Modus Ponens and Derivation from Horn Formulas

Zeilnc?k. 1. math. Lo&k mnd Grunddauen d. Math. Bd. 13, S. 33-54 (1967)

MODUS PONENS AND DERIVATION FROM HORN FORMULAS

by WILLIAM CRAIG in Berkeley, California (U. S. A.)l) In Q 1 of this paper we investigate some problems concerning modus ponens and,

to a much lesser extent, similar problems concerning a rule of syllogism that general- izes modus ponens. Most of the questions were suggested by the work of PAUL HERTZ on sequents and by GENTZEN’S extension of HERTZ’S notion which allows antecedents and succedents to be empty.2) We show that those sets H of hypotheses for a proof which we consider possess a natural partial ordering and that proofs from H largely reflect this ordering. In Q 2 we show that if we restrict ourselves to formulas of certain kinds, then implications provable in the classical first-order predicate calculus with equality are also provable intuitionistically. Two initial results of 9 1 are used in 2; otherwise, the two sections are independent.

Q 1. Modus ponens We let m , n, p , . , . be arbitrary nonnegative integers. We consider throughout

a fixed set of propositional variables, often simply called variables. We let a , p , a,, . . . be arbitrary variables, @, Y , 0 , . . . arbitrary finite sets of such variables, and @ 3 Y , Y, 3 O,, . . . ordered pairs of such sets such that the second term, or succedent, of each pair contains at most one element and hence either is the empty set 0 or is some {a}. The first term @ of an ordered pair @ -+ Y of this kind shall be its antecedent and the pair @ 3 Y itself a sequent.

For an assignment z of truth-values T or F to each a , we let z+(@ + Y) = T if either z (a ) = F for some a in CP or z(a) = T for some a in Y, and we let -G+(@ -+ Y) = F otherwise. Evidently, for every such assignment z and every 01,

z+({a} + {a}) = T, z+(0 30) = F , and r+(0 + {01}) = T if and only if z(a) = T. Also, z+({a,, . . ., a,} -+ 0) = T if and only if z(aJ = F for some i 5 m.

We let H , J , K , . . . be arbitrary sets of sequents. We let H I= 0 + D if and only if for every assignment z of truth-values to each a , either z+(@ + Y ) = F for some @ +Y in H or z+(O 3Q) = T. If z+(@ + Y ) = T for each @ +Y in H and if T + (0 -+ 0) = F , then z shall be a counterinstance to H j= 0 + SZ . We let H be satisfiable if and only if not H 1=0 -+ 0 .

A generalized syllogism, or G S L , is an ordered triple such that, for some CPo, . . ., Qn, and for some Yo,. . ., Yn, 0 containing at most one variable, the

1) This paper was written while the autor was Research Professor at the Miller Institute for Basic Research in Science at Berkeley. Work was also partly supported by Grant GP-1850 of the U. S. National Foundation. The autor should like to acknowledge both forms of support with gratitude.

2) See [7], [8], and [5]. The extension occurs in [el. For an excellent account of the motivation behind HERTZ’S theory and its relationship both to TARSKI’S axiomatic treatment of the notion of consequence and to GENTZEN’S calculi of sequents see pp. 1-5 and others of BERNAYS [l]. The article also contains useful references to the literature.

3 Ztschr. f . math. Log.

34 WILLIAM CRAIG

first term of the triple is the set {@, + Yo, . . . , QIB -< Yn} , 11 hose elements are called the minor premises of the GSL, the second term or major premise is Yo LJ - . u Yn - 0 , and the third term or conclusion is 0, u . . . u Qn --f 0 .

If the antecedent of its conclusion is empty, then a GSL shall be a generalized modus ponens, or GRIP. Thus a GMP is a GSL such that, for some Yo, . . . , ul,, 0 containing at most, one variable, the minor premises are 0 - Yo, . . . , 0 +. Y n , the major premise is Yo LJ - . w Yn +. 0 , and the conclusion is 0 + 0. Thus the major premise of a GMP uniquely determines the conclusion. If two GMP have the same major premise then their sets of minor premises differ a t most by 0 + 0 .

