# What are the names given to statements that can be true or false?

While compiling a set of notes for high school students about logic and proof, I wrote

Let $$A$$ and $$B$$ be statements. Then $$A \Rightarrow B$$ means ''if $$A$$ is true then $$B$$ is true.''

This was my definition of a statement:

A (mathematical) statement is a sentence that is either definitely true or definitely false. (It cannot, therefore represent an opinion, like "apples taste better than bananas" or events that may or may not be true, like "It will rain tomorrow in London".) Statements can therefore be proved/disproved by providing an argument that rigorously demonstrates that they are true/false.

By the above definition, $$A$$ and $$B$$ don't have to be statements for $$A \Rightarrow B$$ to make sense. For instance, for real $$x$$, "$$x>4 \Rightarrow x>2$$" makes sense (and is true) and yet it isn't possible to assign a definite true/false value to these sub-statements "$$x>4$$" and "$$x>2$$" in isolation.

This leads me to my question: is there a proper name for things like "$$x>2$$" and "$$x>4$$", where it makes grammatical sense to call them true or false, but which cannot truly be called either, without being given more information?

I was thinking of calling them boolean phrases, but I would rather not just invent a name for something if a proper name is already in use. Another option is just to redefine what I call a (mathematical) statement to include this kind of thing, but then add another definition like "decidable statement" to describe sentences that are either definitely true or definitely false.

This question also applies to other examples. What name should be given to sentences like the below?

• "This sentence is false" (seems like it can be true or false, but both lead to contradictions)
• "$$x$$ is even" (We cannot decide if it's true or false without more information about $$x$$)
• ''$x >4 \Rightarrow x>2$" is not a statement because its value of truth cannot be established as $x$ is not specified. Instead ''$\forall x \in \Bbb R, x >4 \Rightarrow x>2$" is a statement, and you can prove it's true. See the difference between predicates and propositions for example at this link Jun 6 at 18:01
• I've purposely avoided using $\forall$ since this discussion is aimed at high-school students. But I see your point. To clarify, are you saying $x>4$ is an example of predicate? Jun 6 at 18:09
• Is ‘statement’ being used in the same way as ‘sentence’? Jun 6 at 18:15
• Yes, $x>4$ is an example of a predicate. The term you are looking for is predicate. Jun 6 at 18:16
• @PW _246 The definition of a statement is given, a "sentence" is a more informal notion of basically a pile of words which makes sense. I'm not sure if that's what you meant Jun 6 at 18:18

for real $$x$$, "$$x>4 \Rightarrow x>2$$" is true yet it isn't possible to assign a definite true/false value to its sub-statements "$$x>4$$" and "$$x>2$$" in isolation. is there a proper name for things like "$$x>2$$" and "$$x>4$$"?

For every real $$x,$$ the predicate $$x>4 \to x>2$$ is true. In other words, the statement $$\forall x\:(x>4 \to x>2)$$ is true. Every predicate contains a free variable, whereas every statement contains none.

A statement is a sentence that is either definitely true or definitely false.

$$(x=x)$$ is not technically a statement, but it is definitely true.

$$(\forall x \;x^2\le0)$$ is a statement, but—being true in the universe of imaginary numbers but false in the universe comprises of real numbers—it is arguably neither definitely true nor definitely false. (To be fair: it definitely is either true or false).

• Thank you! Would I be correct in saying that with the definition that I used, $\forall x, \, x^2\leq 0$ is still a predicate since $x$ is still not properly specified; it needs to say $\forall x \in$[some set]? Jul 10 at 15:18
• @James $(\forall x \;x^2\le0)$ is not a predicate simply because it contains no free variable; your "[some set]" is just the domain of discourse, which is always at least tacitly dependent on the context; if your concern is about "definitely" assigning a truth value, then notice that $(\forall x \;x^2\le0)$'s truth value actually depends also on how we interpret the non-logical symbol $≤.$ Sep 10 at 1:55

You're being sabotaged here by your book's imprecise language. $$A$$ and $$B$$ are statements. But the symbols$$A$$” and “$$B$$” are not statements, they are symbols that represent statements.

This is similar to algebra: “$$x$$” is not a number. It is a letter ‘x’, used as a symbol that represents a number. The number itself is $$x$$. Or a more familiar example: you are not “James”, because “James” is a five-letter word, and you are not a word, you are a person. You are James, not “James”. “James” is only your name.

When the book says:

Let $$A$$ and $$B$$ be statements. Then $$A⇒B$$ means ''if $$A$$ is true then $$B$$ is true''

it is, strictly speaking, incorrect, since, as you observed, $$A$$ and $$B$$ are not statements. A more careful writer would have put it like this:

Let $$A$$ and $$B$$ be statements. Then the formula consisting of the statement $$A$$, followed by the symbol “$$⇒$$”, followed by the statement $$B$$, means that if $$A$$ is true then $$B$$ is true.

Maybe you can see why this is rarely done. (Even this is not correct. Instead of “statement” it should say “formula”.)

In contexts where the author really does want to deal carefully with these distinctions, they often adopt a notation called quasiquotation in which the long phrase

the formula consisting of the statement $$A$$, followed by the symbol “$$⇒$$”, followed by the statement $$B$$

is abbreviated to

$$⸢A⇒B⸣$$

with the square quote notation having been defined to have the desired meaning. You can find this, for example, in Quine's book Mathematical Logic. There might also be something about it in SEP, but I did not check.

I don't know if this helps you in writing your notes for high school students, but you should at least be clear in your mind about the distinction between names of things and things themselves, or else you will pass on your confusion to your students.

I'm reluctant to complicate the explanation even more, but there is another layer of confusion, which is that $$A$$ and $$B$$ should be said to be formulas, not statements. Formulas are sequences of symbols that, if well-formed, may represent statements. The neither-true-nor-false formulas you mention, such as “$$x<1$$”, are called open formulas. As you observed, an open formula is not a statement. Open formulas can be turned into closed formulas that do represent statements, by binding the free variables. In “$$x<1$$" the free variable is “$$x$$”. When we bind $$x$$ we get a formula like

$$\forall x. x<1$$,

which represents the statement “Every $$x$$ is less than $$1$$”, or

$$\exists x. x<1$$,

which represents the statement “There is some $$x$$ that is less than $$1$$

• I think this is the most precise (and correct) response to the question, but I equally doubt that most high school students would understand it... A bit toooo subtle, and does not refer to real issues involving things connected to their normal lives. Still, saying "you are not (the word) 'James', you're James" might catch their fancy. Jul 8 at 3:48
• (Btw, I did really enjoy your Perl book! :) Jul 8 at 3:49
• If I were in your position, I would avoid the “$A\to B$ is not a formula” issue completely, as almost everyone does. Instead I would try hard to use the terminology right, between formulas (well-formed sequences of symbols), sentences (closed formulas), and statements (true or false claims that can be asserted by sentences). Using these correctly and uniformly would help everyone understand better. When they got confused, you would be able to use the terminology to help sort them out. And thanks for your kind words about the book!
– MJD
Jul 9 at 3:37