The exercises and theorems referred to by number come from the end of Chapter 3 in the Gries & Schneider text (a copy of which is on reserve in the library). In proving a theorem, you may make use of only lower-numbered theorems (or "metatheorems"). Note that some of the exercises may have hints that appear in the book, but are not repeated here. Don't forget the precedences of operators, which can be found on the textbook's inside front cover (in the hardback version at least) and here.
In exercises involving conjunction, expect to apply the Golden Rule (3.35) quite frequently and (3.32) occasionally.
1. Suppose that you took Axioms (3.24) through (3.28) and replaced each occurrence of ∨ by ∘. Call the resulting equations (3.24'), (3.25'), ..., (3.28'). (The idea is that ∘ could stand for any of the sixteen functions with signature bool × bool ⟶ bool, as are described on page 26 of Gries & Schneider (and here.)
Make a convincing argument that, among those sixteen functions, the only one that makes all of (3.24'), (3.25'), ..., (3.28') into tautologies is the one that we call disjunction. (In other words, show that if we let ∘ stand for any of the fifteen functions other than disjunction, at least one among (3.24'), (3.25'), ..., (3.28') is not a tautology.)
(b) Now find a smallest subset of {(3.24'), (3.25'), ..., (3.28')} such that, if all members of that subset are tautologies, then ∘ necessarily stands for disjunction. (In effect, you are being asked to identify a smallest set of axioms, among (3.24), ..., (3.28), that suffice to uniquely identify disjunction among all sixteen two-argument boolean functions.)
2. Do Exercise 3.17, which is to prove that false is the identity element of disjunction:
Hint: As a first step, use (3.15) by replacing false by ¬p ≡ p.
For each scenario described, make the strongest statement you can regarding the status (knight or knave) of each of Bill and Carol (or the relationship between their statuses).
Use proposition variable b to correspond to the proposition Bill is a knight. Similarly, use c to mean Carol is a knight.
8. A visitor to the island approached Bill and Carol and asked Carol what kind of person she was. Carol answered the question, but the visitor could not hear her answer. Bill then said, loud enough for anyone to hear, "Carol said that she is a knave."
9. A visitor to the island approached Bill and Carol and asked them what kind of persons they were. Bill replied, "We are of opposite kinds."
10.
A visitor to the island approached Bill and Carol and asked them
what kind of persons they were.
Bill replied, "At least one of us is a knave."
Hint: Use Theorem (3.32)