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Functional Predicate
In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term. Functional predicates are also sometimes called mappings, but that term has additional meanings in mathematics. In a model, a function symbol will be modelled by a function. Specifically, the symbol ''F'' in a formal language is a functional symbol if, given any symbol ''X'' representing an object in the language, ''F''(''X'') is again a symbol representing an object in that language. In typed logic, ''F'' is a functional symbol with ''domain'' type T and ''codomain'' type U if, given any symbol ''X'' representing an object of type T, ''F''(''X'') is a symbol representing an object of type U. One can similarly define function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in zero variables is simply a constant symbol. Now consider a mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal 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] |
There Exists
There may refer to: * ''There'' (film), a 2009 Turkish film (Turkish title: ''Orada'') * ''There'' (virtual world) *''there'', a deictic adverb in English *''there'', an English pronoun used in phrases such as '' there is'' and ''there are'' {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Symbols
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a mathematical formula, formula or a mathematical expression. More formally, a ''mathematical symbol'' is any grapheme used in mathematical formulas and expressions. As formulas and expressions are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics. The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. The decimal digits are used for representing numbers through the Hindu–Arabic numeral system. Historically, upper-case letters were used for representing point (geometry), points in geometry, and lower-case letters were used for variable (mathematics), variables and constant (mathematics), constants. Letters are used for representin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo–Fraenkel Set Theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Schema Of Replacement
In set theory, the axiom schema of replacement is a Axiom schema, schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image (mathematics), image of any Set (mathematics), set under any definable functional predicate, mapping is also a set. It is necessary for the construction of certain infinite sets in ZF. The axiom schema is motivated by the idea that whether a class (set theory), class is a set depends only on the cardinality of the class, not on the rank (set theory), rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining Well-formed formula, formulas. Statement Suppose P is a definable binary relation (mathematics), relation (which may be a proper class) such that f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Material Conditional
The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol \to is interpreted as material implication, a formula P \to Q is true unless P is true and Q is false. Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language. Notation In logic and related fields, the material conditional is customarily notated with an infix operator \to. The material conditional is also notated using the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universally Quantify
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable. It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("", "", or sometimes by "" alone). Universal quantification is distinct from ''existential'' quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain. Quantification in general is covered in the article on quantification (logic). The universal quantifier is encoded as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schema (logic)
In logic, the logical form of a statement is a precisely specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language. The logical form of an argument is called the argument form of the argument. History The importance of the concept of form to logic was already recognized in ancient times. Aristotle, in the '' Prior Analytics'', was one of the first people to employ variable letters to represent valid inferences. Therefore, Jan Łukasiewicz claims that the introduction of variables was "one of Aristotle's greatest inventions." ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
For All
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable. It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("", "", or sometimes by "" alone). Universal quantification is distinct from ''existential'' quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain. Quantification in general is covered in the article on quantification (logic). The universal quantifier is e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Codomain
In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to refer to either the codomain or the ''image'' of a function. A codomain is part of a function if is defined as a triple where is called the '' domain'' of , its ''codomain'', and its '' graph''. The set of all elements of the form , where ranges over the elements of the domain , is called the ''image'' of . The image of a function is a subset of its codomain so it might not coincide with it. Namely, a function that is not surjective has elements in its codomain for which the equation does not have a solution. A codomain is not part of a function if is defined as just a graph. For example in set theory it is desirable to permit the domain of a function to be a proper class , in which case there is formally no such thin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metalogic
Metalogic is the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived ''about'' the languages and systems that are used to express truths. The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and the study of deductive systems is the branch that is known as proof theory. Overview Formal language A ''formal language'' is an organized set of symbols, the symbols of which precisely define it by shape and place. Such a language therefore can be defined without reference to the meanings of its expressions; it can exist before any interpretation is assigned to it—that is, befo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
