# Sentence (mathematical logic)

﻿
Sentence (mathematical logic)

:"This article is a technical mathematical article in the area of predicate logic. For the ordinary English language meaning see Sentence, for a less technical introductory article see Statement (logic)."

In mathematical logic, a sentence of a predicate logic is a well formed formula with no free variables. A sentence is viewed by some as expressing a proposition. It makes an assertion, potentially concerning any structure of "L". This assertion has a fixed truth value with respect to the structure. In contrast, the truth value of a formula (with free variables) may be indeterminate with respect to any structure. As the free variables of a formula can range over several values (which could be members of a universe, relations or functions), its truth value may vary.

Example

The following example is in first-order logic.

:$forall yexists x \left(x^2=y\right)$

"is" a sentence. This sentence is true in the positive real numbers but false in the real 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

:$exists x\left(x^2=y\right)$

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 "y" = 2, but is false if "y" = –2.

ee also

* Ground expression
* Atomic sentence
* Open sentence
* Statement (logic)
* Proposition

References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Mathematical logic — (also known as symbolic logic) is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic.[1] The field includes both the mathematical study of logic and the… …   Wikipedia

• Theory (mathematical logic) — This article is about theories in a formal language, as studied in mathematical logic. For other uses, see Theory (disambiguation). In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. Usually… …   Wikipedia

• 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. Universal algebra studies structures that generalize the algebraic structures such as… …   Wikipedia

• Absoluteness (mathematical logic) — In mathematical logic, a formula is said to be absolute if it has the same truth value in each of some class of structures (also called models). Theorems about absoluteness typically show that each of a large syntactic class of formulas is… …   Wikipedia

• Independence (mathematical logic) — In mathematical logic, a sentence sigma; is called independent of a given first order theory T if T neither proves nor refutes sigma;; that is, it is impossible to prove sigma; from T , and it is also impossible to prove from T that sigma; is… …   Wikipedia

• Sentence — or sentencing may refer to:* Sentence (linguistics), a grammatical unit of language * Sentence (mathematical logic), a formula with no free variables * Sentence (music), the smallest period in a musical composition * Sentence (law), the final act …   Wikipedia

• Logic — For other uses, see Logic (disambiguation). Philosophy …   Wikipedia

• logic, philosophy of — Philosophical study of the nature and scope of logic. Examples of questions raised in the philosophy of logic are: In virtue of what features of reality are the laws of logic true? ; How do we know the truths of logic? ; and Could the laws of… …   Universalium

• logic, history of — Introduction       the history of the discipline from its origins among the ancient Greeks to the present time. Origins of logic in the West Precursors of ancient logic       There was a medieval tradition according to which the Greek philosopher …   Universalium

• Outline of logic — The following outline is provided as an overview of and topical guide to logic: Logic – formal science of using reason, considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and… …   Wikipedia