
This article is a technical mathematical article in the area of predicate logic. For the ordinary English language meaning see Sentence (linguistics), for a less technical introductory article see Statement (logic).
In mathematical logic, a sentence of a predicate logic is a booleanvalued wellformed 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 connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic sentences by applying connectives and quantifiers.
A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For firstorder theories, interpretations are commonly called structures. 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 problem.
Example
The following example is in firstorder logic.

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

\exists x(x^2=y)
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.
See also
References

Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters.

.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.