proof theory

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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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 mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. This is made precise in various ways, several of which have a related notion of completion. It should be noted that "complete" here is just a term that takes on specific meanings in specific situations, and not every situation in which a type of "completion" occurs is called a "completion". See, for example, algebraically closed field, compactification, or Gödel's incompleteness theorem. A metric spac

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Weak agnosticism, or empirical agnosticism (also negative agnosticism), is the belief that the existence or nonexistence of deities is currently unknown, but is not necessarily unknowable. Weak agnosticism is in contrast to strong agnosticism, in which the agnostic believes that the existence of any gods is not only unknown, but is also unknowable to humanity. Neither type of agnosticism is fully irreconcilable with theism (belief in a deity or deities) nor strong atheism. A weak agnostic who also considers themse ...

Including:

• Weak agnosticism - Weak agnosticism and symbolic logic
• Weak agnosticism - Weak agnosticism vs indecision
• Weak agnosticism - Criticism against weak agnosticism and response
• Weak agnosticism - Bibliography

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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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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 ...

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• Calculus ratiocinator - Philosophical Implications

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Proof may refer to: a rigorous, compelling argument, including: a mathematical proof (see also proof theory). a logical proof a legal proof. a scientific proof Proofreading Alcoholic proof, a measure of how much ethanol is in an alcoholic beverage Proof coinage, a coin made as an example of a particular strike. Proof (play), a play by David Auburn Proof (1991 film), an Australian film by Jocelyn Moorhouse starring Hu ...

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In everyday use, the correctness of a statement is determined by whether or not it matches reality. People can think a statement is correct and be wrong. When scoring tests of knowledge where there is only one accepted answer for each problem, a device or person marks an answer as correct if it matches what the test designer has determined the testee should answer. It is common for one or more answers, thought at the time by the designer of a test to match reality, and to, at a later date, be shown to not match what can be observed, a ...

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The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g., addition, subtraction, mul ...

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Mathematics, 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 - Mathematical tools, Mathematics - Common misconceptions

Read more here: » Mathematics: Encyclopedia II - Mathematics - History

An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possible view. For a fuller treatment, see areas of mathematics or the list of mathematics lists. Mathematics - Quantity. This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. See also:

Mathematics, 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

Read more here: » Mathematics: Encyclopedia II - Mathematics - Major themes in mathematics

Kurt Gödel - Childhood. Kurt Gödel was born April 28, 1906, in Brünn (now Brno), Moravia, Austria-Hungary (now the Czech Republic) to Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (née Handschuh). In his German-speaking family, young Kurt was known as Der Herr Warum ("Mr. Why"). He attended German language primary and secondary school in Brno and completed them with honors in 1923. Although Kurt had first excelled in languages, he later became more interested in history and mathe ...

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Kurt Gödel, Kurt Gödel - Short biography, Kurt Gödel - Childhood, Kurt Gödel - Studying in Vienna, Kurt Gödel - Working in Vienna, Kurt Gödel - Visiting the USA, Kurt Gödel - Working in Princeton, Kurt Gödel - Psychological disorder, Kurt Gödel - Death, Kurt Gödel - Legacy, Kurt Gödel - Anecdotes, Kurt Gödel - Important publications, Kurt Gödel - Links and references, Kurt Gödel - Further reading, Kurt Gödel - External links

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Mathematical logic - First-order languages and structures. Definition. A first-order language is a collection of distinct typographical symbols classified as follows: The equality symbol ; the connectives , ; the universal quantifier and the parentheses , . A countable set of variable symbols . A set of constant symbols . A set of function symbol ...

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Mathematical logic, Mathematical logic - History, Mathematical logic - Topics in mathematical logic, Mathematical logic - Some fundamental results, Mathematical logic - Technical reference, Mathematical logic - First-order languages and structures, Mathematical logic - Terms formulas and sentences, Mathematical logic - Assignment functions, Mathematical logic - Logical satisfaction, Mathematical logic - Logical implication and truth, Mathematical logic - Variable substitution, Mathematical logic - Substitutability

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Because of its fundamental role in philosophy, the nature of logic has been the object of intense dispute: it is not possible clearly to delineate the bounds of logic in terms acceptable to all rival viewpoints. Despite that controversy, the study of logic has been very coherent and technically grounded. In this article, we first characterise logic by introducing fundamental ideas about form, then by outlining some schools of thought, as well as by giving a brief overview of logic's history, an account of its relationship to other sciences, and finally, an exposition of some of logic's essential concepts. Logic - I ...

