Zeitschr. f. math. Logik und Rrundlngen d. Math. Bd. 16, S. 439-446 (1970)

A PAIR OF PRIMITIVE RULES FOR THE SENTENTIAL CALCULUS

by P ~ I P WEBB in Oxford (England)

1. I n setting up a sentential calculus logicians are ususlly content with four or more primitive ruled) using two constants, and tend to assume that any set of rules is as good as another, provided it is reasonably simple. It is true that as long ago as 1917 NICOD~) showed that using SHEFFER'S stroke one axiom and two rules of inference are sufficient, but his axiom is long and by no means obvious, and proofs in his system are long and very difficult,. Perhaps it is this that has deterred logicians from exploring further the possibilities of the stroke function.

In this paper I shall show that there is in fact one simplest set of primitive rules from which a complete sentential calculus can be derivecl. In 2. a simple and sym- metrical pair of rules (Ra , @) are stated, and the steps by which WHITEHEAD and RUSSELL'S system may be derived from them outlined; further simple complete sets of rules which are variants of Ra,B appear a t the end of the section. I n 3. I prove, in effect, the uniqueness of Ra, ,5: with a few small variations they are the only complete pair, and t'hus give us the simplest possible calculus.

2. Our rules form a natural-deductive system, and we use the bar-and-arm in the conditional-proof rule to mark the introduction and discharge of the assump- tion. The proofs are very simple, and we merely list the derived rules and theorem in the order in which they are to be proved. The stroke is denial of conjunction,3) and to avoid parentheses we adopt the convention that the four ways of writing the constant are, in order from least scope to greatest: 1, 1 , .I., I. We define: P -+ q = D f PIP/!? and P v q = D f P/PIq /q*

The primitive rules are

R@:4) P I alr P

r t - q

from which we derive DR 1: Plrr 1 d P

P, PIP t- B DR 2 :

1) An axiom being a rule with zero premisses. 2) J. G. P. NICOD, Proc. Cambridge Philos. SOC. 19 (1917-1920), 32-41. 3) Similar, but less elegant, rules can be devised for the stroke as denial of disjunction. 4) Strictly, in using RP, we should write both lines of the conclusion every time, but we may

adopt the convention of writing only one line if we do not need the other. No problem will arise in the proofs. We might have called the rules /I, /E.

440 PHILIP WEBB

k P + 4

and from these WHITEHEAD and RUSSELL’S four independent axioms and modus ponens follow a t once (a rule of uniform substitution for variables is not necessary in a natural-deductive system).

R a , /? express the self-evident principle that p entails q and r iff from -1, we can derive q and r.1) There are also other complete pairs which are variants of them, especially

R6: P I qlr P

t r

Ry is hypothetical syllogism, and this pair corresponds to NICOD’S system.a) Also

With only two distinct variables we need three rules

RE: as above MP: PI!& Com: p / q P t a 1 P l P r P l P

t q RE: as above MT: Pl!l DN : PIP I PlP

q (or Plrr I P/T) k 24-1, F P

These give simple four-rule systems for use with two const,ants

R2: p - t q R3: p R4: ---1, - q t - - p k-1,

R1: L t P + q 1 - P ~

1) Perhaps if the Pythagoreans had known of these rules they would have called R a the

2, Puzzle for a rainy afternoon: is there an axiom corresponding to Ra , making a complete female and RP the male.

symmetrical system with R c.

A PAIR O F PRIMITIVE RULES FOR THE SENTENTIAL CALCULUS 441

R1: R2 : - ( P & q ) R3,4: as above 4

1 - P I- - (3, 4 ) Or we may write - p v q for p + q. It would be difficult to improve on the sim- plicity of these sets of rules, and the proofs they give are equally short and simple.

3. I now want to prove informally that the only two-rule systems that satisfy the conditions in (1) below and are complete are Rol, and certain kinds of variant on Rac, ,B.’) Which variants are possible will appear a t the end of the proof (Ry, 6, E , 5 naturally being among them).

It is easy to prove that other two-rule systems are incomplete for derived rules; i.e. that for a t least one WFF E that is entailed by another WFE’ D, it is not pos- sible to derive E from D by the two rules. But such a system is incomplete for tautologies only if one of the rules is a rule of detachment, e.g. RP, which would allow the derived rule to be got by using the corresponding tautology D I EIE. But we must then prove that a system which is complete for tautologies must include a rule of detachment, and it is as easy to prove directly that the other systems are incomplete for tautologies. To do this we must show that there is some tautology which cannot be derived from them.

