Zc&c)ir. 1. math. Logik und Grundlagen d . Nath. jW. 21, S. I41 - 1 4 4 (1975)

A NOTE ON THE EXISTENCE OF TAUTOLOGIES WITHOUT CONSTANTS

by ALAN ROSE in Nottingham (Great Britain)

I t has recently been shown') that the 2-valued propositional calculus whose only primitive symbols are the propositional variables p, q, r , . . . and the variable functors of 2 arguments S, E , . . , possesses tautologies if the set of possible values of the variable funcltors is one of the sets2)

{C, C', E, J > , {C, C', E , S> , and that, if the 6 trivial binary functors are ignored, no tautologies can correspond to any other sets except proper subsets of the above sets, since

H f f = T f ( H E { A , B, B', E', K ) ) anti the functors J, S are duals of eachothcr. Clcarly any set may be extended to include thcb trivial functor V whose truth-table is such that the formula Vpq is a tautology, but not its dual. Since the remaining 4 functors are self-dual no set may be extended to contain any of these.

lye shall show in this note that the above results generalise to the case of variable functors of n arguments (n = 2, 3, . . .) in a straightforward way, though the proof appears to require a more complicated argument. Let Grill, a,,,,, . . ., a,&*, Glrh2 (k = k(n)) denote the functors G of n arguments such that

( i ) G' is not self-dual,

( i i ) (iii) Ilct denote the set . . . , Ql,,k> whew

the dual of G,,hl is GIla2 (a = 1, . . ., k), G'p . . . p =1' N p .

k j = 1 + z 2'-'(/&

i = 1

X , , denote the set of functors H of 71 arguments

d'r l j = A?',, w 3 1 , j (j =

- 1))

1, . . ., 2 9 .

such tha t Hp . . . p = T t and

Vi(vwly, if Sl1 is a set of functors of n arguments and there does not exist an integer j ( 1 s j 5 2 k ) such that Sn is a (proper or improper) subset of &',,, then the propositional c~alculus whose only primitive symbols are the propositional variables p, q, . . . and the variable functors of n arguments d,, , E,, , . . . possesses no tautologies with respect to Sr,. However, such sets are thc only sets without corresponding tautologie~.~) It will be sufficient t,o prove the following

l ) ROSE, A., Tautologies sans constantes. Compb mndus (Paris) 272 (1971), 1617-1619. 2, We use the letters B, B', C', E', J , S to denote non-implication, converse non-implication,

3, It follows easily that if a single tautology P is obtained then 6,6,P . . . P . . . 6,P . . . P is also converse implication, non-equivalence, joint denial and incompatibility respectively.

8 t;rutology.

142 ALAN ROSE

T h e o r e m . The propositional calculus whose only primitive symbols are the proposilional variables p , q, r , . . . and the variable functors of n argum,ents 6,, , E l , , . . . possesses a tau- tology P,,, with respect to the set &,,, (j = 1, . . . , 2"; n = 2, 3, . . .).

. . . , QllaS,./ (7 = y ( n , a, pa)) deIiot,e those formulae Qt,ap,R1 . . . H,, other than G , , , , ~ . . . p such t,hat

Let,

R,, . . . , R,, E { f , p } and G,,aa,R, . . . A, = T N p .

We may suppose, without loss of generality, that , if LX < t and a, t E { 1 , . . . , k) then

(A 1 1.2"-1 - 1 - 3'(?1, a, pa>l 5 1211-1 - 1 - y(n, z, &)I. M'c not,e that GllaBaRl . . . A,, = 1' h'p if and only if the formula Gl lapa l i~ . . . RZ takes t,he t.ruth-value F, where R: is f or t according as R, is f or p ( i = 1, . . ., n). Thus, even if y ( n , a, pa) = y(n, t, P r ) !

(W {Qria~,1 9 * * * 9 QnaB,y} * {QtcrBtl 9 * * * 9 QtlrprY) (a 7 ; 1 x 9 t E { 1 , . - k}) - We shall prove, by induction on i, tha t there exists a tautology symbols other than h,, and p , with respect to the set

cont.aining no

XI, {G,ll,l, - * . > a,,,,,) ( i = 1, * * - 9 k).

If i = 1 then, since is not self-dual and

~ ~ t l l , , P * * ' P = T N P ?

there exist formulae R;, . . ., R:, of the set { p , G t I l a l p . . . p } such t.hat, the formula t2111,1R~ . . . R,; is a tautology or an absurdity. I n the latter case we may take P,,,, t o be t.he formula S,,6,1R~ . . . Hh . . . 6,& . . . R:, (which we shall abbreviate by S,,.) and, in t tic former case, we may take PI,,, to be t h c formula S,,S,, . . . S,,j

\Ye now assume the result. for i - 1 and deduce it, for i . If Pti,, , ,-l becomes a tautology when all occurrences of 6,, are replaced by a,,ia, then we may take P,,-,i to be Pll, , . i- l . If Pll,j,j-l takes the same t,ruth-value as p when S,, takes the value let us con- struct, by met.hods similar t o t,hose used to construct P,,jl , a formula U,,,,, which takcs t h e trut,h-value T whenever S,, takes the value a,,;,,. We may then take P,,j, t o bc t.hc formula obtained from by replacing all occurrences of p by UIr ig , . If takes the same truth-value as N p whenever d,, takes the value G,,igi we replace p by 6,,U,,,,,. . . Ull;p, instead of replacing i t by Unip,.

