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 first-order logic#Formation rules, term. In model theory, atomic formulas are merely string (computer science), strings of symbols with a given signature ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability 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 th ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Term Algebra
Term may refer to: Language *Terminology, context-specific nouns or compound words **Technical term (or ''term of art''), used by specialists in a field ***Scientific terminology, used by scientists *Term (argumentation), part of an argument in debate theory Law and finance *Contractual term, a provision in a contract **Credit repayment terms **Payment terms, "net ''D''" on a trade invoice **Purchase order#Legal, Purchase order, invoice terms more generally *Term life insurance Lengths of time *Academic term, part of a year at school or university *Term of office, a set period a person serves in an elected office *Term of patent, the period of enforcement of patent rights *Term of a pregnancy *Prison sentence Mathematics and physics *Term (logic), a component of a logical or mathematical expression (not to be confused with term logic, or Aristotelian logic) **Ground term, a term with no variables *Term (arithmetic), or addend, an operand to the addition operator **Term of a summ ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Predicate Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, 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. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. 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, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Atomic Sentence
In logic and analytic philosophy, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences. For example, "The dog ran" is atomic whereas "The dog ran and the cat hid" is molecular in natural language. From a logical analysis point of view, the truth of a sentence is determined by only two things: * the logical form of the sentence. * the truth of its underlying atomic sentences. That is to say, for example, that the truth of the sentence "John is Greek and John is happy" is a function of the meaning of " and", and the truth values of the atomic sentences "John is Greek" and "John is happy". However, the truth of an atomic sentence is not a matter that is within the scope of logic itself, but rather whatever art or science the content of the atomic sentence happens to be talking about. Logic has developed artificial languages, ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Focusing (proof Theory)
Focusing may refer to: * Adjusting an optical system to minimize defocus aberration * Focusing (psychotherapy), a psychotherapeutic technique See also *Focus (optics) In geometrical optics, a focus, also called an image point, is a point where ray (optics), light rays originating from a point on the object vergence (optics), converge. Although the focus is conceptually a point, physically the focus has a s ... * Focus (other) {{dab ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Polarity (proof Theory)
Polarity may refer to: Science *Electrical polarity, direction of electrical current *Polarity (mutual inductance), the relationship between components such as transformer windings *Polarity (projective geometry), in mathematics, a duality of order two * Polarity in embryogenesis, the animal and vegetal poles within a blastula *Cell polarity, differences in the shape, structure, and function of cells *Chemical polarity, in chemistry, a separation of electric charge * Magnetic polarity, north or south poles of a magnet *Polar reciprocation, a concept in geometry also known as polarity * Trilinear polarity, a concept in geometry of the triangle *Polarity of a literal, in mathematical logic Humanities *Polarity (international relations), a description of the distribution of power within the international system * Polarity of gender, when a word takes the opposite grammatical gender than expected *Polarity item, in linguistics, the sensitiveness of some expression to negative or affir ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Proof Theory
Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Structure (mathematical Logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures of first-order theories with no relation symbols. Model theory has a different scope that encompasses more arbitrary first-order theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic, cf. also Tarski's theory of truth or Tarskian semantics. For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a '' semantic model'' when one discusses the notion in the more general setting of mathematical models. Log ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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 most commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic: each can be defined in terms of the other 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 ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Predicate (mathematics)
In logic, a predicate is a symbol that represents a property or a relation. For instance, in the first-order formula P(a), the symbol P is a predicate that applies to the individual constant a. Similarly, in the formula R(a,b), the symbol R is a predicate that applies to the individual constants a and b. According to Gottlob Frege, the meaning of a predicate is exactly a function from the domain of objects to the truth values "true" and "false". In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula R(a,b) would be true on an interpretation if the entities denoted by a and b stand in the relation denoted by R. Since predicates are non-logical symbols, they can denote different relations depending on the interpretation given to them. While first-order logic only includes predicates that apply to individual objects, other logics may allow predicates that apply to collections of objects defined ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Tuple
In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is only one 0-tuple, called the ''empty tuple''. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term ''"infinite tuple"'' is occasionally used for ''"infinite sequences"''. Tuples are usually written by listing the elements within parentheses "" and separated by commas; for example, denotes a 5-tuple. Other types of brackets are sometimes used, although they may have a different meaning. An -tuple can be formally defined as the image of a function that has the set of the first natural numbers as its domain. Tuples may be also defined from ordered pairs by a recurrence starting from an ordered pair; indeed, an -tuple can be identified with the ordered pair of its first elements and its t ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Domain Of Discourse
In the formal sciences, the domain of discourse or universe of discourse (borrowing from the mathematical concept of ''universe'') is the set of entities over which certain variables of interest in some formal treatment may range. It is also defined as the collection of objects being discussed in a specific discourse. In model-theoretical semantics, a universe of discourse is the set of entities that a model is based on. The domain of discourse is usually identified in the preliminaries, so that there is no need in the further treatment to specify each time the range of the relevant variables. Many logicians distinguish, sometimes only tacitly, between the ''domain of a science'' and the ''universe of discourse of a formalization of the science''. Etymology The concept ''universe of discourse'' was used for the first time by George Boole (1854) on page 42 of his '' Laws of Thought'': The concept, probably discovered independently by Boole in 1847, played a crucial role i ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |