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I was wondering when is safe to cancel out two negatives in a sentence?

E.g.,

No X are not Y, can we cancel out to X->Y

But if we present a statement

If not X then not Y, then it would invalid to cancel out the two statements to X->Y.

So is it the rule that we cannot cancel out conditional negatives? but we cancel out the negatives in "is" and "are", are there any more rules like this?

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Max324 wrote:When you have words like "All" or "Every" being used to describe elements, diagram by making the clause attached to "All/Every" sufficient, and the remaining clause necessary.

Example:
All dogs go to heaven.
Dogs -> Go to heaven
Every student in the back row wears a green shirt.
Student in the back row -> Wears a green shirt

When you have words like "No" or "None" being used to describe elements, diagram by making the clause attached to "No/None" sufficient, and the negation of the remaining clause necessary.

Example:
No dogs allowed on the lawn.
Dogs -> ~Allowed on the lawn
None of the students in the back row can see the chalkboard.
The students in the back row -> ~Can see the chalkboard

Your first example:
No X are not Y
"X" becomes sufficient, negation of "not Y" becomes necessary.
X -> Y

When you have a conditional such as ~X -> ~Y, as in your second example, form the contrapositive: Y -> X (which you should be doing anyways).

Not exactly what I was hoping for, but thanks anyways.

jimmierock wrote:I was wondering when is safe to cancel out two negatives in a sentence?

E.g.,

No X are not Y, can we cancel out to X->Y

But if we present a statement

If not X then not Y, then it would invalid to cancel out the two statements to X->Y.

So is it the rule that we cannot cancel out conditional negatives? but we cancel out the negatives in "is" and "are", are there any more rules like this?

It is only safe to "cancel" negatives when a conditional statement creates a double negative, as Max234 explains in his third example. The two negatives don't "cancel" in your second example (If not X, then not Y) because the second example does not require you to negate an already negative entity.

"No X are not Y" is a double negative because it applies two "negatives" to Y. The "No ___ are ___" format translates to "All ___ are NOT ___." In this scenario, the X is never negated and the Y is negated twice. It is confusing because it appears like the "No" applies to the X, but it does not.

For example:
No lizards are mammals = All lizards are not mammals.
lizard --> ~mammal
As you see here, the "no" is applied to the second entity, and the first entity is not negated.

So, if you add the second negative, you create a double negative that "cancels" out:
No lions are not mammals = All lions are not not mammals
All lions are not not mammals = All lions are mammals
lion --> mammal

As you can see, the first entity in the "No ___ are ___" is not negative. The "no" applies to the second entity. If the second entity is already negative, then it becomes positive.

The second example (If not X, then not Y) contains two negatives, but not a double negative. Each entity is negated once, so you cannot cancel out any of the negatives. I don't know if there's any trick or rule involved here, you just need to understand what the sentence is telling you in terms of what is sufficient, what is necessary, and what is negated. I'm not sure if I'm explaining myself well, but hopefully this will help.

flippacious wrote:

jimmierock wrote:I was wondering when is safe to cancel out two negatives in a sentence?

E.g.,

No X are not Y, can we cancel out to X->Y

But if we present a statement

If not X then not Y, then it would invalid to cancel out the two statements to X->Y.

So is it the rule that we cannot cancel out conditional negatives? but we cancel out the negatives in "is" and "are", are there any more rules like this?

It is only safe to "cancel" negatives when a conditional statement creates a double negative, as Max234 explains in his third example. The two negatives don't "cancel" in your second example (If not X, then not Y) because the second example does not require you to negate an already negative entity.

"No X are not Y" is a double negative because it applies two "negatives" to Y. The "No ___ are ___" format translates to "All ___ are NOT ___." In this scenario, the X is never negated and the Y is negated twice. It is confusing because it appears like the "No" applies to the X, but it does not.

For example:
No lizards are mammals = All lizards are not mammals.
lizard --> ~mammal
As you see here, the "no" is applied to the second entity, and the first entity is not negated.

So, if you add the second negative, you create a double negative that "cancels" out:
No lions are not mammals = All lions are not not mammals
All lions are not not mammals = All lions are mammals
lion --> mammal

As you can see, the first entity in the "No ___ are ___" is not negative. The "no" applies to the second entity. If the second entity is already negative, then it becomes positive.

The second example (If not X, then not Y) contains two negatives, but not a double negative. Each entity is negated once, so you cannot cancel out any of the negatives. I don't know if there's any trick or rule involved here, you just need to understand what the sentence is telling you in terms of what is sufficient, what is necessary, and what is negated. I'm not sure if I'm explaining myself well, but hopefully this will help.

That makes alot of sense, perfect thanks!

Do you know any other scenarios that creates this "double negative"?