Negation in natural language

Question

"In every supermarket, if we can buy fish then the supermarket has a refrigerator"

Is the negation of this sentence

1) There exists a supermarket where we can buy fish and it doesn't have a refrigerator

or

2) There exists a supermarket where if you can buy fish, then the supermarket does not have a refrigerator.

I would write the first sentence as 'In every supermarket that we can buy fish there is a refrigerator'. The phrase "if we can buy fish" or "that we can buy fish" implies that the supermarket sells fish and at the very least that you have enough money/goods to exchange for the fish. In that case 'There exists a supermarket that sells fish, that doesn't have a refrigerator' is not a negation of the first sentence. Is that really what you meant to say? — James Arathoon, Nov 13 '17 at 14:52

score 1 · Accepted Answer · answered Nov 13 '17 at 06:58

1

1 is correct. 2 is wrong because it is already satisfied if there is a supermarket where you cannot buy fish.

answered Nov 13 '17 at 06:58

Thank you. Number 2 I came up with it yesterday, I was also more confident on number 1 but I needed to be reassured. – Disjoint Nov 13 '17 at 07:12

score 0 · Answer 2 · answered Nov 13 '17 at 07:15

You should be able to tell from the words, but translating to predicate logic might make it clearer.

"In every supermarket, if we can buy fish then the supermarket has a refrigerator"

$ \def\inn{\hspace{1pt}{\in}\hspace{1pt}}\forall s\inn S~(B(s)\to R(s))$

Is the negation of this sentence 1) or 2)?

1) There exists a supermarket where we can buy fish and it doesn't have a refrigerator

$\exists s\inn S~(B(s)\wedge\neg R(s))$

2) There exists a supermarket where if you can buy fish, then the supermarket does not have a refrigerator.

$\exists s\inn S~(B(s)\to\neg R(s))$

2 Answers2