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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?

user539262
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    To me they're the same thing, but I believe some people restrict propositions to be atomic statements, i.e. ' John is tall' would be a propositin and a statement, but 'John is tall and Mary is strong' would be a statement but mot a proposition. – Bram28 Mar 11 '18 at 22:12
  • Is an atomic statement a statement that is not compound? No logical connectives? – user539262 Mar 11 '18 at 22:32
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    @user539262 Yes, but expanding on what Bram28 said, the terminology is chaotic here. Many people use different words to indicate often somewhat technical distinctions. The problem is, people often don't care about the distinctions or are making other distinctions and so they use the terms in a looser sense or with totally different distinctions in mind. And then different groups just prefer different terms. You will ultimately need to find the definition the author is using to know. You can't rely a single, universal definition except to the vaguest extent. – Derek Elkins left SE Mar 11 '18 at 22:55
  • @DerekElkins So people use the words statement, expression, proposition, atomic, formula, predicate, etc, differently? – user539262 Mar 11 '18 at 23:21
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    @user539262 The amount of spread in meaning varies with each of those words, but yes, absolutely. For example, contrasting to Graham Kemp's answer, I don't typically use the term "statement" at all and would use "proposition" but for me "proposition" means a formula with no free variables as contrasted to a predicate but others use "sentence" for that and/or have a narrower meaning for "predicate". I also limit "expression" to terms generally, so I would not usually call a proposition an expression (though in some cases propositions and terms get identified). – Derek Elkins left SE Mar 12 '18 at 00:03
  • "Formula with no free variables"? Is a variable not free if it gets used with for-all, exists, or in? – user539262 Mar 12 '18 at 00:04
  • I wouldn't go so far as to say that propositions or statements must evaluate to true or false ( unless explaining to someone who is a really early beginner ). Some can't be evaluated at all, some logics don't actually have a concept of false, some logics have multiple "false" type concepts. – DanielV Mar 12 '18 at 04:19
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    Tao uses "statement" in the same way others use "proposition". – Mauro ALLEGRANZA Mar 12 '18 at 07:13

2 Answers2

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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.

Community
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Mauro ALLEGRANZA
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  • Is a variable free even if it is quantified? – user539262 Mar 12 '18 at 16:31
  • @user539262 - NO: in $(x=0)$ $x$ is free. In $\forall x \ (x=0)$ $x$ is bound. – Mauro ALLEGRANZA Mar 12 '18 at 16:33
  • So formulas with bound variables are the only ones that can be true or false? (going from your sentence "formulas with no free variables are either true or false") – user539262 Mar 12 '18 at 16:42
  • @user539262 - YES; to stay at Tao's approach (whic is the usual one) in predicate logic only a closed formula (i.e. a formula with no free occurences of variables, also caleld sentence) is a statement. – Mauro ALLEGRANZA Mar 12 '18 at 16:44
  • I don't quite understand how $x=0$ has $x$ free but $\forall x(x=0)$ now it's bound. In both cases it seems like "$x$ is $0$" – user539262 Mar 12 '18 at 16:49
  • @user539262 - because the second one has a quantifier in front. The first says: "a (unspecified) number is equal to $0$"; sometimes it is true, sometimes it is false. The second one says: "every number is equal to $0$", which is plainly false. – Mauro ALLEGRANZA Mar 12 '18 at 16:52
  • Is this true then? "Socrates is a man" is a statement with no free variables and is true. "X is a man" is a predicate with a free variable and is not true/false by itself, as this depends on what X is. But "X is a man" for all X in the universe "Spain" would be false since the variable is bound to something we can assess (X may be female). And "There exists an X in Spain that is man" would be true. – – user539262 Mar 12 '18 at 17:59
  • @user539262 - YES: in "Socrates is a man" there are no variables: neither free nor bound and YES: it is a sentence with a definite truth value, and thus a statement according to Tao's criteria. – Mauro ALLEGRANZA Mar 12 '18 at 18:10
  • "formulas with no free variables are either true or false". A formula that express the paralel axiom in a formal system of absolute geometry is either true or false? – Eduard Apr 24 '22 at 12:58
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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).

Graham Kemp
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  • But according to Tao's Analysis, a statement is composed of expressions, which are different from statements? – user539262 Mar 11 '18 at 23:02
  • @user539262: rather than focusing on the words in this situation, it may be better to focus on the underlying concepts. Different authors use different words for the same thing in logic, particularly in elementary mathematical logic – Carl Mummert Mar 12 '18 at 12:53