Zeilschr. 1. math. Log& und Crundluoetz d. Saib . Rd. 22, 8. 177-186 (1976)

FORMALISATIONS WITH NONSTANDARD DEGREES OF COMPLETENESS

by ALAN ROSE in Nottingham (Great Britain)

Most formalisations of the 2-valued propositional calculus which use only finitely many axioms (i.e. which do not use axiom schemes) and are weakly complete are also strongly complete [l]. These formalisations have 121 degrec of completeness 2. They normally employ substitution and modus p n e m as the primitive rules of procedure, but the latter rule cannot, of course, be formulated if material implication is not defin- able in terms of the primitive functors. Propositional calculi, such as the system with material equivalence and negation as the primitive functors, normally use the rule of modified madus ponens (if P and EPQ then Q) in place of the conventional rule of modus pnens and have [3] degree of completeness 3. The object of the present paper is to show that the situation may be reversed, weakly complete formalisations of func- tionally complete systems having degree of completeness 3 and a weakly complete formalisation of the E - N propositional calculus being strongly complete. Finally it will be shown that, for the general case of a H E N ~ W [4] fragment (i.e. a system-not necessarily functionally complete-in which material implication is definable in terms of the primitive functors) with 6 almost arbitrary primitive functors, by using a suf- ficiently large (but finite) number of primitive rules related to the conventional rule of modus pnens , there can be set up a. weakly complete formalisation 9,, having degree of completeness b + 2 (6 = 1 ,2 , . . .).

We consider first a formalisation d of the 2-valued E-N propositional calculus. The formalisation uses the following 2 axioms and 5 primitive rules of procedure:

Al . Em, A2. EpEqEqp.

R l . If P is a formula, Q is a propositional variable occurring in P and the result of replacing all occurrences of Q in P by a formula R is the formula S then if P t>hen 8.

R2. If P and EPQ then Q. R3. If an occurrence of p in the formula P is immediately followed (reading from

left to right) by an occurrence of E and Q is obtained from P by interchanging these two occurrences then if P then Q.

R4. If an occurrence of p or of E in a formula P is immediately preceded or im- mediately followed by an occurrence of N , the occurrence, if of p , is not the last oc- currence (reading from left to right) of a symbol in P and Q is obtained from P by interchanging these two occurrences then if P then &.

R5. If the result of replacing one or more occurrences of p in P by N p i s Q and the result of replacing exactly these occurrences of p in P by R is S then if 9 and& then 8.

Theorem 1. The formalisation d is weakly complete. 12 Zt9chr. f. math. Logik

178 ALAN ROSE

Lemma. If P is a tautology which contains no propositional variables other than p t h n i-P.

Let p , N occur in P 2x,21/ times respectively. By the EESNSEWSKI-MIHAJLESCU The- orem x, y are integers. Let R be obtained from P by replacing each sub-formula of the form N S , starting from the innermost, by S.

By A2, R l and R2 R6. Y t EQEQP.

By A1 and R1 (A) i-(Ep)O ENvpNvp. By R 6

(B) E N g p N y p I- (EP)*’+~ EWpiV’p (i = 0,1, . . .). BY (A) and (B) (C) I - ( E P ) ~ , ~ - ~ EN’pNgp. By (C) and R 4 (D) l-N2U(Ep)”-z F A P P .

By (D) and R 3 (E) l-NWR. By (E) and R 4 it follows that t-P.

Proof of t h e Main Theorem. Let U, be a tautology containing exactly n pro- positional variables and let us denote these by PI, . . . , p,, . Let Ui be obtained from U , by replacing all occurrences of P,, . . . P,,-.i by p (i = 0, . . . , n - 1). By the Lemma

(4 i-uo- We shall prove, by induction on,& that, for all U,, i-Ui (i := 0 , . . .? n). If i = 0 the result is (A). We now assume the result for i and deduce it for i + 1. Let V,li be ob- tained from U , by replacing all occurrences of Pn-i by NP,-i. Let V j i be obtained from V,,; in the same way as Ui was obtained from 77,. By the induction hypothesis (used with respect to U, and V,,i) (B) i-Ui. Pi;. By R5 (C) ui, vii i- ui+l*

By (B) and (C), tU i+ , . Thus, for all U,,, i-U; (i = 0, . . ., n ) and, in particular, kU,. Hence the theorem is proved.

