I have a point to make about the English. It involves a very common mistake. When using a gerund it should be modified by an adjective. From the examples above: It has been a day without him eating is not correct, it should be; It has been a day without his eating. He came without your knowing. I won`t leave without your kissing me goodbye. etc.
Bonjour Eric - Thank you for pointing this out, it's fixed.
I'm not entirely sure it needed to be fixed. See https://languagelog.ldc.upenn.edu/nll/?p=2637
Some of the examples seem confusing to me, at least in English. What does "They didn't speak without your encouraging them." actually mean?
1. They didn't speak until you encouraged them.
2. They only spoke when (while) you encouraged them.
3. They wouldn't have spoken if you hadn't encouraged them.
4. They didn't speak because you didn't encourage them.
5. Every time they spoke, you encouraged them.
Maybe the French sentences are correct, but require some other translation, perhaps not using "without".
Because some of the English sentences sound a bit strange to me, I find it difficult to say whether a possessive is appropriate in each case.
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