# 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*)))