QUESTION $8$

$[6]$

In this question you have to show that the validity of a sequent cannot be proved by finding a model where all formulas to the left of $F$ evaluate to $T$ but the formula to the right of $|--||$ evaluates to $F.$

Question $8\mathrm{.}1$

Show that the validity of the following sequent

AAx$(R(x)Q(x)),$AAAAx$(R(x)vvQ(x))$

cannot be proved by finding a mathematical model where the formula to the left of $|--||$ evaluates to $T$ but the formula to the right of $F$ evaluates to $F.$

Question $8\mathrm{.}2$

Show that the validity of the following sequent

AAxAAy$(S(x,y)$notS$(y,x)),|--$AAxS$(x,x)||$

cannot be proved by finding a non$-$mathematical model where both formulas to the left of $F$ evaluate to $T$ but the formula to the right of $F$ evaluates to $F.$