 Fragments of \(\mathbf{IPC}\) missing one or more logical connectives restrict the language and incidentally the logic, since the intuitionistic connectives \(\oldand,\) \(\vee,\) \(\rightarrow,\) \(\neg\) are logically independent over \(\mathbf{IPC}.\) Rose [1953] proved that the implicationless fragment (without \(\rightarrow\)) is complete with respect to realizability, in the sense that if every arithmetical substitution instance of a propositional formula \(E\) without \(\rightarrow\) is (number)-realizable then \(E\) is a theorem of \(\mathbf{IPC}.\) This result contrasts with: