Zeilschr. f . math, Logik und arudlagen d. Math. Bd. 2 , S. 166-192 (1956)

P 2 3

AN ALTERNATIVE FORMALISATION OF SOBOCINSKI~S THREE-VALUED IMPLICATIONAL PROPOSITIONAL CALCULUS

1 1 3 3 3 1 2 3 2 1 1 1 1

By ALAN ROSE in Nottingham

1 1 3 2 3 1 3 3 3

SOBOCI~SKI has formalisedl) the 3-valued propositional calculus with two designated truth-values having as primitives the functors C and N whose truth- tables are given below, 1 and 2 being the designated truth-values:

3 3 1

1 N P CPQ I 1 2 3

Zeilschr. f . math, Logik und arudlagen d. Math. Bd. 2 , S. 166-192 (1956)

AN ALTERNATIVE FORMALISATION OF SOBOCINSKI~S THREE-VALUED IMPLICATIONAL PROPOSITIONAL CALCULUS

By ALAN ROSE in Nottingham

SOBOCI~SKI has formalisedl) the 3-valued propositional calculus with two designated truth-values having as primitives the functors C and N whose truth- tables are given below, 1 and 2 being the designated truth-values:

He raised the question2) whether the system with C as the only primitive could be formalised and a formalisation was subsequently found3), hut this formalisation used a primitive rule of procedure other than substitution and modus ponens. The object of the present paper is to formalise the system using only substitution and modus ponens as primitive rules of procedure4).

We first define the ternary functors V , I , P as follows:

Hence if Q takes the truth-value 1 then V P Q R , I P Q R, P P Q R all take the truth- value 1. If Q takes the truth-value 2 and R does not take the truth-value 2 or Q takes the truth-value 3 then CCRRQ takes the truth-value 3. Hence, in these cases, the truth-values of VPQR, I P Q R , F P Q R are determined as follows:

l) BOLESLAW SOBOCI~SKI, Axioniatization of a partial system of three-value calculus of

2) Op. cit. 3, ALAN ROSE, A formalization of Sobociriski’s three-valued implicational propositional

4, This problem was suggested to the author by S O B O C I ~ K I .

propositions. J. Computing Systems 1, 23-55 (1952).

calculus. J. Computing Systems I , 165-168 (1953).

A FORMALISATION O F SOBOCIkSRI'S THREE-VALUED IMPLICATIONAL PROPOSITIONAL CALCULUS 167

If Q and R both take the truth-value 2 then the truth-values of V P Q R, I P Q R , F P Q R are determined as follows:

1 1 3 2 2 2 3 3 3

3 2 1

Now if every propositional variable which occurs in P also occurs in R then P takes the truth-value 2 whenever R does. Thus, in this case, PPQ R, I P Q R, V P Q R all take the truth-value 2 whenever Q and R both take the truth-value 2. It follows from this that formulae such A1-6 (given below) always take designated truth- values. Let us now consider the axioms:

A 1 CVqrCpCqsVCpqrCpCqs A 2 C F p r C p C q s V C p q r C p C q s A 3 C V p r C p C q s C I q r C p C q s P C p p r C p C q s A 4 C V p r C p C q s C F q r C p C q s F C p q r C p C q s A 5 C I p r C p C q s C I q r C p C q s I C p q r C p C p s A 6 CIprCpCqsCFqrCpCqsFCpqrCpCqs A 7 CFpqCrsCFrqCrsFpqCrs A 8 C C C p q C p q C C q p C q p A 9 C C C p p C q q C C C p r C p r C C q r C q r

A 11 CCCpqCpqCCpCqqCpCqq A 12 CCCCpqCrsCCpqCrsCCCsqCrpCCsqCrp A 1 3 C p C C p q q A 14 C C p q C C q r C p r A 15 C C V ~ q q r C C I p q q r C C P p q q r r

A 10, C C C P c q q c p c q q c C P q c P q

A 1 6 C V m m A17 C I P P P P A 18 C C p C p q C p q A19 C p p A 20 C F p q C r s C I r q C r s F p q C r s A 21 C F p q C r s C V r q C r s F p q C r s .

