:''This article is a technical mathematical article in the area of predicate logic. For the ordinary English language meaning see

Sentence (linguistics) In linguistics and grammar, a sentence is a linguistic expression, such as the English example "The quick brown fox jumps over the lazy dog." In traditional grammar, it is typically defined as a string of words that expresses a complete thought, ...

, for a less technical introductory article see Statement (logic).'' In

mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, a sentence (or closed formula)Edgar Morscher, "Logical Truth and Logical Form", ''Grazer Philosophische Studien'' 82(1), pp. 77–90. of a

predicate 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 quantifie ...

is a Boolean-valued

well-formed formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can ...

with no

free variables In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...

. A sentence can be viewed as expressing a

proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

, something that ''must'' be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed

truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Computing In some pro ...

s: 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 In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

s or quantifiers in them are known as

atomic sentence In logic and analytic philosophy, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences ...

s; by analogy to

atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime 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 subformu ...

. Sentences are then built up out of atomic formulas by applying connectives and quantifiers. A set of sentences is called a

theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...

; thus, individual sentences may be called

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

s. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called

structures A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...

. Given a structure or interpretation, a sentence will have a fixed

. A theory is satisfiable when it is possible to present an interpretation in which 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 In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involvi ...

problem.

Example

The following example 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 quantifie ...

\forall y\exists x (x^2=y)

a sentence. This sentence is true in the

positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...

ℝ⁺, false in the real numbers ℝ, and true in the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s ℂ. (In plain English, this sentence is interpreted to mean that every member of the structure concerned is the

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

of a member of that particular structure.) On the other hand, the formula :

\exists x(x^2=y)

is 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. It is the presence of a free variable, rather than the inconstant truth value, that is important; for example, even in the structure of the complex numbers, where the statement is always true, it is still not considered a sentence. Such a formula may be called a

predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...

instead.

References

* * . {{logic-stub Predicate logic Propositions fr:Proposition (logique mathématique)

Example

See also

References