Finally we consider the case where P,, 1 . , 3 , - 1 becomes an absurdity when all occurrences of a,, are replaced by GlI iDi . Let us denote by U , , , . . . , Ui, (y = y(n, i, pi), /I; = 3 - pi) t h t h formulae obtained from Q n i B i l , . . ., Q , l i S I y by replacing all occurrences of f by / i n , 8nP,,, j , ; - l . . . P,, j , i - i respectively.') Thus, whenever S,, take8 the value Cll,, ,

u,l, . . ., u;, = T N P

and, by (A) and (B)a), if 2"-l - 1 - y(n, i , p , ) 2 - 0 then, whenever S,, takes one of the values Qnlal , . . . , GI,, ,-1,8,-1, a t least one of the formulae U , , , . . . , U i y does not

1) This latter formula becomes an absurdity when dn is replaced by any of the functore

2 ) We apply (B) with respect to a eet 8,. (j' + j) considered below, since, by ( i i ) , QnlB,* * * - 9 Qn. i - iJ i -1 ' Qn.i.8-b.

G/3, @ (Q"lS, 9 * - 9 Ga,,-l*S,-ll

A NOTE ON T H E EXISTENCE OF TAUTOLOGIES WITHOUT CONSTANTS 143

always take the same truth-value as Np. Sincc

oirlfil? * * Gn, i - 1, o,-l 8 Gn, i , a-b, E gn,j#

for some integer j' (1 5 j' 2k) , these latter formulae take the tmth-value T irre- spevtiw of the truth-value of p. Hence if Wi, is obtained from U , , by replacing all occoii~~ences of p in U , , by U,, ( x = 1 , . . ., y ) then, if 6,, takes the value Gll,i,3-pi,

ij7i1, . . , i f ' i y = T p and if b,, takes one of t h c valucs Gltlo,, . . . , Oil, ( 1 5 - i.

Let. .Yi, be IYil and X i , be obtained from if';, by replacing all occurrenccs of p bv Xi, ( x = 2, . . . , y ) . Thus X , , = T t when b,, t,akcs any of the values Glllbl, . . . , Gll, ,- l ,bi-l and S;, = T p when d,, takes the value GI,, ,,3-o,.

If p - 1 - 1 - y ( ~ , i, 8,) < 0 then , since there are exactly 2" - 2 - y ( n , A , pa) forniiilae R;. . . Kl ot,her than Q,,,oap . . . p such that' RT, . . ., R,t E { t , p } and

or-1, then there exists an integer y ) such that

iViA ='I' I .

G'iiab,R,' . . R,t = T N p , we may, by methods similar to those used above, construct a formiila X i e (0 = 2'8 - 2 - y ) which t'akes the truth-value F when 6,, takes any of t h c values G n l p , ' . * 7 Qn,i-l, and which takes the same t,ruth-value as p when 6, takcs the valiic. G,, , i ,3-b, . Thus the formula B,,X,:, . . . X i e (which we shall abbreviate by X:@) takths the truth-values of 1 , Np in the respcctive cases and we may obtain Xi, (with thc same truth-table properties as Xi, above) from by replacing all occurrmces of p by X L .

Sin(*(. p is self-dual, if follows that if

- y ( n , i, p1, 2 ( < ) 0 , 211-1 - 1

Xi, (.Vie) takes the same truth-value as p when 6, takes the value obtttin a tautology PZi with respect, to {aIllg,, . . . , all oc.cwrrences of p by Ul,io,.

and we may from X,, ( X i e ) by replacing

We may take to be NdN6Plf j i , where NaP = d f S,P . . . P.

Wc note in conclusion tjhat we may obtain') the following

( 'orol lary. There exist a functor QImba of gI,, (a = ~ y ( j ) , ,!la = Ba(j)) and a formuh Yllabm containing no symbols other than 6,,, pl , . . . , p,, such that (i)

(ii) (iii)

if 6,, takes the value G,laba then Y,ja,a = T Gnabapl . . . p i , , if 6, taka any other value 01 d'nj then Y,,pm = T t , the propositional calculus whose only primitive functor is Qruxam is funct;onaUy complete.

We shall Rhow that the formula NbNdX,#(NdNdXk#,) and the functor Udpk (y = y(n , k, 3 - pk), @ = @(n, k, 3 - Pk)) are suitable choices for Ymba, Qua, re- spectively, where Xk#y @,#,) is obtained from &, (&) by replacing all occurrence8 of p by drip, . . . p , .

l ) This result will enable ua to consider, in a subsequent paper, the problem of formalisation.

144 U N ROSE

(i)

(G) ~afik#,(Nfl,X,#,) = t for these k - 1 + 2"-a valus of 6, belonging to Q,, . (iii) By@),

Shce &, (&) = T p when a,, takes the Value Qnkpk it follows at once that N,,N6X&

Since xky (&) = t when S, takes any other value of gI,, it follows at once that (Nflf iA) = T @&Jkpl * - p n in this case.

I1 I 1

GflkpkP1. . . P, = T N C Pi or Q n k p k P 1 . . . PI, = T N I7 P , . I = 1 i = l

Thus, in the respective cases, we may make the definitions

JPQ =df QtApkPQ * * Q, SPQ = d f Gtthpkf'Q * * * Q and the result follows at once.

(Eingegangen am 26. November 1973)