Zeitsehr, f . muth. Logik und Orundlaqen d. Malh. Bd. 11, S, 277-289 (1965)

A DECISION PROCEDURE FOR THE SYSTEM E, OF ENTAILMENT WITH NEGATION1)

by NUEL D. BELNAP, JR., in Pittsburgh, Pennsylvania (U.S.A.)

and JOHN R. WALLACE a t Oxford (England)

1. The system E has been proposed as an analysis of the intuitive concept of entail- ment (see ANDERSON [l] and BELNAP [2]). The problem of finding a decision pro- cedure for E has proven refract,ory; as a contribution toward the solution of the decision problem for E, we consider the decision problem for that fragment of E determined by the axioms involving only the signs for entailment and for negation : we shall show that the system Ei of entailment with negation is decidable.

This paper is an extension of the methods and results of KRIPRE [9]. The GENTZEN cut-free rules for entailment given below, and the fundamental combinat- orial theorem (9.4 below) needed to establish the decidability of a system based on these rules are both due to him. KRIPKE was able to establish the equivalence of his GENTZEN formulation of pure entailment with the usual axiomatic form- ulation; hence, applying his combinatorial theorem, he established the decidability of the pure calculus Ej of entailment. When standard rules for negation are added to the GENTZEN system of entailment, the proofs of the eliminat,ion theorem and the decidability both remain valid ; but unfortunately, KRIPKE’S proof of the equivalence of his GENTZEN formulation to an axiomatic formulation does not appear to extend to the systems with negation. We overcome this difficulty and prow that the GENTZEN formulation with negation is equivalent to the axiomatic formulation. KRIPKE’S methods then apply here, just as in the pure entailment case.

2. As primitives we take propositional variables p l , p 2 , . . . , - for negation and -+ for entailment. Well-formed formulas (wffs) are as usual. A , B , C , etc., range over wffs. Parantheses are omitted in accordance with the conventions of CHTJRCH.~) “&4 = C” means that A has the form C .

The system E j consists of bhe axiom schemata

E f 1 . A + A + B + B .

EF2. A + B + . B + C + . A + C ,

l) This research was supported in part by the Office of Naval Research, Group Psychology Branch, Contract No. SAR/Nonr-609 (16).

’) CHUROH [4], pp. 74-77.

278 NUEL D. BELNAP AND JOHN R. WALLACE

E i 3 . ( A + . A + B ) + . A + B ,

E i 4 . A - t B + . B + A ,

E i 6. A + -+ A ,

and the rule:

(a) From A and A -t B to infer B .

The proof of decidability makes use of two equivalent GENTZEN Sequenzenkalkul LET, and LE;; LET, is the Sequenzenkalkiil most readily proved equivalent to the axiomatic system E j , while LEi is the system most easily shown decidable.

Pos tu l a t e s for LET, a n d L E i .

M , /?, y , 6, [ a , j], [a, ,9], [ a , ,!?I range over (possibly null) sequences of wffs, [wffs of the form 21, [wffs of the form C -+ 03, [wffs of the form C --f 03. An expression of the form M

+ + ? ?

/? is called a sequent.

The sys tem LEj , .

Axiom schemata.

t 2 , A k Z , A + B , B

StJructural rules of inference.

C

Logical rules of inference.

? - t + 1.- a , A , B I- " t - a , A

- t - Z , A - + B k M , W

The sys tem LEf.')

Axiom schema.

A k - A

l) The essential ideas of this formulation are due to KRIPRE [Y l .

A DECISION PROCEDURE BOR TEE SYSTEM E? OB ENTAILMENT WITH NEQATION 279

Logical rules of inference (generalized to give the effect of structural rules). +

E + a , A k B +I- at- A , y B ,B t - 6 a t - A + B [LY.,P,A -+m t- ["/,I +

where [a , #I, A + B] is any permutation and contraction of a , P , A -+ B , within the following limits: if a wff other than A -+ B occurs m times in a and n times in /I, it occurs no fewer than max(m , n ) in [a , P , A -+ B] ; if A -+ B occurs m times in OL and n times in B , then it occurs no fewer than max(m, n , 1) times in [a, /I, A --f B] . [ y , 61 is any permutation and contraction of y , 6 with the same limits imposed as were imposed for wffs other than A -+ B in [a, 8, A + B ] .

where [A, / I ] is any permutation of 2, /I with the following contraction permitted on A: where occurs n times in 8, A-occurs no fewer than max(n, 1) times in

[A, PI * -I- t A , @

[ & , A ] t- P where the notational conventions are exactly like those for the preceding rule.

method Tinker-to-Evers-to-Chance. Sections 3-6 next following prove the equivalence of E i , LEi and LE,, by the

Sections 7-9 show LE', (hence E i ) decidable.