Theorem 2. The formalisation d is strongly complete. Lemma 1. If P is mixed then, for all &, P i- Q. This follows by methods similar to those used in a previous paper [3] of the author.

Lemnia 2. If P i s an absurdity then there exists a mixed formula Q such t h t P t- Q. Let R denote a propositional variable occurring in P and let S denote the the for-

mula obtained from P by replacing all occurrences of R by p. By R1 (A) P I S .

FORMALISATIONS WITH NON-STANDARD DEGREES OF COMPLETENESS 179

Let U denote a formula obtained from S by replacing one occurrence of p by Np. Since P is an absurdity, 8 is an absurdity and U is a tautology. Thus, by Theorem 1, (B) FU. Let V denote a propositional variable not occurring in S and let Q denote the result of replacing the above occurrence of p by V . By R5 (C) S, li F Q . By (A), (B), ( C ) it follows that P I- Q, and, since F occurs exactly once in&, Q is mixed. Thus the Lemma is proved. The Theorem follows at once from Lemmas 1 and 2.

We consider next a formalisation D of the 2-valued A-N propositional calculus, in which we use the abbreviations E, K defined by the equations

A 1. AApNpq, As. AEEpqEqpr, A 2. AEApqAqpr , AD. A EApEAEpqrAqrs, A3. AEApKqrKApqAps , A 10. AEAEpEAprAqrs, A 4. AEA pAqrA A pgrs, A 11. AEEpqEEqrEp8, A5. EAppp, A12. AEpKpAAqNqrs , A6. AEApppq, A 13. A EYNppq . A”. AEEpqENpNqr , R1. As for do5’. R2. If P and EPQ then Q. Theorem 3. The formalisation 3 is weakly complete.

Lemma 1. If t P then, for all Q, I-APQ. Let us suppose that., in the derivation of P , Ri is used li times (i = 1,2) and let

1 = 1, + 1,. We shall prove the Lemma by strong induction on 1. If 1 = 0 then P is one of A1 -13. If P is A 5 the result follows a t once from A6 and R1. If P is one of the remaining axioms then P is of the form AUV where V denotes a propositional variable not occurring in U. Applying R1 to P , (1) I-AUAVQ. By A4 and R1 ( 2 ) FAEApAqrAApqrEApAqrAAp. By (2), A5, R1 and R2 (3) tEApAqrAApqr . By ( I ) , (3), I11 and R2 it follows that t-AAUVQ, i.e. FAPQ.

We now assume the Lemma for 1, . . . , 1 - 1 and deduce it for 1. Let S denote a propositional variable not occurring in either of the formulae X, P. If P is an immediate consequence of X by R1 then Z(X) = Z(P) - 1 < 1(P) and,

by our induction hypothesis, FAXS. Hence, by R1, tAPS and, by R1 again, FAPQ. If P is an immediate consequence of X , EXP by R2 then Z(X), Z(EXP) < l (P) and,

by our induction hypothesis,

(4) I-AXS, AEXPS.

CPQ =df ANPQ, EPQ =df CCPNQNAPQ, KPQ =df NANPNQ.

12*

180 ALAK ROSE

By an argument similar to that used in the proof of (3) we infer (using A9) (5) I-EAprEAEpqrAqr . By (4), (5), R1 and R 2 (6) I-SPS. By (6) and R1 it follows that FAPQ. Thus the Lemma is proved.

Lemma 2. If Q denotes a propsitionul variable not occurring in P then APQ I- P. The proof is similar to that used in the derivation of (3) of the previous Lemma.

Lemma 3. I f Q is obtained from P by replacing one or m r e occurrences of a sub- formula R of P by S and I-ERX then FEPQ.

Let Z(P) denote the number of (not necessarily distinct) symbols occurring in P and let 9 ( P , R ) = Z(P) - l (R) . We shall prove the Lemma, for the case where one occurrence of R is replaced by S, by strong induction on 9. The general result then follows at once using A l l , Lemma 2, R1 and R2.

