Zcilrchr. fib ma#. Lwik und Qrundlagen d. Math. Bd. 9, S. 177-108 ( 1 9 6 3 )

A FORMALISATION OF THE PROPOSITIONAL CALCULUS CORRESPONDIXG TO WANG’S CALCULUS O F PARTIAL PREDICATES

by ALAN ROSE in Nottinghani (England)

WANC has given recentfly’) a formalisation of a calculus of partial prpdicates in which all tautologies of the corresponding propositional calculus are axioms. The object of the present paper is to simplify this formalisation hy showing that the corresponding propositional calculus can be formalised by means of a finite number of axioms and primitive rules of procedure.

A forniisla of this propositional calcnlus is defined as follows:

1. Every propositional variable is a formula.

2. If P is a formula and the symbol “+” does not occur in P then -P is a for-

3. Tf P, Q are forrnulae and the symbol I ‘ + ” does not occur in either of these mula.

formulae then P v Q , P & Q , P + Q are forniulae.

4. An expresqion cannot be a formula except in virtue of 1. 2 and 3.

The functors v , &, - are those of the 3-valued logic of EUKASIEWICZ~) and the formula P + Q takes the truth-value 1 if and only if Q takes the truth-value 1 whenever P takes the truth-value 1. We shall show that a formula R is provable in our formalisation if and only if R always takes the truth-value 1. We shall use the following 12 axioms and 4 primitive rules of procedure.

The symbol ‘ I - ” is regarded as “stronger” than “&”, ”&” stronger than “v” and “v” stronger than “ -+”. Thus the forniulrs

” P k q V P +!?

((-(PI 8: (I) v P ) -+ q . is an abbreviation for

A1 p + p v q

A2 p v q + q v p

A3 P & q - + P

I ) WAKO, H., The calculus of partial predicates and its extension to set theory I, thia Zeitachr.

*) See, for example, MSSER, J. B. and TURQUETTE, A. R., Many-valued Logics, Amsterdam, ‘I (1961), 283-288.

1952, pp. 15-17.

12 Ztachr. f . math. Loglk

178 ALAN ROSE

A 4 p & q + q & p

AS -- P + P A 6 p + - - P A 7 p & ( q v r ) + p & q v p & r

A8 - ( p & q ) + - p v -9

A9 - p v - q + - ( p & q )

A10 - ( P v ~ ) - > - p 8 : w q

A l l N P & -9 - ( p v 9)

A12 p & - p + p .

R1 R 2

R 3 R 4

If P - -? R and Q --? R then P v Q --f R. .If P + Q and P + K t h e n P + Q ! & R . If P + Q and Q -+ R then P +R. If Q is a propositional variable which oeours in P, R is a formula h

which tho eymbol ‘‘. >’’ does not oaour and the result of replacing all ocourrences of Q in P by R is S then if P then S.

The plausibility of our formalisation follows a t once fiince A1-12 all take the truth-value 1 and the conclusion of eaoh instance of R1-4 takes the truth-value 1 whenever the corresponding premhs(es) all take the truth-value 1.

Wt? shall now establish the weak completeness of our formalisation. From A6, A6 and R 3 we deduce

F1 p + p .

From A 1, A 2 and R 3 we deduce

F2 p + q v p .

Prom A 3 and It4 we deduce

F 3 q 9 C p - - Q .

From A 4 , F3 ltricl R 3 we deduce

P4 p & q + q .

From F 1 and R I we deduce

F5 p v p - t p .

From k’ 1 and R 2 we deduce

E’6 p + p & p .

We next derivo R5 If P + Q then P v R + Q v R .

A BORbfALISATION OF THE PROPOSTTIONAL CALCCLUS 179

By A1 and R 4 (1) b Q + Q v 11.

By (1) and R 3

By F:! and R 4 (3) k R - + Q v R .

By (2), (3) and R1

(2) P + Q F P - P Q v R .

P -+Q P v R + & v R .

We omit the proof of

R6 If P + Q then R v P - + R v Q

since it is similar to the proof of R5.

We next derive

R7

By A3 and R 4 (1) I - P & R - + P . By (1) and R3 (2) By F 4 and R4

By (2), (3) and R2

If P -+Q then P & R + Q & H .

P - & t P & Iz + Q .

(3) k P & R -7 R.

P - h Q F P & R - + Q & R .

We omit the proof of

R8 If P -+Q then R & P + l l & Q

since it is similar to the proof of lt7.

By A1 and R4

F7 q - t q v r .

By F7 and R8

By F2 and R 4

By F9 and R8

By F8. F10 and R1

F8 p & q + p $ ( q v r ) .

F 9 r - t q v r .

