Negation

[DavidH327]

Hello

I have a question regarding negation of "without"

For example, a statement says: Hartley could not have written the book without the passage in question.

Does negation becomes:
Hartley could have written the book without the passage in question?
Or
Hartley could not have written the book with the passage in question?


Moreover, is it possible to cancel could not and without to make into could and with?
For example, if I said "Harley could have written the book with the passage in question,"
is it equivalent with "Hartley could not have written the book without the passage in question?"


Lastly, in a statement where "without" is present, I believe what comes after "without" is a necessary condition.
For example A without B represented as -A --> B
What if question said A with B? Does "with" suggest any conditional chain?

Thank you!

[ohthatpatrick]

Negating a claim just means, "contradicting it, in the most minimal fashion possible".

If you say "Hartley could not have written the book without the passage in question."
and I say "Nuh-uh!"
then what have I just committed myself to?

I'm saying he COULD have written the book without that passage.

The original statement could be paraphrased as "Hartley NEEDED that passage to write that book"
so the negation is, "No he didn't. He didn't need it. He could have written the book without it."

If someone said, "You couldn't have gotten into Harvard without your Grandpa's massive donation"
then negating is saying, "Yes I could have. I could have gotten in without my Grandpa's donation."

If you want a more structural understanding, the word "without" creates a conditional (usually). The original statements was
"If no passage, then couldn't write book"

Negating a conditional always means saying, "It's possible that the LEFT SIDE is true but the RIGHT SIDE is false."
Thus, "it's possible that there was no passage, but he COULD write the book."

YOU ASKED:
if I said "Harley could have written the book with the passage in question,"
is it equivalent with "Hartley could not have written the book without the passage in question?"

No, those don't mean the same thing.

"I could have gone to get money from the ATM with a fork in my pocket"
is not the same as
"I couldn't have gone to get money from the ATM without a fork in my pocket"

When we hear, "I could have gone to get money with a fork in my pocket", it's just as likely to be true that "I could have gone to get money without a fork in my pocket".

Your last two examples aren't enough of a sentence or idea for me to know what you're saying.
"A without B"?

If we said, "Janet gets sad without her puppy", it would be
If no puppy, then sad

If we said, "Janet gets happy with her puppy", it's debatably conditional (LSAT would be more obvious about wording that in a way that indicated a permanent connection). Assuming they said "Janet always gets happy with her puppy", it would be
If puppy, then happy

In general, "with" by itself has never been used to indicate a conditional.