80 LOGIC, PART I: THE PROPOSITIONAL CALCULUS Chapter 2decrease in exports is halted. A drop in interest rates is necessary to weaken the
dollar. Hence a drop in interest rates is sufficient to cause unemployment to fall.- If I study all night for the abstract algebra examination, I will be tired. If I'm not
energetic, I will not do the assignment for Fourier analysis. Therefore in order
both to pass the exam and do the assignment, it is necessary that I not study all
night.
8. Given that p -+ q is a tautology, in order for q --+ p to be a tautology, it is neces-
sary and sufficient that p q be a tautology. We know that p -, q is a tautology
and that p t, q is not a tautology. Hence q -+ p is not a tautology.
- If Dawson did not meet James last night, then either Dawson is the murderer or
James was out of town. If Dawson was not the murderer, then James did not meet
Dawson last night and the murder took place in the hotel. If the murder took place
in the hotel, then either Dawson was the murderer or James was out of town. But
Dawson met James last night and James was not out of town. Therefore Dawson
was the murderer.