MODAL LOGIC EXERCISES FOR HOMEWORK
1. SENTENCE TRANSLATION
For the problems below, translate the given sentences into modal logic form using the following symbols
◊=it is possible that
□=it is necessary that
r=Bob is a redneck
b=Bob is bigoted
g=Bob has a goatee
(1) It is possible that Bob is not bigoted
(2) It is possible that Bob is both a redneck and has a goatee
(3) It is necessary that if Bob is a redneck then Bob is bigoted
(4) It is impossible that Bob is both bigoted and not a redneck
(5) Bob is a redneck but it is possible that he could not have a goatee
2. POSSIBLE WORLD SEMANTICS
For the problems below, determine the truth values of ◊p and □p in each of the possible world scenarios below
Example
Consider the following serial and reflexive relationships between three (and only three) possible worlds. In which of these worlds is □p true and in which is ◊p true:
↷ ↷
{w1 p}———→ {w2 ~p}
↘——→ {w3 p}
↺
Answer: ◊p is true in w1 and w3, □p is true only in w3
(6) Consider the following serial and reflexive relationships between four (and only four) possible worlds. In which of these worlds is □p true and in which is ◊p true:
↷ ↷ ↷
{w1 ~p}———→ {w2 p} ———→ {w4 p}
↘——→ {w3 p}
↺
(7) Consider the following reflexive and reflexive relationships involving two (and only two) possible worlds. In which of these worlds is □p true and in which is ◊p true:
↷ ↷
{w1 ~p} {w2 p}
(8) Consider the following serial and reflexive relationships between two (and only two) possible worlds. In which of these worlds is □p true and in which is ◊p true:
↷ ↷
{w1 ~p}———→ {w2 p}
(9) Consider the following symmetrical and reflexive relationships between three (and only three) possible worlds. In which of these worlds is □p true and in which is ◊p true:
↷ ↷
{w1 ~p} ←———→ {w2 p}
↖————→ {w3 ~p}
(10) Consider the following serial, reflexive and transitive relationships between four (and only four) possible worlds. In which of these worlds is □p true and in which is ◊p true:
↷ ↷ ↷
{w1 p} ———→ {w2 ~p} ———→ {w3 p}
↘————————————————↗
(11) Consider the following reflexive, symmetrical, and transitive relationships between three (and only three) possible worlds. In which of these worlds is □p true and in which is ◊p true:
↷ ↷ ↷ ↷
{w1 ~p}←——→ {w2 ~p} ←——→ {w3 ~p}←——→ {w4 p}
↖———————————↗
↖—————————————↗
↖———————————————————↗
(12) Consider the following reflexive, symmetrical, and transitive relationships between three (and only three) possible worlds. In which of these worlds is □p true and in which is ◊p true:
↷ ↷ ↷ ↷
{w1 p}←——→ {w2 ~p} ←——→ {w3 p}←——→ {w4 p}
↖———————————↗
↖—————————————↗
↖———————————————————↗
(13) Consider the following reflexive, symmetrical and transitive relationships between three (and only three) possible worlds. In which of these worlds is □p true, ◊p true, □q true, and ◊q:
↷ ↷ ↷ ↷
{w1 p, q}←——→ {w2 p, ~q} ←——→ {w3 p, ~q}←——→ {w4 p, ~q}
↖————————————————↗
↖————————————————↗
↖———————————————————————↗
3. MODAL PROOFS
For the problems below, use the rules for propositional calculus and the following modal logic axioms and rules:
Axioms
A1: ◊p ↔ ~□~p
A2: □(p →q) → (□p → □q)
A3: □p → p
A4: ◊p → □◊p
Rules
Necessitation (NEC): if wff A is a proved theorem (i.e., truth table tautology such as “p v ~p”), then we may infer □A
Change Modal Operator (CMO)
◊p :: ~□~p
□p :: ~◊~p
~□p :: ◊~p
□~p :: ~◊p
In the problem below, supply the rules for each statement.
(14)
1. ˫ C → ◊C
2. □~C→ ~C
3. ~~C → ~□~C
4. C → ~□~C
5. ◊C ↔ ~□~C
6. ~□~C → ◊C
7. ˫ C → ◊C
Adding three statements to the premises will produce a formal proof of validity. Supply these statements and indicate the rules used.
(15)
1. ˫ ◊G v ~G
2. □~G→ ~G [AS3]
3. ~~G → ~□~G [2 TRANS]
4. G → ~□~G [3 DN]
5.
6.
7. ˫