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Prenex
A formula of the predicate calculus is in prenex normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. Together with the normal forms in propositional logic (e.g. disjunctive normal form or conjunctive normal form), it provides a canonical normal form useful in automated theorem proving. Every formula in classical logic is equivalent to a formula in prenex normal form. For example, if \phi(y), \psi(z), and \rho(x) are quantifier-free formulas with the free variables shown then :\forall x \exists y \forall z (\phi(y) \lor (\psi(z) \rightarrow \rho(x))) is in prenex normal form with matrix \phi(y) \lor (\psi(z) \rightarrow \rho(x)), while :\forall x ((\exists y \phi(y)) \lor ((\exists z \psi(z) ) \rightarrow \rho(x))) is logically equivalent but not in prenex normal form. Conversion to prenex form Every first-order formula is logically equivalent (in classical logic) to some f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantifier (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The mostly commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic, they are interdefinable using negation. They can also be used to define more complex quantifiers, as in the formula \neg \exists x P(x) which expresses that nothing has the property P. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantifier (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The mostly commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic, they are interdefinable using negation. They can also be used to define more complex quantifiers, as in the formula \neg \exists x P(x) which expresses that nothing has the property P. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Of Quantification
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The mostly commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic, they are interdefinable using negation. They can also be used to define more complex quantifiers, as in the formula \neg \exists x P(x) which expresses that nothing has the property P. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Predicate Calculus
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists''"'' is a quantifier, while ''x'' is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formula (mathematical Logic)
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic. Introduction A key use of formulas is in propositional logic and predicate logic such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven. Although the term "formula" may be used for written marks (for instance, on a piece of paper or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Variable
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol. In computer programming, the term free variable refers to variables used in a function that are neither local variables nor parameters of that function. The term non-local variable is often a synonym in this context. A bound variable, in contrast, is a variable that has been ''bound'' to a specific value or range of values in the domain of discourse or universe. This may be achieved through the use of logical quantifiers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytical Hierarchy
In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers, \mathbb, and over functions from \mathbb to \mathbb. The analytical hierarchy of sets classifies sets by the formulas that can be used to define them; it is the lightface version of the projective hierarchy. The analytical hierarchy of formulas The notation \Sigma^1_0 = \Pi^1_0 = \Delta^1_0 indicates the class of formulas in the language of second-order arithmetic with number quantifiers but no set quantifiers. This language does not contain set parameters. The Greek letters here are lightface symbols, which indicate this choice of language. Each corresponding boldface symbol denotes the corresponding class of formulas in the extended language with a parameter for each real; see projective hierarch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetical Hierarchy
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic. The Tarski–Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets. The arithmetical hierarchy of formulas The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted \Sigma^0_n and \Pi^0_n for natural numbers ''n'' (inclu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Calculus
In mathematical logic, a proof calculus or a proof system is built to prove statements. Overview A proof system includes the components: * Language: The set ''L'' of formulas admitted by the system, for example, propositional logic or first-order logic. * Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems. * Axioms: Formulas in ''L'' assumed to be valid. All theorems are derived from axioms. Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determined and can be used for radically different logics. For example, a paradigmatic case is the sequent calculus, which can be used to express the consequence relations of both intuitionistic logic and relevance logic. Thus, loosely speaking, a proof calculus is a template or design pattern, characterized by a certain style of formal inference, that may be specialized to produce specific formal systems, namely by specifying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BHK Interpretation
BHK is a three-letter abbreviation that may refer to: * BHK interpretation of intuitionistic predicate logic * Baby hamster kidney cells used in molecular biology * Bachelor of Human Kinetics (BHk) degree. * Baltische Historische Kommission, organization dealing with history of Baltic Germans * ''Biblia Hebraica'' (Kittel), by Rudolf Kittel * Bush Hill Park railway station Bush Hill Park is a London Overground station on the branch of the Lea Valley lines, serving the neighbourhood of Bush Hill Park in the London Borough of Enfield, north London. It is down the line from London Liverpool Street and is situate ..., London, UK, National Rail station code * Bukhara International Airport, Uzbekistan, IATA code * Prosperous Armenia, Armenian political party * ''Bedroom - Hall - Kitchen'', as used in India to describe apartments: 2 BHK, 3 BHK… {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intuitionistic Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scope (logic)
In logic, the scope of a quantifier or a quantification is the range in the formula where the quantifier "engages in". It is put right after the quantifier, often in parentheses. Some authors describe this as including the variable put right after the forall or exists symbol. In the formula , for example, (or ) is the scope of the quantifier (or ). A variable in the formula is free, if and only if it does not occur in the scope of any quantifier for that variable. A term is free for a variable in the formula (i.e. free to substitute that variable that occurs free), if and only if that variable does not occur free in the scope of any quantifier for any variable in the term. See also * Modal scope fallacy A fallacy of necessity is a fallacy in the logic of a syllogism whereby a degree of unwarranted necessity is placed in the conclusion. Example :a) ''Bachelors are necessarily unmarried.'' :b) ''John is a bachelor.'' :Therefore, c) ''John cannot ... Notes {{reflist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
