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. ˫