First we need some explanations and definitions and a lemma. (1) I n the systems we are corvcerned with the only constant is the stroke, for

denial of conjunction; neither rule is an axiom (i.e. a rule with zero premisses); and one rule is a conditional-proof rule, i.e. a rule which tells us that if from an assumption of a certain form we can derive by the rules a line or lines of a certain form, then we can write some other line or lines of a certain form. We shall call the conditional-proof rule R i and the other RE. We allow any finite number of lines in the conclusions of both rules, but no proper subset of the premisses of Rii entails any line of its conclusion (otherwise the other premisses are irrelevant or we have two rules in one). We allow more than one line in the conclusions of the rules partly for generality, and partly to be able to treat Rol, /3 as two symmetrical rules. It will seem less odd if a rule is seen as an outline picture of part of a correct proof.

(2) To make the proof easier to state and follow we shall assign the values T and F to the variables so that DIE is P iff D, E are T, and we shall use the terms tautology (tt), contradjction (cd), contingent WFF (ct), valid, consistent, etc. in their usual senses. We shall use + for material implication and ‘ D + E’ for ‘ D -+ E is tt’. We shall use the stroke to name itself, with ‘1s’ for the plural where necessary; and we shall abbreviate ‘lies within the scope of ?z i s ’ to ‘lies within Is ’ . A WFF in which all 1s are written ‘1 ’ is to be read by grouping to the right (e.g.

We shall use several sets of variables. A , B and C, in some cases with primes, subscripts or superscripts, will be WFFs consisting of primitive variables (repre- sented by Cy, etc.), and having special forms defined below. D, . . ., M (but not B) are WFFs consisting of the same primitive variables, but not having a specially

P l q i r b = P I Q I .is).

l) Since neither ruleis complete on its own, it follows there is no complete one-rule system. 29 Ztechr. f. math. Logik

442 PHILIP WEBB

defined form. p , q , r , . . . will be the variables in terms of which actual complete pairs (e.g. RK, p ) are stated. N , . . . , Q are WFFs consisting of p , q , r , . . .. 01 is any WFF (consisting of p , q , r , . . . and I) that might occur as the assumption in the statement of R i ; B is any set of WFFs that Ri might require us to derive from the assumption to be able to infer the conclusion; y is any set of WFFs that might constitute the conclusion of R i ; 6 , c , 7 will be used for other WFFs that migth occur in places in R i or Rii to be defined below.

Commutation is ignored throughout : the proof works equally well if we write q / p for p/q and so on.

(3) For any WFF D we can generate an infinite sequence of WFFs that entail D by starting with D and substituting DIE I DIG for one or more occurrences of D in an earlier member of the sequence; and we can generate an infinite sequence of WFFs that are entailed by D by starting with D and substituting DIDIE for one or more occurrences of D in an earlier member of the sequence.

(4) Defini t ions. A chain of applications of a rule is a sequence of applications such that in every pair of consecutive members the WFF substituted for the con- clusion in the first is substituted for a premiss in the second.

If D is tt (cd, inconsistent), let ' @ ( D , E , G)' be 'if D' is constructed from D by replacing E , G by E', G' such that - (E' 3 G ) , then D' is not tt (cd, incon- sistent)' (where D , D' for 'inconsistent' are sets).

Let 2 be the number of Is within which lie(s) the variable(s) that lie(s) within the most Is in any line of Ri,ii.

Let C, C' be arbitrary cases Ci, Cfi of C1, (72,. . ., C'l, CI2, . . ., and assume i is even (if i is odd the proof is similar).

Let C consist of the WFFs C, , C,, . . . , C,, , where the subscript represents the number of 1s within which Ci (for all i) lies except that C, lies within n - 1 Is, and let there correspond to each Ci a distinct variable Cy such that for all i , either C j = Cy or if i is odd Cf -+ Ci and Ci is constructed from Cp by the rule in (3), or if i is zero or even Ci 3 Cf and Ci is constructed from CY by the rule in (3); provided that no Ci is tt or cd. Let Cl be constructed similarly from Ci , Ci , . . . , C,!, , except that if i is odd Ci 4 CF and if i is zero or even C:" 4 Ci. And let no variable occur more than once in C, C', unless either for some i Cy = CF, or some Cf or Cp is repeated as required in constructing Ci or 6': by the rules in (3).