S O B O C T ~ K I hap shown1) that from A 13, A 14 and A 18 we can deduce

F 1 CCqrCCpqCpr F 2 C C p C q r C q C p r F 3 C C p C q r C C p q C p r .

From F 1 and A 14 we can deduce, in the usual way, the rule of substitutivity of equivalence.

1) Op. cit.

168 ALAN ROSE

Applying the substitution rule to A 1 4 and F 3 we deduce

F 4 CCpqCCqCrsCpCrs F 5 C C p C r s C C p r C p s .

From F 5 and F 1 we deduce CCCqCrsCpCrsCCqCrsCCprCps. Using F 1 again we deduce

Applying modus ponens to F 4 and F 6 we deduce

F 6 CCCpqCCqCrsCpCrsCCpqCCqCrsCCprCps.

F 7 CCpqCCqCrsCCprCps . From F 2, F 7 and the rule of substitutivity of equivalence we deduce

F 8 CCqCrsCCpqCCprCps . Applying the substitution rule to AT, A20 and A21 we deduce

F 9 C I p q C r s C P r q C r s I p q C r s F 10 CVpqCrsCPrqCrsVpqCrs F 11 C I p q C r s C I r q C r s I p q C r s F 12 C Vpq Cr s C I r qCr s VpqCrs F 13 C I p q C r s C V r q C r s I p q C r s F 14 CVpqCrsCVrqCrsVpqCrs.

From F 12 and the substitution rule we deduce

F 15 C V C p q r C p C ~ s C I p r C p C q s V C p q r C p C q s .

F 16 CIprCpCqsCVqrCpCqsVCp¶rCpCqs. From F 15, A 14, A 1 and F 2 we deduce

Similarly we can prove, using F 2 and substitutivity of equivalence if necessary,

F 17 GVprCpCqsCVqrCpCqsVCpqrCpCqs F 18 C F p r C p C q e C V q r C p C q s V ~ p q r C p C q s E” 19 CFprCpCqsCIqrCpCqYVC~qrCpCqs F 20 CF pr C p CqsC Fqr C p CqsV Cpqr C p Cqs .

We shall now prove the Theor em. T h e formalisation i s weakly complete. We first define a C-sum of P, , P,, . . . , P, as follows ; (i) CPP is a C-sum of P. (ii) If i , , i,, . . , , in is a permutation of the integers 1, 2, . . . , n, CQQ, CRR are

C-sums of Pi,, Pi,, . . ., Pi,; Pi,+l, Pi,+a, . . . , Pi, respectively then CCQRCQR is a C-sum of P,, P,, . . ., P,.

(iii) If @(PI, P,, , . ., P,, P,,,) is a C-sum of P,, P,, . . ., Pnf l then @ ( P I , P,. . . ., P,, P,) is a C-sum of P,, P,, . . ., P, (i = 1 , 2 , . . . , n ) .

(iv) No formula can be a C-sum of P,, P,, . . . , P, except in virtue of (i), (ii), (iii). Lemma 1. If &, R are C-sums of P,, P,, . . . , P, and each of the formulae

P, , P,, . . . , P, occurs exactly twice in Q and exactly twice in R then CQR i s provable.

A FORMALISATION OF SOBOCI;YSKI'S THREE-VALUED IMPLIKATIONAL PROPOSITIONAL CALCULUS 169

We shall prove the Lemma by strong induction on n. If n = 1 the Lemma follows from A 19 and the substitution rule. We now assume the Lemma for 1, 2, . . . , n and prove it for n + 1.

Then Q, R are of the forms C C S T C S T , CCUVCUV respectively. We shall divide the proof of the induction step into seven cases and refer to P, , P,, . . , as P-formulae.

