QUESTION $1$

Question $1\mathrm{.}1$

Consider the following propositional symbols and their intended meanings:

$p$: $Heis$ poor

$q$: $Heis$ clever

$r$: $Heis$ honest

$($i$)$ Express the following declarative sentence in propositional logic using the propositional symbols

as given above:

It is false that he is poor or clever but not honest.

$($ii$)$ Express the following propositional logic formula in English where the propositional symbols

have the meanings given above:

$p(qvvr)$

Question $1\mathrm{.}2$

Use the basic natural deduction rules for propositional logic to prove the validity of the following

sequents:

$($i$),|--((pq{)}^{{?}^{?}}(notr$notq$))(pr)$

$($ii$),p{r}^{{?}^{?}}q,$notr,$s^{{?}^{?}}t|--not{p}^{{?}^{?}}t$

Question $1\mathrm{.}3$

Show that the following sequent is not valid by giving an appropriate valuation.

$pq,$notq$|--$notp

Explain why your valuation proves that the sequent is not valid.