|
Clause (logic)
In logic, a clause is a propositional formula formed from a finite collection of literals (atoms or their negations) and logical connectives. A clause is true either whenever at least one of the literals that form it is true (a disjunctive clause, the most common use of the term), or when all of the literals that form it are true (a conjunctive clause, a less common use of the term). That is, it is a finite disjunction or conjunction of literals, depending on the context. Clauses are usually written as follows, where the symbols l_i are literals: :l_1 \vee \cdots \vee l_n Empty clauses A clause can be empty (defined from an empty set of literals). The empty clause is denoted by various symbols such as \empty, \bot, or \Box. The truth evaluation of an empty disjunctive clause is always false. This is justified by considering that false is the neutral element of the monoid (\, \vee). The truth evaluation of an empty conjunctive clause is always true. This is related to the concept ... [...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] |
Truth Assignment
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics. The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate ''T'' (for "tall") and assign it the extension (for "Abraham Lincoln"). Note that all our interpretation does is assign the extension to the non-logical constant ''T'', and does not make a claim about whether ''T'' is to stand for tall and 'a' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or Quantifier (logic), quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" (logical conjunction, conjunction), "or" (lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunctive Normal Form
In boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions; it can also be described as an OR of ANDs, a sum of products, or (in philosophical logic) a ''cluster concept''. As a normal form, it is useful in automated theorem proving. Definition A logical formula is considered to be in DNF if it is a disjunction of one or more conjunctions of one or more literals. A DNF formula is in full disjunctive normal form if each of its variables appears exactly once in every conjunction. As in conjunctive normal form (CNF), the only propositional operators in DNF are and (\wedge), or (\vee), and not (\neg). The ''not'' operator can only be used as part of a literal, which means that it can only precede a propositional variable. The following is a context-free grammar for DNF: # ''DNF'' → (''Conjunction'') \vee ''DNF'' # ''DNF'' → (''Conjunction'') # ''Conjunction'' → ''Literal'' \wedge ''Conju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunctive Normal Form
In Boolean logic, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs. As a canonical normal form, it is useful in automated theorem proving and circuit theory. All conjunctions of literals and all disjunctions of literals are in CNF, as they can be seen as conjunctions of one-literal clauses and conjunctions of a single clause, respectively. As in the disjunctive normal form (DNF), the only propositional connectives a formula in CNF can contain are and, or, and not. The not operator can only be used as part of a literal, which means that it can only precede a propositional variable or a predicate symbol. In automated theorem proving, the notion "''clausal normal form''" is often used in a narrower sense, meaning a particular representation of a CNF formula as a set of sets of literals. Examples and non-examples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horn Clause
In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form which gives it useful properties for use in logic programming, formal specification, and model theory. Horn clauses are named for the logician Alfred Horn, who first pointed out their significance in 1951. Definition A Horn clause is a clause (a disjunction of literals) with at most one positive, i.e. unnegated, literal. Conversely, a disjunction of literals with at most one negated literal is called a dual-Horn clause. A Horn clause with exactly one positive literal is a definite clause or a strict Horn clause; a definite clause with no negative literals is a unit clause, and a unit clause without variables is a fact;. A Horn clause without a positive literal is a goal clause. Note that the empty clause, consisting of no literals (which is equivalent to ''false'') is a goal clause. These three kinds of Horn clauses are illustrated in the following propositional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prolog
Prolog is a logic programming language associated with artificial intelligence and computational linguistics. Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is intended primarily as a declarative programming language: the program logic is expressed in terms of relations, represented as facts and rules. A computation is initiated by running a ''query'' over these relations. The language was developed and implemented in Marseille, France, in 1972 by Alain Colmerauer with Philippe Roussel, based on Robert Kowalski's procedural interpretation of Horn clauses at University of Edinburgh. Prolog was one of the first logic programming languages and remains the most popular such language today, with several free and commercial implementations available. The language has been used for theorem proving, expert systems, term rewriting, type systems, and automated planning, as well as its original intended field of use, nat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Programming
Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem domain. Major logic programming language families include Prolog, answer set programming (ASP) and Datalog. In all of these languages, rules are written in the form of ''clauses'': :H :- B1, …, Bn. and are read declaratively as logical implications: :H if B1 and … and Bn. H is called the ''head'' of the rule and B1, ..., Bn is called the ''body''. Facts are rules that have no body, and are written in the simplified form: :H. In the simplest case in which H, B1, ..., Bn are all atomic formulae, these clauses are called definite clauses or Horn clauses. However, there are many extensions of this simple case, the most important one being the case in which conditions in the body of a clause can also be negations of atomic formulas. Logic programming languag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Conjunction
In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this operator is typically written as \wedge or . A \land B is true if and only if A is true and B is true, otherwise it is false. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English "and". * In programming languages, the short-circuit and control structure. * In set theory, intersection. * In lattice theory, logical conjunction ( greatest lower bound). * In predicate logic, universal quantification. Notation And is usually denoted by an infix operator: in mathematics and logic, it is denoted by \wedge, or ; in electronics, ; and in programming languages, &, &&, or and. In Jan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Formula
In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula. A propositional formula is constructed from simple propositions, such as "five is greater than three" or propositional variables such as ''p'' and ''q'', using connectives or logical operators such as NOT, AND, OR, or IMPLIES; for example: : (''p'' AND NOT ''q'') IMPLIES (''p'' OR ''q''). In mathematics, a propositional formula is often more briefly referred to as a "proposition", but, more precisely, a propositional formula is not a proposition but a formal expression that ''denotes'' a proposition, a formal object under discussion, just like an expression such as "" is not a value, but denotes a value. In some contexts, maintaining the distincti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Implication
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?Beall, JC and Restall, Greg, Logical Consequence' The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.). All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical consequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuous Truth
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she does not own a cell phone" will imply that the statement "all of her cell phones are turned off" will be assigned a truth value. Also, the statement "all of her cell phones are turned ''on''" would also be vacuously true, as would the conjunction of the two: "all of her cell phones are turned on ''and'' turned off", which would otherwise be incoherent and false. For that reason, it is sometimes said that a statement is vacuously true because it is meaningless. More formally, a relatively well-defined usage refers to a conditional statement (or a universal conditional statement) with a false antecedent. One example of such a statement is "if Tokyo is in France, then the Eiffel Tower is in Bolivia". Such statements are considered vacuous t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
