Logical Structures Exercises
Monday 6/24
The Drinkers Problem
Suppose that the universe is the people in the conference.
Prove that the following occur:
- Someone is a light drinker: Whenever everyone else
is drinking, this person is also drinking.
- Someone is a heavy drinker: Whenever anyone else
is drinking, this person is also drinking.
- Someone is a drinking leader: Whenever this person drinks,
everyone else also drinks.
- Someone is a non-drinking leader: Whenever this person does
not drink, no one else drinks either.
In logical notation, let D(n) be the proposition that
n is drinking, then the theorems are:
- ∃a((∃x.D(x))→D(a))
- ∃a((∀x.D(x))→D(a))
- ∃a(D(a)→(∀x.D(x)))
- ∃a(D(a)→(∃x.D(x)))