Is "$x$ is an integer, $1 < x < 3$, and $x^2 = 4$" a proposition?

Question

A proposition is frequently defined as a statement that has one and only one truth value.

By themselves, "$x$ is an integer," "$1 < x < 3$," and "$x^2 = 4$" are not propositions because they may be true or false depending on the value of $x.$

Can someone explain why the conjunction of these three is considered a proposition?

Many thanks!

Answer 1

One thing that may be considered a proposition is the slightly different combination

If $x$ is an integer and $1<x<3$, then $x^2=4$.

Formally, at least if we interpret "if--then" as denoting material implication, this is still a claim with a free variable, and therefore would not count as a "proposition" under the definition you quote -- no matter that its truth value happens to be "true" no matter what you plug in for $x$.

On the other hand, in informal mathematical speech, "if--then" not only encodes material implication but also tends to give you a universal quantification for free. (But it's up to the reader/listener to figure out which variables are quantified if there are more than one that appears in both the assumption and the conclusion). In this way, the above sentence can be used as an abbreviated form of

For every $x$, if $x$ is an integer and $1<x<3$, then $x^2=4$.

This has no free variables and would therefore satisfy your definition of "proposition".

Answer 2

I'd say it isn't a proposition, merely a compound, non-conditional statement. Just as the truth of the three component statements depends on the identity of $x,$ the truth of the compound statement does, as well. In particular, it is true precisely when $x=2,$ but just the name $x$ is the only information we have, outside of that given in the statement, so there's no reason to suspect that $x=2,$ but it certainly could be true. Hence, the statement has undetermined truth-value, devoid of other context.

There are a few ways we could fix it. One would be to add context, such as:

Suppose $x=2.$ The following are true:

• $x$ is an integer.

• $1<x<3.$

• $x^2=4.$

Another is to combine the three component statements into a conditional, such as:

If $x$ is an integer and $1<x<3,$ then $x^2=4.$

There may be other ways to "fix" it, but the upshot is that it needs "fixing" to be a proposition.