3. El imina t ion Theorem for L E i . If a t- y and P t- 6 are both provable in L E i , then so is [a, B*] t [y*, 61, where B* [?*I is the result of deleting one or wwre occurrences of some formula A from B [ y ] , and with the number of occurrences of wffs limited as in the rules of LE;.

This can be proved after the manner of GENTZEN [S].

4. Ei i s contained in LE;, in the seme that if A is a theorem of E r , then t. A is a theorem of LE;.

Proof. Analogues of the axioms of E i may be proved easily in LE; , and that an analogue of the rule (a) is available follows from the Elimination Theorem for LE;.

5. In this section we shall show (corollary to 5.11) that LET, is contained in E f in the sense that if t A is a theorem of LEi,, then A is a theorem of E i .

5.1 For convenience we list here some theorems and derived rules of Ei needed in the proof. We abbreviate wffs of the form C , +. C, -+. . +. C , -+ B by p + B or ly -+ B . We allow n = 0; i.e., y~ + B may be B. A - B means that) A + B and B - + A both hold.

280 NUEL D. B W A P AND JOHN a. WALLAUn

5.1.1 . A -+ B - B-+ 2; 5.1.2 A +B+. B +A; 5.1.3 2 + B -+. + A ;

5.1.4 A -A; 5.1.5

5.1.6

5.1.7 (6)

(trans) from q + B and B + C to infer g~ -+ C ;

(perm) from q~ +. B -+ C -+ D to infer B + C +. p + D ; from A +. B + C and B to infer A -+ C ;

5.1.8 (A, + . A , +. * * * + A , -+. (B -+ C ) -+ D ) + . A , 3. A , +. +,

(B + C ) +.An -+ D ; 5.1.9 5.1.10 (A , +. Atn-l +. * * * + . A , -+, D -+ E ) +. (B, +. Bn,l +. - *

(A, + . A , -+. * * * -+. B + . B + C ) +. A , +. A , - + . * a * +.B + C ; +. B, +.

E -+ F ) +. (em+, +. Cm+,+1 +. . * - +. C, -+. D + F ) ;

where Ci is some Ai or some Bk , and where if Ci is Aj [Bk] and Ci. is A, [Bk'] then (i > it) if and only if (j > y) [(k > k')] .

5.1.1-5.1.6 may be proved easily using the subproof format E; of ET (see ANDERSON [l]). 5.1.7 may be proved by observing that if B is a theorem of E j so is B 3 C -+ C. 5.1.8-5.1.10 are proved in BELNAP [2], p. 57.

6.2 We'turn now to the problem of providing an interpretation of sequents of

(a) Ai( l 5 i 5 n) is a partial interpretation of r. (b) If B and C are partial interpretations of r so is

LET, in E j : Given a sequent r = A, , . . . , A, of LET, we say that

-+ C . Let A be a partial interpretation of I: Among the well formed parts of A we select some which will be said to occur in A or which A will be said to contain. We will say that A contains D, or D occurs in A , if D is an Ai [under clause (a)] or a B or a C [under clause (b)] used in the construction of A . Notice that there may be several occurrences of a given Ai in a partial interpretation of r.

6.3 We turn now to the development of some properties of partial interpret- ations, returning to the proof of equivalence in 5.10.

