Definition:Statement
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Definition
A statement is a sentence which has objective and logical meaning.
In a non-mathematical / logical context, the term statement has a wider and looser meaning than this.
Equivalent terms for statement are:
- Assertion
- Declarative 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.
- In Aristotelian logic all statements have a truth value that is either true or false.
- In multi-value logic, it is admissible for a statement to have a truth value other than those two values.
Symbolic Logic
In the various branches of symbolic logic, statements are assigned symbols:
A statement label is a symbol which is assigned to a particular statement, so that it can be identified without the need to write it out in full.
A statement variable is a symbol which is used to stand for arbitrary and unspecified statements.
The citing of a statement label or variable can be interpreted as an assertion that the statement represented by that symbol is true.
That is:
- $p$
means
- $p \text { is true}$
Simple and Compound
A statement may be either:
- simple, consisting of just a subject and a predicate, or
- compound, which is a statement that consists of one or more simple statements joined together by logical connectives.
Axioms, Theorems and Assumptions
In order to determine the truth value of statements, one subjects them to the process of argument.
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:
- An assumption is a statement, introduced into an argument, whose truth value (temporarily) accepted as true.
- A premise (sometimes spelt "premiss") is an assumption that is used as a basis from which to start to construct an argument.
- A conclusion is a statement that is obtained as the result of the process of an argument.
- A theorem is a statement which can be shown to be the conclusion of an argument which can be obtained as the result of no premises.
The word hypothesis is 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. (See, for example, the famous Riemann Hypothesis.)
There are other sorts 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?"
- Exhortations:
- "May the powers that be strike me down here and now if six multiplied by nine isn't forty-two in base thirteen!"
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.
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 2$: The Axiom of Specification
- Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{I}: \S 1$
- E.J. Lemmon: Beginning Logic (1965): $\S 1.2$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 1$
- Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.1$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 3$
- D.J. O'Connor and Betty Powell: Elementary Logic (1980): $\S 1.1$
- M. Ben-Ari: Mathematical Logic for Computer Science (1993): $\S 1.2$
- Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000): $\S 1.1$