F10 p & r - + p & ( q v r ) .

3'11 p & p v p & r - + p & ( q v v ) . 12'

180 ALAN ROSE

By F7 and R 6

F12 p ' i y - + p v ( q v r ) .

By F2 and R4

F13 q v r - + p v ( q v r ) .

By FQ, F13 and R3 F14 r p v (q v r ) .

By F12, F14 and R1 F15 ( p v q ) v r --f p v (q v r ) .

By A1 and R4 F16 p v q + ( p v q ) v r .

By A l , P16 arid R3

F17 p -+ ( p v q) v r .

By F2 and R4

F18 ' p v p .

By Fl8 and R5

F19 q v r - + ( p v y ) v r .

By P17, F19 and R1

By A3 and R 4

By A3, F21 and R 3

F20 p v (p v I ) --f ( p v q ) v r .

F21 ( p & q ) & r - + p & q .

P22 ( p & q ) & r + p .

By Y4 and R7 F23 ( p & q ) & ~ - t q & r .

By F22, F23 and R 2

F24 ( p & p ) & r - + p & ( q & ? ) . We omit the proof of

F25 p & (q & I ) + ( p & y) & I

since it is similar to the proof of F24.

By A6 and R4

By A6 and R4 F26 --- P + - P.

F27 - p -+ - - - p .

A FORMALISATION OF THE PROPOSITIONAIi QbUTzTLU.9

By A10, A4, R4, A l l and RR

F28 “(P v 9) -+ - (q v PI. By A$, A2, R4, A9 and R 3

F29 - ( p 8: q ) -+ - (q & p ) .

By A10 and R4

By A10 and R7

By F24 and R4

By A l l and R4

F30 - ( ( p v q) v r ) -+ - ( p v q ) & - r .

F31 - ( p v q) & - r -+ ( - p & -q) & -7.

F32 ( - p & - q ) & -7 -+ - p & ( - q & - r ) .

F33 -q & -7 + - (q v 2).

By F38 and R 8 F34 - p & ( -q& - r ) -+ - p & - (q v r ) .

By A l l and R4 F35 - p & - (q v r ) -+ - ( p v (q v r ) ) .

By F30, F31, F32, F34, F35 and R 3 F36 - ( ( p \,’ q ) v r ) -+ - ( p v (q v 2)).

F37 -b v (q v r ) ) -+ - ( ( p v q ) v r )

F38 -UP & q ) & f-) -+ - (P (q d6 r ) ) F39 - ( p & (q & t ) ) + - ( ( p & q) & r )

We omit the proofs of

sinoe they are similar to the proof of F36. By A3, R4 and R6

F40 p v q & r - + p v q .

By F4, R4 and R6 P41 p v q & r + p v r .

By F40, F41 and R2 F42 p v q &, r + ( p v q ) & ( p v r ) .

By A8 and R4 F43 - ( p & (q v r ) ) + - p v - (q v r ) .

By A10, R4 and R6 F44 - p v - (q v r ) + - p v - q & - r .

181

182 N A N ROSE

By F42 and R-l

By A9, R4, R7. H.8 and It3

F-15 - p ’ / - q $ - r + ( - p v - q ) & ( - p v - r ) .

F46 ( - p ~ - q ) & ( - p ~ - r ) + - ( p & y ) & - ( p & ~ ) .

By A l l and R 4

Hy F43, F44, F45, F46, F47 and R 3 F47 - ( p & q ) & - - p & r ) - - ( p & q v p & r ) .

F48 “(p 8: (q r ) ) + - ( p 6: q v p 86 r ) .

R 9 If P - Q then Y - Y & Q . We next derive

By F1, R 4 and R 2

R10 If P -+ N P V Q then P - Q . By R 9 (1) P + - P v Q I- P +- P6: (-P v Q ) .

By A 7 and R 4

By ( l ) , (2) and R 3

(2) t - P & ( - P v Q ) + P & - P v P & ( $ .

(3) P + N P V Q k P -+ P & -Pv P & Q .

By A12 and R 4 (4 1 t -PP&-P-+Q. By P4 and R4

By (4), (5) and R 1

(5) t - P P & Q + Q .

(6) t- P & -P v P & Q + Q .

By [3), (6) and R 3

By A7 and R4

P + - P v Q F P - Q .

F49 (P v q ) & (P v r ) -+ IP v Ir) P v (P v q )

F50 (P v d 8~ p v ( p v q ) & r - P & (P v q ) v r

r .

By A4, R4, R5, R 6 and R.3

( p v q ) .

By A7, R4, R5, R6 and R 3 F61 p & ( p v q ) v r & (a v q ) --+ ( p & y v p & q ) v ( r & p v Y & q ) .

