CdaChT. 1. math. Logik clnd Orundlagen d . Nath.. Bd. 1, 5.198-191 (1966)

A SINGLE AXIOM FOR A PARTIAL SYSTEM OF THE PROPOSITIONAL CALCULVS

By ALAN ROSE in Nottingham, England

calculus with equivalence and non-equivalence as primitives. There has recently been foundl) a formalisation of the 2-valued propositional

E p q I T F T 1;; P

This formalisation uses the two axioms

(8) E E p q E E r q E p r (b) E E p q E E' rq E' pr

together with a substitution rule and the rule ,,If P and E PQ then Q". These axioms have been shown to be independent.2)

The object of the present paper is to show that if instead of E and E' we take E and the constant 0 (denoting a false statement) as primitives then the formalisa- tion which uses only the axiom (a) and the above rules of procedure is complete. Except for formulae which contain no propositional variables, every function which can be defined in the present system can also be defined in the system of Mrs. RASIOWA and vice-versa. In the present system we can define E'PQ by

E ' P Q E E E P Q O .

In the system of Mrs. RASIOWA any formula of the form

@ ( p l , p 8 , . . . , p n , 0) has the same truth-table as @Po1, p z , . . . , p n , E ' p l p l ) .

Since (b) becomes identical with (a) if we replace E' by 2, (a) is sufficient for the deduction of any identical formula in whioh the only symbols used are E

1) IT. RASIOWA, ,,Axiomatisation d'un systhme partiel de la thborie de la d6duction".

3) loo. cit. Comph rendus (Warsaw), Chsse 111 40 (1947). 22-37.

A SINGLE AXIOM FOR A PARTIAL SYSTEM OF THE PROPOSITIONAL CALCULUS 197

and propositional variables. Thus we can deduce from (a):

(F 1) E E p q E E E r q s E E p r s , (F 2) E E p q E E r s E E p r E q s , (F 3) EEEmq!I *

Substituting 0 for s in (F 1) we deduce

(F 4) EEpqEE'rqE'pr.

Thus we can deduce from (a) any identical formula in which the only functions are E and E', E' being defined as above.

We now consider the other identical formulae. To any formula P there corre- sponds a formula P', such that E P ' P is identically true and Pi contains no functions other than E and E', E' being defined as above. We will now prove that, for a suitable choice of P', E P ' P can be deduced hom (a).

We prove this by strong induction on the number n , of (not necessarily &tinct) symbols occurring in P. If n = 1 then P is either 0 or a single propositional variable, let us say p . Thus P' is E'pp in the first case and p in the second. Hence the respective forms of E P' P are EEEppOO and E p p . The first of these is deduced by applying the substitution rule t o (F 3) and the second is derivable since the only symbols occurring in it are E and propositional variables.

We now assume the result for 1 , 2 , . . . , n and prove it for n -j- 1 . Then P is of the form EQR where neither Q nor R contains more than n symbols. By our induction hypothesis we can prove EQ'Q and E R'R. From these and (F 2 ) we deduce EEQ'R'EQR, i. e. E P ' P .

Hence if P is a tautology then P' is a provable tautology and E P P is provable. From P' and E P' P we deduce P . Thus the formalisation is complete.

(Eingegangen am I . November 1966)