The permutation principle in quantificational logic

The story goes back to 1940, with the publication of Quine's Mathematical Logic [5]. He there presents a system of quantificational logic in which only sentences or closed formulas are theorems. In that system, the sole rule of inference is Modus Ponens and the axioms are the universal closures of the following formulas: (1) $, for @ tautologous; (2) Wvb) 1 vcwv~~); (3) qb 3 Vcuj, for o not free in 4; (4) VCY~(LY) > I#@), with usual restrictions; (5) vdv@p 3 v'pvcv#J. Suppose the variables, in alphabetic order, are x1, x2,. .. . Then Quine takes the universal closure of a formula I#J to be the result of prefixing the quantifiers Va, for (II free in 9, to that formula in their reverse of anti-alphabetic order. The presence of the Permutation Principle (5) somewhat mars the elegance of the system; but Berry [l] was able to show how to do without it. Alter the definition of closure: prefer the universal quantifiers in their alphabetic, not their anti-alphabetic, order. The change is small and seemingly insignificant. But taking as axioms the alphabetic closures of (l)-(4) suffices to prove all theorems of the original system, including Permutation. This change was incorporated into the second edition of Mathematical Logic [6]. But the question remained as to whether it was essential Could axiom-scheme (5) also be dropped from the original system without any loss of theorems? The answer is no. Let us define a certain transform (@)' of a formula 9. Call the universal formula Vo$ non-vacuous if 01 occurs free in $. Given a subformula occurrence $ of 4, say that $ is bound in $ if a free variable (Y of J, is bound by a quantifier Va outside of J/ but in 4. Now call the sub-formula occurrence I) of $ replaceable if(i) it is a non-vacuous formula of the form Vx, x (note the specific variable), (ii) it is bound in 4, and (iii) it is