Zt8Chr. 1. math. Logik und Orundlagen d. Math., Bd. 2, S. 204-209 (1968)

SOME FORMALISATIONS OF no-VALUED PROPOSITIONAL CALCULI

By ALAN ROSE in Nottingham, England

In 1930 EUKASIEWICZ developed1) an no-valued propositional calculus with two primitive functors called implication and negation. The truth-values were all rational numbers satisfying 0 5 x 5 1, 1 being the designated truth-value. If the truth-values of P , Q , N P , CPQ are x , y , n (x) , c (x , y) respectively then

n(x) = 1 - x, c ( z , y) = min(1, 1 - x + y). Alternation and conjunction were defined by

A P Q ; CCPQQ, K P Q N A N P N Q . Thus if the truth-values of A P Q , KPQ are a ( x , y) , k(x, y) respectively then

a ( x , y) = max(x, y), k(x, y) = min(x, y). We define a second alternation functor and a second conjunction functor re- spectively by

BPQ C N P Q , LPQ as NCPNQ.

Thus if BPQ, LPQ take the truth-values b(x, y), l ( x , y) respectively then

b(x, y) = min(1, x + y) , 1(x, y) = max(0, x + y - 1).

We shall also make use of the functor H ) such that the truth-value of H P is half that of P . This functor is not definable2) in terms of C and N . Although no ,,standard condition" implication3) functor is definable in the EUKASIEWICZ calculus is has been shown4) that if P , Q are formulae of this calculus and Q takes the truth- value 1 whenever P takes the truth-value 1, then for some integer a, the for- mula (C P)"Q always takes the truth-value 1 . The above notation is in accordance with an abbreviation convention previously used by the author 6). This result can easily be extended to the system with C, N and H as primitives. It can also be shown that if Q takes the truth-value 1 whenever PI, P,, . . ., P , all take the truth-value 1 then, for some integer u ,

(Q=i pi)" Q always takes the truth-value 1. This follows from the previous result and the

1) J. LUKASIEWICZ-A. TARSKI, Untersuchungen iiber den Aussagenkalkul, C. r. SOC. Sci.

2) R. MCNAUOHTON, A theorem about infinite-valued sentential logic, J. Symbolic Logic

3) J. B. ROSSER and .A. R. TURQUETTE, Many-valued logic^, Amsterdam 1952. 4, A. ROSE, The degree of completeness of Eukasiewicz's No-valued propositional calculus,

K, A. ROSE, A formalisation of an N,-valued propositional calculus, Proc. Cambridge Phil.

Lett. Varsovie, Classe 111, 23, 1-21 (1930).

16, 1-13 (1951).

J. London Math. Soc. 28, 176-184 (1953).

SOC. 49, 367-376 (1953).

SOME FORMALISATIONS OF NO-VALUED PROPOSITIONAL CALCULI 205

fact that, if I7 denotes summation by L, rr=l Ui V has the same truth-value as CnT=n=, Ui V . Here r is the summation operator of ROSSER and TURQUETTE~).

The first published formalisations) for an N,-valued propositional calculus was for the system whose primitives were C, N , H and the logical constant 1 . This formalisation used 21 axioms and three primitive rules of procedure. Much simpler formalisations have since been found 3, for the C - N and C calculi. These use only four axioms and no primitive rules of procedure other than substitution and modus ponens. The object of the present paper is to Rhow that the last two formalisations lead t o a much simpler formalisation of the C - N - H - 1 calculus and also to formalisations of the C - N - H , C - H and C - H - 1 calculi. It would seem to be of interest to develop such formalisations since, contrary to the behaviour of the C - N and C calculi4), they are strongly complete.

A proof that every weakly complete formalisation of the C - N - H - 1 cal- culus which has substitution and modus ponens as (primitive or derived) rules of procedure is also strongly complete, in effect, already been given5) and the proof is easily adaptable to the C - N - H calculus. It is only necessary to replace, in the substituted logical constants, 1 by C p p . The proof can also be adapted to the C - H calculus for, since c and h are continuous when x and y are regarded as real variables, it follows that if p is defined in terms of c and h and there exist xl , x2, . . . , x, (0 5 x1 5 1 , 0 5 x2 5 1 , . . . , 0 5 xfl 5 1) such that p(xl, x2, . . . , xn) $- 1 , then there exist finite binary fractions yl, y2, . . . , yn such that yl, yZ, . . ., yn =t= 0 and y(y1, y2, . . ., Y,) =i= 1 .