If 3' = 0 then R, S are P, Q respectively and the Lemma follows a t once. We now assume the Lemma for 1, . . . , Y - 1 and deduce it for 2.

If P is of the form XU and the result of replacing the relevant occurrence of R in U by S is V then Q is NV and Y ( U , R ) = 9 ( P , R) - 1 e Y ( P , R ) . By our induction hypothesis, I-EUV. Hence, by A7, Lemma 2, R1 and R2, C-ENUNV, i.e. I-EPQ.

If P is of the form AUW where the relevant occurrence of R is as a sub-formula of U and V is defined as above then P ( U , R) < 9 ( P , R ) and, by our induction hypo- thesis, I-EUV. Hence, by Lemma 1, FAEUPW. Thus, by A10, Lemma 2, R1 and R2 follows FEAUWAVW, i.e. I-EPQ.

If P is of the form AWU then, by the above (1) I-EAUWAVW. By Lemma 2, A2 and R 1 (2) I-EPAUW and (3) I-EAVWQ. By (I) , (2), (3)) A l l , Lemma 2, R1 and R2 it follows that tEPQ. Thus the Lemma is proved.

Lemma 4. If P , Q, R, S are as in Lemma 3 and I-ERS then P F Q.

This follows a t once from Lemma 3 and R2.

Proof of t h e Main Theorem. We define the concept of conjunctive normal

R3. EPQ I- EQP. By A12 and Lemma 2 (1) I-EpKpAAqNqr . We now show, by induction on n, that

form in the usual way. By A8 and Lemma 2, I-EEpqEqp. Hence, by R1 and R2,

tKn-lAAplNplql . . . AAp,Np,p;, (n = 1,2 , , . .).

FORBIALISATIONS WITH XOS-STANDARD DEGREES OF CONPLETENESS 181

If n = 1 the result f o l l o ~ at once from A 1 and R 1. We now assume the result for I I and deduce it for n + 1.

By (1) and R1 (2) kEKrl-lAApliVplq, . . . AAplINpl ,qnKK1~- lAApl~p lq , . . .

- * ,4ApI,NpnqnAApn+,N~~+~qn+l-

By our induction hypothesis, (2) and R2, FK1~AAplNplq, . . . AApn+lNp,+lqn+l. Thus

( 3 ) FK'l-'AAplNp,ql . . . AApIaNpnqn (n = 1,2, . . .).

Let Pi E {AApiNpiq;, ApiNpi} (i = 1, . . . , 1 ~ ) and let 8 denote the set of those values of i for which Pi is ApiNpi . Substituting ApiNpi for qi by R1 (i E 8) in the formula of (3) and then using A5, R1 and Lemma 4,

(3A) tK'b-lP, . . . P,,.

Let S denote a tautology which is in conjunctive normal form. Since S is a tautology it. will, for some values of n, 8, be a conjunction of formulae U,, . . ., U,, where U , , . . ., Un are disjunctions of ax,. . ., a, formulae respectively, a,, . . .,a,, 2 3 and M~ = 2 if and only if i E B (i = 1, . . ., n). Let the disjunct.s of 77; be Xi, iVXi if i E 8 and Xi, N X i , Y i l , . . ., YY,(& 2 1) if i 96 (i = 1, . . ., ra). We apply R1 to the formula of (3A), substituting X i for pi ( i = 1, . . ., n) and ABi-lYil . . . Yip, for qi ( i E (1, . . . , nf - 6). Let us denote the formula thus derived by Kn-l&, . . . Q,, . We note that the formula 8 may be (informally) obtained from Kn-lQ, . . . Q, by applica- tions of the equations

APQ = T AQP. KPQ = T KQP, AAPQR = *r APAQR, KKPQR = T KPKQR.

By -4 13 and Lemma 2

(4) FENNpp. By A2 and Lemma 2

(5) FEApqAqp. By (5 ) and R 1

(6) FEANpNqANqNp. By A7 and Lemma 2

(7) I- EEpqENpNq.

By (6), (7), R1 and R2, tENANpNqNANqNp, i.e.

