QUESTION $3$

Question $3\mathrm{.}1$

$38$

Consider the following Kripke model with worlds $x_1,x_2$ and $x_3$ :

$[$TURN OVER$]$

$5$

COS$3761$

Jan$/$Feb $2024$

a$)$ For each of the following relations, determine whether it holds in the above Kripke model and give reasons for your answer: Please note : Reason carries $2$ marks

$($i$),x_1||diamondp$

$($ii$),x_2||pq$

$($iii$)x_3||(pp)$diamond$(pp)$

b$)$ Find a Kripke model that does not satisfy the modal logic formula below. Also explain why the Kripke model does not satisfy the formula. $($Assume there are two worlds $x_1$ and $x_2)$

diamondp$q$

Question $3\mathrm{.}2$

Suppose we want to re$-$engineer basic modal logic to fit the following reading of $$ :

$$:$,$ Agent believes $$

For each of the following two formulas, state whether it should be valid or not and explain your answer in both cases.

$($i$)vvnot$

$($ii$)$ not$not$

Question $3\mathrm{.}3$

Say we interpret the modal operators $$ and diamond to represent the notions "necessary" and "possible" respectively, and say we interpret the propositional letters p and q$,$ to mean $"$we have ingredients for a cake" and $"$we know how to make a cake" respectively.

$($i$)$ Express the following modal logic formula in English:

$p$diamondq

$[$TURN OVER$]$