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What is negation of All birds can fly.

The question seems bit funny but i don't know which of the following two sentences is correct:

  1. Some birds can not fly
  2. There is at least one bird which can not fly.

Both the sentence seems almost logically same. But which of the following is true.

In book correct option out of four is sentence 1.

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What would be the difference between "Some birds cannot fly" and "Some bird cannot fly"? The second is equivalent to "Some bird cannot fly." –  Thomas Andrews Sep 1 at 17:20

5 Answers 5

up vote 4 down vote accepted

$B(x):$ x is a bird.

$F(x):$ x can fly.

All birds can fly: $$\forall x(B(x)\rightarrow F(x))$$

Negation of the above $$\lnot \forall x(B(x)\rightarrow F(x))\equiv \exists x \Big(\lnot(\lnot B(x) \lor F(x)\Big) \equiv \exists x (B(x) \land \lnot F(x)$$

You can certainly translate the statement as you did, and your translations are equivalent: mathematically, some is equivalent to at least one. So you may as well stick with "some".

An alternative translation of the negation would be simply the negation symbolized at the very left: "Not all birds can fly."

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ultimately here, the important thing about the translation, symbolically or in "natural language", is the assertion of existence: we don't know whether only one or 1 million birds can't fly. But if the statement is true, we do know that such flightless bird(s) exists. –  amWhy Sep 1 at 17:36

Translated in to set-language: "Every element of the set of birds $B$ belongs to the set of animals that fly $F$". So this translates as $B \subset F$. Negating this would be the equivalent of saying $B \nsubseteq F$. So in set language this would translate:"At least one bird cannot fly"(there is an element of $B$ that does not ly in $F$).

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Both the sentence are correct.

I have come through many questions like these in which two answer are correct, when i asked to my teacher about it teacher he told me to mark first correct option.

So if it is asked as multiple choice question then you should mark which ever comes first.

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1  
Depends. (2) Is equivalent to "Some bird cannot fly." (1)s usage of "birds" plural is a grey area in English, I think. –  Thomas Andrews Sep 1 at 17:21

If "some birds" is synonymous with "at least one bird", the two are equivalent. Whether the plural form implies "at least two" rather than "at least one" may be debatable, but this is a question of English rather than mathematics. "There is at least one bird which can not fly" is unambiguous.

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At least 1 bird mean $1 \ge$ birds which cannot fly. –  Freddy Sep 1 at 17:25
    
'some' is usually more than 'a couple', and less than 'several'. –  Pete Kirkham Sep 1 at 21:09

I would say only 2 is the negation of All birds can fly.

All birds can fly is only true if all birds can do it.

So if not all birds can fly it would be a negation.

There is at least one bird which can not fly, is equal to not all birds can fly

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