E.g. basically C = Cy/Cz/C;;lCt. . . IC;, and C' similarly. And instead of Cy we might write C,"/CIlD, and so on for C:, C:, etc.; and for C; we might write CllD I ClIE or CilG I CiIH. I . D I Cl/E, etc. ; and similarly for C' (except that Ci 3 C y , etc.). But e.g. C: =+ C z , and C y =+ D or E or any variable in D or E .

Further let C =# C'; let n be odd, and let n - 1 > 2; let C, = Ci'; for some i, j (x < i, j 5 n) let Ci = Cp = C:, and let Ci not be a variable while Cj is; let CA-z = C?-,._,, and similarly for n - 1, n . And for all i, j (i =+ j) let no variable that occurs in Ci occur also in Cj, and similarly for C'i, C' j .

Let C*, C'* be the arbitrary cases of Cl*, Cz*, . . ., C'1*, C'2*, . . . ; let Ci* be constructed from Ci by the second rule in (3) if i is odd and by the first d e in (3) if i is even, and let C"* be constructed from C'i by the first rule in (3) if i is odd

A PAIR OF PRIMITIVE RULES FOR THE SENTENTTAL CALCULUS 443

and by the second rule in (3) if i is even; let C**, C'** be constructed from C1*, C2*, . . ., C'1*, C'2*, . . . as C, C' are from C,, . . ., Ci, . . .. For all i let Ci, C'i lie within more than x 1s in Ci*, Ci*; and for some i let C'*, C''* lie within more than x 1s in C**, C'**.

Let B, B', Bi*, etc. be constructed like C, C', etc. and if B = B,/ . . . /B, and C = CJ . . . /C, let m = n ; but for some i let BP $: BY (whence -(B + B') by the lemma below).

Let A be tt, and contain C**, C'** such that @ ( A , C**, C'**). (It follows that C** 3 C'**, C'l* + C1*, C2* 3 C'2*, C'l+ C1, C2 4 (so C 3 C', and by the lemma for all i CP = C?) .)

Finally suppose that a new variable is used whenever possible in WFFs and proofs.

(5 ) Lemma. If C -5 C', then for all i Cy = Cp. If C < C', then

(a)

(b)

(c; & c;/ . . . /CA) 3 c,

(c; & c;/ . . . /CL) 3 c,/ . . . /c*. and

Suppose Cy =+ Cp. Then when Cy is F , Cp is T and suitable values are assigned to the other variables in C, and C' (which is possible by definition) Ci and Cg/ . . . /CA are both T and C, is F , which contradicts (a). Suppose C!j' =j= C:. Then when C? is T, C; is T, C? is F and suitable values are assigned to the other variables in C , C' (which is possible by definition) Cl and Ci / . . . ICA are both T and C2/ . . . /C, is F , which contradicts (b). Therefore Cy = Cp and Cg = CF, and by similar argu- ment for all i, Cy = C r .

We may now state and prove our metatheorem.

Metatheorem. A pair of rules satisfying the conditions in (1) above is complete (or certain variants described at the end of the proof, inc. Ry, 6 and

If Ri, ii is complete there is a proof of A , the last line of which follows by Ri or by Rii. Suppose by Rii. Then there is a set of chains of applications of Rii such that in each chain every WFF substituted for a premiss in the first member is derived by Ri , and every WFF substituted for a premiss in every other member is derived either by R i or by an application of Rii t,hat is the preceding member of some chain.

A is substituted for the conclusion in the last member of each chain. Then every application of the chain (Dl substituted for the conclusion, D,, D,, . . . , Dj, . . . for the premisses) is such that either

only if it is Ra, RE, 5 ) .

(i) (ii) D1 contains C**, C'** similarly, Dj contains C*, C'* such t,hat @(Dj, C*, C'*)

(iii) neither of these.

D,, Dj contain C**, C'** such that @(Dl (Dj) , C**, C'**) , or

or @(Dj, C'*, C*), or

29*

444 PHILIP WEBB

Suppose some application is (iii). Then there is an application satisfying (iii) such that @(Dl, C**, C’**) (or similarly for C*, C’*). Construct another application replacing C**, C’** by B**, B’**, when DI replaces D, and is not tt (lemma), but all DJ are tt. But then the system is inconsistent. Then either every application satisfies (i) or some satisfies (ii).

Suppose the former. Then if Ri, ii is complete it is possible to derive by R i Dj such that @(Dj, C**, C‘**) ; whence @ (Dj , C, C’) . Suppose the latter. Then if Ri, ii is complete it is possible to derive by R i Dj such that @ (Dj , C* (C*) , C‘* (C*)) ; whence @(Dj, C, C’) . (Rii cannot license us to build up from simpler parts both C**, C’** and C*, C‘*.) Thus this case reduces to the case where A is got directly by Ri , to which we turn.

