# Term (logic)

**termstermfirst-order termsconstantexpressionexpressionsfinite termsfirst-order termliterally similarsubterm**

In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact.wikipedia

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### Variable (mathematics)

**variablesvariableunknown**

A first-order term is recursively constructed from constant symbols, variables and function symbols.

In mathematical logic, a variable is either a symbol representing an unspecified term of the theory, or a basic object of the theory, which is manipulated without referring to its possible intuitive interpretation.

### Uninterpreted function

**empty theoryfree theoryfunction letter**

A first-order term is recursively constructed from constant symbols, variables and function symbols.

Function symbols are used, together with constants and variables, to form terms.

### First-order logic

**predicate logicfirst-orderpredicate calculus**

A first-order term is recursively constructed from constant symbols, variables and function symbols.

also term structure vs. representation.

### Well-formed formula

**formulamathematical formulaformulas**

In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact.

### Coefficient

**coefficientsleading coefficientfactor**

In the context of polynomials, sometimes term is used for a monomial with a coefficient: to 'collect like terms' in a polynomial is the operation of making it a linear combination of distinct monomials.

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression.

### Polynomial

**polynomial functionpolynomialsmultivariate polynomial**

In the context of polynomials, sometimes term is used for a monomial with a coefficient: to 'collect like terms' in a polynomial is the operation of making it a linear combination of distinct monomials.

That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms.

### Series (mathematics)

**infinite seriesseriespartial sum**

A series is often represented as the sum of a sequence of terms.

In modern terminology, any (ordered) infinite sequence of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the a_i one after the other.

### Like terms

**Combining like termsgrouping**

In the context of polynomials, sometimes term is used for a monomial with a coefficient: to 'collect like terms' in a polynomial is the operation of making it a linear combination of distinct monomials.

In algebra, like terms are terms that have the same variables and powers.

### Rewriting

**term rewritingterm rewriting systemrewrite system**

Besides in logic, terms play important roles in universal algebra, and rewriting systems.

To this end, each such number has to be encoded as a term.

### Ground expression

**ground termgroundground atom**

A term that doesn't contain any variables is called a ground term; a term that doesn't contain multiple occurrences of a variable is called a linear term.

In mathematical logic, a ground term of a formal system is a term that does not contain any free variables.

### Unification (computer science)

**unificationmost general unifierE-unification**

The syntactic first-order unification problem { y = cons(2,y) } has no solution over the set of finite terms; however, it has the single solution { y ↦ cons(2,cons(2,cons(2,...))) } over the set of infinite trees.

### Substitution (logic)

**substitutionsubstitution instancesubstituting**

In contrast, a term t is called a renaming, or a variant, of a term u if the latter resulted from consistently renaming all variables of the former, i.e. if u = tσ for some renaming substitution σ.

A substitution is called a ground substitution if it maps all variables of its domain to ground, i.e. variable-free, terms.

### Encompassment ordering

**encompassment preorder**

the containment, or encompassment, preorder on the set of terms, is defined by :s ≤ t if a subterm of t is a substitution instance of s.

### Expression (mathematics)

**expressionmathematical expressionexpressions**

### Equation

**equationsmathematical equationunknown**

### Noun phrase

**noun phrasesNPnominal phrase**

In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact.

### Sentence (linguistics)

**sentencesentencesdeclarative sentence**

In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact.

### Mathematical logic

**formal logicsymbolic logiclogic**

### Recursive definition

**inductive definitionrecursively definedinductively defined**

A first-order term is recursively constructed from constant symbols, variables and function symbols.

### Predicate (mathematical logic)

**predicatepredicatespredication**

An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation.

### Atomic formula

**atomatomicatomic expressions**

An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation.

### Truth

**trueTruth theorytheory of truth**

An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation.

### False (logic)

**falseFalsityfalsehood**

### Principle of bivalence

**bivalentBivalencetwo-valued logic**

### Interpretation (logic)

**interpretationintended interpretationinterpretations**