If B and c! are partial interpretations of r which contain exactly the same number of occurrences of each Ai, and if in Ez B - C , then relatively to r we will say B = C . We have the following properties of this relation, where A , B , B', C , and C ) are partial interpretations of r:

5.3.1 B = B ; 5.3.2 if B = C then C = B ; 5.3.3 if A = B and B = C then A = C ;

A DECISION PROCEDURE FOR THE SYSTEM E~ OP ENTAILMENT WITH NEQATION 281

5.3.4 if B = B', B occurs in C , and C' is the result of replacing B by B' in C,

5.3.5 if C occurs in B then there is a partial interpretation B' of r such that

5.3.6 if C occurs in B then there is a partial interpretation B' of r such that

then C = C';

B ' = p - t C a n d B = B';

B' = B ; and (unless B = C ) B' = p -+. C --f E . -

Proof. 5.3.1, 5.3.2, and 5.3.3 follow immediately from the definition of A = B and the properties of entailment.

5.3.4 is an immediate consequence of the replacement theorem [if A - B then (. . . A . , .) - (. . . B . . .)] which is provable for Er by the usual inductive pro- cedure.

We prove 5.3.5 by induction on the length of B. If B has length one (i.e., B is a propositional variable), then B = C and we take B' = B. Suppose inductively that 5.3.5 is true of B' with length less than n and let B have length n . B has the form 5 -t E , where D and E are partial interpretations of P which have length less than n . C occurs in at least one of D and E ; hence by the inductive hypothesis we have either D = y -+ C or E = y + C , for some y. In the first case we have B = y -+ C -+ E , by 5.3.4, whence by contraposition and 5.3.3 we have B = E +. y + C ; wetakeB'- E +. y -+ C . In the second case we have B = D +. y --f C by 5.3.4; we take B' = D +. y + C . We remark that the transformation of B into p + C (where B contains C ) is secured simply by a series of applications of 5.1.3.

We prove 5.3.6 as follows: from 5.3.5 we obtain a B" such that B" = B and BPI = p -+ C . Unless B = C , in which case p is empty, B" = g?' -+. E --f C . By contraposition and the replacement theorem we have B I ~ = 9;' -+. C -+ E ; hence by 5.3.3 we have B =

- - -

-

-

+. C -+ E . pi +. C +. E is the B' sought.

This completes the proof of 5.3.1-5.3.6. 5.4 If B and C are partial interpretations of a sequent r which contain occur-

rences of exactly the same Ai's, if no Ai occurs more times in C than in B, and if B -+ C in E j , then we will say, relatively to I', B 2 C. We rematk that this reln- tion is transitive and that if B = C then B 2 C .

We also have the following theorems concerning the relation 2. 5.6 Theorem. If C contains occurrences of B, B 2 B', and CI i s the result of

replacing (zero or more) occurrences of B by B' in C , then C 2 C'.

Proof. The proof is by induction on the number of occurrences of B replaced by B' in C . If no B's are replaced the theorem is trivial. Suppose inductively that the theorem is true for n replacements of B by B' in C , and let C' be obtained from C by replacing n -+ 1 occurrences of B by B'. The n f 1 occurrences of B replaced will be called the distinguished B's of C . Let C" be the result of replacing n of the

282 NUEL D. BELNAP AND JOHN B. WALLACE

distinguished occurrences of B in C by B'. By the hypothesis of the induction C 2 C" and by 5.3.5 there is a D such that C" = D and D = pl -+ B (where B is the remaining distinguished B). Since B 2 B', we have D 2 pl -+ B' by (trans), whence C1' 2 9 + B'. Now by performing on pl +- B' in reverse order the opera- tions which led from C" to D (in the proof of 5.3.5) we obtain C' - the result of replacing all distinguished B's in C by B'. It is easy to see from the proof of 5.3.5 that C' = pl -+ B'. Hence we have C 2 C1', C" 2 pl -+ B', C' = pl + Bt; therefore we have C 2 C', as was to be proved.

6.6 Theorem. If B is a partial interpretation of r containing occurrences of Ai but not containing any A j ( j =I= i ) , then B 2 Ai .

Proof. The proof is by strong induction on the length of B. Suppose inductively that the theorem is true for B' of length less than n and let B have length n . If B has length one, then B is Ai and the theorem is trivial. Otherwise B has the form C --f D, where C and D have length less than n ; hence by the induction hypothesis, C 2 Ai and D 2 Ai . Therefore, by two uses of 5.5 we have B 2 -& + A i , whence B 2 Ai by axiom E i 6, etc.