By A 3 and R 4 F62 p & p + p .

A FORMALISATION OF TEE PROPOSITIONAL CAU2ULUS 183

By F 4 and R4

F 5 3 r & p + p .

By A 4 and R4

F54 r & q + q & r .

By F52, A l , R4 and R3

F55 p & p + p v i & r .

By A3, A l , R4 and It3

F 5 6 p & q - + p v q & r .

By P53, A l , R4 and R3

F57 r & p - + p v q & r .

By FM, F 2 , R4 and R3

F58 r & q + p v q & r .

By F65, F56, F57, F58 and R1 P59 ( p & p v p & q ) v (r & p v r & q ) + p v q & r .

By F49, P50, F61, F59 and R 3

F60 ( p v q ) & ( p v r ) - + p v q & r .

By A10 and R4

F61 - ( p & q v p & r ) -> - ( p & q) & - ( p & 7 ) .

By A 8 , R4, R7, R8 and R 3

F62 - ( p & q ) & - ( p & r ) + ( - p v - q ) & ( - p v - r )

By F60 and R4

F63 ( - p v ~ q ) & ( - p v ~ r ) ~ ~ p v ~ ~ & - r . By A l l , R4 and R6

F64 - p v - p l & - r + - p v ~ ( q v r ) .

By A 9 and R4

F 6 5 - y v - (q v r ) -+ - ( p & (q v 7 ) ) .

By F61, F62, F63, F64, F66 and R3 F66 - ( p & q v p & r ) + - ( p & ( q v r ) ) .

By A 6 and R4 F67 - - ( p & q ) + p & q .

Ry A6, R4, R7, R8 and R 3 F 6 8 p & q + ” p & - w q .

I84 &AN ROSE

By A 11 and R4 F69 - - p & N -q -> - ( ~p v “ 9 ) .

By F67, F68, F69 and R 3

F70 N - ( p & q ) -+ - ( - P v -a ) . We omit the proofs of

F71 - ( - p v -q ) - > - - ( p & q)

F72 - - ( p v q ) + - ( - P& -d F73 - ( - p & -q) + “ - ( p v q )

since they we zimilar t o the proof of F70.

We next derive

R11 If -P+-Qthen - ( P v l i ! ) - > ~ ( Q v R ) .

By A10 and R4

( 1 ) l- - ( P v R ) + -P& -R. By R7

By A l l and R4

By (I) , (21, (3) and R 3

(2) -P --f -Q l- -P& - R -+ - Q & --.H.

(3) I- - & & - H + - ( Q v H ) .

NP -+ N-& I- N ( ~ v R ) --> w ( Q V R).

We onlit the proofs of

R12 If --P -+ -Q then -(R v P) + - (H v Q).

R 13 If --P + -8 then - (P & R) -3 N (Q & R) . R14 I f --P --f -Q then N ( R & P) --f - ( R & Q ) .

Rince they are bimilar to tlie proof of R 1 1 .

We arc now in a position to derivc

I t 16 If P is a subformula of R, R does not ooiitRiil the symbol ‘. 4’’ and the result of replacing one or more occurrences of P in R by Q is S then if P -+ Q, Q -* P, -P --> -Q and -Q -+ -P then R + 8, S -+ R , NR + -S und -8 -+ -R.

Sicice R3 is a primitive rule of prooedure i t will bc sufficient to consider the cam where only one occurrence of P in R is replaced by Q . Let the lengths’) of P, R be

l) We define the length of a formula to be the total number of variables and functor8 occurring in it, repeated Rymbols being reckoned aocording to their multiplicity. Thus, for example, ( p v q) v - p is of length 6.

A BORMALISATION OF THE PROF'OSITIONAI, CALCULUS 185

?n, n respectively and let 1 (= l (P, R ) ) be defined by the equatioii

l = n - m . We shall derive the rule by strong induction on 1 .

If 1 = 0 then R, -R, S, -S are P, -P, Q , -Q respeatively and there is nothing to prove. We now assume the rule for all non-negative integers less than 1 and derive it for 1 . Since, now, 12 1 we have

n = 1 + 911 2 2

and R cannot be a propositional variable. Thus R is of one of the forms

where,

(i) if P is a subforlrula of C,

and,

N U , uv v , U & v

l ( P , U ) < 1(P, R)

(ii) if P is a subformula of V ,

l (P , V ) < l (P, 11). In uase (i) ((ii)) let the result of replacing P by Q in U ( V ) be W ( X ) .

