There are many kinds of statement that are not theorems: A statement can be an axiom, that is, something that is assumed to be true without proof. It is usually self-evident, but like Euclid's parallel postulate, need not be. A statement need not be true in all circumstances - for example, A*B = B*A (commutativity) is not necessarily true for matrix multiplication. A statement can be false. A statement can be self-contradictory for example, "This statement is false".
false
To prove a statement false, you need ONE example of when it is not true.To prove it true, you need to show it is ALWAYS true.
It is often a tautology - a statement which is true bit adds nothing to anyone's knowledge. A recent example is the fatuous remark by the British Prime Minister "Brexit is Brexit".
You make a statement that is either a true one or a false one. Simple as that. For example, you might say "The moon is made of green cheese" as a false statement and most people know that's wrong.
true
An example of a true statement in algebra is x=x
Circular logic would be a statement or series of statements that are true because of another statement, which is true because of the first. For example, statement A is true because statement B is true. Statement B is true because statement A is true
One classic example of a paradox is the "liar paradox," which revolves around a statement that cannot consistently be true or false. An example would be the statement "This statement is false." If the statement is true, then it must be false, but if it is false, then it must be true, creating a paradoxical situation.
In writing, there is a difference between telling statements and showing statements. A telling statement states a fact, such as that it was cold. A showing statement would illustrate that by, for example, describing how everyone was shivering.
The below statement is false. The above statement is true. I am lying. I am lying when I say I am lying.
There cannot be a proof because the statement need not be true.
counter example
true
"All human beings are animals" is a true statement. All animals are not human beings.
There are many kinds of statement that are not theorems: A statement can be an axiom, that is, something that is assumed to be true without proof. It is usually self-evident, but like Euclid's parallel postulate, need not be. A statement need not be true in all circumstances - for example, A*B = B*A (commutativity) is not necessarily true for matrix multiplication. A statement can be false. A statement can be self-contradictory for example, "This statement is false".
The statement is true.