# What's the negation of “One of the sentence is false”?

I think is "Any of the sentence is wrong" but I'm not sure, maybe "Any of the sentences is right"?

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Is it "Only one of the sentence is false"? – hkmather802 Feb 23 at 12:11
This depends on whether "one" means "at least one" or "one and only one". – BrianO Feb 23 at 13:18
Keeping about the same level of ambiguity: "Not one of the sentences is false". – Yves Daoust Feb 23 at 15:57
“It is not the case that one of the sentences is false.” – MJD Feb 23 at 16:05
i think this could go for a healthy dollop of grammar – MichaelChirico Feb 23 at 16:42

If we interpret the word "one" as meaning "exactly one" then negating "one of the sentences is false" means that we should not have exactly one of the sentences false thus we should have the sentence

$$\text{All sentences are true or more than one sentence is false}$$

We could of course interpret "one" as meaning "at least one" (something which may be argued to be the most common mathematical interpretation) in which case we would translate the sentence to $$\text{All sentences are true }$$ Which is what you answered (formulated a bit differently).

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I'm taking "One of the sentences is false" to mean "There is a false sentence".

"One of the sentences is false" is negated to "It is not the case that one of the sentences is false": that is, "there are no sentences which are false".

"There exists a sentence $S$ such that $S$ is false" is negated to "For all sentences $S$, have $S$ is not false": i.e. "For all sentences $S$, $S$ is true".

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The negation of
"one of the sentences is false"
is
"no sentence is false or more than one sentences are false"

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If we take "One of the sentences is false" as "There exists a sentence that is false", it can be written as:

$∃(x) \text{ false} (x)$

And its negation would be:

$¬∃(x) \text{ false} (x)$

From here, the $¬$ can be pushed inwards.

$∀(x)¬\text{ false}$

This is basically saying "All of the sentences are not false".

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Let $F(x)$ mean that the sentence $x$ is false.

Then we have that $$\exists x : F(x)$$

Its negation is $$\forall x : \neg F(x)$$

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To negate a sentence, we negate the primary operator. For example, to negate "$\exists x. f(x)=4$", we negate both the quantifier and the "=" operator but leave $f$ alone:

Original: $\exists x. f(x)=4$
Negated: $\forall x. f(x)\ne4$

The quantifier in your example is "one of", that is, a 'there exists' quantifier. The primary operator in the sentence is "is". To negate the sentence, negate the quantifier to get 'for all' and the operator to get "is not" (or "are not"):

Original: One of the sentences is false.
Negated: All of the sentences are not false.

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This is wrong; both statements, as you have stated them, can be true at the same time (e.g., one sentence is true and one sentence is false) or they can both be false (e.g., there are no sentences, or, depending on your interpretation, two sentences are true and two sentences are false). Thus they cannot be each other's negations. – Frxstrem Feb 23 at 17:48
@Frxstrem Yes, I left out the quantifier. Fixed now. – Lawrence Feb 23 at 23:34