# Negation of a sentence

Consider the following statement:

All cats have tails.(1)

The negation of this statement is

Some cats do not have tails. (2)

Now, someone can arrive at this statement by thinking that, 'hey, my cat does not have a tail'. So this statement can be negated by assuming that there exists atleast one cat which does not have a tail. My first question is, is the statement some cats have tails equivalent to the statement (2)? Why? Can one assume that all cats have tails based on the statement 'some cats have tails'?. I think not. So can I write it as a negation of (1) ?

Consider this statement :

Not all cats have tails

How does this differ from (2)?

Like I said, my cat does not have a tail. So can't I say that not all cats have tails since the first statement was based on the assumption that all cats have tails?

And finally, for a statement to be true, should it be absolutely true? Consider my example. All cats necessarily don't possess a tail. It has been scientifically proved that a cat has a tail, but due to some reasons (eg:a fight), it may lose a tail. Does my statement cease to exist as a statement at that precise moment since a statement should be either completely true or completely false, but not both? Also by keeping the above sentences in mind, can I say that the sentence Spiderman is a Superhero a statement?

PS: I don't have much experience in this . So could you kindly avoid the usage of symbols while explaining this? ie. Would you use plain english?

Thank you

• The negation of "All are P" is "Some are not-P" that - in general - does not mean that "Some are P". The meaning of "Some" is "At least one (maybe more)" Commented May 20, 2021 at 11:38
• And "Not all Cats have Tails" is exactly the same ad "Some Cat has no tail". In general $\lnot \forall x Px$ is equivalent to $\exists x \lnot Px$ Commented May 20, 2021 at 11:39
• "for a statement to be true, should it be absolutely true? " What is "ablsotely" true? Commented May 20, 2021 at 11:40
• If some are not P isn't equivalent to some are P? Commented May 20, 2021 at 16:41
• Consider a cat born with no tails. Obviously i can use its presence to negate my first statement. What I meant is, for a statement to be true, should it be universally acceptable? Does it permit exceptions? Commented May 20, 2021 at 16:45