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Assignment (mathematical Logic)
In logic and model theory, a valuation can be: *In propositional logic, an assignment of truth values to propositional variables, with a corresponding assignment of truth values to all propositional formulas with those variables. *In first-order logic and higher-order logics, a structure, (the interpretation) and the corresponding assignment of a truth value to each sentence in the language for that structure (the valuation proper). The interpretation must be a homomorphism, while valuation is simply a function. Mathematical logic In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. Valuations are also called truth assignments. In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives. In this context, a valuation begins with an assignment of a truth value to each propositional variable. This assignment can be uniquely e ... [...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 study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T-schema
The T-schema ("truth schema", not to be confused with " Convention T") is used to check if an inductive definition of truth is valid, which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth. Some authors refer to it as the "Equivalence Schema", a synonym introduced by Michael Dummett. The T-schema is often expressed in natural language, but it can be formalized in many-sorted predicate logic or modal logic; such a formalisation is called a "T-theory." T-theories form the basis of much fundamental work in philosophical logic, where they are applied in several important controversies in analytic philosophy. As expressed in semi-natural language (where 'S' is the name of the sentence abbreviated to S): 'S' is true if and only if S. Example: 'snow is white' is true if and only if snow is white. The inductive definition By using the schema one can give an inductive definition for the truth of compound sentences. Atomic sentences are assigned truth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semantic Units
Semantics is the study of linguistic meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction between sense and reference. Sense is given by the ideas and concepts associated with an expression while reference is the object to which an expression points. Semantics contrasts with syntax, which studies the rules that dictate how to create grammatically correct sentences, and pragmatics, which investigates how people use language in communication. Lexical semantics is the branch of semantics that studies word meaning. It examines whether words have one or several meanings and in what lexical relations they stand to one another. Phrasal semantics studies the meaning of sentences by exploring the phenomenon of compositionality or how new meanings can be created by arranging words. Formal semantics relies on logic and mathematics to provide precise frameworks o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Semantics (mathematical Logic)
In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, boolean algebras with an interior operator. Other modal logics are characterized by various other algebras with operators. The class of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositional intuitionistic logic. MV-algebras are the algebraic semantics of Łukasiewicz logic. See also * Algebraic semantics (computer science) * Lindenbaum–Tarski algebra Further reading * (2nd published by ASL in 2009open accessat Project Euclid * * * Good introduction for readers with prior exposure to non-classical logics but without much background in order theory and/or universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Variables
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. A ''free variable'' is a Mathematical notation, notation (symbol) that specifies places in an expression (mathematics), expression where Substitution (logic), substitution may take place and is not a parameter of this or any container expression. 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 variable (programming), variables used in a function (computer science), function that are neither local variables nor parameter (computer programming), parameters of that function. The term non-local variable is often a synonym in this co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sentence (mathematical Logic)
In mathematical logic, a sentence (or closed formula)Edgar Morscher, "Logical Truth and Logical Form", ''Grazer Philosophische Studien'' 82(1), pp. 77–90. of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that ''must'' be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary. Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic sentences by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
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] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. 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. History of the function concept, 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 function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
