Let $($A$,<)$ be a partial order with two minimal elements a$,$ b and a maximum element c$.$ Let p: A $->\{$true$,$ False$\}$ be a predicate defined on a$.$ Suppose that P$($a$)=$ true, p$($b$)=$ false and p$($a$)=>$p$($b$)$ for all satisying a $<$ b$,$ where $=>$ stands for logical implication. Which of the following statement is false