Sequents { R } --f {a} shall be logical axioms. An ordered pair (@ --f Y , @’ --f Y’) of sequents shall be a thinning in the antecedent, or TA, if and only if @ 2 @’ and Y = Y’, and a thinning in the succedeni, or TS, if and only if (I, = @’ and y’. The first term shall be the premise and the second term the conclusion of the TA or TS respectively. Since each succeclent of a sequent contains a t most one element, if the conclusion of a TS differs from the premise, then trhe succedent of the premise is empty.

We henceforth let S , T, . . . be finite nonempty sequences of sequents. The last term of a sequence S shall be its end sequent. We now turn to four notions of proof. Consider any H and S = (0, --f Do, . . . , 0, -+ 52,) . We let H be a set of GMP- hypotheses for S, or S a GMP-proof from H to the end sequent of S, if and only if, for each k (= m , either (i)* Ok +. fik is the conclusion of a GMP all premises of which are in (0, --f Q,, . . . , @k-- l --f Qk-l}, or (ii) @k -+ Q k is in H . We let H be a set of GSL-hypotheses for S, or S a GSL-proof frowb H to the end sequent of S, if and onIy if, for each k 5 m , eithcr (i) Ok Qk is the conclusion of a GSL all premises of which are in (0, -+ Q,, . . . , 0 k - I -+ Q k - - l } , or (ii) --f Q k is in H , or (iii) 0, 3 Qk is a logical axiom. We define {GSL, TA}-proof similarly, except that in addition to (i), (ii), and (iii) we also allow the alternative: (iv) Ok --f Q k is the conclusion of a TA whose premise is in (0, 3 no, . . . , Ok--l -> Q k - 1 ) . Finally we define {GSL, T}-proof similarly, except that in addition to (i), (ii), (iii), and (iv) we also allow the alternative: (v) Ok Q k is the conclusion of a TS whose premise is in (0, + SZ,, . . . , OkFi -+ Qkdl}.

61 + L? if and only if there is a GMP-proof from H to 0 -+ 52 , and we Iet IFsL, issL, TB, and ksI,, be defined similarly.

One verifies easily that if H hsL,T 0 -+ Q, then H I= 0 - D. Hence also if H I ~ ~ L , T ~ 0 ~ Q , H I ~ ~ ~ 0 ~ 5 2 ) o r H I ~ ~ P 0 - f 5 2 , t h e n H I = O - ~ Q . T l l u s e a c h of our four notions of provability is sound from a truth-functional point of view. We now show that each is also adequate to a certain extent.

To express the four corresponding notions of provability, we let H

Theorem 1. (a) I f HI=@ +Q, then H b B L , T O - 5 2 . (b) I f H I= 0 --f Q, then either H bsL 0’ --f 0 for some 0’ 2 0 and hence

23 bsL,TA 0 + 0 or else H k s L 0’ + 52 for s m e 0’2 0 and hence H ksL, T-3. 0 + 9. (c) I f HI=@ + Q ) then either H loMpO + 0 or H b M p O + Q .

MODUS PONENS AND DERIVATION EaOM HORN FORMULAS 35

Proof.1) We first prove (b). Note that if H /zsL Q0 -+ Yo, . . . , H lcyL@n -+ Vl,, and H bsLYo u - . - u Y, -+ Y , then H hsL@, u. . LJ @$% -+ Y. Now let any 0 3 Q be given. First, we observe that if @ -+ 0 is in H and if for each a in @ fhere is some 0,L 0 such that H l~~~ 0, -+ {a}, then there is some 0' 2 0 such that H bsL 0' --* 0. For if @ + 0 = 0 + 0 is in H , then evidently H laSL0' -* II for some 0' 2 0 . Now consider any @ + B = {ao, . . . , an} + 0 in H and assume that H l~~~ 0; --f {a,}, . . ., H hsL 0; 3 {a,} for some subsets 0;) . . .) 0; of 0. Then H bsL {ao,. . .) a,} +0, hence H ksL 0; u - u0; + 0 , and hence H hsL 0' -+ 0 for some 0' 2 0. Second, we observe similarly that if @ a {/3} is in H and if for each a in @ there is some 0,L 0 such that H 1~~~ 0, + {a), then there is some 0' 2 0 such that H bsL 0' --f {,9>.

