If a logical argument doesn't lead to a false conclusion, can we be certain that all of its premises must be true?

Over the weekend, Mistress decided to host a barbecue. She made me a list of rules that I need to follow for the barbecue to be a success. The first rule was, "If it rains, the barbecue will be canceled." The second rule was that if the grill is unreliable, then the barbecue will be canceled.

"Yes, Mistress," I said.

On the day of the barbecue, it didn't rain, and Mistress was so happy that it went on as planned. However, Mistress was worried about another problem: her unreliable grill. Throughout the day, the grill worked perfectly, as I slaved away on preparing the best bbq chicken in the South, and the barbecue was a great success.

Later, Mistress reflected on the day's events and shared her divine observations. "If the barbecue had been canceled," she said, "we would have known it was either because it rained or the grill malfunctioned."

"But the barbecue wasn't canceled, was it slave?"

"No, Mistress."

"Then we can be sure the grill was in perfect working order all along?"

She slapped her riding whip in her leathered hand, as if to challenge whether I was telling the truth. I could feel the tears welling in my eyes.

Her wisdom puzzle me. Why was she interrogating me? Why did she seem to challenge logical reasoning itself, not just her slave pig. Didn't the absence of a false outcome guarantee that I had followed all of her rules?

modus tollens
modus tollens
If a logical argument doesn't lead to a false conclusion, can we be certain that all of its premises must be true?
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