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Logic, 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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Lorenzen came 1962 to University of Erlangen (south germany) and founded the school of constructivist philosophy there. He wrote with Kamlah the famous book Logical Propaedeutic ("Logische Propädeutik") and worked on game semantics ("Dialogische Logik") with Kuno Lorenz. With Peter Janich he invented protophysics of time and space. He developed, Constructivist logic, Constructivist type theory and Constructivist analysis. Lorenzen took modal (incl. normative) logic as a base of techn ...

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Paul Lorenzen, Paul Lorenzen - Biography, Paul Lorenzen - Theory, Paul Lorenzen - Major works

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Semantics is a subfield of linguistics that is traditionally defined as the study of meaning of (parts of) words, phrases, sentences, and texts. Semantics can be approached from a theoretical as well as an empirical (for example psycholinguistic and neuroscientific) point of view. The decompositional perspective towards meaning holds that the meaning of words can be analyzed by defining meaning atoms or primitives, which establish a language of thought. An area of study is the meaning of compounds, another is the study o ...

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Semantics, Semantics - In linguistics, Semantics - In mathematics and computer science, Semantics - In logic

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In a set of rules, an inference rule could be redundant in the sense that it is admissible or derivable. A derivable rule is one whose conclusion can be derived from its premises using the other rules. An admissible rule is one whose conclusion holds whenever the premises hold. All derivable rules are admissible. To appreciate the difference, consider the following set of rules for defining the natural numbers (the judgment asserts the fact that ...

See also:

Rule of inference, Rule of inference - Admissibility and Derivability, Rule of inference - Other Considerations

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When predicate logic was introduced to the mathematical community by Frege (and independently — and more influentially — by Peirce, who coined the term Second-order logic), he did use different variables to distinguish quantification over objects from quantification over properties and sets; but he did not see himself as doing two different kinds of logic. After the discovery of Russell's paradox it was realized that something was wrong with his system. Eventually logicians found that restricting Frege's logic in various ways—to ...

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Second-order logic, Second-order logic - Why second-order logic is not reducible to first-order logic, Second-order logic - Second-order logic and metalogical results, Second-order logic - The history and disputed value of second-order logic, Second-order logic - Power of the existential fragment on finite structures, Second-order logic - Applications to complexity

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A model is formally defined in the context of some language L, which consists of a set of constant symbols, a set of relation symbols each of valence some positive integer, and a set of function symbols each of valence some positive integer. A model of the language L consists of several things: A universe set A which contains all the objects of interest (the "domain of discourse"), and An element of A for each constant symbol of L. A function from ASee also:

Model theory, Model theory - Definition, Model theory - Theorems of model theory

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In classical logic, we often discuss the truth values that a formula can take. The values are usually chosen as the members of a Boolean algebra. The meet and join operations in the Boolean algebra are identified with the ∧ and ∨ logical connectives, so that the value of a formula of the form A ∧ B is the meet of the value of A and the value of B in the Boolean algebra. Then we have the useful theorem that a formula is a valid sentence of classical logic if and only if its value is 1 for every v ...

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Intuitionistic logic, Intuitionistic logic - Intuitionistic logic as a paradigm for logical reasoning, Intuitionistic logic - Intuitionistic logic as a formal logical calculus, Intuitionistic logic - Heyting algebra semantics, Intuitionistic logic - Kripke semantics

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For an explanation of Gödel's original proof of the theorem, see Original proof of Gödel's completeness theorem. In modern logic texts, Gödel's completeness theorem is usually proved with Henkin's proof rather than with Gödel's original proof. ...

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Gödel's completeness theorem, Gödel's completeness theorem - Proofs, Gödel's completeness theorem - External link

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