The present note gives a reduction to E.G(?)tlind's axiom system forthe propositional calculus:(1)p∨p(?)p,(2)p(?)p∨q,(3)p(?)p,(4)(p(?)r)(?)(q∨p(?)r∨q).which he showed in a paper to be sufficient but with the independenceproblem left open.(I learn this only through a review of his paper byB.Jónsson(Math.Reviews,9(1948),p.1).)In I.it is shown that(3)is a consequence of(1),(2)and(4)(using-and ∨ as primitive junction symbols)and hence is superfluous.In Ⅱ.a proof of the mutual independence of(1),(2)and(4)is ap-pended;the proof of their sufficiency is easy(omitted);hence(1),(2)and(4)constitute a new axiom system for the propositional calculus.Finally,comparison of the new system with Hilbert's axiom systemsuggests a frequently occuring phenomenon,namely,the omission of thecommutative laws(or symmetric laws)contained in an axiom system can frequently be effected as a result of suitably permuting the variables con-tained in other axioms of the system.As another(new)example of thisphenomenon is given(ithout proof)the following result,which is asolution to Problem 64 in G.Birkhoff's Lattice Theory(revised edition,(1949)p.138):Birkhoff's axioms on a ternary operation(abc)for defin-ing a distributive lattice with O,I:(14),(15),(16)and(17)(see his bookp.137),in which(16)is a commutative law,are equivalent to the follow-ing(mutually independent.)ones:(14),(15)and(17')(d(abc)e)=((ebd)(dae)c).