(8) EKpqK9P. By A4 and Lemma 2

(9) tEApAqrAApqr. By (9), (7), R1 and R3 (10) FENANpANq-VrNAANpNqNr . By (lo), Lemma4,A13, A8,Lemma2 and R1,i.e. F E N A N p ~ ~ ~ N q N r N A N N A N p N ~ N ~ ~ (11) FEKpKqrKKpqr.

182 ALAN ROSE

By A 8 and Lemma 2

(12) C-EEpqEqp. If S is a tautology in conjunctive normal form then, as shown above

(13) KI1-lQ1 . . . Q,. By (13), (5), (8 ) ) (9)) (11)) R1, R 3 and Lemma 4

(14) I-8.

If P is a tautology then, by informal application of the equations

NAPQ = T KNPNQ, A PKQR = T KAPQAPR ,

NKPQ = T ANPNQ, N N P = T P, APQ = T AQP

we may establish that, for some tautology S (= S(P) ) in conjunctive normal form, P (15) FENApqKNpNq. By (4) and R1, FENNANpNqANpiVq, i.e.

(16) kENKpqANpNq. By A3 and Lemma 2

(17) FEApKqrKApqApr. By (14), (15), (16), (4)) (17), (5). R1, R 3 and Lemma 4 it follows that I-P. Thus the formalisation is weakly complete.

S. By (4), R3, R 1 and Lemma 3, t-Eh‘ApqNANNpNNq, i.e.

Theorem 4. The formlisation him degree of completeness 3. If P is an absurdity then, as in a previous paper [3], P F 8 if and only if Q is a tau-

tology or an absurdity. Thus it will be sufficient to establish that if P is mixed then, for all Q, P t &.

Let us suppose that the mixed formula P contains exactly n propositional variables, which we shall denote by P, , . . . ) P , and let P (which we shall denote by @ ( P , , . . . , P,)) t,ake the truth-value q(xl , . . ., x,) when P I , . . .) P , take the truth-values xl, . . .) x,, respectively. Since P is mixed there exist truth-values x, , . . + ) x,, yl , . . . , y, such that

Q)(XI 9 * * ., ~ n ) = T, Q ) ( Y ~ * * * 9 ?/n) = F* We may assume, without loss of generality, that, for some integers a, b, y (0 5 01 5 p 5 y n ) ,

21, . . ., XP = T; X P + ~ , . . ., X, = F ; Ya+l,. * - 3 YD) Y y t l ? * * * > ~n = F .

~ 1 , . . ., ya, y p + l , . . ., yy = 7’;

Let us apply R 1 to P , substituting A p N p for p l , . . ., pa; 29 for pa+,, . . ., p 8 ; N p for pD+,, . . ., p y and NApNp for p,,+,, . . .) pn. We shall deiiote the resulting formula

by Y ( p ) . Thus @(ApNp, . . -, ApNp, p , . . ., p , N p , - . ., N p , NApXp, . - ., NApNp)

(18) p w4, (l9) y ( p ) = T p -

FORHALISATIONS WITH KON-STANDARD DEGREES OP COMPLETEXESS 183

By (19) and Theorem 3

(20) FEY(P)P. By (18), (20), R2 and R1, for all Q, P k Q. Thus the Theorem is proved.

More generally we shall now consider any fragment of the 2-valued propositional calculus in which material implication is definable. We note that we cannot delete the separate requirement of definability of equivalence by defining the pseudo equix- alence functor E* such that

E*PQR = 1‘ AEPQR by t.hc equations

APQ = d l CCPQQ, N”PQ = d f CPQ, E*PQR = d f CCPN*QRN*APQR, replacing R2 by the rule

R*2. If R denotes a propositional variable not occurring in P Q then if P and E*PQR then Q since from a mixed formula we could, applying the substitution rule as in a pre- vious paper of the author [l]: infer a formula Y ( p ) such that Y(p) = 1’ p or Y ( p ) is an absurdity. In the latter case the “fragment” would be functionally complete (since N P = C P Y ( P ) ) and methods similar to those used above would be applicable. In the former case we could, in a weakly complete formalisation g, prove the formula

E*ul(P) Pq from which p follows by R*2, strong completeness of V then following from R1.

