I have got problems to understand what a statement is. We have not really defined what a statement is, rather we have given an introductory remark about statements:
By a statement or proposition we mean any sentence (any sequence of symbols) that can reasonably be assigned a truth value, i.e. a value of either true, abbreviated T, or false, abbreviated F.
As an example of what statements are we have said:
Every dog is an animal
2+3=4
3 is even
So from what I can see statements have a truthvalue which is already determined.
However there are also statements like that
In a group of kids we have a kid that is called Sasha
we then have two sentences (bold because I don't understand why they should be statements).
1) Sasha is a girl 2) In the group there is a girl
In our lecturenotes (http://www.math.lmu.de/~philip/publications/lectureNotes/philipPeter_LinearAlgebra1.pdf) (p7) we wrote
Suppose we know Sasha to be a member of a group of children. Then
the statement A “Sasha is a girl.” implies the statement B “There is at least one girl
in the group.” A priori, we might not know if Sasha is a girl or a boy, but if we can
establish Sasha to be a girl, then we also know B to be true. If we find Sasha to be a
boy, then we do not know, whether B is true or false.
My professore said the senences 1) and 2) are statements. But I don't see how this is consistent with our introductory remark about statements. Namely a statement is a secuence of sysmbols that can reasonably be assigned to a truth value, but we cannot assign 1) and 2) a truthvalue because we don't know whether it is true or not. The whole idea of assuming statements to be true or not does not make sense to me, we cannot compare the sentence "Sasha is a girl" with "Every dog is an animal" the first sentence might be true or false but the second sentence is true and thus a statement. I have concluded that my understanding of a statement must be false, i.e. it is not something that is definitely either true or false but it also can be something that can be assumed true or false.
My request to you is to give me a definition of a statement such that a sentence like 1) is a well-defined statement like "2+3=4"
Best Answer
The informal definition says just that a sentence is a statement just when it
That says nothing about whether it's true or false or whether you know it's true or false or can or can't be one or the other.
"The moon is made of green cheese" is a statement. It was a statement even before Apollo 11 showed it was false.
Both "$2+2 =4$" and "$2+2=5$" are statements.
"There are infinitely many Mersenne primes" is a statement.
All this is informal. When you study formal logic you encounter rules for constructing statements from symbols that try to capture what we say in English and in everyday mathematics. Then you have to formalize what "reasonably assign a truth value" means.
Edit in answer to the comment in which you ask whether you can say
As written that's not a sentence. You can say that the sentence
is a statement - one that happens to be true.
When you put a "suppose" in front of that sentence to get the sentence
you no longer have a statement, since You can't assign a truth value to the act of supposing, just to the statement being supposed. The purpose of a sentence like that one (beginning with a supposition) is to proceed with an analysis of a universe in which the the statement being supposed (every dog is (an) animal) happens to be true. If you want your universe to include hot dots the statement being supposed is false.
Last word: I think you are overthinking this. As you read and write mathematical arguments you will become comfortable with statements and suppositions.