5.7 Theorem. If B is a partial interpretation of T which contains occurrences of A ( , and also occurrences of at lmt one Ai ( j $= i ) , then there is a B' such that B 2 B' and B' = pl -+. c + A i , where C does not contain Ai .

Proof. The proof is by strong induction on the length of B. Suppose inductively that the theorem is true for B' of length less than n and let B have length n . In virtue of 5.3.5, there is a Bf / such that B = B" and BI1 = y +. F-+ D, where D contains occurrences only of A i , and where C contains an occurrence of some A j ( j $= i ) . There are two cases according as C contains occurrences of A, or not.

Case 1. C contains occurrences of Ai . By the hypothesis of the induction, C 2 C', where C' = pl +. 2 -+ Ai and Ai does not occur in E. We have B" = y +, D + C by contraposition and the replacement theorem. By 5.5 we have y -+. D + C 2 y +. D +. pl +. E -+ Ai. Hence we have B 2 y +. 5 +. pl +. E + A i , which is of the required form.

Case 2. C does not contain occurrences of A i . By 5.6, D 2 A i , hence by 5.5 y +. c+ D 2 y +. 54 A < . Therefore B 2 y +. c + A i . y~ +. C + Ai is of the form required.

5.8 Theorem. If B is a partial interpretation of r containing occurrences of A; and Ai ( i =+ j) but no occurrences of any A k ( k + i , k + j), then B 2 A: + A j .

Proof. The proof is by strong induction on the length of B. If B is of minimum length (i.e., B has length 2) the theorem is trivial (see the basis of 5.7). Suppose inductively that the theorem is true for B' of length less than n and let B have length n . B is of the form + D, where C and D are of length less than n . There are three cases.

-

-

-

- -

-

A DECISION PROCEDURE FOR TEE SYSTEM fii OF ENTAILMENT WITH NEGATION 283

Case 1. C contains occurrences of Ai only. By 5.6 C 2 Ai; hence by 5.5 B 2 Ai --f D. If D contains only occurrences of Ai then D 2 Ai by 5.6, and B 2 Ai -+ A j by 5.5. If D contains occurrences of both Ai and Ai then by the hypothesis of the induction D 2 Ai + Aj , hence B 2 xi -+. Ai -+ Ai. Therefore by axiom E i 3 B 2 Ai + Ai.

__

- c

Case 2. C contains occurrences of A j only. Similar t o Case 1. Case 3. C contains occurrences of Ai and A,. By the hypothesis of the induction

C 2 xi + A , , And by the same argument as that used in Case 1, either D 2 & --f A, or D 2 Ai or D >= Ai. Hence by two applications of 5.5 we have either Ai --f A, +. Ai -+ Aj or Ai --f Ai --f Ai or Ai --f Ai + Ai. We apply axiom E i 6 [contra- position and E , 31 [contraposition twice, (trans), axiom E i 3 , and again contra- position] to obtain Ai --f A j .

5.9 Theorem. Let B be a partial interpretation of r which contains occurrences of

several negated entailments Ei + Fi , 1 5 i n , and also occurrences of other wffs. Then there is a B such that B 2 B' and Bf = p + G , where G contains no occurrences of any Ei -+ Fi , 1 5 i 2 n , and all occurrences of the other wffs in B' are in G .

Proof. The proof is by strong induction on the number of occurrences of the several negated entailments Ei --f Fi , 1 5 i 2 n , in B. We suppose inductively that the theorem is true for B* with less than n occurrences of negated entailments, and let B have n occurrences of negated entailments. Select one of the occurrences

of a negated entailment in B, say Ek -+ F k . By 5.3.6 B = p -+. Ek + Fk + D , and by 5.1.4, the replacement theorem, and repeated uses of (perm) we have

y 3. E, + Fk --f D 2 E, --f Fk -+. p + D . p + D contains one fewer occurrences of negated entailments than B; hence if B contained only one occurrence of a negated entailment we are done. Otherwise we apply the inductive hypothesis t o y 3 D , obtaining a B* such that 9 --f B 2 B* and B* = p f + G , where G contains no occurrences of any Ei + F i , 1 5 i 5 n and all occurrences of the other wffs.

By 5.5 we have B 2 Ek + Fk +, p' --f G . Ek -+ Fk -+. p f + G is the form required by the theorem.