Even if the functor E were definable we could, except in the case of a system con- taining absurdities, in & formalisation %“ using R1, R2 as primitive rules of procedure, derive a formula Y ( p ) such that Y ( p ) = P I p and, by weak completeness of V‘. derive the formula E!P(p) p , strong completeness of v‘ then following as for %.

We may, however, obtain results of a very different kind for these fragments by making a more radical alteration to the primitive rules of procedure. Let the primitive functors be the functors F, , , . . , F,, of m,, . . . , nb arguments respectively. implication being definable by an equation of the form

CPQ = d t F6@1(p, Q ) * . @nb(Pj Q ) where G1(p, q), . . ., Qnb(p, qf contaiii no propositional variables other than p , q and p , q both occur in the formula F6Gs,(p, q ) . . . Gnb(p, q), the case where nb = 2 and C is Fb being included. We shall assume that the truth-tables of the functors Fl . . . , F6 are such that none of the formulae F , p I . . . pnl, . . ., F b p 1 . . . pnb is an absurdity.

Theorem 5. There exists a tautology Ti whose pincipal connective i s F , (i = 1 , + . , , b ) .

Let F,p . . . p take the truth-values x , y when p takes the truth-values T , F re- spectively. If L = T then we may take T , to be F,Cpp. . . Cpp. If x = F , y = !!’ we may make the definition

We may then take T , to be F,NCpp . . . NCpp. If x , y = F we may make the definition

N P F,P . . . P.

N p =df c P F , P . . P .

184 ALAN ROSE

Since F,pl . . . pnl is not an absurdity there exist formulae Q1, . . . , Q., such that Q 1 , . . ., Q,,, ~ { C p p , NCpp) and F,Q, . . . Q,; is a tautology. Thus the Theorem is proved.

Let &‘ be a weakly complete formalisation [4] of this propositional calculus using as axioms the formulae PI, . . ., P,. in which p does not occur1), and the primitive rules of procedure of substitution and modus p e n s . Let gb be the formalisation obtained from &‘ by using the axioms

A i. CpP, (i = 1, . * ., I c ) ,

A(L + 1). CCppCCpCqrCpr , A(k + 1 + i). T, (i = 1, . . ., b ) ,

the usual substitution rule (denoted by R1) and the following rules of modus ponens R(i + 1). If F,P, . . . P,,, and CF,P, . .’. P,,IF,QI. . . Qlli then F,Ql. . . Q,,,

(i = 1, . . ., b ) . We shall use the “yields” symbols “ t* ”, “ t ” with respect to &‘> g b respectively.

Theorem 6. The formalisation 9(, is u w k l y complete; Lemma. If t-*S then, for all formulae U , kCUS.

Let us suppose that, in the above derivation of S , %he rules of substitution and modus ponens are used I , , 1, times respectively and that 1 = 1, + 1,. We shall prove the Lemma by strong induction on 1. If 1 = 0 then 8 is P, for some integer i (1 6 i 5 k) and CpS is Ai. Thus by R1, kCUS.

We now assume the Lemma for 0, . . ., 1 - 1 and deduce it for 1. If the last step in the proof of S is an application of the substitution rule to a formula V then, by our induction hypothesis,

t-CU’ v, where U‘ denotes a propositional variable not occurring in V , S. Hence, by R1, t-CU’S and, using R 1 again, tCUS.

i 5 b) , the application of R 2 to two formulae 2, CZS then, by our induction hypothesis,

kCUZ and I-CUCZS

Hence, by A(k + l), R1 and R(b + 1) (twice), t-CUS. Thus the Lemma is proved. Proof of t h e Main Theorem. If S is a tautology then, since iV is weakly com-

plete, t*S. Thus, by the Lemma, FCUS, where U denotes a propositional variable not occurring in 8. Hence, by R1, t-CTiS, where i is the integer such that the principal connective of S is F , . Thus, by A(k + 1 + i) and R(i + l), tS.

Theorem 7. The formalisation Bb has degree of completeness b + 2 ( b = 1, 2, . . .).

If the last step is, for some integer i (1

l) If p occurs in some axiom W of a weakly complete formalisation W , let X denote a proposi- tional variable which does not occur in W and let Y denote the formula obtained from W by replac- ing all occurrences of p by X. &’ is obtained from &” by replacing each such axiom W by the cor- responding axiom Y . Since, by the substitution rule, Y k W the completeness of 2 follopra from that of W .

