MODAL LOGIC IN-CLASS EXERCISES
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
b=Joe is a bachelor
m=Joe is married
h=Joe is happy
(1) It is possible that Joe is unmarried ◊~m
(2) It is possible that Joe is both a bachelor and happy ◊(b & h)
(3) It is impossible that Joe is both married and a bachelor ~◊(m & b)
(4) Joe is a bachelor but it is possible that he could have been married b & ◊m
(5) It is necessary that if Joe is a bachelor then Joe is unmarried □(b → ~m)
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 only 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}
↺
Answer: ◊p is true only in w1, w2 and w3; □p is true only in w1 and w3
(7) 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}
Answer: ◊p is true in w1 and w2, □p is true in w1 and w2
(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}——→{w3 ~p}
Answer: ◊p is true in w1 and w2; □p is true only in w1
(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}
↺
Answer: ◊p is true in w1 and w2, □p is true only in w2
(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}
↘ —————————————————↗
Answer: ◊p is true in w1, w2, and w3; □p is true only in w2 and w3
(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}
↖———————————↗
↖—————————————↗
↖———————————————————↗
Answer: ◊p is true in w1-4; □p is not true in any world
(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}
↖———————————↗
↖—————————————↗
↖———————————————————↗
Answer: ◊p is true in w1-4; □p is true in w1-4
(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}
↖————————————————↗
↖————————————————↗
↖———————————————————————↗
Answer: ◊p and ◊q are each true in w1-4; neither □p nor □q are true in any world
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 , then we may infer □A
Example: from the truth table tautology “p v ~p”, we can infer □( p v ~p)
Change Modal Operator (CMO)
◊p ↔ ~□~p
□p ↔ ~◊~p
~□p ↔ ◊~p
□~p ↔ ~◊p
In the problem below, supply the rules for each statement.
(14)
1. ˫ □□H → H
2. □H → H [A3]
3. □(□H → H) [2 NEC]
4. □(□H→H) → (□□H → □H) [A2]
5. □□H → □H [4,3 MP (→E)]
6. ˫ □□H → H [5, 2 HS]
Adding two statements to the premises will produce a formal proof of validity. Supply these statements and indicate the rules used.
(15) Use conditional proof (CP) in the following proof.
1. ˫ J → ◊J
2. □~J → ~J [A3]
3. ~~J → ~□~J [2 TRANS]
4. J → ~□~J [3 DN]
5. | J [assumption CP]
6. | ~□~J [4, 5 MP]
7. | ◊J [6 CMO]
8. ˫ J → ◊J [5-7 CP]