Let us abbreviate the set of formulae

{ P + Q , Q + P , - P + - Q , - - Q + - P }

by P = Q . In case (i) we have, by our induotion hypothesis,

(1) P E Q I - U = W By AS, A6, R4 and R3 (2) U E W k N U G - W . By R5 and R11

By R7 and R13 (3) U E W k U V V ~ W \ I V .

(4) U = W k U & V ~ W & V .

Since the symbol "+)) does not occur in R, one of (2), (3), (4) may be rewritten

(5 )

The rule follows at once from (1) and (5 ) . We omit the prodf for cme (ii) since it is similar to that for case (i).

U 3 W I- R E S .

We now derive

R 16 If P is a subfornula of R and the rosult of replacing one or more ocourrences of Pin R by Q is S then if P + Q , Q - + P , -P + -Q, -Q - -P and R -+ U then S --f U .

186 ALAN ROSE

By R15 P + Q , Q + P , - P + - J & , - Q + - P I - S - t R .

Hence, by M 3, P +&, Q -+ P , NP + -Q, -Q + -P, R + U k S -> V .

We omit the proof of

R 17 If P is a subformula of R and the result of replacing one or more occurrences of P i n H b y & i s S then if P + Q , Q A P , -P -+ - Q , -Q + -P and U + R then U + S

since it is similar to the proof of R16.

We are now in a position to establish the weak completeness of the formahation. TRt) U - F V be’) a formula which take8 the truth-value 1. Ry means of the equationsa)

(1) ,,p= T p

(2) P & Q = T Q & P

(3) - ( P & u ) = T -P\’ -&

(4)

(5)

(6)

- ( P V Q ) - T -P & -& P & (Q v R ) = T P & Q v P & H Y v (Q v R ) = T ( P v Q) L R

( 7 ) P & (Q& It) =rp ( P & Q ) & R

we can find formulae X, Y of the respective forms

Z ’ f = * ( 1 7 ~ ~ 1 P i r & I l ~ = i ” Q i r ) (z,mi + n 1 , . . . , m I + n l L I ) ,

where P , , , . . . , PI?,,,, . . . , P11, . . . , PI,,,,, Qll , . . . , Qln,, . . 011, . . . ? Qln,, 111 I : . . . ) R i M , , . . . , RL ablcs and associatkon is to the left such that

. . . , RL ML, 8, , . . . , S1 N,r . . . , SLi, . . . , SL ML are propositional vari-

X = T u , Y = T v . (Tf ni = 0 the formula njmiPii & np= - Qir denotes nzlPi,, similarly in the cases mi = 0, M i = 0, N i = 0 . ) By A6, F26, A6, F27, A4, F29, AS, F70, A9, F71, A10, F72, A l l , F73, A7, F48, F11, F66, F15, F36, F20, F37, F24, F38, F25, F39, R4, R16 and R17 it follows that it is suffioient to derive the formula x -f Y.

I) Since -P, P v Q, P & Q all take the truth-value 2 when P, Q take tho truth-value 2 we

*) See, for example, H I L B ~ T , D. and Ac~mnaam. W.. Principles of Mathematical Logic, need coneider only formulae of the form U + V .

New York, 1960, pp. 11-13.

A PORMALSSATION OF THE PROPOSITIONAL CALCULUS 187

a I ) Since X Y takes the truth-value 1 it follows that to each integer a (1 such that the set

E , = {p,~, . . . * Pum,! { Q u l , * * > Qun,}

is empty there corresponds an integer, ( = p ( a ) , 1 5 #? 5 L ) such that

{Pu l , ~ * . , P a m , , ~ & u ~ , . . * , N Q a n u } 2 { h ! p l . . . . , H p M p ’ ~ S p l , . . . , -So Np} (A) .

Ifit us suppose that there exists an integer a (1 I; a case and such that E , is empty. Then, for each integer i (1 5 i an integer y i such that 1 5 yi of Y does not belong to the set

1) for which this is not the L) , t,here exists

Mi + N , and the yith conjunct of the ith disjunct

{ P a 1 , * * * , p a m u , ~ Q a ~ v . . . t - & a n , } .

Henm, if P U 1 , . . ., Pamu take the truth-value I , Q U l , . , ., Qana take the truth- value 3 and, for all i (1 5 i 5 L), the propositional variable occurring in the yith conjunct of the ith disjunct of Y takes the truth-value 2 whenever it is not one of the variables PU1, . . . , P,,,,,, Q a l , . . ., Q,,, the formiila X takes the truth-value 1 and the formula Y does not take the truth-value I . Thus our supposition that such an integer a exists is false.

We are now in a position to derive t,he formulae

n!nu 9 - 1 Puj&nn,”,, N Q . k + n ~ l R p j & n ~ ~ , - s p k ( 1 s a 5 1 , E u empty).

