mathematical logic

The following table shows some of the classes of problems (or languages, or grammars) that are considered in complexity theory. If class X is a strict subset of Y, then X is shown below Y, with a dark line connecting them. If X is a subset, but it is unknown whether they are equal sets, then the line is lighter and is dotted. Technically the breakdown into solvable and unsolvable belongs more in Computabi ...

See also:

Complexity class, Complexity class - Relationships Between Complexity Classes

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Alfred Tarski (January 14, 1901 in Warsaw – October 26, 1983 in Berkeley, USA) was a Polish mathematician, and widely considered one of the four greatest logicians of all time, along with Aristotle, Gottlob Frege, and Kurt Gödel. Tarski wrote on algebra, algebraic logic, measure theory, mathematical logic, set theory, and metamathematics. See Truth for a brief description of the "Convention T" (see also T-schema) standard in his "inductive definition of truth". This was an important contribution to symbol ...

Including:

• Alfred Tarski - The concept of truth in formalized languages
• Alfred Tarski - On the concept of logical consequence
• Alfred Tarski - What are logical notions?

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Antinomy (Greek αντι-, against, plus νομος, law, literally, the mutual incompatibility, real or apparent, of two laws) is a term used in logic and epistemology, which, loosely, means a paradox or unresolvable contradiction. The term acquired a special significance in the philosophy of Immanuel Kant, who used it to describe the equally rational but contradictory results of applying to the universe of pure thought the categories or criteria of understanding proper to the universe of sensible perception or experience (phe ...

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Arend Heyting (May 9, 1898 – July 9, 1980) was a Dutch mathematician and logician. He was a student of L. E. J. Brouwer, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic (which in a definite sense ran counter to some of the initial intentions of its founder). He was born in Amsterdam, Netherlands, and died in Lugano, Switzerland. See also. Heyting algebra Heyting arithmetic Hey ...

Including:

• Arend Heyting - External link

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Reality in everyday usage means "everything that exists." The term "Reality," in its most liberal sense, includes everything that is, whether or not it is observable, accessible or understandable by science, philosophy, theology or any other system of analysis. Reality in this sense may include both being and nothingness, whereas "existence" is often restricted to being. (Compare with nature). In the strict sense of European-German philosophy, there are levels or gradation to the nature and conception of reality. These levels include, from the most subjective to the most rigorous: Phenomenological reality ...

Including:

• Reality - Simple reality
• Reality - Phenomenological reality
• Reality - Truth
• Reality - Fact
• Reality - Axiom
• Reality - What reality might not be
• Reality - Reality world views and theories of reality
• Reality - Philosophical views of reality

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Inference is the act or process of drawing a conclusion based solely on what one already knows. Suppose you see rain on your window - you can infer from that, quite trivially, that the sky is grey. Looking out the window would have yielded the same fact, but through a process of perception, not inference (note however that perception itself can be viewed as an inferential process). Inference is studied within several different fields. Human inference (i.e. how humans draw conclusions) is traditionally studied within the field o ...

Including:

• Inference - The accuracy of inductive and deductive inferences
• Inference - Valid inferences
• Inference - An example: the classic syllogism
• Inference - Automatic logical inference
• Inference - An example: inference using Prolog
• Inference - Inference and uncertainty
• Inference - Common sense and uncertain reasoning
• Inference - Bayesian statistics and probability logic
• Inference - Frequentist statistical inference
• Inference - Fuzzy logic

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In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. Not all epistemologists agree that any axioms, understood in that sense, exist. In mathematics, an axiom is not necessarily a self-evident truth but rather, a formal logical expression used in a deduction to yield further results. Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms. Axiom - Etymology. The word axiomIncluding:

• Axiom - Etymology
• Axiom - Mathematics
• Axiom - Logical axioms
• Axiom - Non-logical axioms
• Axiom - Role in mathematical logic
• Axiom - Further discussion

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The Right Honourable Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was an influential British logician, philosopher, and mathematician, working mostly in the 20th century. A prolific writer, Bertrand Russell was also a populariser of philosophy and a commentator on a large variety of topics, ranging from very serious issues to the mundane. Continuing a family tradition in political affairs, he was a prominent liberal as well as a socialist and anti-war activist for most of his long life. ...