The axioms for the C - N calculus are:

A2 CCpqCCqrCpr

A 4 CCNpNqCqp

A1 CFCPP

A 3 CAP4AW

We first show that if t o the formalisation of the C - N calculus we adjoin the axioms :

A5 CpBHpHp A6 CBHpHpp A7 CHpNHp

then we obtain a complete formalisation of the C - N - H calculus. Let O(pl , p z , . . ., p J be a formula of the C - N - H calculus which always

takes the truth-value 1 and in which the functor H occurs m times. We shall show,

l) See p.204, Note 3. 2) See p. 204, Note 5. 3, The formalisation of the C - N calculus will be developed in a joint paper by the author

and J. B. ROSSER. Five axioms were used originally, but MEREDITII and CHANG have shown that the axiom A C p q C p p can be deduced from the others. The proof for the C calculus will appear in a forthcoming paper by the author.

*) See p. 204, Note 4. 6 ) See p.204, Note 5.

206 ALAN ROSE

by induction on m, that @ is provable. If m = 0 then 0 is derivable from A1-4. We now assume the result for m and prove it for rn + 1 .

Then @(pl, pz, . . ., pn) is of the form Y ( p 1 , ~ 2 , . * - 7 l>n, H P Z , . .

where q occurs exactly once in !P(pl, p2,'. . ., p,, q) and the functor H does not occur in A ( p l , p2, , . . , p,) . Let us now consider the formulae

The formula KKCABppCBppACpNp takes the truth-value 1 if and only if A s m i n ( l , 2 x ) , A Z m i n ( l , 2 x ) , x g l - z ,

where x, 1 are the truth-values of p , A respectively. These conditions reduce to x = 4 A . But, when x = t A , the formula Y(pl, p 2 , . . ., pn, p ) takes the same truth-value as Y(pl, p 2 , . . ., pn, H A ( p l , p 2 , . . ., pn)) , i. e. it takes the truth- value 1 . Hence, for some integer a , the formula (A) always takes the truth-value 1. Thus, since the functor H occurs in this formula exactly m times, it follows from our induction hypothesis that (A) is provable. Substituting H A for p in (A) and A for p in A5-7 and then using the provable formula CpCqCrKKpqr and applying the rule of modus ponens M. + 3 times we deduce @. Thus the formalisation is complete.

In order to establish the independence of A5-7 we use the truth-tables of C and N as interpretation tables for these respective functors and, in the respective cases, define the interpretation tables for the functor H by

(A) (CKKCABp@BppACpNp)" Y(P, 9 ~ 2 , . * * 9 Pn, I)) *

1 h(1) = 1 1 ' h(x) = $ x (z =/= 1)( ' W X ) = $ x (5) t ) h(x) = x (x 5 4) h ( 4 = $2,

The interpretation tables in the respective cases then satisfy A6 and A7 but not A5, A5 and A7 but not A6, A5 and A6 but not A7. The tables do, of course, satisfy substitution and modus ponens.

If we adjoin the further axiom: A8 1

then we obtain a complete formalisation of the C - N - H - 1 calculus. Let P be a formula of this calculus which $ways takes the truth-value 1 and let Q be the formula obtained by replacing each occurrence of 1 in P by a propositional variable p which does not occur in P. Then Q takes the truth-value 1 whenever p takes the truth-value 1. Hence, for some integer a , the formula (Cp).& always takes the truth-value 1 and is derivable from A1-7. Substituting 1 for p in (Cp)"& and applying modus ponens iy times we derive P. Thus the formalisation is complete.

The axioms for the C calculus are A1-3 and A4C ACpqCqp.

We shall now show that a complete formalisation of the C - H calculus is obtained if to the formalisation of the C calculus we adjoin the axioms:

A5 C CCHpHqCCHpHqCpq A6 C CCCHpHqCpqCHpHq A7C CHpp

SOME FORMALISATIONS OB NO-VALUED PROPOSITIONAL CALCULI 207

and that a complete formalisation of the C - H - 1 calculus is then obtained by adjoining A8.

Lemma. If the propositional variables occurring in the formulae P,, P,, . . . , P,, Q,, Q,, . . ., Q,, R are p , , p , , . . ., p,,& and R takes the truth-value 1 whenever the truth-values of Q,, Q2, . . ., Q, are half those of P,, P,, . . ., P, respectively then, whenever the formulae

CPP (1) QQiPi (i = 1 , 2 , . , ., n) (2) %Pi ( i = 1 , 2 , . . ., m) (3) CpPi (i = 1 , 2 , . . ., n) (4) CCQiqCCQiqCPip ( i = 1, 2 , . , ., n) ( 5 ) CCCQiqCPtpCQiq (i = 1 , 2 , . . . , n) (6)

all take the truth-value 1, R takes the truth-value 1. We first notel) that if a formula @(pl, p , , . . ., p,) of the C calculus takes the

truth-value p(xl, xz, . . . , xn) when p , , p , , . . . , p , take the truth-values xl, x2,. . . , X, respectively then, if k < l , x * = k + ( l - k ) x and (min(x,,x,, . . . , x , ) )*20 then

~ ( 4 , x,*, . - * , = (p(x1, 2 2 , * * * , xn))** (7 1 Also, for k < 1, it follows from the definition of x* that x* < y* if and only if x < y .