BY (A) Rf iA‘ {Pa l r* . . )Pum, ) (2 = 1, . . . , Np)

and -Sp,,E { N Q . I , . . ., %Qzn, } (p= 1 , . . . ? Np).

Hence, by A3, F4, R4 and R3, we can derive the formulae’)

17,”~,Puj & I I i 2 1 N Q a k +

n m a Puj& L Q 1 w Q , ~ + ~ s p ,

( A = 1, * . ., M p ) ,

( p = 1, . . . , N p ) . j=1

Using R 2 M , + N P - 1 times we can then derive the formulae

nt.. 3 = 1 Pai & n n a k = l - Q U k --* nZIRpi & n f ~ ~ N Sp,

From A l , F2 and R 3 we deduce

(1 5 a 5 I , E , empty).

njM=8,Rpi& nN@ k = l “ s , t l k -cf=l(n;2;‘lRij&nfi1 - S i k ) .

Using R3 again we deduce the formulae

nmu 3 = 1 Paj & n;:, - Q,r + 2;- 1 (n2lRi j & nfLl N S i k ) (1 2 a f; 1 , E, emptty)

1) If Hp = 0 or Np = 0 Borne formulae are, of corn, omitted.

188 ALAN ROSE

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

1 2 3 2 2 2 3 2 3

A FORMALISATION OF T H E PROPOSITIONAL CALCULUS

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

189

1 2 3 2 2 2 3 3 3

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

1 1 1 3 1 2 3 3 3 3

1 1 1 1 1 2 1 2 2 . 2 3 1 3 3 3 P

1 1 3 2 2 3 3 3 3

P p & q - - f y & p 1 1 2 2 2 1

1 3 2 3 3 1

The independence of A 5 and A0 follows by means of the interpretaticn tables for negation given below.

A 3

1 3 2 3 3 3

P + P 1 3 2 2

N N

A 6

P + - - P 1 3 3 1

190 ALAN ROSE

1 1 1 1 1 1 1 1 2 1 2 1 1 1 3 2 3 1 1 3 1 1 3 3 4 1 1 1 4 1 4 4 5 1 1 1 1 5 5 5 6 1 2 3 4 5 6 6

1 2 3 4 5 6 6 2 2 6 6 6 6 5 3 6 3 6 6 6 4 4 6 6 4 6 6 3 5 6 6 6 5 6 2 6 6 6 6 6 6 1

1 2 3

A 9

1 2 3 2 3 3 3 3 3

P & Q I 1 2 3 9 1 2 3

1 2 3 2 2 3 3 3 2

P The independence of A10 and A l l is established by means of the interpretation

tables for disjunction given below.

1 2 3

1 1 1 1 3 2 1 2 3

A FORMALISATION OF THE PROPOSITIONAL CALCULUS

1 2 3

191

1 I. 1 1 2 2 1 2 2

All P v Q l 1 2 3 Q

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

1 2 3 2 3 3 3 3 3

"P 1 3 1 1 3 2 3 2 3

.-Y -+ - (P v Q )

2 3

The independence of A 12 is established by assigning the value 1 to P -+ Q if and only if the value of Q i 4 never nunicrically greater than that of P.

P & -P - + Q 2 2 2 2 3

The independence of R1 follows by means of the interpretation tables for dis- junction and conjunction given below.

I 1 1 1 1 2 2 1 2 2

A1-12 all take the value 1 and, if the premiss(cs) of any instance of any of R 2 4 take the value 1, so does the conclusion. Thus, if R1 is omitted, F5 is unprovable.

P " P - + P 2 1 2 2

It follows similarly, by means of the interpretation tables for disjunction and conjunction given below, that if R2 is omitted F6 is unprovable.

p & Q 1 1 2 3 P

1 2 3 Q

2 2 3 2 2 3 3 3 3

P - t P k P 1 121

In order to establish theindependence of R3 we note that if P is a premiss of an instance of Ri, Q is the conclusion of that instance of Ri And the lengths of P, Q are 2, y respectively then

Z I y ( i - 1,2,4).

192 ALAN ROSE

Hence, if It3 is omitted, no provable formula has length less than the length of the short.est axiom. Thus, since F 1 is of length 3 ond no axiom is of length less than 6, F1 is unprovable in the resulting formalisation.

Tn order to establish the indepcndence of R4 we note that the formula

P 74 (P v qk v ( r v 8 ) -' (7, v 4 v (q " 8 )

takes the truth-value 1 and is therefore, by the weak completencsa, provable. W e note also that if P, , P, are the premisses of an instance of R i and Q is the conclusion of that instance of R i then every propositional variable which occurs in Q occurs in at least onc of the formulae P, , P , ( i =- 1 , 2 , 3 ) . Thus, since the propositional variable "3" docs not occur in any of AI-12, F74 is unprovable if R 4 is omitted.