6.10 We give the following definitions of interpretation and valid interpretation. If R is a partial interpretation of r which contains a t least one occurrence of every Ai in r, then B is an interpretation of r. If an interpretation B of r is a theorem of E i , then B is a valid interpretation of r. Not every interpretation of every provable sequent of LET, is a valid interpretation, but we have the following

5.11 Theorem. If P is a provable sequent of LE?,, then there is a valid inter- pretation B of T.

We remark that 2 and = preserve valid interpretationhood. And by 5.6 we have a

Corollary. If

- P

-

.~

___

-~

____

-- _ _ _

A is provable in LEi, , then A is a theorern of E j .

284 NUEL D. BELNAP AND JOHN R. WALLACE

Proof of t h e theorem. The proof is by induction on the proof of r in LET,.

Basis. If Tis an axiom we set B = A -+A or B = A + B + . A + B , which can

Induction. We show that if the theorem is true of the premiases of rules C, W,

Cases 1 and 2. I' comes from I" by rule C, or W where B' is a valid interpretation

Case 3. r comes from I" and P' by rule Cut. The proof that there is a valid inter-

Lemma 1. Let C be an interpretation of k a ,A and let D be an interpretation

- - - -

be easily shown to be theorems of E i .

Cut, k +, and k N-, then it is true of the conclusions. There are five cases.

of I". B' is also a valid interpretation of I', so we let B 3 B'.

pretation of r makes use of three lemmas.

of k /I. Then there is a C' which is an interpretation of t a , /I such that - A + D + . C +C' in E,.

Proof. The proof is by strong induction on the number of occurrencesaof A-in C . Suppose inductively that the theorem is true for C" with less than n occurrences of A, and let C have n occurrences of 2. By 5.7 we have C 2 cp --f. E -+ A. Whence by the theorem of E j , (rp +. E + A) -+. 2 --f D +. rp -+. D + E , and (trans) we have C + .A + D + . p +. D --f E ; from this by (perm) we have 2- D -+.

c +. rp +. D + E NOW rp +. D + E contains a t least one less occurrence of 3 than C . If it contains no occurrences of A, it is an interpretation of 01 ) /I, in which case we let C' = rp +. D + E . Otherwise rp +. D + E is an interpretation of a , b , A, and we may apply the inductive hypothesis to it, obtaining a C' which is an interpretation of

- - -.

- - - -

- - - -

01 ) b ) /I) hence t- a , /I ) such that -

A -+ D +. (p -+. 5 -+ 2) + C'.

From this together with 2. -+ D -+. C -+. a, +. 5 + E we have A -+ D +. C + C' by 5.1.8, 5.1.9, and 5.1.10.

Lemma 2. Let C be a valid interpretation of t- a , A and let D be an interpretation

-

of t- p . Then there is a C' which is an interpretation of k a , /I such that - A + D -+Cr in E i .

Proof, By lemma 1 we have x-+ D +. C -+ C' where C' is an interpretation of t- 01 , P, whence by (6) and the theoremhood of C,

-

A + D -+ C' holds in E i .

Lemma 3. Let q~ -+. z-+ D be a valid interpretation of k a , A and let D be an interpretation of y , where every member of y occurs in 01 . Let C' be an interpretation of + b , y and let A -+ D -+ G" be provable in E j . Then rp + 0' is a valid interpreta- tion of k a , /I) A (if A mum more than once in a, +. A -+ D) or 'r a , (otherwise).

A DECISION PROCEDURE FOR THE SYSTEM q OF ENTAILMENT WITH NEGATION 286

Proof. From P] +. A -+ D and 2 + D + C' by (trans) P] 4 C1 is provable and by definition is an interpretation of t- a , j3, A (if A occurs more than once in q +. 2 -+ D) or

We return now to the proof of Case 3. Let I" be j- a , A with valid interpretation B' and let P' be k j3, with valid interpretation B". We need to show that there is a valid interpretation of t- a , j3. There are four subcases according as a and /I are empty or not.