Now assume that neither H hsL 0' --f 0 for some 0' 5 0 nor H bsL 0' + 52 for some 0' 2 0. Let z(a) = T if and only if H [ F ~ ~ 0' -+ {a} for some 0' 2 0. By our two observations, z+ (@ + Y) = T for each @ --jr F i n H . Also 0 tgsL (a} -+ {a}, so that if a is in 0 then there is some 0' 2 0 such that H bsL 0' -+ {a} and hence z(a) = T, while if a is in Q, then z(a) = F. Hence z + ( 0 + Q) = F . Hence z is a counterinstance to H I= 0 + 52. Contraposition then completes the proof of (13).

To an intuitionist, our argument so far only shows that if neither H hsLO' - M for some 0' 2 0 nor H bsL 0' -+ SZ for some 0' C - 0 , then the assumption that T is not a counterinstance to H j= 0 -+ 52 leads to a contradiction. There are however methods of deciding for finite H and for arbitrary 0 -to and a whether or not respectively H I= 0 -+ Q, H IxsL 0' --f 0 for some 0' 2 0 , or H bsL 0' + {a} for some 0' 2 0. For bsL this can be seen by noting that for constructing poss- ible proofs it is sufficient to consider sequents in which no variables occur other than those occurring in 0 --f {a} or in some sequent in H . Hence for finite H the argument for (b) also seems acceptable intuitionistically.

Condition (a) of theorem 1 is an immediate consequence of (b). One can also give a classical and, for finite H , intuitionistic proof similar to that for (b), except that one lets z (a) = T if and only if H bsL,T 0 -+ {a}. The proof of (c) is also similar, except that one lets z(a) = T if and only if H haIp 0 + {a). Q.E.D.

In Q 2 we shall also use the following lemma for the case where Q = 0. For finite H , our argument is acceptable intuitionistically.

Lemma 1. I f H bMpO +Q, then J hMpO -+SZ for some J 2 H such that no distinct sequents in J have the same succedent.

Proof. Assume that HbMP 0 --f Q. Then there is a finite set KO = (@, -+ Yo, . . . , QP --f Yp} 2 H such that KO IxMp 0 + Q. Let Ki+l = K , n - {Di + !Pi> if either K , n - {@< + !Pi} hMp 0 + !Pi or not Ki hMp 0 -+ Yi, and let Ki+, = K j

l) The proof slightly modifies and extends the proof given by HERTZ in [8] 3 3 and in [7]. Related considerations occur in GENTZEN [5], p. 335. For his analogue of (b), HERTZ [7] is able to avoid the use of TA by taking as logical axioms dl sequents @+ {a}, a E 0. When empty antecedents are admitted, this no longer works. For example, if 4j A Q = 0, then there is no proof without TA from {0 + Q} to 0 -+ Q.

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36 WILLIAM CRAIQ

otherwise, i p . Then Ki+, lzvp 0 + 0 whenever K i l~~~ 0 -+ 0, i 5 p . By induction, if Qi -+ Yi is in Ki+j+l, then Ki+j+l 1 ~ ~ ~ 0 -+ Y i and therefore if

=/= !Pi. Now let J = K,+,. Then J L H , J

The rest of 0 1 is not needed for 9 2. Let H be GMP-minimal for K if and only if H 0 --f Q for each 0 + D in K and there is no proper subset J of H such that J 0 -+ D, We now turn to problems which involve the condition that H be GMP-minimal for H.

Consider any binary relation R. The field of R shall be the set of those x such that for some y either x R y or y R x . We let RO be the set of pairs (2, x) such that x is in the field of R, and we let Rn+I be the set of pairs (x, z ) such that, for some y , xRlly and y R z . Then Rn is the set of pairs (x, y ) for which there is a

( z , , . . ., z,+J such that x = x , , y = z,+], and z ~ R ~ ~ + ~ for each k (= m . We let R be well-founded if and only if there is no infinite sequence (z, , zl, . . .) such that xk+ I Rz, for each k . We let R be finite-valued if and only if for each y the number of x, if any, such that xR y is finite. An argument similar to the usual proof of KONIG’S infinity lemma for trees shows: If R is well-founded and finite-valued, then U Rn is finite-valued.

We henceforth let E be that binary relation between sequents such that @+YE@’-+Y’ if and only if YnOf+O. We let H x H = { ( x , y ) : x E H ,

Theorem 2, Let ( I ) , . . ., (5) be the conuiitions listed below. Then the condition that H is GMP-minimal for H is equivalent to (l), (2), and (3) together, to (1) and (4) together, and to (5 ) , (2), and (3) together. Also (2) and (3) together imply (4), and (1) implies (5 ) .

