaluated to true. It follows that is not functionally complete. However, if one adds a nullary connective ⊥ for falsity, then one can define all other truth functions. Formulas over the resulting set of connectives are called f-implicational. If ''P'' and ''Q'' are proposit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Argonne National Laboratory
Argonne National Laboratory is a science and engineering research United States Department of Energy National Labs, national laboratory operated by University of Chicago, UChicago Argonne LLC for the United States Department of Energy. The facility is located in Lemont, Illinois, outside of Chicago, and is the largest national laboratory by size and scope in the Midwest. Argonne had its beginnings in the Metallurgical Laboratory of the University of Chicago, formed in part to carry out Enrico Fermi's work on nuclear reactors for the Manhattan Project during World War II. After the war, it was designated as the first national laboratory in the United States on July 1, 1946. In the post-war era the lab focused primarily on non-weapon related nuclear physics, designing and building the first power-producing nuclear reactors, helping design the reactors used by the United States' nuclear navy, and a wide variety of similar projects. In 1994, the lab's nuclear mission ended, and today ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Nicod
Jean George Pierre Nicod (1 June 1893, in France – 16 February 1924, in Geneva, Switzerland) was a French philosopher and logician, best known for his work on propositional logic and inductive reasoning, induction. Biography Nicod's main contribution to formal logic was to show that Classical logic, classical propositional calculus could be axiomatized with only one axiom - which is now known as Nicod's axiom - and one rule of inference, both formulated using the Sheffer stroke as only Logical connective, connective. In Inductive reasoning, inductive logic and Scientific method, confirmation theory, he famously proposed Nicod's criterion, according to which a conditional hypothesis is confirmed by all and only its positive instances. This principle plays a central role in the derivation of Carl Hempel's raven paradox. Nicod died at the age of 30 from tuberculosis. Legacy The Institut Jean Nicod (Paris) — a branch of the French ''Centre National de la Recherche Scientif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheffer's Stroke
In Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called nand ("not and") or the alternative denial, since it says in effect that at least one of its operands is false. In digital electronics, it corresponds to the NAND gate. It is named after Henry M. Sheffer and written as ↑ or as , (but not as , , , often used to represent disjunction). In Bocheński notation it can be written as D''pq''. Its dual is the NOR operator (also known as the Peirce arrow or Quine dagger). Like its dual, NAND can be used by itself, without any other logical operator, to constitute a logical formal system (making NAND functionally complete). This property makes the NAND gate crucial to modern digital electronics, including its use in computer processor design. Definition The NAND operation is a logical operation on two logica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, the Church–Turing thesis, proving the unsolvability of the Entscheidungsproblem, the Frege–Church ontology, and the Church–Rosser theorem. He also worked on philosophy of language (see e.g. Church 1970). Alongside his student Alan Turing, Church is considered one of the founders of computer science. Life Alonzo Church was born on June 14, 1903, in Washington, D.C., where his father, Samuel Robbins Church, was a Justice of the Peace and the judge of the Municipal Court for the District of Columbia. He was the grandson of Alonzo Webster Church (1829-1909), United States Senate Librarian from 1881-1901, and great grandson of Alonzo Church, a Professor of Mathematics and Astronomy and 6th Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