Subcase 1. a and j3 are both non-empty. We will show that there is a valid inter- pretation of a , j3 by strong induction on the number of occurrences of A in B'. Suppose inductively that the case is true for B'* with less than n occurrences of A and let B' have n occurrences of A . By 5.7

B ' Z p - + . A + D

a , j3 (otherwise).

where P] +. + D contains no more than n occurrences of 2 and D is an inter- pretation of some k y , where every member of y occurs in a . Since B" is a valid interpretation of j3, 2 and D is an interpretation of t- y , by lemma 2 there is a B"' which is an interpretation of k j3, y such that

- A + D + B1".

p + B"' Hence by lemma 3 we have that

is a valid interpretation of I- a , j3 or a , j3, A . In the first case we are done. Otherwise, by the construction in the proof of lemma 3, P] + B"' contains fewer occurrences of A than B'; hence we may apply the inductive hypothesis to P] + B"' and B" to obtain an E which is a valid interpretation of t a , /3.

Subcase 2. a is empty and j3 is non-empty. We shall show that there is a valid interpretation of a , j3 by strong induction on the number of occurrences of A in B". Suppose inductively that the case is true for B"* with less than n occur- rences of 2, and let B" have n occurrences of 2. By 6.6

B ' z A ,

and since B' is provable, so is A . By 5.7 we have

B " 2 y + . A + D

where 9 +. A -+ D contains no more than n occurrences of A and D is an inter- pretation of some y , where every member of y occurs in j3. From A we have A and by (8) we have

- -

~ -

-

-+D,

which is a valid interpretation of t- a , j3 or t a , j3, A . In the first case we are done. Otherwise we apply the inductive hypothesis to y + D (which contains less than n occurrences of 2) to obtain an E which is a valid interpretation of 1 a, /?.

286 NDEL D. BELNAP AND JOHN R. WALLACE

Subcase 3. OL is non-empty and /? is empty. Almost exactly symmetrical to Sub-

Subcase 4. OL is empty and /? is empty. By 5.6 we have B' 2 A and Bff 2 2. But since B' and Bjf are both provable (by hypothesis) we should have A and 2 in Ei ) contrary to the fact that Ei is consistent. Therefore this Subcase cannot arise.

case 2.

This completes the proof of Case 3.

Case 4. r = t El +. F,, . . ., En --f F,, C -+ D comes from Tf t- El +. F , , . . . , En + F % , c, D by rule k -+) where Bf is a valid interpretation of P.

By 5.9 B' 2 p --f 6' ) where 6' contains no occurrences of Ei --f Pi, 1 i 5 n , and all occurrences of the other wffs of rf ; i. e., 6' contains all and only occurrences

____

- of c and D. By 6.8 -

By 5.5

By 5.1.4 and the replacement theorem we have

p l + . C + D

which is by definition a valid interpretation of I'. A , , . . ., Ai, . . . , A ,

by the rule t- NN, where B' is a valid interpretation of P. Let B be the result

of replacing every occurrence of Ai in Bf by Ai. Since Ai +. A i , B is a valid inter- pretation of F in virtue of the replacement theorem for E i .

This concludes the proof of the theorem and the proof that LEit is contained in E i .

6. LE-', is contained in LEir , in the sense that if dc t /? is provable in L E i , then k &, ,9 is provable in LE,. This is immediate for the axiom schema of LE; and for all the rules except + ; for the latter, use the axiom k A , A + B , B and the rule Cut.

- - Subcase 5. F = A , , . . ., A i , . . .) A , comes from rf =

- - -

- -

By sections 4-6 we have the following:

Theorem. A is provable in E i if and only if

In the remaining sections we show that LEi is decidable; from which, by thc Theorem, it follows that E i is decidable.

7. We say that a sequent Fl reduces to a sequent r, (rl red r,) if can be obtained from rl by zero or more applications of the rules W and C (for LErt) or their analogues for the left aide. We have concerning LEf the following:

Lemma. If a sequent P is provable in L E f , then there is a proof of r containing no P such that P red T.

This follows from the fact, easily provable by induction as in CURRY [5 ] , that if rf red r and P is provable in m steps, then r is provable in 5 m steps.

A is provable in LEf.