+ is in Kifj+,, then 13 -+ D, and no two sequents in J have the same succedent. Q.E.D.

0 + D .1) Let H be the set of 0 -+ D such that H

U o<n<o

O s n < w

Y E H ) .

(1) For any @ + Y and CDr 3 !P’ in H , if Y = Y’, then @ = 0’. (2) For any Y -+ 0 in H , if Y $. 0 , then there are @, -+ Yo, . . . , @, + Yn in H

(3) E n ( H x H ) is well-founded. (4) I / o -+ sz is in H , then 0 -+ Q i s in i. 2)

(5 ) For any 0 --f SZ in H there i s at most one J 5 H such that J i s GMP-minimal

Proof. Assume that H is GMP-minimal for H. If @ + @’ and if @ + Y and @’ - Y were both in H , then by the proof of lemma 1, H n - {@ Y} 1 ~ ~ ~ 0 -+ Q for each 0 + Q in G, so that H would not be GMP-minimal for E. Hence (1). Now let Y‘ + Yff be in H and let Y‘ $. 0 , so that Y’ = {a,, . . . , a,} for some a,, . . . , a,.

such that Y = Yo u . ’ v Ym.

for (0 3 D} .”)

l) GSL-minimal for K can be defined similarly. However we have no resultfa for this notion. 2, Evidently for any H , if 0 - 9 is in I?, then some 0 3 R is in H . s, Evidently for any H and any 0 + 0 in 5 there is a t least one finite J 5 H such that J

is GMP-minimal for {0 -+ Q}.

MODUS PONENS AND DERIVATION FROM HORN FORMULAS 37

Suppose that for some i 5 n there is no !Pi such that !Pi -+ {a i } is in H . Then 0 + {a i } is not in k. Then Y’ + Y” is not the major premise of any GMP whosc minor premises are all in H . Then H n - {Y’ +- Y”) l~~~ 0 -+ 0 for each 0 --f Q in H so that H would not be GMP-minimal for G. Hence ( 2 ) . Now suppose that E n ( H x H ) is not well-founded. Then there is an infinite sequence (@, --f Yo, 0 1 -+PI, . . .> such that, for each k , @, + Y, E H and @,+, + Y,+,ECD~ + Y,. Assume first that there is some k such that H n - (Ok + Y,} (zwp 0 -+ Yk . Then H n - {@& -+ Yk} Imp 0 -+ Q for each 0 --f Q in fi. Then H is not GMP-minimal for i. Assume now that there is no k such that H n - {Qi(. -+ Y,} lalp 0 + Yk. Consider any j and any GMP-proof from H to 0 -+ !Pi. Then there is some k and an initial segment (0, --f Q,, . . . , 0, + Q m ) of the proof such that 0, --f Q,,L =

= 0 -+ Y, and such that no 0 -+ !Pi is in (0, -+ Go, . . . , ern-, + l2,-,}. Since H n - {Dk --f Yk} j~,, 0 + Yk does not hold, there is a GMP each of whose premises is in (0, -+ Q,, . . . , Omdl -+ Qm-,} and among whose premises is @ a -+ Y/, . Since Q k + , -+ !Pk+lE@k -+ Yk and hence Y k + , n @, =+ 0, @, + Y, is the major premise of this GMP and 0 + Yk+, is among the minor premises. This is impossible, since no 0 + !Pi is in (0, +Go, . . ., Om--, + Q,-,}. Thus there is no GMP-proof from H to any 0 -+ Y,. Hence H n - +- Yo, Q1 .+ !PI, . . .} lzvp 0 -+ Q for each 0 +9 in I?. Hence H is not GMP-minimal for i?. Hence (3). This concludes the proof that (l), (2), and (3) are necessary conditions for H to be GMP-minimal for g.

