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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 veri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of 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 analysis. In the early 20th century it was shaped by David Hilbert's 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 proving consistency. Work in set theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpretation (logic)
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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Constant
In logic, a logical constant of a language \mathcal is a symbol that has the same semantic value under every interpretation of \mathcal. Two important types of logical constants are logical connectives and quantifiers. The equality predicate (usually written '=') is also treated as a logical constant in many systems of logic. One of the fundamental questions in the philosophy of logic is "What is a logical constant?"; that is, what special feature of certain constants makes them ''logical'' in nature? Some symbols that are commonly treated as logical constants are: Many of these logical constants are sometimes denoted by alternate symbols (''e.g.'', the use of the symbol "&" rather than "∧" to denote the logical and). Defining logical constants is a major part of the work of Gottlob Frege and Bertrand Russell. Russell returned to the subject of logical constants in the preface to the second edition (1937) of '' The Principles of Mathematics'' noting that logic becomes lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of ... [...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] |
Ground Atom
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] |
Syntax
In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure ( constituency), agreement, the nature of crosslinguistic variation, and the relationship between form and meaning (semantics). There are numerous approaches to syntax that differ in their central assumptions and goals. Etymology The word ''syntax'' comes from Ancient Greek roots: "coordination", which consists of ''syn'', "together", and ''táxis'', "ordering". Topics The field of syntax contains a number of various topics that a syntactic theory is often designed to handle. The relation between the topics is treated differently in different theories, and some of them may not be considered to be distinct but instead to be derived from one another (i.e. word order can be seen as the result of movement rules derived from grammatical relations). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Satisfiable
In mathematical logic, a formula is ''satisfiable'' if it is true under some assignment of values to its variables. For example, the formula x+3=y is satisfiable because it is true when x=3 and y=6, while the formula x+1=x is not satisfiable over the integers. The dual concept to satisfiability is validity; a formula is ''valid'' if every assignment of values to its variables makes the formula true. For example, x+3=3+x is valid over the integers, but x+3=y is not. Formally, satisfiability is studied with respect to a fixed logic defining the syntax of allowed symbols, such as first-order logic, second-order logic or propositional logic. Rather than being syntactic, however, satisfiability is a semantic property because it relates to the ''meaning'' of the symbols, for example, the meaning of + in a formula such as x+1=x. Formally, we define an interpretation (or model) to be an assignment of values to the variables and an assignment of meaning to all other non-logical sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Herbrand's Theorem
Herbrand's theorem is a fundamental result of mathematical logic obtained by Jacques Herbrand (1930). It essentially allows a certain kind of reduction of first-order logic to propositional logic. Although Herbrand originally proved his theorem for arbitrary formulas of first-order logic, the simpler version shown here, restricted to formulas in prenex form containing only existential quantifiers, became more popular. Statement Let :(\exists y_1,\ldots,y_n)F(y_1,\ldots,y_n) be a formula of first-order logic with F(y_1,\ldots,y_n) quantifier-free, though it may contain additional free variables. This version of Herbrand's theorem states that the above formula is valid if and only if there exists a finite sequence of terms t_, possibly in an expansion of the language, with :1 \le i \le k and 1 \le j \le n, such that :F(t_,\ldots,t_) \vee \ldots \vee F(t_,\ldots,t_) is valid. If it is valid, it is called a ''Herbrand disjunction'' for :(\exists y_1,\ldots,y_n)F(y_1,\ldots ... [...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] |
Jacques Herbrand
Jacques Herbrand (12 February 1908 – 27 July 1931) was a French mathematician. Although he died at age 23, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse and Richard Courant. He worked in mathematical logic and class field theory. He introduced recursive functions. '' Herbrand's theorem'' refers to either of two completely different theorems. One is a result from his doctoral thesis in proof theory, and the other one half of the Herbrand–Ribet theorem. The Herbrand quotient is a type of Euler characteristic, used in homological algebra. He contributed to Hilbert's program in the foundations of mathematics by providing a constructive consistency proof for a weak system of arithmetic. The proof uses the above-mentioned, proof-theoretic Herbrand's theorem. Biography Herbrand finished his doctorate at École Normale Supérieure in Paris under Ernest Vessiot in 1929. He joined the army in October 1929, ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
