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}

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(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}

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↖————————————————

↖———————————————————————

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. ˫