Assume ( 2 ) and (3). Let J be the set of those 0 -+ l2 in H such that 0 + Q is in H. If 0 -+ Q is in H and 0 = 0 , then 0 + Q is in J . Now consider any 0 + Q in H such that 0 $. 0 and assume as inductive hypothesis that every 0‘ + 9’ such that (0‘ + Q’, 0 + Q) E E n ( H x H ) is in J . By (2), there are @, --z Yo, . . . , @% + Yn in H such that Yo u . u Yn = 0 and such that Yo $- 0, . . . , Y,+0. Then ( @ i + y / i , O - + Q ) i s i n E n ( H x H ) for each i S n . Hence by our assumption Qi -+ !Pi is in J and hence 0 + YB is in H for each i n . Then 0 -+ Q is the conclusion of a GMP whose minor premises are in H and whose major premise is in H . Hence 0 + 9 is in H and therefore 0 + Q is in J . Hence for any 0 .+ Q in II , whether or not 0 + 0 , the inductive hypothesis implies that 0 - 9 is in J . From (3) i t follows that J = H . Hence (4).

Assume (1). First, consider any 0 -+ Q in H for which there is a GMP-proof from H of length 1. Then 0 + Q is in H . Then (0 + L2} C J holds for any J 2 H such that J l ~ ~ , 0 + L?. Moreover (0 Q} Icxfp 0 --> 9. Hence (0 --t f-’) is the unique J 2 H such that J is GMP-minimal for 0 + 9. Now assume as inductive hypothesis that (5) holds for all those 0 -> Q’ in fi for which there is a GMP-proof from H of length less than n + 1, and consider any 0 .+ Q in H for which the shortest GMP-proof from H is of length n + 1. By ( l ) , there is only one 0 --f 9 in H . Also 0 =+ 0 so that 0 = {a,, . . ., cum} for some cue, . . . , a,. Thus 0 -+ Q is the conclusion of a GMP whose major premise is 0 + 9 and whose minor premises are 0 + {a,}, . . . , 0 --f {a,}. By the inductive hypothesis, there are unique

-

- -

38 WILLIAM CRAIG

J o L H , . . ., J,,, 2 H such that J i is GMP-minimal for 0 --f {ai} , i 5 m . Hence {@ +a} v J , u. . . u J n E L J holds for any J s I1 such that J 1 ~ ~ ~ 0 +L?. Moreover (0 ~ 9 } v J , v ~ ~ ~ v J , ~ ~ V , 0 - t 9 . Hence ( 0 + Q } w J O w . . . u J , is the unique J c= H such that J is GMP-minimal for 0 + L?. By induction therefore (5) .

Assume (1) and (4). Consider any 0 --r f2 in H . By (4), 0 + D is in H . By (l) , there is no 0’ =+ 0 such that 0’ - 9 is in H . Hence H n - (0 + Q} Izxp 0 --f Q fails. Hence t>here is no proper subset J of H such that J IF^^ 0 --f Q’ for each 1.) +9‘ in i. Hence H is GMP-minimal for fi.

With one exception all implications asserted by the theorem now follow immed- iately from those already proved. It only remains to show that ( 5 ) , ( 2 ) , and (3) together imply that H is GMP-minimal for &. For this purpose we now assume that (2) and (3) hold, and that (1) fails. It will he sufficient to show that (5) fails, since (l), (2), and (3) together imply that H is GMP-minimal for Z?. Since (1) fails, there are 0 - f2 and 0’ --f f*, in H such that 0 $; 0’. Let J , = (0 --f Q). Assume now that J , has been formed and is a finite subset of H . If there are no Y + Y’ in J , such that Y =+ 0 , let Jm+l = 0 . If there are Y + Y‘ in J , such that Y += 0 , then let Jmil be any finite subset of H such that, if !P --+ Y‘ is in J , and Y =+ 0 , then !P = !Po v . . . u Yn for some a, --f Yo, . . . , @ n + ul, in Jmil and such that, if @ --f @‘ is in J,+l, then @ ’ A Y =+ 0 for some Y --f Y‘ in J,. By (2) and the assumption that J , is a finite subset of H , there is such a finite subset Jm+l of H . There is some k such that Jk+T = 0 and hence such that each Jk+m+t + 0, for otherwise a construction similar to that for KONIC’S lemma would yield an infinite sequence (0, - Yo, Q1 -?. Y l , . . .) such that dim + Y, E J , and Ym+, n @, += 0 for each tr), which would be incompatible with (3). By induction on k - m , J , V . , . U J , I ~ ~ , ~ + ~ ; ~ ’ for each 0 ‘ ~ f 2 ‘ in J k u V . ~ u J , . Hence J , w . . . u J , b&Ip B - 9. By (3), there is no 0” + 9 in J J k w . . u J , such that 0” + 0. Hence c-) - R is in each J J,. w . + w J , such that J kxp 0 -+ 9. Finally, sincc J , u . v J , is finite and J , w - 9 v J , kHp 0 + Q, there is some J 2 J , u . . w J , , such that J is GMP-minimal for (0 --f 9}. Similarly, there is a subset J’ of H such that 0’ + 9 is in J‘, such that no other 0” + f2 is in J’, and such that J’ is GMP-minimal for (0 - Q}. Henre (5) fails. Hence (2), (3), antl (5) together imply (1 ) .1 ) Q.E.D.