A DECISION PROCEDURE FOR THE SYSTEM ~7 OF ENTAILMENT WITH NEGATION 287

8. By a tree we mean a configuration of elements arranged in levels, which has a first (or bottom) level containing only one element and such that lines are drawn between elements of consecutive levels in such a way that every element in every level (except the first) is connected to exactly one element in the preceding (lower) level. A branch of the tree is a sequence of elements al, aa, u3, . . . (which may or may not terminate), such that ocl is the element a t the first level anda i i s some member of the i th level which is connected to aiw1 by a line. A branch may terminate in an element a, such that no line connects a, to an element of the next higher level; or a branch may be infinite. We say that a tree is finite if the number of elements in it is finite.

A tree has the finite fork property if each level of the tree has only a finite number of elements; a tree has the finite branch property if every branch of the tree is finite.

8.1 Konig’s Lemma. A tree is finite if and only if it has both the finite fork property and the finite branch property (KONIQ, [S]).

8.2 Now we turn our attention from trees to finitary logical systems having wffs, axioms and rules. Let S be such a system. Evidently any proof in S can be written in the form of a finite tree. We put the wff proved a t the bottom; above each wff D of the kth level we put the wffs from which D follows by means of one of the rules, with lines drawn from these wffs to D. Such a tree will be called a proof tree. All branches of a proof tree must terminate in axioms.

By a proof search tree for (a wff) C we shall mean a tree whose elements are wffs of S, which has C a t the bottom level. The idea is that given a proof search tree for C we may hunt for a proof of C among the subtrees (where “subtree” is defined in the obvious way) of the proof search tree. By a wmplete proof search tree for (a wff) C we shall mean a proof search tree for C which has a proof of C as a sub- tree, if a proof of C exists.

8.3 For an arbitrary finitary logical system S there is clearly an effective pro- cedure which when applied to , an arbitrary wff C yields a complete proof search tree for C ; such a complete proof search tree will not, in general, be finite. On the other hand, if for a system S there is an effective procedure which when applied to an arbitrary wff C yields a finite complete proof search tree for C, then S is decidable. This follows from the fact that a finite complete proof search tree has only a finite number of subtrees - one of which is a proof of C , if such a proof exists.

8.4 It follows from 8.3 and theorem 8.1 that if for a finitary logical system S there is an effective procedure which when applied to an arbitrary wff C yields a complete proof search tree having both the finite fork property and the finite branch property, then S is decidable.

9. We shall show that LEi is decidable by describing an effective procedure which when applied to an arbitrary sequent r of LEi yields a proof search tree for r which

(i) is complete,

(ii) has the finite fork property, (iii) has the finite branch property.

288 NUEL D. BELNAP AND JOHN B. WALLAOE

Given a sequent F we form the tree (called the distinguished proof search tree

(i) we put I' a t the bottom, (ii) above every sequent I'/ occurring in the kth level we put (a) nothing, if TI

is an axiom, (b) otherwise, all P I such that (i) P can serve as a premiss for P, (ii) I'll does not reduce to P nor to any sequent below in the same branch as P. 9.1 That the distinguished proof search tree for I' is complete is a consequence

of the lemma of 7, which shows that if there is a proof of I', then there is a proof of I' which satisfies condition (ii).

9.2 That the distinguished proof search tree for r has the finite fork property is a result of the following property of LEi : there are only a finite number of wffs in r which can serve as principal constituent, and the choice of principal constituent determines a finite set of premisses from which I' can follow. (The restrictions on the number of wffs occurring in the premisses and conclusions of applications of rules of LE; were designed to secure this property.)

9.3 It remains to show that ' the distinguished proof search tree'for I' has the finite branch property. Following KL~NNB [7], we will say that two sequents OL /- p and are cognate if exactly the same wffs (not counting multiplicity) occur in a[p] as in a l v ] . We call the class of sequents which are cognate with a given sequent the cognation class of I'.

for T) as follows:

We define the class of sub-formulas of a given wff as follows:

(1) If A is a wff, then A is a sub-formula of A .

(2) If C is a sub-formula of A or B, then C is a sub-formula of A + B ; if C je

a sub-formula of A , then C is a sub-formula of 2. (3) All the sub-formulas of a wff are given by (1)-(2).

Sub-formula theorem. If LX is a provable sequent in LE; then any wff occurring in any sequent in the proof of LX P is a eub-formuh of some wff occurring i n a P.