Including:

• Bertrand Russell - Biography
• Bertrand Russell - Russell's philosophical work
• Bertrand Russell - Analytic philosophy
• Bertrand Russell - Epistemology
• Bertrand Russell - Ethics
• Bertrand Russell - Logical atomism
• Bertrand Russell - Logic and mathematics
• Bertrand Russell - Philosophy of language
• Bertrand Russell - Philosophy of science
• Bertrand Russell - Religion and theology
• Bertrand Russell - Influence on philosophy
• Bertrand Russell - Russell's activism
• Bertrand Russell - Pacifism war and nuclear weapons
• Bertrand Russell - Communism and socialism
• Bertrand Russell - Women's suffrage
• Bertrand Russell - Sexuality
• Bertrand Russell - Eugenics and race
• Bertrand Russell - Russell summing up his life
• Bertrand Russell - Comments about Russell
• Bertrand Russell - As a man
• Bertrand Russell - As a philosopher
• Bertrand Russell - As a writer and his place in history
• Bertrand Russell - As a mathematician and logician
• Bertrand Russell - As an activist
• Bertrand Russell - As a recipient of the Nobel Prize for Literature
• Bertrand Russell - From a daughter
• Bertrand Russell - Quotes
• Bertrand Russell - Asides
• Bertrand Russell - Succession

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Gottfried Wilhelm von Leibniz (also Leibnitz) (July 1 (June 21 Old Style) 1646, Leipzig – November 14, 1716, Hanover) was a German polymath, deemed a genius in his lifetime and since, and the last true polyhistor. Trained as a lawyer and active as a diplomat and librarian, he wrote on philosophy, science, mathematics, theology, history, and comparative philology, even writing verse. Through his service to two major German noble houses, he played a major role in the European ...

Including:

• Gottfried Leibniz - Life
• Gottfried Leibniz - Early life and education
• Gottfried Leibniz - Career
• Gottfried Leibniz - Writings
• Gottfried Leibniz - Posthumous reputation
• Gottfried Leibniz - Philosopher
• Gottfried Leibniz - Metaphysics
• Gottfried Leibniz - Theodicy and optimism
• Gottfried Leibniz - Symbolic thought
• Gottfried Leibniz - Characteristica Universalis Universal characteristic and Calculus Ratiocinator
• Gottfried Leibniz - Formal logic
• Gottfried Leibniz - Mathematician
• Gottfried Leibniz - Topology
• Gottfried Leibniz - The dispute over who first invented the calculus
• Gottfried Leibniz - Science and technology
• Gottfried Leibniz - The vis viva
• Gottfried Leibniz - Information technology
• Gottfried Leibniz - Philology
• Gottfried Leibniz - The Sinophile
• Gottfried Leibniz - Works
• Gottfried Leibniz - Secondary literature
• Gottfried Leibniz - Quotes

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Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. However the subject is grounded, the task of the logician is the same: to advance an account of valid and fallacious inference to allow ...

Including:

• Logic - Nature of logic
• Logic - Informal formal and symbolic logic
• Logic - Rival conceptions of logic
• Logic - History of logic
• Logic - Relation to other sciences
• Logic - Deductive and inductive reasoning
• Logic - Topics in logic
• Logic - Syllogistic logic
• Logic - Predicate logic
• Logic - Modal logic
• Logic - Deduction and reasoning
• Logic - Mathematical logic
• Logic - Philosophical logic
• Logic - Logic and computation
• Logic - Controversies in logic
• Logic - Bivalence and the law of the excluded middle
• Logic - Implication: strict or material?
• Logic - Tolerating the impossible
• Logic - Is logic empirical?

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Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...

Including:

• Mathematics - History
• Mathematics - Inspiration pure and applied mathematics and aesthetics
• Mathematics - Notation language and rigor
• Mathematics - Is mathematics a science?
• Mathematics - Overview of fields of mathematics
• Mathematics - Major themes in mathematics
• Mathematics - Quantity
• Mathematics - Structure
• Mathematics - Space
• Mathematics - Change
• Mathematics - Foundations and methods
• Mathematics - Discrete mathematics
• Mathematics - Applied mathematics
• Mathematics - Important theorems
• Mathematics - Important conjectures
• Mathematics - History and the world of mathematicians
• Mathematics - Mathematics and other fields
• Mathematics - Common misconceptions

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In computational complexity theory, a complexity class is a set of problems of related complexity. A typical complexity class has a definition of the form: the set of problems that can be solved by abstract machine M using O(f(n)) of resource R (n is the size of the input) For example, the class NP is the set of decision problems that can be solved by a non-deterministic Turing machine in polynomial time, while the class PSPACE is the set of decision problems that can be solved by a deterministic Turing machine in polynomial space. Some comple ...