-4 (relational) structure (of similarity type ( 2 ) ) shall be any ( A , R) such that R 2 A x A =+= 0. In particular, for each nonempty H , ( H , E n ( H x H ) ) is a relational structure. To different structures ( H , E n ( H x H ) ) there correspond different H . For each mapping or function f and each set A , let f r A be the restriction of f to the intersection of A with its domain, so that y = ( f P A ) (2) if and only if y = f (.c) antl x t A . We now give a simple 1-1 correspondence between the set MS of structures (If, E P (H x H ) ) such that His GMP-minimal for fi and H is satisfiable and another set FW of structures which has a definition that is more transparent.

Let H consist of the three sequents 0 + {a), {a) -+ { p } , and (@) -+ {m} where a =# 8. Then (2), (4), and (5 ) hold but (1) fails.

YODUS PONENS AND DERIVATION FROM HORN FORMULAS 39

Theorem 3. Let MS be the set of structures ( H , E n ( H x H ) ) such that H i s GMP-minimal for H and H i s satisfiable. Let FW be the set of structures <A, R ) such that A is a set of variables and such that R is finite-valued and well-founded. For each finite-valued binary relation R between variables, let 13f-R be the mapping which assigns to each B the sequent {a : aRp} + { B } . Let sv be the mapping which assigns to each CD -+ {/I} the variable p . Then for any ( A , R) in FW, the image of ( A , R) under pr, r A is a structure ( H , E n ( H x H ) ) in M S and sv I’ H is the inverse of prR A . And for any ( H , E n (H x H ) ) in M S , the image of ( H , E n ( H x H ) ) under sv I’ H is a structure <A, R) in FW and prB r A is the inverse of sv r H .

Proof. Let any ( A , R ) in FW be given. Evidently, prR [’ A is a 1-1 mapping of A onto a nonempty set H of sequents with nonempty succedents. Evidently, B + 0 is not in k so that H is satisfiable, su H is the inverse of p R j’ A , and no distinct sequents in H have the same succedent. Also, if Y + 0 is in H and Y += 0 , then Y = { P o , . . .,B,} for some P o , . . .,B, in A , and hence there are @, -+ {Po}, . . ., CD, -+ {p,} in H . Hence (2) of theorem 2 holds. For any CD + {a} and @’ -+ {a’} in H , the conditions that CD -+ {a} E 0’ --f {a’}, that a E CD‘, and that aRa‘ are equivalent. Hence PTR r A maps R onto E n ( H x H ) . Since R is well-founded, so is E n ( H x H ) . Thus, (l), (2), and (3) of theorem 2 hold, and therefore ( H ) E n ( H x H ) ) is in MS.

Now let any ( H , E n ( H x H ) ) in MS be given. Since H is satisfiable, 0 --f 0 is not in H and hence, by (4), no CD -+ Y in H has an empty succedent. Hence H is included in the domain of s v , and sv H is a mapping of H onto a nonempty set A of variables. By (l) , sv H is 1-1. Let R be the image of E n ( H x H ) under su r H . By (3), R is well-founded. By (l) , R is finite-valued. Hence ( A , R ) is in FW. By (l), if 0 --f { B } is in H , then there is no a such that aRP, and hence l)rR I’ A (tw [‘ H(O + {B} ) ) = 0 +B. By (2), if {a,, . . ., a,} --t {B } is in H , then there are 0, -+ {a,}) . . . , -+ {a,} in H , hence, by (l), aRB if and only if a E {a,) . . . , u ~ } ) and hence prR Hence prR r A is the inverse of sv r H . Q.E.D.

Theorem 4. Por any H there i s some J 5 H such that J is GMP-minimal for H . Moreover, J can be chosen so that H n fi 2 J.