A is substituted for y i , and since C, C’ lie within more than x /s in A (Dj) E, E’ containing C, C‘ respectively are each substfitfuted a t least once for a variable in y i ; G is substituted for a , H I , , , ,, for p . Then either

(i) C or C’ occurs in neither G nor H I , . . , , r , or both occur in H I , ..., and neither in G, or

(ii) both occur in G and neither in H,, , . , , r , or (iii) one occurs in G and the other in H I , , , ,, r .

Suppose (i). By construct’ing another application of R i replacing C, C’ by B, B’ and using the lemma, we can show the system is inconsistent (as in (iii) below). Suppose (ii). If is consistent, we can show the system is inconsistent as for (i); so ,8 is inconsistent. But p is inconsistent only if there are two occurrences of a variable 6 in /3 such that if one occurrence is replaced by 6, and the other by 6, (6, + 6,) the resulting p‘ is consistent. But H I , . . . , r is substituted for ,8 and C =+ C’, so -@(HI , . . , , r , C, C’), when if there is a proof of H I , . . ., from G we can show the system inconsistent as for (i). Therefore (iii); suppose C occurs in G and C‘ in H I , , , ,,

Suppose that in the derivation of H I * . , ,, I . from G there is for some Cg = CF no chain of applications of Ri, ii such that

(a) in every member there is one occurrence of Cf (or C?) in the premiss given by the previous member and one of Cy (C?) in the conclusion such that if new variables J , J’ ( J $5 J’) are substituted for Cf (Cp) in the premiss and conclusion respectively then the resulting conclusion does not follow from the resuking prem- isses, and

(b) in the first member the operative premiss is G and in the last the conclusion is Hi. Now in the proof of A replace Cp by Cp (or vice versa) such that in each chain (satisfying (a) or (b)) only Ci’ or only C? occurs in the operative places. The result is still a proof of A . Now construct a similar proof, replacing Cf, C y by B, B‘. By the lemma, the resuIting A’ is not tautologous, but every inference between G and H I , . , . , r still holds, so that the system is inconsistent. Thus for every Cp = C p there is a chain of applications of Ri, ii satisfying (a, b).

(the other case is similar).

A PAIR OF PRIMITIVE RULES FOR THE SENTENTIAL CALCULUS 445

But for all i Cp is not derived from Cy by one step of deriving C' from C, since Cp lies within more than x 1s in C (if it could be, we could use B, B' and the lemma to show the system inconsistent as above). Hence it must be possible for all i to derive WFFs in which Ci lies within successively less I s until it lies within x or less than x I s . But n - 1 > x, so it must be possible to drop some of Cl, , , ,,n,

i.e. to derive from L containing C M containing a WFF like C except that it does not contain some one or more of Cl, , , .,n: call it C ,- C,, where Cj is the first of the omitted WFFs reading from the left.

If j = n, we cannot derive M from L for n - 1 > x (unless the system is in- consistent, as above); so Cj is one of Cl, , . ., Then there are parts of C, C ,- Cj as follows :

(where may = C,c etc., but no WFF in C ,- C, = C j ; and where Cj-l is omitted if j = 1).

But then if i is odd and C j , C, are F and C, is T - (C --f C - C j ) , and if C,, Cj+l are F and Cj is T - (C ,- Cj -+ C) ; and if j is even and Cj is T and Cj+l, C, are P - (C -+ C - Cj) , and if C, is T and C j , C,,, are F - (C N Cj -+ C ) . But then i f L + M o n l y i f e i t h e r C + C N C j o r C - C j + C , - ( L + M ) ; a n d i f L + M if both - (C 3 C ,- C j ) and - (C ,- Cj 3 C) , then the system is inconsistent.

as above. But since they are truth-functionally independent by definition, we must have one or more other premisses which entail that one or more of Ci , j+l, /(, ,,, are T or F so as to ensure that C 3 C - Cj or C N Cj 4 C as appropriate. Now either j = 1 or j = 2 , . . . , n - 1.

Suppose j = 2 , . . . , n - 1. Then if we have premisses which entail that C,, C, etc. are T or F so that C+C-C, or C , - C j 4 C as appropriate (2 5 p , q etc. 2 n - 1) , all Ci which occur to the right of C, , C, etc. are irrelevant t'o the inference and the system is inconsistent. (There is just one such Ci in C or in C N Cj if j = n - 1.)

