In mathematics and other

formal sciences Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, gam ...

, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of higher degree", or * "without self-reference", as in

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quanti ...

and other logic uses, where it is contrasted with "allowing some self-reference" (higher-order logic) In detail, it may refer to:

Mathematics

* First-order approximation * First-order arithmetic *

First-order condition In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information abo ...

First-order hold First-order hold (FOH) is a mathematical model of the practical reconstruction of sampled signals that could be done by a conventional digital-to-analog converter (DAC) and an analog circuit called an integrator. For FOH, the signal is reconstructe ...

, a mathematical model of the practical reconstruction of sampled signals *

First-order inclusion probability In statistics, in the theory relating to sampling from finite populations, the sampling probability (also known as inclusion probability) of an element or member of the population, is its probability of becoming part of the sample during the dra ...

First Order Inductive Learner In machine learning, first-order inductive learner (FOIL) is a rule-based learning algorithm. Background Developed in 1990 by Ross Quinlan,J.R. Quinlan. Learning Logical Definitions from Relations. Machine Learning, Volume 5, Number 3, 1990/ref> F ...

, a rule-based learning algorithm * First-order reduction, a very weak type of reduction between two computational problems *

First-order resolution In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a Completeness (logic)#Refutation completeness, refutation complete theorem-proving technique for sentences in propositional logic and first-order lo ...

* First-order stochastic dominance * First order stream

Differential equations

Exact first-order ordinary differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

First-order differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

First-order differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...

First-order linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = ...

First-order non-singular perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...

* First-order partial differential equation, a partial differential equation that involves only first derivatives of the unknown function of ''n'' variables * Order of accuracy

Logic

First-order language First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantif ...

First-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quanti ...

, a formal logical system used in mathematics, philosophy, linguistics, and computer science * First-order predicate, a predicate that takes only individual(s) constants or variables as argument(s) *

First-order predicate calculus First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantif ...

First-order theorem provers Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a maj ...

First-order theory First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

Monadic first-order logic In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols in the signature are monadic (that is, they take only one argument), and there are no function symbo ...

Chemistry

First-order fluid A first-order fluid is another name for a power-law fluid with exponential dependence of viscosity on temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measu ...

, another name for a power-law fluid with exponential dependence of viscosity on temperature * First-order reaction a first-order chemical reaction *

First-order transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...

Computer science

First-order abstract syntax In computer science, the abstract syntax of data is its structure described as a data type (possibly, but not necessarily, an abstract data type), independent of any particular representation or encoding. This is particularly used in the representa ...

First-order function In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following: * takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itse ...

First-order query In descriptive complexity, a query is a mapping from structures of one signature to structures of another vocabulary. Neil Immerman, in his book Descriptive Complexity, "use the concept of query as the fundamental paradigm of computation" (p. ...

Other uses

First-order desire Higher-order volitions (or higher-order desire), as opposed to action-determining volitions, are volitions about volitions. Higher-order volitions are potentially more often guided by long-term beliefs and reasoning. A first-order volition is a ...

First-order election ''Second-order election'' is a term that appeared for the first time in Karlheinz Reif and Hermann Schmitt's "Nine second-order national elections – A conceptual framework for the analysis of European election results" article for the uropean Jou ...

, in political science, the relative importance of certain elections *