Proof. Let 5 be a limit ordinal, let = (0 -+ Yo, . . .) 0 -+ Y E , . . .}e<c and for each 5 < C let K , be the set of 0 -+ Y, in l? such that rj < f . Let J, = 0 and, if 5 is a limit ordinal, let J , = U J,. Now suppose 5 = 7 + 1 for some 7 . Then

there is a finite set Jk 2 H n - J , , possibly empty, such that J , u J; IF^^ 0 -+ Y, and such that if J: is a proper subset of J i ) then not J , u JY l~~~ G1 + Y,,. Then we take that Jh of this kind which is earliest according to some preassigned well- ordering of the finite subsets of H , and we let J 6 = J, u J i .

and assume that K , 5 j , , J, 2 H , and J , is GMP-minimal for y,, for each 7 < 5 . Let 5 be a limit ordinal. Then K E S j , and J , L H . By theorem 2 , no distinct sequents in J, have the same succedent, q < 6. Hence no distinct sequents in J ,

A ( s v r H({a , , . . ., a,} -+ { B } ) ) = {a,, . . ., a,} -+ { P } .

Il<e

Evidently, I<, 2 .?,, J, H , and J , is GMP-minimal for J,. Now let 0 < 5 2

40 WILLIAM CRAIG

have the same succedent. Also, by theorem 2, if 0 --f 52 is in J,, then 0 + 9 is in jq, q < 4 . Hence if 0 3 !2 is in J , , then 0 + 9 is in j t . Hence, by theorem 2, J, is GMP-minimal for f, , Now let = 7 + 1 for some q . Then K , 2 i, and J, 2 H . By our choice of Jh, no distinct sequents in Jh have the same succedent. Since c) + 0 is in J , for each 0 + !2 in J , , Ji also contains no sequent whose succedent is the same as that of a sequent in J,. Hence no two distinct sequents in J, = J, w J; have the same succedent. By our choice of J,!,, if 0 -> 0 is in J;, then J , v

~ J ~ I ~ ~ ~ 0 + ! 2 . H e n c e i f O - S L , i s i n J E = J , w J ; , t h e n O + 9 i s i n J G . H e n c e , by theorem 2, J , is GMP-minimal for j c . It follows by induction that K , 2 j,, J , g H , and J, is GMP-minimal for j t . For 6 = (, K , = H so that k2 ic 2 I?. Finally, the last clause of theorem 4 follows from the fact that if {0 + !Po, . . . , 0 -+YE,. . .}t<So is H n H and 5, is a limit ordinal, then Jco = H n l?. Q.E.D.

According to theorem 4, some J 2 H which is GMP-minimal for k can be chosen such that H n fi 2 J and hence such that J n - & is a subset of K - H whenever K 2 H and I? = H. The situation for J n H is different, as is shown by thc following example related to one in HERTZ [ 8 ] , page 469. Let ao, al, . . . be distinct and let H consist of the sequents 0 + {a i } and {ai+k+l} - {az } . (Note that if H ksL d, -+ Y then @ + !P is in H and @ contains a t most one variable.) Consider an arbitrary J 2 H such that = G. Then J contains infinitely many 0 + { a i } . Choose any 0 + {cx~} in J. There is some 0 --f {a i+k+l} in J . Lct

K = ( J - (0 -+ {ail-}) {@,+k+l} --> {a,>> *

Then K 2 H and K = k, yet K n H is a proper subset of J A G . For the same H , a similar argument shows that for any J 2 H v H , if J u J =

= H v h, then there is some proper subset K of J such that K w K = H w H. Thus no subset of H u fi is GMP-minimal for I3 u H . I )

IJ ( E n ( H x H))" . Then dH is always reflexive and

transitive. If E n ( H x H ) is well-founded, then & is also antisymmetric and hence a partial ordering of H . For sets H that are GMP-minimal for i?, we extend this partial ordering of H to a partial ordering of H v H by making each 0 3 YJ in H n - H the immediate successor of that unique sequent in H whose succedent is Y. More precisely, for any @j0 Yo and + Yl in H LJ H we let Q0 --f Yo S U v ~ --> !PI if and only i f , for some @; and @;, @A + 'Po dH @; + !PI and either Yo $; !PI or @;

Theorem 5. II is QMP-minimal for (0 +L?} if and only if H is GMP-minimal for fi and

- -

For any H , let & be O s n < w

- -

-

@A.

