In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, a sentence of a predicate logic

Predicate logic

In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...

 is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may 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 connective

Logical connective

In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...

s or quantifiers in them are known as atomic sentence

Atomic sentence

In logic, an atomic sentence is a type of declarative sentence which is either true or false and which cannot be broken down into other simpler sentences...

s; by analogy to atomic formula

Atomic formula

In mathematical logic, an atomic 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...

. Sentences are then built up out of atomic sentences by applying connectives and quantifiers.

A set of sentences is called a theory

Theory (mathematical logic)

In mathematical logic, a theory is a set of sentences in a formal language. Usually a deductive system is understood from context. An element \phi\in T of a theory T is then called an axiom of the theory, and any sentence that follows from the axioms is called a theorem of the theory. Every axiom...

; thus, individual sentences may be called theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation

Interpretation (logic)

An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...

 of the theory. For first-order theories, interpretations are commonly called structures

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 which are defined on it....

. Given a structure or interpretation, a sentence will have a fixed truth value. A theory is satisfiable when all of its sentences are true. The study of algorithms to automatically discover interpretations of theories that render all sentences as being true is known as the satisfiability modulo theories

Satisfiability Modulo Theories

In computer science, the Satisfiability Modulo Theories problem is a decision problem for logical formulas with respect to combinations of background theories expressed in classical first-order logic with equality...

 problem.

Example

The following example is in first-order logic

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

.


is a sentence. This sentence is true in the positive real numbers, false in the real numbers, and true in the complex numbers. (In plain English, this sentence is interpreted to mean that every member of the structure concerned is the square of a member of that particular structure.) On the other hand, the formula


is not a sentence, because of the presence of the free variable y. In the structure of the real numbers, this formula is true if we substitute (arbitrarily) y = 2, but is false if y = –2.