What is the difference between a statement and proposition?

Question

According to Tao's Analysis, a statement is usually made up of expressions and must evaluate to either true or false.

But to me a proposition seems to be the exact same thing. Is there a difference?

Answer 1

Tao "informal" exposition in Appendix A: the basics of mathematical logic, can be easily formalized.

We have a language $\mathcal L$, that is a collection of symbols: variables: $x_1,x_2,\ldots$, constants, predicate symbols, connectives: $\lnot \to$, equality: =, quantifiers.

Any finite sequence of symbols is an expression: some are "meaningless", like e.g. $= 2++4 = − = 2$, some are meaningful: $2 + 2 = 4$.

The meaningful ones we call it statements: they "are either true or false".

See dictionary: statement: "1. something stated [...] 3. a single sentence or assertion".

In order to discriminate between them, we define precise syntactical rules for constructing meaningful expressions, like e.g.

if $t,s$ are terms (either variables or constants), then $t=s$ is an (atomic) formula;

if $\varphi, \psi$ are formulas, then $(\varphi \to \psi)$ is a formula;

and so on.

Thus, we define well-formed the expressions that satisfy the formation rules, and ill-formed the expressions that do not satisy them.

Finally: we call statements the well-formed expressions.

In the footnote, Tao specifies that "statements with no free variables are either true or false".

This must be read more precisely as: "formulas with no free variables are either true or false".

A formula with a free variable, like e.g. $(x=0)$, is similar to the expression "it is red"; we cannot assign to it a truth value until we do not specify what the pronoun "it" (the variable $x$) refers to.

---

The term ‘proposition’ has a broad use in contemporary philosophy.

Thus, in a mathematical context, someone prefers to avoid it and speak of "lingustic" entities, like symbols, expressions and statements.

Answer 2

Only in usage.   Propositions are things of which statements are constructed (along with logical connectives), while statements are things of which propositions construct.   Other than that, statements are propositions.

A proposition is a truth-vauled expression.   An atomic proposition consists of a truth-valued expression that contains no logical operators.

A statement is a truth-valued expression composed of one or more atomic propositions connected by logical operators.   A compound statement is a truth-valued expression that is constructed of more than one atomic proposition (and thus some logical connectives).