The meaning of: a program P posseses a property R
I cannot see the article and have to guess. The text makes clear that they are using the symbol $\supset$ as notation for implication. So you could read it as 


  In order to prove that a program $P$ possesses a property $R$, one has only to prove the implication 
  $W(P) \implies R$.


See the Wikipedia page on material implication for other notation for implication.