A Note on Modified Abstraction Principles

The abstraction principle is for K any formula not involving x . This is well known to be inconsistent and in [1] Hintikka proposed two modified versions of (0) which we briefly describe. First, he suggested where K is any formula not involving x , and K + is obtained from K by replacing subformulae of the form ∃ zL by ∃ z ( z ≠ x ∧ L ) and those of the form ∀ zL by ∀ z ( z ≠ x → L ), etc. The second version is where K + is a formula of the type described for (1) and z 1 …, z k are all the free variables in (2).