Case I. The P-formulae occurring in S , T are the same as those occurring in U , V respectively. We can, by our induction hypothesis, prove CCSSC U U and CCTTCVV. From the first of these formulae and A 9 we deduce

CCCSTCSTCC UTCUT,

. e. CQCQUTCUT. From the other formula and A 9 we deduce CCCT U C T UCCVUCVU. From this, A 8 and the rule of substitutivity of equi- valence we deduce CCCUTCUTCCUVCUV, i. e. CCCUTCUTR. From this, CQGCUTCUT and A14 we deduce CQR.

Case 11. The P-formulae occurring in S, T are the same as those occurring in 8, U respectively. Proceeding as in Case I we can prove CQCCV UCV U. From this, A 8 and substitutivity of equivalence we deduce CQCCUVCUV, i .e. CQR.

Case 111. The class of P-formulae occurring in S is a proper subclass of the class of P formulae occurring in U . We may suppose, without loss of generality, that the members of the two classes are P,, P,, . . ., Pa; P,, Pz, . . . , P a + b respectively. Then the P-formulae occurring in T', V are Pa+l , P.+z, . . ., Pn+l; Pa+b+1, Pa+8+2, . , ., P,+l respectively.

We now define, following ROSSER and TURQUETTE~), the symbol P by

Let us abbreviate the formula I';'fT: Pi Pa+, by W . Then it follows from our induc- tion hypothesis that we can prove CCS W C S WC U U and CCTTCG V WCV W. From these formulae, A 8, A 9 and substitutivity of equivalence we deduce CCCVCS WCVCS WCCUVCUV and CCCSTCSTCCXCVWCSCVW. From the last two formulae, F 2, substitutivity of equivalence and A 14 we deduce CQR.

Case IV. The class of P-formulae occurring in S is a proper subclass of the class of P-formulae occurring in V . We deduce the result for this case from that of Case I11 in the same way as the result for Case I1 was deduced from that for Case I.

Case V. The class of P-formulae occurring in T is a proper subclass of the class of P-formulae occurring in U . Proceeding as in Case 111 we deduce CCCTSCTSR. From this, A 8 and substitutivity of equivalence we deduce C C C S T C S T R , i .e. CQR.

Case VI. The class of P-formulae occurring in T is a proper subclass of the class of P-formulae occurring in V . We deduce the result for this case from that of Case V in the same way as the result for Case I1 was deduced from that for Case I.

1) J. B. ROSSER and A. R. T ~ Q U E T T E , Many-valued Logics. Amsterdam 1952.

12 Ztschr. f . math. Logik

170 ALAN ROSE

Case VII. None of the classes of P-formulae occurring in S , T , TJ, V is a subclass (proper or improper) of any of the other three classes. We may suppose, without loss of generdit'y, that the P-formulae occurring in S, T , U , V are

P p + i ~ P p + z ~ * . * ) Pa, P y + i , Py+n, . , ., P n + 1 O < P < a < y < n+ 1 . Let C W W , C X X , C P Y , C Z B beC-sums of P,, P a , . . . , PB; Pi.l+l, PBiz, . . ., Pa; 1'a+1, P a + z j . . .> Py; Py+l, Py+z,. . P n + 1 respectively. Then, by our induction hypothesis we can prove

'1, '2, ' . .> Pa; f ' a + i , Pa+a, . ' . $ Pa+,; PI, P, , . . ., Pa, Pa+l, Pa+2, , . ., r,; where

(1) C C S S C C W X C W X , (2) C C T T C C Y Z C Y Z , (3) C C c r v Y c W Y C U u , (4) C C C X Z C X Z C V V .

The proof of CQR tJhen proceeds as follows:

By (1) and A 9 By ( 5 ) , A 8 and s. of e. By (2) and A 9 By (6), (7) and A14 By (3) and A 9 By (9), A 8 and s. of e. By (4) and A 9 By (lo), (11) and A 14 By (12), A 12 and s. of e. By (€9, (13) and A 14

( 5 ) C Q C C C W X T C C W X T (6) C Q C C T C W X C T C W X (7) C C C T C W X C T C W X C C C Y Z C W X C C Y Z C W X ( 8 ) CQ C C C Y Z C W X CC Y Z C W X (9) C C C C W Y V C C W Y V R

(10) C C C V C W Y C V C W Y R

(12) C C C C X Z C W Y C C X Z C W Y R (13) C C C C Y Z C W X C C Y Z C W X R (14) C Q R .