Let the truth-values of p , q , Pi, Qi, pi be x, y , ui, vi, wj respectively ( i = 1 , 2 , . . ., n ; j = 1, 2 , . . ., m ) . Then, by (l), (2), (3), (4)

Y S X (8) 9,s ug (9) y g w i ( i = 1 , 2 , . . . , m) (10) x 5 u i ( i = 1 , 2 , . . . , n) . (11)

y I_ vi (12)

(i = 1 , 2 , . * . , n)

By (10) (i = 1 , 2 , . . . , n).

By (11) and (12) the truth-values of CQiq, CPip are 1 - vi + y, 1 - ui + x respectively. But, by (6), the formula CCPipCQiq takes the truth-value 1. Hence

1 - ui + x 5 1 - vi + y and the formula CCQiqCPip takes the truth-value

1 - (1 - vi + y ) + (1 - ui + 2) = 1 + v< - ui + x - y

1 - vi + y = 1 + V$ - ui + x - y. Hence, by (5) and (6)

1) A.RosE, Le degr6 de saturation du calcul propositionnel implicatif ii m valeurs de Lukasiewicz, Comptes rendus 240, 2280-2281 (1955).

208 ALAN ROSE

Hence

and

Let us now put

2(Vi - y) = Ui - x

2(v; - y") = 2 C t - x*.

k = (2y - x)/(2y - 2 - 1).

Then, by (S), k < 1 . Also, the lowest truth-value considered (other than starred values) is y and

But

Hence, by (13), ut = 2vt. Hence, when p l , p , , . . ., p m , p , q take the truth- values w:, w,*, . . ., w;, x*, y* respectively the truth-values of Q1, &,, . . ., &, are half those of Pl, P,, . . ., P, respectively and R takes the trut'h-value 1. Hence, by (7), R must also take the truth-value 1 when p I , p , , . . ., p m , p , q take the truth-values wl, wz, . . ., w,, x , y respectively.

We now define the ( p , q) translations of a formula of the C - H calculus as follows :

(I) If the functor H does not occur in P the only ( p , q ) translation of P is P. (11) If (rE, U i ) N S ( M 2 0 , N 2 1) is a ( p , q) translation of P, the pro-

positional variable r is distinct from ql , p,, . . . , q,,, , p , q , no propositional variables other than q l , q,, . . . , qw, p , q occur in ( I 'EIUi)NS and p , q do not occur in P then (r2 U,CCqpCCqrCCrXriw, ,Cqq,CCpSCCCrqCCrqCSpCCCCrqCXpCrq)N r is a ( p , q ) translation of HP. We shall call r the associated variable of HP.

(111) If (TE1 U i ) N X , (I'EIV,)NT are ( p , q) translations of P, Q respectively and no associated variable occurs in both the former formulae then (r& Ui rE1 Vi)NCST is a ( p , q ) translation of C P Q .

(IV) A formula cannot be a translation of another formula except in virtue

It follows from the Lemma that if P is a formuIa of the C - H calculus which always takes the truth-value 1 then there is a ( p , q) translation of P which always takes the truth-value 1. Hence this translation is derivable from A 1-3, A4C. Let us now apply the substitution rule to this ( p , q) translation, substituting H p for q , and substituting for each associated varible the formula of which it is the associated variable. We thus derive a formula of the form I'j=lWiP where each Wi is either derivable from one of A5C-A7C by the substitution rule or is of one of the forms

where X is a formula such that HX is a subformula of P and pl, p , , . , ., pm are the propositional variables occurring in P. Using A1-3, A4C, A5C we then derive a formula of the form

y* = (2y - x - y)/(2y - x - 1) = (y - x)/(y - 2 + y - 1) > 0.

x* = (2y - 2 - 2)/(2y - x - 1) = 2y*.

of (I), (11)) (111).

C H p p i , C H p H X , C p X

CHppi r{=, CHpHXi)PP.

SOME FORMALISATIONS OF NO-VALUED PROPOSITIONAL CALCULI 209

Substitut,ing pl, p , , . . ., pm, XI, X , , . . ., X , in turn for p and using A7C and the fact that all formulae of the form

Ti.=, (CrL pi q,v) 4? are derivable from A1-3, A4C we deduce P. Hence the formalisation is complete. The treatment of the introduction of A 8 for the C - H - 1 calculus is the same as that given above.

Proceeding as above, we can establish the independence of A5C-A7C by setting h(x) = ix, h(x) = x, h ( z ) = i z + 4 respectively.

(Eingegengen em 24. August 1966)