Lecture 6(B): Logic 2 (Strategy for proofs)

Vacuously true statements and their use in proofs.

There's an issue in Parts B and C of Lecture 6 due to the sequencing of the lectures. These two parts of Lecture 6 use open sets in some examples -- and that's a problem, because open sets aren't introduced until Lecture 10, which I had recorded *before* the Logic lectures. Until I can fix this, I suggest that before doing Parts B and C of Lecture 6 you look ahead to Lecture 10 on open sets

[ozgurkaya]

I dont know whether its same thing you mention but how I do think on these is like that; my father 50 years old now. but still the statement that 'if my father born in 2000 he would be 20 years old in 2020' is true. Is my father born in 2000? nope. Is my father 20 years old? nope. but still the satatement 'if my father born in 2000 he would be 20 years old in 2020' is true. because the statement does not claim neither he is 20 years old nor his born in 2000. but the connection between properities. In fact the subject(my father) does not have any importance. what ever you put in the slot of subject statement keep its truth value.

[m.emreturan]

I was watching the videos for my upcoming Math Camp. Thus, I watch your videos in order. And I really appreciate your work. The problem is that, am I mistaken or you did not give the definition of an open ball in your earlier videos, right? It is the first time we are seeing that. I know the defn from my MA but I think it is omitted from your earlier videos in these lecture series.

[FrancoisLichtenstein]

Would we not be able to prove that the empty set is a closed set using the same kind of vacuous proof and thus contradict ourselves? And couldn't we make many nonsensical arguments using this technique. For example, if the argument in the video is okay, then what's wrong with arguing that for every x in the set of natural numbers, x in the empty set implies there exists a y in the empty set equal to x + 1, showing that it's vacuously true that the empty set is an infinite set?