# Definition:Statement

## Contents

## Definition

A **statement** is a sentence which has objective and logical meaning.

## Also defined as

In a non-mathematical / logical context, the term **statement** has a wider and looser meaning than this.

In the field of computer science, where it is more usual to encounter **commands** and **questions**, the term **statement** is generally used to encompass *all* types of sentence; what we refer to as a **statement** tends to be given the term **assertion**.

In other fields of science, the term **statement** is usually tacitly understood as being a **true statement**, and in such a context such statements can be referred to as **laws of asserted statements**.

Some sources, in their definition of **statement**, specifically invoke Law of Excluded Middle and Principle of Non-Contradiction, and word the definition as:

- A
**statement**(or**proposition**) is a sentence which is either true or false and cannot be both true and false.

However, it needs to be understood that this definition restricts discussion to Aristotelian logic and does not encompass logic of the intuitionist school.

## Examples

Examples of statement include:

- $3 \times 4 = 11$ (which is false)
- $2^{19 \,937} - 1$ is a prime number (which is true)

## Also known as

Equivalent terms for **statement** are:

**Assertion****Declarative sentence****Indicative sentence****Declaration****Expression**(used in a wider context, and has a less precise interpretation)**Boolean expression**(used in the specific context of mathematical logic)

The term **proposition** is often seen for **statement**, but modern usage prefers to reserve the term proposition for something more specific.

Some sources use the word **sentence**, but that word is considered nowadays to have too wide a range of meanings to be precise enough in this context.

The term **relation** can sometimes be seen, to be used to mean a **statement composed of mathematical signs and objects**, but this usage of the word **relation** is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some sources, not having developed the necessary linguistic terms as successfully as the mathematical ideas behind them, use vague terms like **sentence of English**, which fails on multiple levels.

## Also see

In the various branches of symbolic logic, statements are assigned symbols:

- Definition:Statement Label: a symbol which is assigned to a particular statement, so that it can be identified without the need to write it out in full.

- Definition:Statement Variable: a symbol which is used to stand for arbitrary and unspecified statements.

During the course of an argument, statements perform different tasks. In this context, a statement is given a name according to what task it is doing, as follows:

- Definition:Axiom: a statement which is
*accepted*as true.

- Definition:Assumption: a statement, introduced into an argument, whose truth value is (temporarily) accepted as true.

- Definition:Premise: an assumption that is used as a basis from which to start to construct an argument.

- Definition:Conclusion: a statement that is obtained as the result of the process of an argument.

- Definition:Theorem: a statement which can be shown to be the conclusion of an argument which can be obtained as the result of no premises.

- Definition:Proposition: a statement whose truth value is about to be investigated.

- Definition:Hypothesis: sometimes used to mean either assumption or premise, but this tends nowadays to mean a statement whose truth is suspected, but has not actually been proven to be true.

- Definition:Aristotelian Logic, in which all statements have a truth value that is either true or false.

- Definition:Multi-Value Logic, in which it is admissible for a statement to have a truth value other than those two values.

## Other types of sentence

There are other types of sentences which may be encountered, for example:

**Questions**: for example:

- "What do you get if you multiply six by nine?"

**Commands**: for example:

- "Multiply six by nine."

Other types of sentence which are also technically **commands** are:

**Instructions**:

- "In order to solve this problem, you need to multiply six by nine."

**Requests**:

- "Would you please kindly multiply six by nine, if it's not
*too*much trouble?" - "Why don't you just sit right down there and multiply six by nine?"

- "Would you please kindly multiply six by nine, if it's not

**Exhortations**:

- "May the powers that be strike me down here and now if six multiplied by nine isn't forty-two in base thirteen!"

## Sources

- 1910: Alfred North Whitehead and Bertrand Russell:
*Principia Mathematica: Volume $\text { 1 }$*... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations - 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (next): $\S 1.1$: Constants and variables - 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 2.1$: Propositions and their Relations - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 2$: The Axiom of Specification - 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 1$ - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 1.2$: Conditionals and Negation - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (next): $\S 1$: Some mathematical language: Connectives - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $1.2$: The Nature of Argument - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 3$: Statements and conditions; quantifiers - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{I}: 1$: The Logic of Statements $(1)$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic - 1988: Alan G. Hamilton:
*Logic for Mathematicians*(2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.1$: Statements and connectives - 1993: M. Ben-Ari:
*Mathematical Logic for Computer Science*(1st ed.) ... (previous) ... (next): $\S 1.2$: Propositional and predicate calculus - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (next): $\S 1.1$: Declarative sentences - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if... - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (next): $\S 2$