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Technically, both of those are right. The existential quantifier ("for some x") mostly just says that there exists AT LEAST ONE instance x, but maybe more, that satisfy the sentence.

So if I say "for some x, x was a U.S. President and x is black," that would be both grammatical and true because exactly one former U.S. president is black.

If I say "for some x, x is a swan and x is black," that is also grammatical and true, because there are many swans that are black.

If I say "for some x, x is a man and x is mortal," that is also grammatical and true because if ALL men are mortal, then there is at least one man that's mortal. Technically it might be better to say, "for all x, if x is a man, then x is mortal", but using the existential quantifier isn't WRONG. It's just not as strong as it could be.

BUT If I say "for some x, x is a prime number and x halves to an integer greater than one," that would be grammatical but false, because by definition, "for all x, if x is a prime number, x does NOT half to an integer greater than one," which is equivalent to saying, "It is NOT the case that for some x, x is a prime number and x halves to an integer greater than one."

Technically, both of those are right. The existential quantifier ("for some x") mostly just says that there exists AT LEAST ONE instance x, but maybe more, that satisfy the sentence.

So if I say "for some x, x was a U.S. President and x is black," that would be both grammatical and true because exactly one former U.S. president is black.

If I say "for some x, x is a swan and x is black," that is also grammatical and true, because there are many swans that are black.

If I say "for some x, x is a man and x is mortal," that is also grammatical and true because if ALL men are mortal, then there is at least one man that's mortal. Technically it might be better to say, "for all x, if x is a man, then x is mortal", but using the existential quantifier isn't WRONG. It's just not as strong as it could be.

BUT If I say "for some x, x is a prime number and x halves to an integer," that would be grammatical but false, because by definition, "for all x, if x is a prime number, x does NOT half to an integer," which is equivalent to saying, "It is NOT the case that for some x, x is a prime number and x halves to an integer."

Technically, both of those are right. The existential quantifier ("for some x") mostly just says that there exists AT LEAST ONE instance x, but maybe more, that satisfy the sentence.

So if I say "for some x, x was a U.S. President and x is black," that would be both grammatical and true because exactly one former U.S. president is black.

If I say "for some x, x is a swan and x is black," that is also grammatical and true, because there are many swans that are black.

If I say "for some x, x is a man and x is mortal," that is also grammatical and true because if ALL men are mortal, then there is at least one man that's mortal. Technically it might be better to say, "for all x, if x is a man, then x is mortal", but using the existential quantifier isn't WRONG. It's just not as strong as it could be.

BUT If I say "for some x, x is a prime number and x halves to an integer greater than one," that would be grammatical but false, because by definition, "for all x, if x is a prime number, x does NOT half to an integer greater than one," which is equivalent to saying, "It is NOT the case that for some x, x is a prime number and x halves to an integer greater than one."

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esquisilo
esquisilo
  • 161
  • 2

Technically, both of those are right. The existential quantifier ("for some x") mostly just says that there exists AT LEAST ONE instance x, but maybe more, that satisfy the sentence.

So if I say "for some x, x was a U.S. President and x is black," that would be both grammatical and true because exactly one former U.S. president is black.

If I say "for some x, x is a swan and x is black," that is also grammatical and true, because there are many swans that are black.

If I say "for some x, x is a man and x is mortal," that is also grammatical and true because if ALL men are mortal, then there is at least one man that's mortal. Technically it might be better to say, "for all x, if x is a man, then x is mortal", but using the existential quantifier isn't WRONG. It's just not as strong as it could be.

BUT If I say "for some x, x is a prime number and x halves to an integer," that would be grammatical but false, because by definition, "for all x, if x is a prime number, x does NOT half to an integer," which is equivalent to saying, "It is NOT the case that for some x, x is a prime number and x halves to an integer."