Is "If $x=5$ then $x^2=25$" a proposition? My teacher said that $x=5$ and $x^2=25$ are not propositions but they are logical predicate with $x$ as a free variable (I don't know what are logical predicate though).
My question is whether "If $x=5$  then $x^2=25$" is a proposition or not?
I think it shouldn't be because the atomic statements aren't propositions themselves.
(And please don't mind any mistake because it's my first time asking a question on StackExchange.)
 A: Usually, in logic, "proposition" has the same meaning of "declarative sentence".
A sentence:

is a meaningful group of words that express a statement, question, exclamation, request, command or suggestion.

In philosophy, there is a sharp contrast between a sentence (a linguistic entity) and the corresponding proposition, i.e. some non-linguistic entity that is expressed by the sentence.
The proposition is taken to be the thing that is in the first instance true or false, while a declarative sentence is true or false derivatively, in virtue of expressing a true or false proposition.
Having said that, a declarative sentence is an expression stating a fact, like: "The rose is red", and we can assign to it a definite meaning and a truth value.
Thus, according to this point of view, an expression with a free variable is not a sentence (proposition) because it lacks a definite meaning.

We can follow Frege's analysis, starting from a sentence: "Socrates is a Philosopher", that we can symbolize with: "$\text{Philosopher}(\text {Socrates})$".
Then erase the name "Socrates", replacing it with a place-holder (a variable) to get:

"$\text{Philosopher}(\xi)$".

The result is an incomplete expression: an open formula.
The incomplete expression has no meaning, unless we replace the variable "$\xi$" occurring in it with a name, and thus it is not a sentence.
The same with the arithmetical sentence "$2 < 5$", from which we get open formula "$x < 5$" replacing the numeral "$2$" (the name of the number two) with the variable "$x$".
As you said, an open formula, i.e. a formula with a free variable (also called propositional function) expresses a predicate.
