How do I write minimal negations for the following statements?

Question

Write minimal negations for the following statements. (In other words, if the statement below is $P,$ what would $¬P$ mean?)

a) Everybody was kung-fu fighting for at least 10 hours.

b) There is a river with at least two tributaries.

c) No baboons wear bowler hats.

I get somewhat confused when it comes to negating the statement. I haven't fully grasped the concept. but this is what I was thinking and kinda feel like its wrong.

A) There is somebody who was kung-fu fighting for not 10 hours

B) There is not a river with at least two tributaries

C) All baboons wear bowler hats

Answer 1

Your reasoning is not quite correct. As it happens all of your answers are unsatisfactory in some way. Let's go through each:

A) There is somebody who was kung-fu fighting for not 10 hours

This is almost correct. If “$\neg$(Everybody is doing X)” then there has to be somebody who is not doing X. However, by saying “not 10 hours” you are mistaken: the correct negation is “less than 10 hours”.

B) There is not a river with at least two tributaries

Whilst this is a technically correct negation, it has not been simplified at all. You're likely expected to produce a sentence of the form “All rivers (...)”. I'll leave you to work this one out yourself; you can probably do so, maybe after reading the tips later on in this answer.

C) All baboons wear bowler hats

This is not a correct negation. To see why, try to negate the false statement “No cats are ginger”. By your logic, the negation, a true statement, should be “All cats are ginger”—a clearly false statement! The correct negation is “There is a baboon that wears a bowler hat”.

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The general rules for solving this sort of problem can be written as so:

1. $\neg(\text{All $X$s have property $Y$}) \Leftrightarrow (\text{There is an $X$ without property $Y$})$.
2. $\neg(\text{There is an $X$ with property $Y$}) \Leftrightarrow (\text{All $X$s do not have property $Y$})$

Clearly, a) falls under rule 1, whereas b) falls under rule 2. It may be hard to see, but c) actually also falls under rule 1—can you see why? This should help you understand this kind of statement better.

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If you'd like more justification, I recommend reading up on logical quantifiers. If these problems are part of a book or course on logic, you will likely encounter these very soon. Logical quantifiers allow rule 1, for instance, to be written as $\neg(\forall x, P(x)) \Leftrightarrow \exists x, \neg P(x)$.