Statement (logic)

In the area of mathematics called symbolic logic a statement is a declarative sentence that is either true or false.

Examples of statements:

*"Socrates is a man."
*"A triangle has three sides."
*"Paris is the capital of England."

The first two statements are true, the third is false.

Examples of things that are not statements:

*"Who are you?"
*"Run!"
*"I had one grunch but the eggplant over there."

The first two examples are not declarative sentences, the third is not a sentence at all, but a meaningless string of words.

Matters of opinion are also usually not considered to be statements.

Examples of declarative sentences that are not statements:

*"Lincoln was the greatest president of the United States."
*"Red is a pretty color."
*"Broccoli tastes good."

Predicates

A predicate is a statement that contains a variable. Whether or not the statement is true or false depends on the value of the variable.

Examples of predicates:

*"The number $x$ is even."
*"Today is Tuesday."
*"I like broccoli."

All three predicates are true if $x$ is four, today is May 27, 2008, and 'I' am the person writing these words. All three predicates are false if $x$ is five, today is May 28, 2008, and 'I' is George Herbert Walker Bush (an American president who stated publicly that he did not like broccoli).

The variable in a predicate may be understood rather than stated.

An example of an understood variable:

*"It is raining."

The truth of this predicate depends on the values of the variables place and time. It is always understood to mean "It is raining in this particular place at this particular time."

Quantifiers

Any predicate may be quantified. The quantifer is understood to bind the variable it quantifies, and a quantified variable is called a "bound variable" while an unquantified variable is called a "free variable". There are three quantifiers, "for all", "for some", and "for none", all of which can be expressed in various ways. A quantified predicate is either true or false, and you cannot replace a bound variable with a value.

Examples of quantified predicates:

*For all even numbers $x$, $x$ is divisible by two.
*For some even numbers $x$, $x$ is divisible by two.
*For none of the even numbers $x$, $x$ is divisible by two.

The first two quantified predicates are true, the third is false. In all three, the variable $x$ is bound, while in the unquantified predicate: "The number $x$ is divisible by two," the variable $x$ is free. The free variable could be replaced by a number, which would turn the unquantified predicate into a statement which would be either true or false. The bound variables in the quantified predicates cannot be replaced by numbers.

Statement as an abstract entity

In some treatments the term "statement" is introduced in order to distinguishing a sentence from its information content. A statement is regarded as the information content of an (information-bearing) sentence. Thus, a sentence is related to the statement it bears like a numeral to the number it refers to. Statements are abstract, logical entities, while sentences are grammatical ones. [harvnb|Rouse] [harvnb|Ruzsa|2000|p=16]

See also

* Sentence (mathematical logic)
* Proposition

Notes

References

*A. G. Hamilton, "Logic for Mathematicians", Cambridge University Press, 1980, ISBN 0521292913.
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