(11) C C C C X Z C W Y C C X Z C W Y C C V C W Y C V C W Y

Lemma 2. Lemma 1 remains true if some or all of P,, P,, . . . , P% occur more than twice in either or both of Q, R.

Let us suppose that Pi occurs 2ai times in Q and 2gi times in R (i = 1 , 2 , . . . , n) . Then a,, az, . . ., a,, PI , Fa, . . ., P, 2 1 . If a; = (i = 1 , 2 , . . ., n) the Lemma follows at once from the substitution rule. We may now suppose, without loss of generality, that ai > P i for i = 1 , 2 , . . ., I., a$ < & for i = A + 1, A -t 2 , . . ., p and ai = p i for i = p + 1 , p + 2 , . . ., 12 where O < A < p 5 1%. Hence al, a,, . . . , ad 2 2 ; pa+,, . . . , P,, 1 2 . If 0 = A < p or 0 < A = ,u the proof is similar, but some steps are omitted. Let C S S be a C-sum in which Pi occurs 2gi times for i = 1 , 2 , . . ., A and 2ai times for i = J. + 1 , A + 2 , . . ., n . Let .Zt=l(ai - pi) = V Q ~ . We shall show, by induction on v, that CQCSS is provable.

If v = 0 then the result has been proved above. We now assume the result for v and prove i t for v + 1 .

Let i he an integer such that 1 5 i 5 A . Then a, 2 2 . Let T be a C-sum of P,, P, , . . ., P, or of PI, P,, . . ., Pi-l, Pi+l, Pi ip , . , ., P, in which Pi occur8 four times less than in Q and in which each of PI, P,, . , . , Pi+l , Pi+z. . , . , P, occurs the same number of times as in Q. Let us abbreviate CCTPiCTPi by U . Then vus = vQs - 1 . Hence, by our induction hypothesis, we can prove C U C S S . Since the Lemma holds for v = 0 we can prove CQCCTCP, Pi CTCPi Pi. Applying the substitution rule to A 10 we deduce CCCTCPiPiCTCPiPiCCTPiCTP,. Prom the last three formulae and A 14 we deduce C Q C S S .

A FORMALISATION OF SOBOCIkSKI’S THREE-VALUED IMPLIKATIONAL PROPOSITIONAL CALCULUS 171

Similarly, using A 11 in place of A 10, we can prove CCSSR. From the last two

Let us now define the symbols G, , C,, G, to be V , I , F respectively.

Lemma 3. If @ ( p , , p z , . . . , ps) takes th.e truth-value q ( x l , x,, . . ., xn) when p l , p,, . . .. p , take the truth-values x,, x 2 , . . ., x, respectively, Q i s a formula in which all the variables p l , p 2 , . . . , p W , q occur and y = y(xl , xz, . . . , 2,) then the 3n formulae

r & l G x l p i r Q a , @ ( p , , 21,: 3 . $ p n ) r Q are provable.

We shall prove the Lemma by strong induction on the number, m, of (not ne- cessarily distinct) propositional variables occurring in @. If m = 1 the Lemma follows from A 19. We now assume the Lemma for 1 , 2 , . . . , m and prove it for m + 1 .

formulae and A 1 4 we deduce CQR. Thus the Lemma is proved.

