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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"

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4 Answers 4

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The informal definition says just that a sentence is a statement just when it

can reasonably be assigned a truth value.

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

"Suppose every dog is animal is false"

As written that's not a sentence. You can say that the sentence

Every dog is (an) animal.

is a statement - one that happens to be true.

When you put a "suppose" in front of that sentence to get the sentence

Suppose every dog is (an) animal.

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.

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  • $\begingroup$ Okey so we have statements where we know the truthvalue already and we have statements where we don't the truthvalue yet. We call the first one facts and the other ones we would call claims. We can suppose truthvalues of claims but not of facts. The assignment of the truthvalue already took place and facts are like constants whereas claims are like variables. That would also mean you cannot assign a new truthvalue to a fact, so you cannot say "Suppose every dog is animal is false", right? Is this justification correct? $\endgroup$
    – New2Math
    Commented Jul 23, 2019 at 22:46
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You say that in your lecture notes you wrote

Suppose we know Sasha to be a member of a group of children.

This English sentence carries the meaning that we are considering an actual group of children, and Sasha is an actual child in that group. Therefore one of the two sentences "Sasha is a boy" or "Sasha is a girl" is actually true, hence the sentence "Sasha is a boy" is a statement.

A similar situation holds in mathematics. We might start a mathematical discussion by saying

Suppose that $a$ is a real number.

This carries the meaning that $a$ is an actual number, what we call in mathematics a constant. So one of the two sentences $a>0$ or $a \le 0$ is actually true. Hence the sentence $a > 0$ is a statement.

Constrast this with a different mathematical discussion:

As $x$ varies over the real numbers, consider the inequality $x>0$.

In this context, the sentence $x>0$ is not a statement, because we are explicitly letting $x$ vary, and the truth value of this sentence varies as $x$ varies.

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  • $\begingroup$ Ah okey I see there is always an implicit context such that in this case the "child" Sasha has a gender-value (we know this value exists altough we don't know what exactly the value is), a context can either name the value of a object or it is left unknown and all we know is that the object is assigned to a value which is in a set of possible values which is provided by the context. If the value is unknown an assumption can be made which is either true or false, if we get a new context we may establish the truthvalue $\endgroup$
    – New2Math
    Commented Jul 23, 2019 at 21:38
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We don't need to know whether it's true to be a statement, only whether it an assignment of true or false makes sense. Sasha is either a girl or not, so an assignment of true or false would be reasonable. For a mathematical example,

All postive even numbers are either prime or the sum of two primes.

would definitely make sense to be either true or false, even though we don't currently know which.

A sentence that isn't a statement would be something like "A child is a girl." Which child? This could be made into a statement by adding a quantifier or by specifying a child. "All children are girls.", "There is a child who's a girl.", and "Sasha is a girl." would all be statements since they have definite truth values. For a mathematical example, "x is even." can't be assigned a truth value because we don't know what x is, but "For all x, x is even.", "There is an x that's even", and "2 is even." all have definite truth values.

This ties into the idea of free variables, which will probably come up soon if it hasn't already. A free variable is one whose identity has not been specified and is not being quantified over, so it could represent anything. A statement has no free variables, so it can in principle be assigned a definite truth value, even if we don't necessarily know what it is at the moment.

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In classical logic, a statement is usually defined as a sentence that has a truth value, which can either be true or false. With this definition, the sentence "Sasha is a girl" is a statement because it certainly is either true or false, even though we don't know which one it is.

There is also another possible definition of statement, which is typical of intuitionistic logic: a statement is a sentence for which we can agree on what will convince us that it is true. With this definition, the sentence "Sasha is a girl" is a statement because, for instance, we could agree that it is true if asking Sasha about their gender we receive the answer "female".

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