Proof . Inspection of the rules of LE; shows that no sub-formulas are lost in passing from premiss(es) to conclusion. From the sub-formula theorem, it follows that the number of cognation classes occurring in any branch is finite.

If we can show that only a finite number of members of each cognation class occurs in any branch, then it will follow that each branch is finite. To show this we introduce the following terminology. A sequence of cognate sequents I',, ) TI , . . . is irredunhnt if for no i , j , j > i , rj red Pi. From condition (ii) in the definition of the complete proof search tree for I' it follows that every sequence of cognate sequent# occurring in any given branch is irredundant. Hence the finite branch property and, in turn, that constructing the distinguished proof search tree for I' constitutes a decision procedure for LEf (and hence for E i ) will follow from the following

A DEOISION PROOEDUI~E BOR THE SYSTEM q OB ENTAILMENT WITH NEQATION 289

9.4 Theorem [due to KRIPKE~)]. If a sequence of cognate sequents is irredundant, it is finite.

Proof. Let Z = r,, r,, . . . be a sequence (finite or infinite) of cognate sequents. We shall show that if there are no i , j, j > i , such that I'j red Ti (i.e., if the sequence Z is irredundant) then the sequence Z is finite. The proof is by induction on the number n of distinct wffs having one or more occurrences in Z. If n = 1 the theorem is obvious. Suppose inductively that the theorem is true for 2 with n wffs and let C have n + 1 wffs. Pick out one of the wffs in Z, say A . I', is critical in a sequence Z* if and only i f , for all m > k, the number of occurrences of A in r, is less than or equal to the number of occurrences of A in r,. Let I'i be the result of deleting all A's in ri. We define a new sequence of sequents, Z! = A , , A , , . . . in the follow- ing manner. Let r, be the first sequent critical in Z. A,, is I'L. Suppose A E has been defined and is Ti'. If ri+, exists find the first critical sequent in the sequence Ti+, ,

. . . ; call it Ti and let A k + , be rj'. Note that 2 terminates only if Z terminates.

Z' is a sequence of cognate sequents which contains occurrences of only n distinct wffs. Hence by the hypothesis of the induction if 2 is irredundant Z' is finite. But by the choice of critical sequents in the construction of 2?, if Z is irredundant Z' is irredundant; and if 2 is finite then ,Z is finite. Hence, if Z is irredundant, Z is finite. This completes the proof of the theorem and the proof that the system EF of entailment with negation is decidable.

Bibliography

[l] ALAN Ross ANDERSON, Completeness theorems for the systems E of entailment and E& of entailment with quantification. This Zeitschr. 6 (1960), 201-216.

[2] NUEL D. BELNAP, JR., A formal analysis of entailment. Technical report No. 7, Office of Naval Research, Group Psychology Branch, Contract SAR/Nonr-609 (16). New Haven 1960.

[3] ALONZO CHURCH, The weak theory of implication. Kontrolliertes Denken. Miinchen 1951. [41 ~ , Introduction to mathematical logic, vol. 1. Princeton 1956. [5] HASKEL B. CURRY, A theory of formal deducibility. Notre Dame 1950. [6] GERFLARD GENTZEN, Untersuchungen iiber das logische SchlieBen. Mathematische Zeitschr.

[7] STEPHEN C. KLEENE, Introduction to Metamathematics. Princeton 1950. [ti] D.KOENIQ, Uber eine SchluDweise aus dem Endlichen ins Unendliche (Punktmengen-

Kartenflrben-Verwandtschaftsbeziehungen-Schachspiel). Acta Litterarum ac Scientiarum (Sectio Scientiarum Mathematicarum) 3 (1927), 121 -130.

[9] SAUL A. KRIPKE, The problem of entailment (abstract). Journal of Symbolic Logic 24 (1959), 324.

39 (1934), 176-210, 406-431.

(Eingegangen am 21. Dezember 1964)

l) This theorem, which is referred to obliquely in KRIPKE [9], was communicated to one of us by KRLPKE in a letter dated September, 1959. We are indebted to KRIPKE for permission to publish its proof for the first time. As noted by KRIPEE [9], the theorem leads to the solution of t,he decision problem for a variety of Sequenzenkalkul.

19 Ztwhr. f. math. Logik