Finally we shall oomider the degree of completeness of the formalisation. We eholl show t.hat the cardinal and ordinal degrees of completeness are 2Ho and f2 respect,ively. T t will be convenient, for this part of tho paper, to use the notation') of EUKI\SIEWICZ. Ifit n, , na, , . . be distinct non-negative integers. We shall show that if i , . i,, . . . are distinct integers,

and thc formulae 2,

ace taken as additional axioms then, if

the formula

remains unprovable. We shall show, by strong induction on N : that if a previously unprovable for-

mula P is derivable in the new formalisation by using R 1 , . . . , R 4 ul, . . . , u, times respectively and

N = Zfp,ui then P is derivable by (possibly repeated) application of R4 to one of the additional axionis or is one of these axioms.

If N y= 0 the result is trivial. We now asslime the result for all non-negative integers less than N and deduce it for N. The last step in the deduction of P mu&, by our induction hypot,hesis, bc eit,her

(i) the application of one of R1-3 to two formulae Q, R a t leest one of which (since P is unprovable in the original fornal system) in either derivable by applica- t iw(s) of R4 to one of the additional axioms or one of these axioms or

(ii) the result of an application of R4 to a formula 8 which is either derivable by application($) of R 4 to one of the additional axioms or one of these axiomn.

1) Sce, for example, ROWER, J. €3. end TURQUETTB, A. R., Many-valued Logics, Ameterdam,

2) A ' k denotes ik consecutive A's.

i k € {n,, n2, . . .}

A i r K p l p z a * * pik,?

(k =I 1 , 2 , . . .)

(k = 1 , 2 , * . .)

2E {.a,, n,, . . .} - {il, d,, . . .},

d'KP,P, * . . P1+2

-. -

1952, pp. 15-16.

A FORMALISATION OR THE PROPOSITIONAL CALCULUS 193

In case (i) the principal connective of a t least one of Q , R cannot be C . Hence this case is vacuous. In case (ii) the result follows at once. Thus the result is proved. It follows at once from a theorem of TAR SKI^) that the cardinal and ordinal degrees of completeness are 2Ne and LR respectively.

Since. in the original formal system, every provable formula has implication as its principal connective, it seems of interest to determine the effect of adjoining to A 1-12 as an axiom an unprovable formula P with implication as its principal con- nective. We shall show that if P is a tautology of the %valued propositional cal- culus a necessary and sufficient condition for Q to be provable in the resulting fornialisation is that Q has implication as its principal connective and is a tautology of the 2-valued propositional calculus. If P is not a tautology of the 2-valued pro- positional calculus we shall show that the condition is that the principal connective of Q is implication.

We note first that, since all 13 axioms and all conclusions of R1-3 are of the form R + S and, if the premiss of an instance of R 4 is of this form so is the cun- clusion, all provable formulae are of this form in both cases.

Let us now consider further the case where P i3 a tautology R --f S of the 2- valued propositional calculus. Since A1-12 and R --f S are tautologies of the 2- valued propaqitional calculus and, if the premisqes) of an instance of any of R 1 4 are tautologies of the 2-valued propo3itional calculus, so is the conclusion, only tautologies of the 2-valued propositional calculus are provable in the resulting for- malisation. We note next2) that we can find formulae X , Y of the respective forms

C f = i ( I 7 ~ ~ i P i j & l 7 ; ‘ = 1 M Q ~ E ) ) n?-l(CiM,’1Ri? v Cf’i - S i r ) , where P i , , . . .,PI,,, . . , P l 1 , . . . , p l m , , Q i i , . . * t Q l n l , . t Q ~ i ) * - * , Q ~ n , t Ri1, * - 9

RIMlr . . . , RLl , . . . , RL M L , S, , . . . , S1 N,, . . . . SL 1 , . . . , SL NL are propositional vari- ables, such that

X ; , R , Y - p S .

In view of the weak completeness we can derive the formulae X -> R , S + Y . Since R -> S is an axiom we can, by R3, derive the formula X + Y . Since X --f Y does not take the truth-value 1 it follows at once that there exist integers w , W (1 w 5 I , 1 5 W 5 L) such that the formula

flyy1Pwj & HZ1 Qwk -+ Z E T R w j v cf=! Swk)

which we shall denote by U --f V , does not take the truth-value 1 and the set

E , = {PWl, * - ., Pwm,) {&la1 9 * * * 3 Q”VJ

is empty. Since X + Y is a tautology of the 2-valued propositional calculus, so is U 4 V . U + V is derivable from X -+ Y by means of A3, F4, A l , F2, R4 and R3.

