Zsifnrhr. f. math. Log* und GruMagmi d. Jlatl i . Bd. Z?, .$. I l l - l l b (1970)

A NOTE ON THE EXISTENCE OF TAUTOLOGIES IN CERTAIIS PROPOSITIONAL CALCULI WITHOUT PROPOSITIONAL VARIABLES

by W ROSE in Nottingham (Great Britain) It has recently been shown that1), in the propositional calculus whose primitive

symbols are the logical constant t an variable functors 8, E , . . . of two arguments, the possible values of these variables being the 10 non-trivial binary functors, the formula

m a t t a d t m

is a tautology and that infinitely many tautologies exist since, if P, Q are (not neces- sarily distinct) tautologies and the result of replacing an occurrence of t in P by Q is R then R is a tautology. The object of this note is to consider the existence of tau- tologies in those propositional calculi g,, whose primitive symbols are the logical constant d and variable functors J,, E,,, . . . of n arguments (n = 2,3, . . .). We shall denote by 8,, the set of possible values of the variable functors of P,, and by F, the functor of n arguments whose truth-table is such that the formula Fnpl . . . pn is an absurdity (n = 2,3, . . .).

If Fn E &,, it follows that Pn possesses no tautologies other than the formula t since every other formula takes the truth-value F when all its variable functors take the value F,. However, in all cases where F, # rZn, P,, possesses tautologies other than the trivial tautology t. Let 9,, denote the set of all the Z2" - 1 functors G, of ?a argu- ments such that the formula G n p l . . . p, is not an absurdity. It will be sufficient to prove the following

Theorem. The. propodionul calculus Pn for which 8, = gn possesses infinitely many tautologies (n = 2,3, . . .).

We shall prove, by induction on n, that Pa possesses a non-trivial tautology contain- ing no variable functors other than d,,. The theorem then follows by considering the tautologies P, Q and the related formula R as above.

In the case where n = 2 let us denote by H , , . , . , H, the functors whose truth- tables are such that

Hlpq = ~ p , H#q = T q , Hspq =TNP, H@q = T N q , HriW = ~ t . It will be sufficient to show that*) if 6 takes the value Hi then the formula ddtdttddtdttt takes the truth-value T (i = 1, . . ., 5) .

If i = 1 then ddtdttddtdttt = T dtdtt = T t . If i = 2 then 88tdttddtdttt = T 88tdttt = T t . If i = 3 then ddtdttddtdttt = T Ndtdtt = T NMt = T t . If i = 4 then ddt8tt66tdttt = NBdttQttt = NNt = T t . If i = 6 the result follows a t once.

l) ROSE, ALAN, Tautologies sam variables propositionnels. C. R. Acad. Sci. Paris 276 (1972),

2) When considering the cme where n = 2 we shall omit the suffix from 6,. 377 - 379.

118 ALAN BOSE

We now assume the result for n and deduce it for n + l.. By our induction hypo- thesis there exists a nontrivial tautology P , of @, containing no variable functors other then 6,. Let Q, denote the formula obtained from P, by replacing all subformulae of the form &R, . . . R,, starting from the innermost, by the corresponding formulae 6,,,,tR,. . . R, and let P,+, be the formula obtained from P, by replacing all sub- formulae of the form 6,R1 . . . R,, starting from the innermost, by the corresponding subformulae Blr+lQnR1. . . R,. We shall divide the proof into two cases, the value G,,, of the variable functor dntl in case I (11) being such that the formula G,+,tpl . . . pn is not (is) an absurdity.

Case I. Let L, be the functor of n arguments such that, for all formulae S,, . . ., S,,

(4 L,S, . . . S, = T G,.+,tS1 . . . S, . Since L, is not F, the formula P, takes the truth-value T when S, takes the value L, . Hence, by (A), Q, takes the truth-value T when a,+, takes the value GQt1. Thus, in this case,

6,+,QnR, . . . R, = Bn+,tRl . . . R,, and PILtl = Q, = t . Case 11. Let Mn be the functor of n arguments such that, for all formulae S,, . . . , S,,

(B) M,,S, . . . 8, = T G,+,fS, . . . 8,%.

Since G,+l is not F,+l the formula Gn+,fpl . . . p , is not an absurdity. Thus 1M, is not F, and P, takes the truth-value T when 6, takes the value M,.

Hence, by (B), M n s , * * * Sn = T Gn+,QnS, * * * sit

and P,,+l takes, in this case, the truth-value taken by P, when 6, takes the value M , , i.e. T .

Alternatively we may prove1) the theorem by showing, by induction on m, that there exists a formula Pnm (P,,,,, += t ) containing no variable functors other than d, , , which takes the truth-value T whenever the value of d,, is one of the functors Gnl,. . ., Gflm ( m = 1 , . . ., 2=% - l), where 9, = {G,,, . . ., G,,,}, a = 2=" - 1 .

= 1 we may takes P,,, to be a,$. . . t since we may assume, without loss of generality, that

We now assume the result for m and deduce it for m + 1.

If P,,,,, takes the truth-value T when 6, takes the value G,,,,, we may take Pl,.I,lT1 to be P,,,,, . In the other case we note that, since G,,,,, is not F,,+,, it follows easily that if Q,,,, is obtained from P,, by replacing all occurrences of d, by G,,,+, and all occurrences of t by distinct propositional variables then there exists an assignment of truth-values to these propositional variables under which Qnn takes the truth-value T7. Hence lJ,l,m+l may be obtained from Q,, by replacing all occurrences of Gl,,,,j+l by b,, and all propositional variables which take the truth-value T (P) under the above assignment by t (Pnna).

If

Gn@I . . . p , = T K"-'P, . . . &.

This second proof is much shorter than the first, though it appears to provide more com- plicated examples of tautologies.

(Eingegangen am 6. Dezember 1973)