Logical is related to Proof. Proof is a synonym of Show. Show is related to Improvement. Improvement is related to Patch. 
Hope is related to Wish. Wish is a type of Request. 
Logical <--RelatedTo--> Proof <--Synonym--> Show <--RelatedTo--> Improvement <--RelatedTo--> Patch 
Hope <--RelatedTo--> Wish --IsA--> Request 