Including:

• Complexity class - Relationships Between Complexity Classes

Read more here: » Complexity class: Encyclopedia - Complexity class

In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the subset of the real numbers consisting of the numbers which can be computed by a finite, terminating algorithm. They can be defined equivalently using the axioms of recursive functions, Turing machines or lambda-calculus. In contrast, the reals require the more powerful axioms of Zermelo-Fraenkel set theory. The computable numbers form a real closed field and can be used in the place of real numbe ...

Including:

• Computable number - Formal definition
• Computable number - Properties
• Computable number - Computing digit strings
• Computable number - Uncomputable numbers
• Computable number - Can computable numbers be used instead of the reals?

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In computer science, computational complexity theory is the branch of the theory of computation that studies the resources required during computation to solve a given problem. The most common resources are time (how many steps it takes to solve a problem) and space (how much memory it takes). Other resources can also be considered, such as how many parallel processors are needed to solve a problem in parallel. Complexity theory differs from the computability theory, which deals with whether a problem can be solved at al ...

Including:

• Computational complexity theory - Overview
• Computational complexity theory - Decision problems
• Computational complexity theory - Complexity classes
• Computational complexity theory - The P = NP question
• Computational complexity theory - Intractability
• Computational complexity theory - Notable researchers

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In mathematical logic, in particular as applied to computer science, a unification of two terms is a join (in the lattice sense) with respect to a specialisation order. That is, we suppose a preorder on a set of terms, for which t* ≤ t means that t* is obtained from t by substituting some term(s) for one or more free variables in t. The unification u of s and t, if it exists, is a term that is a substitution instance of both s and t. Any common substitution instance of s ...

Including:

• Unification - Unification in Prolog
• Unification - Examples of unification

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David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. He established his reputation in a broad range of fields including invariant theory, the axiomization of geometry and the foundations of functional analysis. Later in life, he became a world leader in mathematics, exemplified by his presentation, in 1900, of a set of probl ...

Including:

• David Hilbert - Major contributions
• David Hilbert - Miscellaneous talks essays and contributions
• David Hilbert - Hilbert's program
• David Hilbert - Later years

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In computer science, computability theory is the branch of the theory of computation that studies which problems are computationally solvable using different models of computation. Computability theory differs from the related discipline of computational complexity theory, which deals with the question of how efficiently a problem can be solved, rather than whether it is solvable at all. Computability theory computation - Introduction. A central question of computer science is to address the limits o ...

Including:

• Computability theory computation - Introduction
• Computability theory computation - Formal models of computation
• Computability theory computation - Power of computational models
• Computability theory computation - Power of finite state machines
• Computability theory computation - Power of push-down automata
• Computability theory computation - Power of Turing machines
• Computability theory computation - Unreasonable models of computation
• Computability theory computation - Infinite execution
• Computability theory computation - Oracle machines
• Computability theory computation - Limits of hypercomputation
• Computability theory computation - History of computability theory

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Computability theory may refer to: A branch of mathematical logic, traditionally called recursion theory. A branch of the theory of computation; see computability theory (computation). Other related archivescomputability theory (computation), mathematical logic, recursion theory, theory of computation

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There are two different understandings of Leibniz's 'Calculus Ratiocinator' in the history of ideas. In analytic philosophy, such as pure logic, the term Calculus Ratiocinator is commonly understood as referring to a formal logical system envisioned by Leibniz, which did not exist in his lifetime. A completely formal, calculational form of logical inference is realized in modern developments of mathematical logic starting with the Be ...

Including:

• Calculus ratiocinator - Philosophical Implications

Read more here: » Calculus ratiocinator: Encyclopedia - Calculus ratiocinator

Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. It has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. Combinatory logic - Combinatory logic in mathematics. Combinatory logic was intended as a simple 'pre-logic' which would clarify the meaning of variables in logical notation, and indeed eliminate the need for ...

Including:

• Combinatory logic - Combinatory logic in mathematics
• Combinatory logic - Combinatory logic in computing
• Combinatory logic - Summary of the lambda calculus
• Combinatory logic - Combinatory calculi
• Combinatory logic - Combinatory terms
• Combinatory logic - Examples of combinators
• Combinatory logic - Completeness of the S-K basis
• Combinatory logic - Simplifications of the transformation
• Combinatory logic - Reverse conversion
• Combinatory logic - Undecidability of combinatorial calculus
• Combinatory logic - Combinatory terms as graphs
• Combinatory logic - Applications
• Combinatory logic - Compilation of functional languages
• Combinatory logic - Logic

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