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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 (pro ... [...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 usually un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
False (logic)
In logic, false or untrue is the state of possessing negative truth value or a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, O''pq''), and the up tack symbol \bot. Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), \bot, is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition, and is often called absurdity. In classical logic and Boolean logic In Boolean logic, each variable denotes a truth value which can be either true (1), or false (0). In a classical propositional calculus, each proposition will be assigned a truth value of either true or false. Some systems of classical l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x \mapsto -x), reciprocation (x \mapsto 1/x), and complex conjugation (z \mapsto \bar z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions ''f'' and ''g'' is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: :a_0 = a_1 = 1 and a_n = a_ + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logically Equivalent
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 usually und ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish Notation
Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators ''precede'' their operands, in contrast to the more common infix notation, in which operators are placed ''between'' operands, as well as reverse Polish notation (RPN), in which operators ''follow'' their operands. It does not need any parentheses as long as each operator has a fixed number of operands. The description "Polish" refers to the nationality of logician Jan Łukasiewicz, who invented Polish notation in 1924. The term ''Polish notation'' is sometimes taken (as the opposite of ''infix notation'') to also include reverse Polish notation. When Polish notation is used as a syntax for mathematical expressions by programming language interpreters, it is readily parsed into abstract syntax trees and can, in fact, define a one-to-one representation for the same. Because of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Semantics (mathematical Logic)
In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, boolean algebras with an interior operator. Other modal logics are characterized by various other algebras with operators. The class of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositional intuitionistic logic. MV-algebras are the algebraic semantics of Łukasiewicz logic. See also * Algebraic semantics (computer science) * Lindenbaum–Tarski algebra Further reading * (2nd published by ASL in 2009open accessat Project Euclid * * * Good introduction for readers with prior exposure to non-classical logics but without much background in order theory and/or universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures thems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heyting Algebra
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of ''implication'' such that (''c'' ∧ ''a'') ≤ ''b'' is equivalent to ''c'' ≤ (''a'' → ''b''). From a logical standpoint, ''A'' → ''B'' is by this definition the weakest proposition for which modus ponens, the inference rule ''A'' → ''B'', ''A'' ⊢ ''B'', is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by to formalize intuitionistic logic. As lattices, Heyting algebras are distributive. Every Boolean algebra is a Heyting algebra when ''a'' → ''b'' is defined as ¬''a'' ∨ ''b'', as is every complete distributive lattice satisfying a one-sided infinite distributive law when ''a'' → ''b'' is taken to be the supremum of the set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (order Theory)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These ''lattice-like'' structures all admit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S , assuming that R abbreviates "it is raining" and S abbreviates "it is snowing". In classical logic, disjunction is given a truth functional semantics according to which a formula \phi \lor \psi is true unless both \phi and \psi are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an ''inclusive'' interpretation of disjunction, in contrast with exclusive disjunction. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. Disjunction has also been given numerous non-classical treatments, motivated by problems including Aristotle's sea battle argument, Heisenberg's uncertainty principle, as well ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