Therefore j = 1. But then for all i in deriving Ci from Ci, either one or both of them occurs as part of a longer WFF or we derive C{ on its own from Ci on its own. But if the former, we cannot derive it by a sequence of applications of Rii; and , we cannot reduce the number of /s within which Cp lies in Ci in the longer WFF till it lies within a small enough number to apply Rii, since the inference would be invalid by the same argument as that above about C, C ,- Cj where j = 2 , . . . , n - 1 . Hence it must be possible to derive C{ on its own from Ci on its own. But as for some i Ci = Cp = Ci , a t least one premiss of Rii must be a single variable, and a t least one line of its conclusion must be a single variable. But Rii cannot be 5 , p t- p , nor p t- p , since for some j Cj is not a variable and Ci is. So Rii is c ,

c = . . . /cj.-l/cj/cj+l/ . . . c N cj = . . . /cj-l/ck/cm/ . . .

Therefore it must be impossible to assign values to C j , j+ l , k ,

P t- !I* (17) .')

l) In the above proof, if we did not use C but a single variable, say p , with p' constructed from p as C, is from C;, when i odd. i t might be that p in a always occurred in p / p I p / p and p' in @ in p'/p' I p'/p', in which case we could not show we must be able to derive p from p' and p' from p on their own on lines.

446 PHILIP WEBB

I n Rii, q must occur in some WFF Nlq in 5 so that it is entailed by c, p but not by c alone. But q is entailed by Nlq iff Nlq is F ; and - (Nip ) is entailed by PI N / q iff PI Nlq is T and either P is T or Nlq 4 P . So if q is entailed by 5 , p then q occurs in PI N / q in some member of c , say 5’ where PI Nlq is T and either P is T or Nlq < P. But P is a single variable, or we cannot substitute into CL to get C (since C, = C,”). So q occurs in rl Nlq in el, when since - ( N / q 3 r ) r is T. But by the same argument, either r I Nlq = ( 1 or it occurs in sIQ .I.r I Nlq in C L , and so on, so that 5‘’ = cn = q , and every 5; where j is odd is entailed by some premiss. But one of these zf’s must be entailed only by p (and not by any other ( J ) . Then the shortest version of Rii is RB or R6.

In R i it cannot be that all yc’s are p / p or p l p / q or p lq /q or p i qlrls or some- thing longer. For if n is odd then in breaking down C to derive C‘ we shall derive Cn-2/Crl-l/C,, on its own on a line. Then after getting CA from C,,, to derive C~-2/C~.-l/CA we must useRi. But this cannot be done i f , as is the case, C,!,-2, C;-l, C:, are distinct variables and all yl’s are one of the five alternatives above. It follows that a t least one y L = plq or p I q / r . Hence p 3 q/q or q . r depending on which form of y L we have. This allows R i to be Re, Ry or a longer version of Ry where e.g. B = s , slr, r / r l q , or R a . And since these give complete systems, there is no need to have other lines in the conclusion of Ri.

We can show, by using the derivation of C‘ from C, that 5 cannot be more than one line, from which it follows that 5‘ = p I N l q . Also it seems likely that oc must be a single variable (it must be at least equivalent to one, e.g. p lp lp lp ) , and that longer WFFs cannot be written for q , r in 8, even though the rule remains valid: but these things are not important enough to deserve proof.

Finally it may be noted that it must be possible to get plq t qlp as a derived rule, in order to derive the tt pIqIqIpIqlp. This probably requires both the twist in the conclusion of Ri , and either that the conclusion of R i is p / q or that q , r can be derived in the conclusion of Rii. But again proof is unnecessary.

4. Since the premiss of R a and the conclusion of RP do not contain I , it might look as if the rules give us a way of defining all truth functions in terms of some- thing else (e.g. a primitive calculus whose WFFs and rules belong to mathematics). But R a , B allow us to define only a WFF whose least variable lies within an even number of I s : the meaning of pIq is not determined, and remains primitive. Only conjunction and implication can be defined by R a , /3 : but these can also be defined directly from a primitive calculus by the rules ‘ p & g iff p , q’ and ‘ p + q iff from p we can derive q’.

Whether Rol , @ or another system is ‘best’ will depend on our interests. R 1-4 will be preferred if we want to remain close to English ; RDL , if we want simplicity and the smallest number of primitives.

. . . where there are n - 1 Is in

(Eingegangen am 22. Februar 1969)