FOR3ULHATIOSS WITH SON-STANDARD DEGREES OF COXPLETENESS 185

Lemma 1. I f i,, . . ., is ~ ( 1 , . . ., b - 11, the p r i w i p l Connective of P, is F L m (a = 1, . . . , 6 ) and P l , . . . , Ps I- Q then Q is a tautology or its principal connectiae is one of the fu,ictors F , l . . . ., F,* .

Let us define 1(Q) in a manner corresponding to the above definition (Lemma of Theorem 6 ) of I ( # ) . We shall prove the Lemma by strong induction on Z(Q). If I = 0 then Q is an axiom (and therefore a tautology) or Q E ( P I , . . . , P,>, in which case its principal connective is one of the functors F l l , . . . , Pi,. We now assume the Lemma for 0, . . . , 1 - 1 and deduce it for 1. If the last step in the proof of Q is an application of R1 to a formula U then, by our induction hypothesis, U is a tautology (and there- fore so is Q ) or the principal Gonnective of U (and therefore of Q ) is one of the functors F,, . . . , F,, . If the last step is the application of R(a + 1) to two formulae F,Pl . . . P,,, CF,P, . . . PnLFSQl . . . Q,, then, by our induction hypothesis, a E ( i l , . . . , i,) or FuPl . . . Pna is a tautology. In the former case, since Q is FaQ, . . . Q,,,, there is noth- ing to prove. I n the latter case we note that by our induction hypothesis, the formula CF,P,. . . P,,,Q (whose principal connective is Fb) is a tautology and the result, fol- lows at once.

Lemma 2 . If the formula F,P, . . . P,,, is not a tautology then, for all Q l , . . ., Q,,, , F , P , . . . PI,& I- F,Q, . . . Q,, (i = 1, . . .,a).

Let us denote the propositional variables occurring in F,Pl . . . PI,+ by S,, . . . , S,, . We may suppose, without loss of generality, that F,P, . . . P,. takes the truth-value F when S, , . . ., 8, take the truth-value T and Sr+,,. . ., S,,, take the truth-value F . We apply R1 to PIP, . . . Pni, substituting Cpp for S, , . . . , S, and p for SU+,, , . . , S,,,, denoting the resulting formula by Y(p). Thus

(1) F,P, . . . Pni I- Y ( p ) . By Theorem 6

(2) t-cR4 P.

(3) FcP(FiQ1. QnJ PzQi . . * Qn, .

By (2) and R1

By (1) and R1

(4) FJ', . . - Pn, I- F(J',Qi * . . Q n J .

By (3), (4) and R ( i + 1) it follows that F,Pl . . . P,,, k F,&, . . . Q,,,. Proof of t h e Main Theorem. Since, for all propositional variables Y , Z,

Y I - Z by R 1, it follon7s from Lemma 2 that there exists no sequence of formulae X, , . . . , X, where t > b + 1 and not X,, . . ., X, I- XX+, (x = 0, . . ., t - 1). 0 1 1 the other hand, by Lemma 1, if the principal connectives of the formulae P, , . . ., Pb are F , , . . ., respectively and Pa+, is a propositional variable then not P , , . . . , P , t- P,,, ( x = 0. . . . , b) and, for all W , P,,, I- W . Thus the degree of completeness is b + 2.

186 ALAN ROSE

References [l] ROSE, A., Strong completeness of fragments of the propositional calculus. J. Symb. Log. 16

(1951), 204. [2] TARS& A., ober einige fundamentale Begriffe der Metamathematik. Comptes rendus (Warsaw)

Claase I11 23 (1930), 22-29; for an English translation see TARSKI, A., Logic, Semantics, Meta- mathematics. Oxford 1956.

[3] ROSE, A., The degree of completeness of a partial system of the 2-valued propositional calculus. Math. Zeitechr. 64 (1952), 181-183.

[a] HENKIN, L., Fragments of the propositional calculus. J. Symb. Log. 14 (1949), 42-48.

(Eingegangen am 10. Januar 1975)