I) Tarrsgz, A,, Logic, Semanhics, Metamathematics, Oxford, 1956, p. 104. 2 ) See footnote 2 on p. 186; also ibid., p. 17.

13 Ztschr. f. math. Logik

194 ALAN ROSE

We shall show next that thcrc exist integers a , t!? (1 5 a M , , 1 S fi 5 Nw) Huoh that

(i) R w . is S W P ,

(ii) Rw,B {P,,, . . .,Pmtc,?Qu.l,. . . j Q w n w } -

In order to establish (i) we note that since U + V docs not take thc truth-valuc I the sets

E, = {P,1!. . ., p w m w } p {Rpvi , . . - 3 R,,,},

E , { Q t o l , . . ., &,,,>.\ { S W l , . - - 3 A%VNw)

are empty. Since El , E2 and E3 arc cmpty it would be possible, if (i) were fahe, to assign the truth-value T to Pu,l, . . . , P,,, Swl, . . . , S w N w and the truth-value 1' to Qwl,. . ., Qwn,, RkvYl , . . ., R W M w . Thus U + V would not be a tautology of the. %valued propositional calculus and we should have it contrildiction.

either

or Tn the latter case

If ( i i ) were false then R,,.E (PU31, * * .3wn,w} Rw.E {&,I, . . . I Q,t,,J.

" S w p c 1 ( - Q t U i , . * . $ -Qtc,,>. Thus, in both cases, U + V would be a tautology of the 2-valued propositional ctrlciilus. The proof of (ii) est,ablishes aleo that R,v, does not occur in U unless

RIV, 4' {SIv1, . . ., Swx,} (7 - 1 , . . . ' Mw). LetusnowapplyR4to U +. V,substitutingpforPwl,. . . .Pu,n,,;qforQ,,,l , . . .,&,,,, and r for all other propositional variables occurring in U - V . Let us denote the forniula thus derived by lJ* - V*. We can, in view of the weak completeness: derive the formula

and also one of t,hc forinu1a.e pP6: " q -+ u*

p7* -> r L' - r ,

V * --> ( r v - r ) v - p ,

V* +. ( r

V* + (t.

- r ) v q , - r ) \.J (y v " p ) .

Thus, usinp R3, we can derive one of the formulae1)

(i\)

( W p $ -q -+ r v - r

p & -q -+ ( r v - r ) v - p

p & -q -+ ( r v - r ) v q

(W p (r, -q . -+ ( r v - r ) v (p \/ - p ) .

1) If m,(n,) = 0 we could. in fact, replace p & - q by - q ( p ) , but them is no need to treat. thie 88 a separate case since the formula p & - q U* iS d ~ k ~ y 8 provable.

A.FORMALIBATLON OF TEE PROPOSITIONAL OALOULUB 198;

We shallnow show thet from (B) we can deduce (A). From (B), F1, R4 and R2 we deduce the formula

p & -q + ( ( r v - r ) v - p ) & ( p BE -9).

By the weak completeness we can prove the formula

( ( r v - r ) v - p ) & (p & -q) - r v - r .

Formula (A) now follows a t onoe from R3. I n an exactly similar way we can de- duoe (A) from (C) or (D). Thus !A) is always provable and, using R4, we oan derive the formula

p & - - p + q v -9.

But, by the weak completeness, we oan derive the formula

p + p & - - p .

Thus, using R3, we deduce the formula

P -+qv “Q. Let us now suppose that R* + S* is a tautology of the 2-valued propositional cal- culus and that the propositional variables occurring in this formula are P, , . . . , Pk. From the formula p --f q v -9, R4, F1 and R 2 we deduoe

R* -+ (nfni1Pi v -Pi) & R*.

By the weak completeness we can prove

Pi v -P,) & R* + S*.

Using R3 we then deduce R* -+ S*.

We note, in particular, that, if S* is a tautology of the 2-valued propositional ualculus and R* is a propositional variable which does not occur in S*, then R* + S* is derivable.

Let us now consider the case where P is a formula R + S which is not a tauto- logy of the 2-valued propositional calculus. We shall show that every formula of the form U + V is provable in the formalisation. Let us denote the propositional variables occurring in P by P I , . . ., P,, respectively. We may suppose, without loss of generality, that there exists an integer N (0 5 h’ 5 n ) such that R, S take the trutfh-vahes 1, 3 respeotively when P I , . . ., PN take the truth-value 1 and P,,,, . . ., P, take the truth-value 3. Let us now apply R4 to P, substituting Q v -Q for P I , . . . , PN and substituting Q & -Q for P,,,, . . . , P,, where Q is a propo3itional variable not occurring in V , and let us denote the formula thus derived by X --f Y. It follows a t once that X takes the truth-value 2 uhen Q takes the truth-value 2 and X takes the truth-value 1 when Q takes either of the truth- values 1, 3. It also follows’) a t onoe that Y never takes the truth-value 1. Henoe,

1) Cf. footnote 1 on p. 186.

