On Test 2b there were two questions on the issue of statements being logically independent that I found myself tripping on a little. Can you help me understand them more clearly?
The first is Test 2b, 11a: “It is later than 1:00 pm. / It is later than 2:00 pm.”
The next is Test 2b, 11c: “Some siblings are twins. / Some siblings are not twins.”
Both are said to be not logically independent. I would appreciate if you could help me see that more clearly than I do.
Two statements are independent when their truth values do not affect each other. In both these pairs of statements, the truth value of one affects the truth value of the other.
“It is later than 1:00 pm. / It is later than 2:00 pm” — If the first is false, the second must be false. You could also argue that if the second statement were true, the first statement would have to be true. Thus their truth values affect each other.
“Some siblings are twins. / Some siblings are not twins” — These statements cannot both be false, for if they were, then “No siblings are twins” and “All siblings are twins” would both be true, which is impossible. If one is false, the other must be true. Thus their truth values affect each other. (They are in fact related by subcontrariety.)