Thread: Is this english sentece a statement or not and why?

Consider:
a)x=2
b)All politicians are honest
c)there exists a real number x such that x^2=x

Can someone please help me determine a way to figure out whether something is a statement? I am correct most of the time, but when my prof. tries to trick he is successful lol. What should I look out for? Can you provide an example of a tricky sentence and then show me how u figured out whether it is a statement or not?


My attempt:
a)IS NOT a statement b/c we do not have proof that it is true for all 'x', Therefore, it can be true for this case, but x can always be a different number.
b)I would say b is a statement. B/c it sounds like it is saying the sentence is true, but how we do we know its true? if it wasn't proven it can be either true or false and a statement that is true or false is NOT a statement, can someone explain?
c)This is a statement as long as I find 1 case where this is correct right?

Originally Posted by matthayzon89

Consider:
a)x=2
b)All politicians are honest
c)there exists a real number x such that x^2=x

Can someone please help me determine a way to figure out whether something is a statement? I am correct most of the time, but when my prof. tries to trick he is successful lol. What should I look out for? Can you provide an example of a tricky sentence and then show me how u figured out whether it is a statement or not?


My attempt:
a)IS NOT a statement b/c we do not have proof that it is true for all 'x', Therefore, it can be true for this case, but x can always be a different number.
b)I would say b is a statement. B/c it sounds like it is saying the sentence is true, but how we do we know its true? if it wasn't proven it can be either true or false and a statement that is true or false is NOT a statement, can someone explain?
c)This is a statement as long as I find 1 case where this is correct right?

I'm not sure there is a universally accepted definition for "statement" in this sense, to tell the truth. Don't you have a definition approved by your teacher that you can test against?

In my experience the words proposition, statement, and sentence are used interchangeably to mean (generally) "an expression that is either true or false," but this may not be the "correct" usage of terms.

In particular, a false statement is still a statement, and I would consider a, b, and c all to be statements.

Originally Posted by undefined

In particular, a false statement is still a statement, and I would consider a, b, and c all to be statements.

A is not a statement. For sure. B, C are statements (I have these problems graded from my previous test but I am trying to understand how to come up with the answer). Also, I am pretty sure that statement or proposition need to be true or false, so if they are either true or false they are statements, but if the answer is infinity or completely unpredictable like x=2 then it is NOT a statement, b/c you don't know whether this is true, b/c I can say x=5, or x=2092 so every time you claim x is a different value you are redefining your definition of true and basically saying your previous definition of x was false. therefore, true and false at the sametime= NOT a statement.

So, I think I know why statement B is true also, because saying "All politicians are honest" is a universal statement. So, universal statements can either be true or false, but not both, its Either the case that "All politicians are honest" is true, OR it is NOT the case that "All politicians are honest" is true. Therefore, since there is an "OR" in the sentence, we can assume it is in fact a statement! if thats OR was an AND then it would be false (not a statement). What do you think?

Originally Posted by matthayzon89

A is not a statement. For sure. B, C are statements (I have these problems graded from my previous test but I am trying to understand how to come up with the answer). Also, I am pretty sure that statement or proposition need to be true or false, so if they are either true or false they are statements, but if the answer is infinity or completely unpredictable like x=2 then it is NOT a statement, b/c you don't know whether this is true, b/c I can say x=5, or x=2092 so every time you claim x is a different value you are redefining your definition of true and basically saying your previous definition of x was false. therefore, true and false at the sametime= NOT a statement.

So, I think I know why statement B is true also, because saying "All politicians are honest" is a universal statement. So, universal statements can either be true or false, but not both, its Either the case that "All politicians are honest" is true, OR it is NOT the case that "All politicians are honest" is true. Therefore, since there is an "OR" in the sentence, we can assume it is in fact a statement! if thats OR was an AND then it would be false (not a statement). What do you think?

Sorry but this all sounds way too overthought to me. Why can't you just get your teacher to give you a simple definition that will unfailingly tell you whether or not something is a statement? Possibly the distinction between "statement" and "non-statement" your teacher wants you to understand is important, but I think it's mainly a definitions thing. Once you know the definition and how to apply it, you're set.

There are all sorts of word games you can play if you don't have definitions set in stone. For example, Let x represent the number of coins in my pocket. In this context, "x = 2" represents "the number of coins in my pocket is 2" which you will agree is a statement.

Please just get a working definition.

In my experience the words proposition, statement, and sentence are used interchangeably to mean (generally) "an expression that is either true or false

I agree. A proposition is something that is either true or false. All information to make it true or false must be contained in the proposition. The expression x = 2 is not a proposition because it is true or false depending on x, which is not given.

Now, whether something is a proposition has nothing to do whether:
(*) it is true
(*) we know it is true or we know it is false

We know that 2 + 2 = 5 is false, and yet it is a proposition. Nobody knows whether Goldbach's conjecture is true or false, but it is a proposition.

In practice, a proposition is a declarative sentence (not a question or command) where all information is provided (there are no unknowns like x).

Originally Posted by undefined

Sorry but this all sounds way too overthought to me. Why can't you just get your teacher to give you a simple definition that will unfailingly tell you whether or not something is a statement? Possibly the distinction between "statement" and "non-statement" your teacher wants you to understand is important, but I think it's mainly a definitions thing. Once you know the definition and how to apply it, you're set.

There are all sorts of word games you can play if you don't have definitions set in stone. For example, Let x represent the number of coins in my pocket. In this context, "x = 2" represents "the number of coins in my pocket is 2" which you will agree is a statement.

Please just get a working definition.

You must of missed it, but this is the definition: "A sentence is a statement if it is either true or false, but not both."

That's all my professor expects us to know to be able to solve these types of problems.

I'm pretty sure my explanations above are correct, I was just hoping that someone that is very good in discrete math can confirm that.

In addition, what I said was not over thought, it was just well-explained.

Originally Posted by emakarov

I agree. A proposition is something that is either true or false. All information to make it true or false must be contained in the proposition. The expression x = 2 is not a proposition because it is true or false depending on x, which is not given....

I agree with you completely. It depends on the statement. If the sentence simply says x=2 then it is NOT a statement. B/c it is not making a claim neither way, true nor false. If the sentence said "x+x=17 and x=2", then it WOULD be a statement (even though it is a false).

Originally Posted by matthayzon89

You must of missed it, but this is the definition: "A sentence is a statement if it is either true or false, but not both."

That's all my professor expects us to know to be able to solve these types of problems.

I'm pretty sure my explanations above are correct, I was just hoping that someone that is very good in discrete math can confirm that.

In addition, what I said was not over thought, it was just well-explained.

I see that emakarov's explanation is what you needed. By calling your reasoning overthought, I wasn't insulting you, merely telling you that I thought it was much simpler than you were making it out to be.

Here's food for thought:

Let A = "This sentence is not a statement."

Is A a statement?

(Edited to add the following)

Then there's the classic paradox:

Let B = "This sentence is false."

Is B a statement?

Note that an equivalent way to state B is:

Let B = "B is false."