Statements

The raw material of mathematical reasoning is the statement. Not every sentence in English qualifies as one, so the first job is to pin down exactly what does.

Definition. A sentence is a mathematically acceptable statement if it is either true or false, but never both, and never neither. If a sentence's truth cannot be pinned down — because people can reasonably disagree about it, or because it depends on information we haven't been given — it is not a statement.

Compare two sentences: "An elephant weighs more than a human being" is a statement — it is unambiguously true. "Women are more intelligent than men" is not a statement, because its truth is a matter of opinion, not fact; different people would answer differently and there is no way to settle it once and for all.

Several everyday kinds of sentences fail the test and are therefore not statements:

• Exclamations, commands and questions — "How beautiful!", "Open the door", "Where are you going?" — these do not assert anything true or false at all.
• Sentences with a free variable — "The sum of x and y is greater than 0" is true for some values of x and y and false for others, so on its own it is not a statement. Adding a quantifier fixes this: "For any natural numbers x and y, the sum of x and y is greater than 0" IS a statement (and a true one), because now it makes one definite claim about every choice of x and y.
• Sentences with variable time, place or unspecified pronouns — "Tomorrow is Friday", "She is a mathematics graduate", "Kashmir is far from here" — these change truth value depending on when, where or who is speaking, so they cannot be pinned to a single true/false verdict.

A sentence can be a statement even if it is always false — "There are 40 days in a month" is a statement (a false one, since no month has more than 31 days) precisely because its falsity does not depend on which month you pick. …