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]