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What would be the negation of

"No dogs have three legs".

I think "Some dogs do not have three legs"

"Some animals don't eat meat"

I think "All animals eat meat"

"I make the bread, or she does not make the bread"

I think "I do not make the bread, and she does make the bread"

Am i right here?

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3 Answers

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The negation of the first statement would be "Some dogs have three legs." You are correct on the second and the third.

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Tip: a good way to start when negating any proposition $P$ is to assert $\lnot P$, i.e., if we have a sentence "P", we can negate it simply by writing "It is not the case that P". Then, if we want to "translate" further from loglish to more customary natural language, we can do so if we choose. I personally find it helpful to translate the initial statement into logic. Then I negate it, perhaps distribute the negation, and "read off" the result back in natural language. I do that simply because negating natural language can be easily side-tracked or awkward to directly negate.

You did fine, overall, except for the first assertion:

(1) We start with the assertion $P$: No dogs have three legs.

• Then we negate it by stating $\lnot P:$ It is NOT the case that (No dogs have three legs).
• If we translate further, we see that $\lnot P$ can be expressed as "There exists one or more dogs with three legs": I.e., "Some dogs have three legs".

If we were to translate the initial statement to "logic" first, with the domain being "dogs": and $T(x)$ meaning $x$ has three legs, then the initial first statement can be espressed as $P: \forall x \lnot T(x)$

The negation would be $\lnot P$: $$\lnot [\forall x \lnot T(x)] \iff \exists x(\lnot \lnot T(x)) \iff \exists x T(x)$$ $$\;\;\text{Negated sentence: Some dogs have three legs}$$

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$(2)$ Your second negation is just fine.

$(3)$ Yes, you used DeMorgan's correctly, and your translation is correct.

P: I make the bread;
Q: she makes the bread

Given $P \lor \lnot Q$, its negation is $\lnot (P\lor \lnot Q) \equiv \lnot P \land Q$, which gives us "I don't make the bread and she makes the bread."

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For a sanity check, try to imagine someone saying these sentences in conversation. What would you have to do to prove that they are lying?

For "no dogs have three legs", you would just have to prove the existence of a dog with three legs (hopefully you can find such a dog, rather than having to produce your own example...)

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