What is the correct understanding of the phrase 'for some x' in logical definitions? For example sometimes I've seen it as 'for some (values of) x' and others in uses such as 'there exists some x such that x+1=2' which seems like 'there is some (single) value of x such that...', and at other times we may see 'f=ma for some mass m and acceleration a ' which in this context implies that it's for 'some x' in the same way as we would say 'some person' which is like 'any' or 'any arbitrary' value, similar to 'any value' or 'any possible value'.
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The correct understanding is that the quantifiers All & Some can be associated with percentages. ALL & NO would correspond to absolutely 100 percent. SOME corresponds to any percentage from 1 percent to 99 percent. SOME is typically expressed as 'at least one' in texts. For instance there is at least one person on Earth that has a Philosophy degree. The statement doesn't express 100 percent of people have a Philosophy degree. We don't KNOW what percentage of the human population have a Philosophy degree based on the claim. All we do know is there is one human as a minimum. One is required.– LogikalCommented Apr 30, 2022 at 16:02
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If I'm a logician and I tell you I ate some of the cookies, that does not preclude the fact that I ate all the cookies! See philosophy.stackexchange.com/questions/87048/… and en.wikipedia.org/wiki/Implicature In conversational language, "some" implies "not all." But in logic, some might include all.– user4894Commented Apr 30, 2022 at 22:11
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1 Answer
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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Your last paragraph is wrong - all prime numbers are odd, except two, which is the oddest one, because it is the only one that is even. Half of the prime number 2 is an integer. Commented May 9, 2022 at 15:46
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