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Strength (mathematical Logic)
The relative strength of two systems of formal logic can be defined via model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat .... Specifically, a logic \alpha is said to be as strong as a logic \beta if every elementary class in \beta is an elementary class in \alpha. Heinz-Dieter Ebbinghaus ''Extended logics: the general framework'' in K. J. Barwise and S. Feferman, editors, ''Model-theoretic logics'', 1985 page 43 See also * Abstract logic * Lindström's theorem References Model theory Mathematical logic Concepts in logic {{mathlogic-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Class
In model theory, a branch of mathematical logic, an elementary class (or axiomatizable class) is a class consisting of all structures satisfying a fixed first-order theory. Definition A class ''K'' of structures of a signature σ is called an elementary class if there is a first-order theory ''T'' of signature σ, such that ''K'' consists of all models of ''T'', i.e., of all σ-structures that satisfy ''T''. If ''T'' can be chosen as a theory consisting of a single first-order sentence, then ''K'' is called a basic elementary class. More generally, ''K'' is a pseudo-elementary class if there is a first-order theory ''T'' of a signature that extends σ, such that ''K'' consists of all σ-structures that are reducts to σ of models of ''T''. In other words, a class ''K'' of σ-structures is pseudo-elementary if and only if there is an elementary class ''K''' such that ''K'' consists of precisely the reducts to σ of the structures in ''K'''. For obvious reasons, elementary cla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinz-Dieter Ebbinghaus
Heinz-Dieter Ebbinghaus (born 22 February 1939 in Hemer, Province of Westphalia) is a German mathematician and logician. He received his PhD in 1967 at the University of Münster under Hans Hermes and Dieter Rödding. Ebbinghaus has written various books on logic, set theory and model theory, including a seminal work on Ernst Zermelo. His book ''Einführung in die mathematische Logik'', joint work with Jörg Flum and Wolfgang Thomas, first appeared in 1978 and became a standard textbook of mathematical logic in the German-speaking area. It is currently in its sixth edition (). An English edition of ''Mathematical Logic'' was published in the Springer-Verlag Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics (UTM) () is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow books of a stand ... series in 1984 (), with a second edi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jon Barwise
Kenneth Jon Barwise (; June 29, 1942 – March 5, 2000) was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used. Education and career He was born in Independence, Missouri, to Kenneth T. and Evelyn Barwise. A pupil of Solomon Feferman at Stanford University, Barwise started his research in infinitary logic. After positions as assistant professor at Yale University and the University of Wisconsin, during which time his interests turned to natural language, he returned to Stanford in 1983 to direct the Center for the Study of Language and Information (CSLI). He began teaching at Indiana University in 1990. He was elected a Fellow of the American Academy of Arts and Sciences in 1999. In his last year, Barwise was invited to give the 2000 Gödel Lecture; he died prior to the lecture. Philosophical and logical work Barwise contended that, by being explicit about the context in which a propos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solomon Feferman
Solomon Feferman (December 13, 1928July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. In addition to his prolific technical work in proof theory, computability theory, and set theory, he was known for his contributions to the history of logic (for instance, via biographical writings on figures such as Kurt Gödel, Alfred Tarski, and Jean van Heijenoort) and as a vocal proponent of the philosophy of mathematics known as predicativism, notably from an anti- platonist stance. Life Solomon Feferman was born in The Bronx in New York City to working-class parents who had immigrated to the United States after World War I and had met and married in New York. Neither parent had any advanced education. The family moved to Los Angeles, where Feferman graduated from high school at age 16. He received his B.S. from the California Institute of Technology in 1948, and in 1957 his Ph.D. in mathematics from the University of California, Berkeley, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Logic
In mathematical logic, an abstract logic is a formal system consisting of a class of sentences and a satisfaction relation with specific properties related to occurrence, expansion, isomorphism, renaming and quantification. Based on Lindström's characterization, first-order logic is, up to equivalence, the only abstract logic that is countably compact and has Löwenheim number ω. C. C. Chang and Jerome Keisler Howard Jerome Keisler (born 3 December 1936) is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis. His Ph.D. advisor was Alfred Tarski a ... ''Model Theory'', 1990 page 132 See also * * * * * References {{mathlogic-stub Mathematical logic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindström's Theorem
In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the '' strongest logic'' (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property. Lindström's theorem is perhaps the best known result of what later became known as abstract model theory, the basic notion of which is an abstract logic; the more general notion of an institution was later introduced, which advances from a set-theoretical notion of model to a category-theoretical one. Lindström had previously obtained a similar result in studying first-order logics extended with Lindström quantifiers. Jouko VäänänenLindström's Theorem/ref> Lindström's theorem has been extended to various other systems of logic, in particular modal logics by Johan van Benthem and Sebastian Enqvist. Notes References * Per Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
