Is statement like "If x>7, then x^2>49" a conditional proposition?

Question

For statements like

x>7

we all know that since we have no idea what value x holds, it isn't a proposition by definition. But we can often see propositions like this:

If x>7, then x^2>49.

which seems to be an conditional proposition p → q, where p is "x>7" and q is "x^2>49". Maybe you'll say that it isn't a compound proposition; rather, it is just an atom proposition in the form of "conditional natural language". But I've read a few textbooks on mathematical logic and find that they all seem to believe that statements like above are indeed compound propositions. For example, in the book Discrete Mathematics and Its Applications (7th Edition), Example 10 gives the following statement as an example of biconditional:

Let p be the statement “You can take the flight,” and let q be the statement “You buy a ticket.” Then p ↔ q is the statement “You can take the flight if and only if you buy a ticket.”

The p, q above, in my opinion, are no different from statements like "x>7". So why are they considered propositions here?

To sum up, I've got these questions:

Is the example from the textbook given above flawed? (Maybe some more correct forms will be given later in the text?) Note that in the same textbook, in Exercise 1 of Section 1.1, it clearly says that "x+2 = 11" is NOT a proposition (so neither would "x>7").
If not, then the "propositions" here are different from what the definition says, which restricts proposition to sentences that are clearly determined as either true/false. So what's wrong here? Could it be that "statements of the same form can mean different things in different contexts?" For instance, although “You can take the flight” isn't a proposition on its own, when connected with the other statement, it will magically become one?
We know that propositional calculus is used for mathematical proofs, and in our daily proofs, forms like "if p, then q", where p and q are not propositions (in the definition's sense), appear quite often. If that is true, we'll rarely see conditional propositions in proofs, so why do we need to study them, instead of studying statements composed of "conditional natural language"?

These questions really frustrate me a lot. Could it be that what I've read so far are just some simplified versions of more convoluted concepts? And is there any novice-friendly textbook that detailedly explains what is going on here? Many thanks!

@MauroALLEGRANZA Many thanks. I've edited the question a little to better demonstrate my confusion. — SuperSaiyanGod, Dec 24 '19 at 18:10
$x > 7$ is an open formula, and thus it is not a sentence. "if $x >7$, then $x^2 > 49$" is often read (see answer below) as $\forall x \ (x >7 \to x^2 > 49)$, in which case it is a sentence : no free variable inside. — Mauro ALLEGRANZA, Dec 24 '19 at 18:19
@MauroALLEGRANZA So what about the English sentence example given above? I don't think the truth values of the clauses can be determined, and thus I see no difference between them and "x>7". — SuperSaiyanGod, Dec 24 '19 at 18:24
$\forall x (x > 7 \to x^2 > 49)$ is not a "logical formula"; it is a formula in the language of arithmetic. If we do not interpret it as an arithmetical formula, it makes little sense : "Socrates > 2" is neither true nor false. — Mauro ALLEGRANZA, Dec 24 '19 at 18:30
For statements expressed with natural language, we have not the choice of interpreting them in different ways: "you buy the ticket" (assumed an obvious context for interpreting "you") is true or false according whether you buy it or not. — Mauro ALLEGRANZA, Dec 24 '19 at 18:32

spaceisdarkgreen · Answer 1 · 2019-12-24T18:35:14.487

0

The concept you are missing is quantifiers. $x>7\to x^2 >49$ just means that it isn’t the case that $x>7$ and $x^2\le 49$ for this particular value of $x$. The proposition you are thinking of is this formula preceded by "for all $x$". This no longer depends on $x$ since it quantifies over it.

edited Dec 24 '19 at 18:35

answered Dec 24 '19 at 17:55

spaceisdarkgreen

58,508

Thank you! This might make sense for the first example, but what about the second one? – SuperSaiyanGod Dec 24 '19 at 18:00
@SuperSaiyanGod I don't understand the issue with the second example. Here $p$ and $q$ are both propositions, as is $p\leftrightarrow q.$ – spaceisdarkgreen Dec 24 '19 at 18:25
@SuperSaiyanGod Ahh, I see reading your comment above that you're concerned about the clauses not having a well-defined truth value. I could try to rationalize this (what we really mean is that they have a truth variable in a given interpretation... and yes variables are treated differently) but instead I'd recommend that you take this for granted since it's just an example, and focus on more mathematically precise things. – spaceisdarkgreen Dec 24 '19 at 18:30
Yeah, thank you so much! So basically, a proposition is "a declarative sentence that can be determined if it is TRUE or FALSE", even if its truth value is only known to the speaker, right? And we, as bystanders, may not be able to determine the truth value, thus making it necessary to consider both possibilities into consideration. – SuperSaiyanGod Apr 22 '21 at 12:53

Is statement like "If x>7, then x^2>49" a conditional proposition?

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