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Variable-free
In mathematical logic, a ground term of a formal system is a term that does not contain any variables. Similarly, a ground formula is a formula that does not contain any variables. In first-order logic with identity, the sentence Q(a) \lor P(b) is a ground formula, with a and b being constant symbols. A ground expression is a ground term or ground formula. Examples Consider the following expressions in first order logic over a signature containing the constant symbols 0 and 1 for the numbers 0 and 1, respectively, a unary function symbol s for the successor function and a binary function symbol + for addition. * s(0), s(s(0)), s(s(s(0))), \ldots are ground terms; * 0 + 1, \; 0 + 1 + 1, \ldots are ground terms; * 0+s(0), \; s(0)+ s(0), \; s(0)+s(s(0))+0 are ground terms; * x + s(1) and s(x) are terms, but not ground terms; * s(0) = 1 and 0 + 0 = 0 are ground formulae. Formal definitions What follows is a formal definition for first-order languages. Let a first-order language ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion 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 the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate Symbol
In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predicate variables include capital roman letters such as P, Q and R, or lower case roman letters, e.g., x. In first-order logic, they can be more properly called metalinguistic variables. In higher-order logic, predicate variables correspond to propositional variables which can stand for well-formed formulas of the same logic, and such variables can be quantified by means of (at least) second-order quantifiers. Notation Predicate variables should be distinguished from predicate constants, which could be represented either with a different (exclusive) set of predicate letters, or by their own symbols which really do have their own specific meaning in their domain of discourse: e.g. =, \ \in , \ \le,\ <, \ \sub,... . If letters are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truth Value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number zero, the empty string, empty lists, and null evaluate to false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called "truthy" and "falsy" / "false". Classical logic In classical logic, with its intended semantics, the truth values are ''true'' (denoted by ''1'' or the verum ⊤), and '' untrue'' or '' false'' (denoted by ''0'' or the falsum ⊥); that is, classical logic is a two-valued logic. This set of two values is also called the Boolean domain. Corresponding semantics of l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbrand Interpretation
In mathematical logic, a Herbrand interpretation is an interpretation in which all constants and function symbols are assigned very simple meanings. Specifically, every constant is interpreted as itself, and every function symbol is interpreted as the function that applies it. The interpretation also defines predicate symbols as denoting a subset of the relevant Herbrand base, effectively specifying which ground atoms are true in the interpretation. This allows the symbols in a set of clauses to be interpreted in a purely syntactic way, separated from any real instantiation. The importance of Herbrand interpretations is that, if any interpretation satisfies a given set of clauses ''S'' then there is a Herbrand interpretation that satisfies them. Moreover, Herbrand's theorem states that if ''S'' is unsatisfiable then there is a finite unsatisfiable set of ground instances from the Herbrand universe defined by ''S''. Since this set is finite, its unsatisfiability can be verified i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbrand Base
In first-order logic, a Herbrand structure ''S'' is a structure over a vocabulary σ that is defined solely by the syntactical properties of σ. The idea is to take the symbols of terms as their values, e.g. the denotation of a constant symbol ''c'' is just "''c''" (the symbol). It is named after Jacques Herbrand. Herbrand structures play an important role in the foundations of logic programming. Herbrand universe Definition The ''Herbrand universe'' serves as the universe in the ''Herbrand structure''. Example Let , be a first-order language with the vocabulary * constant symbols: ''c'' * function symbols: ''f''(·), ''g''(·) then the Herbrand universe of (or ) is . Notice that the relation symbols are not relevant for a Herbrand universe. Herbrand structure A ''Herbrand structure'' interprets terms on top of a ''Herbrand universe''. Definition Let ''S'' be a structure, with vocabulary σ and universe ''U''. Let ''W'' be the set of all terms over σ and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atomic Formula
In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives. The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, a propositional variable is often more briefly referred to as an "atomic formula", but, more precisely, a propositional variable is not an atomic formula but a formal expression that denotes an atomic formula. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. In model theory, atomic formulas are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbrand Universe
In first-order logic, a Herbrand structure ''S'' is a structure over a vocabulary σ that is defined solely by the syntactical properties of σ. The idea is to take the symbols of terms as their values, e.g. the denotation of a constant symbol ''c'' is just "''c''" (the symbol). It is named after Jacques Herbrand. Herbrand structures play an important role in the foundations of logic programming. Herbrand universe Definition The ''Herbrand universe'' serves as the universe in the ''Herbrand structure''. Example Let , be a first-order language with the vocabulary * constant symbols: ''c'' * function symbols: ''f''(·), ''g''(·) then the Herbrand universe of (or ) is . Notice that the relation symbols are not relevant for a Herbrand universe. Herbrand structure A ''Herbrand structure'' interprets terms on top of a ''Herbrand universe''. Definition Let ''S'' be a structure, with vocabulary σ and universe ''U''. Let ''W'' be the set of all terms over σ and ''W' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Language
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 ax ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal System
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined abstraction, system of abstract thought. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Background Each formal system is described by primitive Symbol (formal), symbols (which collectively form an Alphabet (computer science), alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation. The system thus consists of valid formulas built up through finite combinations of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signature (mathematical Logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic. Definition Formally, a (single-sorted) signature can be defined as a 4-tuple , where ''S''func and ''S''rel are disjoint sets not containing any other basic logical symbols, called respectively * ''function symbols'' (examples: +, ×, 0, 1), * ''relation symbols'' or ''predicates'' (examples: ≤, ∈), * ''constant symbols'' (examples: 0, 1), and a function ar: ''S''func \cup ''S''rel → \mathbb N which assigns a natural number called ''arity'' to every function or relation symbol. A function or relation symbol is called ''n''-ary if its arity is ''n''. Some authors define a nullary (0-ary) function symbol as ''constant s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Order Logic
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] |
