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

 

(2) It is possible that Joe is both a bachelor and happy

 

(3) It is impossible that Joe is both married and a bachelor

 

(4) Joe is a bachelor but it is possible that he could have been married

 

(5) It is necessary that if Joe is a bachelor then Joe is unmarried

 

 

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 relationship 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, □p is not true in any world

 

(6) Consider the following serial relationship 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 relationship 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}

 

(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 relationship 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 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 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

AS1: ◊P ↔ ~□~P

AS2: □(P→Q) → (□P → □Q)

AS3: □P→ P

AS4: ◊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 missing premises.

 

(14)

1. ˫ □□P → P

2. □P → P [AS3]

3. □(□P → P) [2 NEC]

4. □(□P→P) → (□□P → □P)

5. □□P → □P

6. ˫ □□P → P

 

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. ˫ P → ◊P

2. □~P → ~P [AS3]

3. ~~P → ~□~P [2 TRANS]

4. P → ~□~P [3 DN]

5. | P [assumption CP]

6. |

7. |

8. ˫