Zeroth-order logic
First-order logic without variables or quantifiers
Arity
In logic, mathematics, and computer science, the arity Examples[edit] The term "arity" is rarely employed in everyday usage. A nullary function takes no arguments.A unary function takes one argument.A binary function takes two arguments.A ternary function takes three arguments.An n-ary function takes n arguments. Nullary[edit] Unary[edit] Binary[edit] Most operators encountered in programming are of the binary form. Ternary[edit] with arbitrary precision. n-ary[edit] From a mathematical point of view, a function of n arguments can always be considered as a function of one single argument which is an element of some product space. The same is true for programming languages, where functions taking several arguments could always be defined as functions taking a single argument of some composite type such as a tuple, or in languages with higher-order functions, by currying. Variable arity[edit] In computer science, a function accepting a variable number of arguments is called variadic.
Structure
Arrangement of interrelated elements in an object/system, or the object/system itself Load-bearing[edit] Buildings, aircraft, skeletons, anthills, beaver dams, bridges and salt domes are all examples of load-bearing structures. The structure elements are combined in structural systems. Load-bearing biological structures such as bones, teeth, shells, and tendons derive their strength from a multilevel hierarchy of structures employing biominerals and proteins, at the bottom of which are collagen fibrils.[4] Biological[edit] In another context, structure can also observed in macromolecules, particularly proteins and nucleic acids.[6] The function of these molecules is determined by their shape as well as their composition, and their structure has multiple levels. Chemical[edit] Chemical structure refers to both molecular geometry and electronic structure. Mathematical[edit] Musical[edit] Social[edit] A social structure is a pattern of relationships. Data[edit] Software[edit] Logical[edit]
Cartesian product
Cartesian product of the sets and The simplest case of a Cartesian product is the Cartesian square, which returns a set from two sets. A Cartesian product of n sets can be represented by an array of n dimensions, where each element is an n-tuple. The Cartesian product is named after René Descartes,[1] whose formulation of analytic geometry gave rise to the concept. Examples[edit] A deck of cards[edit] An illustrative example is the standard 52-card deck. Ranks × Suits returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}. Suits × Ranks returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}. A two-dimensional coordinate system[edit] An example in analytic geometry is the Cartesian plane. Most common implementation (set theory)[edit] A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. . , where For example: Similarly
Model theory
This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang and Keisler (1990):[1] universal algebra + logic = model theory. Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997): although model theorists are also interested in the study of fields. In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. Branches of model theory[edit] This article focuses on finitary first order model theory of infinite structures. During the last several decades applied model theory has repeatedly merged with the more pure stability theory. and or
Signature (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. Signatures play the same role in mathematics as type signatures in computer programming. They are rarely made explicit in more philosophical treatments of logic. Formally, a (single-sorted) signature can be defined as a triple σ = (Sfunc, Srel, ar), where Sfunc and Srel are disjoint sets not containing any other basic logical symbols, called respectively function symbols (examples: +, ×, 0, 1) andrelation symbols or predicates (examples: ≤, ∈), and a function ar: Sfunc Srel → which assigns a non-negative integer called arity to every function or relation symbol. A signature with no function symbols is called a relational signature, and a signature with no relation symbols is called an algebraic signature. Symbol types S.
Higher-order logic
Formal system of logic The term "higher-order logic", abbreviated as HOL, is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types (see Type theory). Quantification scope[edit] First-order logic quantifies only variables that range over individuals; second-order logic, in addition, also quantifies over sets; third-order logic also quantifies over sets of sets, and so on. Higher-order logic is the union of first-, second-, third-, ..., nth-order logic; i.e., higher-order logic admits quantification over sets that are nested arbitrarily deeply. Semantics[edit] There are two possible semantics for higher-order logic. In the standard or full semantics, quantifiers over higher-type objects range over all possible objects of that type. In Henkin semantics, a separate domain is included in each interpretation for each higher-order type. Properties[edit] See also[edit]
First-order logic
A theory about some topic is usually first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions which map from that domain into it, finitely many predicates defined on that domain, and a recursive set of axioms which are believed to hold for those things. Sometimes "theory" is understood in a more formal sense, which is just a set of sentences in first-order logic. The adjective "first-order" distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted.[1] In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets. First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Introduction[edit] . x in .
