|
Equational Logic
First-order equational logic consists of Quantification (logic), quantifier-free terms of ordinary first-order logic, with equality as the only predicate symbol. The model theory of this logic was developed into universal algebra by Garrett Birkhoff, Birkhoff, George Grätzer, Grätzer, and Paul Cohn, Cohn. It was later made into a branch of category theory by William Lawvere, Lawvere ("algebraic theories").equational logic. (n.d.). The Free On-line Dictionary of Computing. Retrieved October 24, 2011, from Dictionary.com website: http://dictionary.reference.com/browse/equational+logic The terms of equational logic are built up from variables and constants using function symbols (or operations). Syllogism Here are the four inference rules of logic. P[x := E] denotes textual substitution of expression E for variable x in expression P. Next, b = c denotes equality, for b and c of the same type, while b \equiv c, or equivalence, is defined only for b and c of type Boolean algebra, boo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (''and'') denoted as ∧, disjunction (''or'') denoted as ∨, and the negation (''not'') denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction and division. So Boolean algebra is a formal way of describing logical operations, in the same way that elementary algebra describes numerical operations. Boolean algebra was introduced by George Boole in his first book ''The Mathematical Analysis of Logic'' (1847), and set forth more fully in his '' An Investigation of the Laws of Thought'' (1854). According to Huntington, the term "Boolean algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false when P is true. Negation is thus a unary logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes ''truth'' to ''falsity'' (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P is the proposition whose proofs are the refutations of P. Definition ''Classical negation'' is an operation on one logical value, typically the value of a proposition, that produces a value of ''true'' when its operand is false, and a value of ''false'' when its operand is true. Thus if statement is true, then \neg P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Logic Symbols
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents, and the LaTeX symbol. Basic logic symbols Advanced and rarely used logical symbols These symbols are sorted by their Unicode value: Usage in various countries Poland and Germany in Poland, the universal quantifier is sometimes written ∧, and the existential quantifier as ∨. The same applies for Germany. Japan The ⇒ symbol is often used in text to mean "result" or "conclusion", as in "We examined whether to sell the product ⇒ We will not sell it". Also, the → symbol is often used to denote "changed to", as in the sentence "The interest rate changed. March 20% → April 21%". See also * Józef M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carel S
Carel is a given name, and may refer to: Arts * Carel Blotkamp, Dutch artist and art historian * Carel de Moor, Dutch etcher and painter * Carel Fabritius, Dutch painter and one of Rembrandt's most gifted pupils * Carel van Mander, Flemish painter, poet and biographer * Carel Vosmaer, Dutch poet and art-critic * Jacques-Philippe Carel (), Parisian cabinet-maker Education * Carel Gabriel Cobet, Dutch classical scholar * Carel van Schaik Carolus Philippus "Carel" van Schaik (born 15 June 1953, Rotterdam) is a Dutch primatologist who since 2004 is professor and director of the Anthropological Institute and Museum at the University of Zürich, Switzerland. Van Schaik studied biolog ..., Dutch professor and director of the Anthropological Institute and Museum at the University of Zürich, Switzerland Other fields * Carel Godin de Beaufort, Dutch nobleman and Formula One driver * Carel Victor Gerritsen (1850–1905), Dutch radical politician * Carel Jan Scheneider, Dutch for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Gries
David Gries (born April 26, 1939 in Flushing, Queens, New York) is an American computer scientist at Cornell University, United States mainly known for his books ''The Science of Programming'' (1981) and ''A Logical Approach to Discrete Math'' (1993, with Fred B. Schneider). He was Associate Dean for Undergraduate Programs in the Cornell University College of Engineering from 2003–2011. His research interests include programming methodology and related areas such as programming languages, related semantics, and logic. His son, Paul Gries, has been a co-author of an introductory textbook to computer programming using the language Python and is a professor teaching Stream in the Department of Computer Science at the University of Toronto. Life Gries earned a Bachelor of Science (B.S.) from Queens College in 1960. He spent the next two years working as a programmer-mathematician for the U.S. Naval Weapons Laboratory, where he met his wife, Elaine. He earned a Master of Scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roland Carl Backhouse
Roland Carl Backhouse (born 18 August 1948) is a British computer scientist and mathematician. , he is Emeritus Professor of Computing Science at the University of Nottingham. Early life and education Backhouse was born and raised in the Thorntree district of Middlesbrough, an industrial town in the north-east of England. In 1959, he won a place at the then all-male Acklam Hall Grammar School before going on to Churchill College, Cambridge, in 1966. His doctorate (Ph.D.) was completed under the supervision of Jim Cunningham at Imperial College London. Career Backhouse's career has included Royal Aircraft Establishment (1969–1970), Heriot-Watt University (1973–1982), University of Essex (1982–1986). He was formerly Professor of Computer Science at the University of Groningen (1986–1990) and Eindhoven University of Technology (1990–1999) in the Netherlands, before his position at the University of Nottingham. He was a member of the International Federation for Informa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Relation
In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive. Definition A homogeneous relation on the set is a ''transitive relation'' if, :for all , if and , then . Or in terms of first-order logic: :\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, where is the infix notation for . Examples As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy, too, is an ancestor of Carrie. On the other hand, "is the birth parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then this does not imply that Alice is the birth parent of Claire. What is more, it is antitransitive: Alice can ''never'' be the birth parent of Claire. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history and philology. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. In addition, he contributed to the field of library science: while serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would have served as a guide for many of Europe's largest libraries. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substitution (logic)
Substitution is a fundamental concept in logic. A substitution is a syntactic transformation on formal expressions. To apply a substitution to an expression means to consistently replace its variable, or placeholder, symbols by other expressions. The resulting expression is called a substitution instance, or instance for short, of the original expression. Propositional logic Definition Where ''ψ'' and ''φ'' represent formulas of propositional logic, ''ψ'' is a substitution instance of ''φ'' if and only if ''ψ'' may be obtained from ''φ'' by substituting formulas for symbols in ''φ'', replacing each occurrence of the same symbol by an occurrence of the same formula. For example: ::(R → S) & (T → S) is a substitution instance of: ::P & Q and ::(A ↔ A) ↔ (A ↔ A) is a substitution instance of: ::(A ↔ A) In some deduction systems for propositional logic, a new expression (a proposition) may be entered on a line of a derivation if it is a substitution instan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inference Rule
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of inference called ''modus ponens'' takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion. Typically, a rule of inference preserves truth, a semantic property. In many-valued logic, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are recursive are important; i.e. rule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