Then we may assume, without loss of generality, that @ is of the form

Q ~ ( P , , P ~ , * * * 7 ~ a ) g ( ~ l , ~ 2 , V e - 7 ~ f i > ~ a + l > p a + 2 , . . * r p n )

where @ (Y 5 n and 1 (Y 5 n . By our induction hypotheNis we can prove

T;= Gxt pi r R G, Y r R , where z = y(x1 , x2, . . . , x a ) , w = 5(xl , x 2 , . . ., xp, x,+~, x,+~, . , . , xn), R is any formula in which the variables pl, p , , . . . ~ p a , q all occur and S is auy formula in which the variables p , , p z , . . ., p p , pa+ , , pa+ , , . , ., pn, q all occur. I n the first of these formulae let us substitute for q a formula in which the vagiables p a + 1 , pa+?, . . . , p , , q all occur and let the result of the substitution be

I d C, pi r S r r s Gx, pi r X G, 9 r X

r;=, G, pi r T G , Y r T .

We now show, by induction on 6, that the formulae

r:+,d,G,,pirTG,YrT ( 6 = 0 , 1, . . . , n-a) are provable.

for 6 and prove it for 6 + I . If 6 = 0 the required formula has been proved above. We now assume the result

By Lemma 2 we can prove formulae of the forms

CCTTCCpa+6+1UCl)=+8+lU, CCCpa+6+lUCPa+8+1UCTT,

where u is a formula containing all the variables p l , p , , . . . , p a , q. From A 7, A 20, A 21 or one of F 9-14 we deduce

CGz p r C P a t 8 4 - 1 P,+&+I r c P a + 6 + 1 UGz YrC13,+&+1 u. Using the rule of substitutivity of equivalence we deduce

CG;: !Pr T CGZa+8+l pa+dfl YTC, ! P r T .

By our induction hypothesis we can prove formula but one we deduce, by repeated use of F 1,

G,$ pi r T G Z Y r T. From the last

C F;:: Gs pi r T G, Yr Tr;+; G , pi r T C GXacd+ ~ . + a + ~ r T C, Y r T. 129

172 ALAN ROSE

Applying the rule of modus ponens to the last two formulae and then using k' 2 and substitutivity of equivalence we deduce

. ly :+ Gx, pi r T G, Y r T . Putting 6 = n - a we deduce

q=,, Gxd pi r TG,Y r T.

Using substitutivity of equivalence and Lemma 2 we then deduce

Similarly we can prove

We now show, by induction on A, that the formulae

~ ; ~ l G x , p i r Q G z Y r Q .

l-rW1 Gxc( pi r Q Gw S r Q .

Cr&l Gxd pi r Q G,Yr Q C r t - l G, pi r Q G, Er Qri, Gx6pi r Q G, @ r Q (2, = 0 , 1, . . . ) n)

are provable. If 1 = 0 the formula becomes C G ~ Y r Q C G ~ , E r Q G y C V ~ r Q . Since the variables

p,, p2, . . .) pn, q all occur in Q we can deduce the required formula from the rule of substitutivity of equivalence, Lemma 2 and one of A3-6, F 16-20. We now assume the result for A and prove it for A + 1 .

From our induction hypothesis and F 8 we deduce

Thus the result is proved. Putting A. = n and applying the rule of modus ponens to the resulting formula and the previous two formulae we deduce

Thus the Lemma is proved. Proof of t h e Main Theorem. Let @ ( p l , p 2 , . . . , p n ) be a formula which

takes designated truth-values exclusively. By Lemma 3 we can prove the 3n formulae

J-;=l Gxd p , r Q Gy @ r Q .

T L Qx, pi r Q Q, @ r Q I

where y = 1 unless xl, xa) . . ., x, = 2, in which case y = 2 . Substituting @, pl for r, q respectively and then using Lemma 2 and the rule of substitutivity of equivalence we deduce

From A 16 or A 17 we then deduce CG, di @ @ @ . From the last two formulae and F 1 we deduce Tin_, G x i p @ @ @ . From the 3" formulae Ti"=, GXipi @ @ @ we deduce @ by repeated use of A15. Thus the Theorem is proved.

r$l G, pi @ @ G, @ @ @ .

(Eingegangen am 27. Juli 1956)