13.

196 ALAN ROSE

in view of the weak oompleteness, we can derive the formulae

Q - X , Y A V .

By R 3 wt’ can then derive the formula

Q + v . Since& does not occur in V we can then deduce thc formula U -+ V by means of R4.

If, instead of regardmg conjunction as a separate primitive, we make the defini- tion

P & Q =df. - ( w P V -Q)

we may, by modifying A3, A4 and R2, omit A S 1 1 and rename A12 8s AS. Thud) our formalisation is now a8 follows:

A1 P - , P V Q

A 3 - ( p v q ) -+ - P

A 4 - ( P v r l ) - + - ( q V P )

A 5 - N P - P

A 6 p + - - P

A 2 p v q - f t l v p

A7

A 8 - ( P V -p) ’ q .

R1 If P + R and Q --f R then P v Q -+ R .

R2 If P - . - Q a n d P - + - R t h e n P + - ( Q v l K ) .

R 3 Tf P -+Q and Q -+A then P -j R.

R 4 Tf Q is a propositional variable which occurs in P, R is a formula in which the symbol “ >’’ does not oocur and the result of rcplacing all occurrences of Q in P by R is S then if P then S.

- ( p v - ( q J r ) ) + - ( p v -q) v - ( p v - r )

By A 3 and R 4

P 1 p&y-+”p*

By F1, A 5 and R 3

F2 p & q - p .

I) A7 end A8 are also simplified slightly, but the formalisation would still be complete if the simplificatione were not carried out.

A FORMALISATION OF THE PROPOSITIONAL OAUJULUS 197

By A4 and R4

F 3 p & q - + q & p .

By A7 and R4

F4 p & ( q v r ) + p & q v p & r .

Hy AS and R4

F6 p & - p + q .

By A5 and R4

F6 - ( p & q ) -+ - P V -q .

By A6 and R4

3-7 - p v - q + - ( ( p & q ) .

By R2 R5 I f P - . - - Q a n d P - + - - N R t h e n P ~ & & R .

By RB, A6, R4 and R 3 R6 If P -+ Q and P -+ R then P -+ Q & H.

By A 3 and R4

F 9 - (pvp) -+ - q .

By A4, F8 and K.3

F9 - ( p v q) -+ -p.

By A3, F9 and R6

F10 - ( p v q ) -+ -p & - q .

By F 2 and R4

F11 - p & -q + - p .

By F3, R4, F11 and R 3

F12 - p & -q + - 9 .

By F11, F12 end R2

P13 - p & - p - + - ( p ~ q ) .

Since Al, A2, AS, A6 are axiom, R1, R3, R4 are primitive rules of prooeduro, F2, F3, F4, F6,F7, F10, F13, F B are provable formdee and R6 is a derived rule the weak oompleteness of the formaliscrtion follows as before. The independenoe of A 1-7, R 1 and R2 follows from the interpretation tableel) used for the axioms and rules of these names in the previous system and the independenoe of A 8 follows

1) Since the formelieetion is weekly complete the formulee p v p + p , - p + - ( p v p ) . p 3 p , ( p v q) v (r v 8) -+ (JI v r ) v (q v 8) ere deriveble.

198 ALAN ROSE

from the interpretation of "+)) used in the proof of the independence of A12 in the previous system. The independence of H.3 and R 4 follows exaotly as before.

A1 A2 A3

P ' P V V Q P V Q + Q " Y 4 P v 9 ) + " P 1 1 2 2 1 1 2 2 2 1 1 2 3 3 2 2

A4 AS A6

P + P P + - - P 1 2 3 3 2 3 2 2 1 3 2 2 1 3 3 1

N h l -(P v 9 ) - - (q " PI

A7 A 8 "(71 v N (q v r ) ) --> - ( p v -y) v N ( p v - r ) - ( p v -p ) + q 5 2 2 6 3 1 4 6 2 1 4 3 6 6 2 1 3 4 2 2 2 2 2 3

R1 P V P - ' P 2 1 2 2

K2

" P -+ " ( P L ' P ) 1 3 2 3 2 3

The result5 ccmcerning the relationship between strong and weak deductive com- pleteness follow as before.

(Eingegangen am 12. Miin 1963)