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. ˫