I

(6) If d, 3 Y is any sequent in H w H , then @ -+ ' Y 5 H v g 0 +!2.

l ) For a negative and s positive result concerning a related problem see parts I1 and I11 of GENTZYN [5].


Proof. Assume that H is GMP-minimal for (0 + a}. Then evidently (1) and (4) of theorem 2 hold so that H is GMP-minimal for H. Then, by (1) and (3), the relation ia is finite-valued. Since 0 3 Q is in H, some 0 + Q is in H . Then the set J = {@ + !P: @ -+ Y dH 0 -+ Q} is finite. By induction, 0 + !P is in 5 for each @ + Y in J . Hence J = H , since otherwise H is not GMP-minimal for (0 -+ Q}. Then @ -+YsHdgO+Q for each @ + ! P i n H U H .

Now assume that H is GMP-minimal for H and that GJ -+ Y s H , , g 0 + Q for each @ + Y in H us. Then, for some 0 , @ + Y -& 0 -+ Q for each @ +!P in H . Refining that part of the proof of theorem 2 which shows that ( 5 ) is a necessary condition for H to be GMP-minimal for H one shows by induction that for each 0' -+ 8' in H the set J' = {@ + Y: GJ -+ Y & 0' + Q'} is the unique subset of H which is GMP-minimal for (0 -+ a'}. For the case where 0' --f Q' is 0 -+ Q this implies that {@ --t Y : @ -> !€' & 0 -+ Q} = H is GMP-minimal for (0 +Q}. Q.E.D.

Theorem 6. Let H be GMP-minimal for (0 --f Q} . Then (0, + Q,, . . . ,Om -+ Q,> is a GMP-proof from H to 0 + 9 if and only if (0, + Q,, . . . , 0, + Q,} = H u H , @ , + Q , = @ + Q , a n d , i f O j + Q n i i s n o t i n H , j s m , Oi+Qi+@,-+Q,,and Oi -+ Qi &vg Oi +Qj , then Oi --f Qi is in (0, + Q,, . . . ,

-

--f Qj-l}.

Proof. When 0 + 0, then the theorem evidently holds. Now let 0 = 0 and let H be GMP-minimal for (0 -+ Q}. Assume first that (0, -+ Q,, . . . , 0, -+ a,) is a GMP-proof from H to 0 + 9 . Evidently, (0, -+ So, . . . , 0, -+ Q,} H u H . If H is not a subset of (0, -+ Q,, . . . , 0, -+ a,}, then (0, -+ Q,, . . . , 0, + 12,) is a GMP-proof from a proper subset of I€ to 0 -+ Q. If 0 -+ 9' E 5 n - H is not in (0, + Q,, . . . , O m -+ 52,) , then, for some GJ', @' + Q' is in H and a subsequence of (0, -+ Q,, . . ., 0, -+ Q,) not containing GJ' + Q' is a GMP-proof from H n - {a' --f Q'} to 0 -+ Q. Since H is GMP-minimal for H, it follows that H u H ~ { 0 , + ~ , , . . . , 0, +Qm}. Nowsuppose t h a t j l m a n d O j + Q f i s n o t in H so that O j -+ Q, is the conclusion of a GMP whose major premise is in H and whose minor premises are in H uk. Then all premises of this GMP are in (0, + Q,, . . . , + Qj-,>. It follows by induction on j that, if Oi -+ Qj is not in H , Oi -+ Qi $; Oi +aj , and Oi -+ Qi S H v g --f Q f , then Oi -+ Qi is in

Now assume that (0, + Q,, . . . ,Om -+ a,} = H v I? and that, if 0, -+Qj is not in H , Oi +Qi += O j +Qi, j 2 m, and Oi - + Q i s H d g 0, -+Q,,thenOj -+ai is in (0, -+ Q,, . . . , @j-l --z Qj-l}. Then, by induction on j, (0, --f Go, . . . , O f + Q j ) is a GMP-proof from H to Oj -+ Qi, j rn. If 0, -+ Q, = 0 + Q, it follows that <@,+Q,, . . . , @,+Q,) i saGMP-prooff romHto@+Q. Q.E.D.

-

(0, - + . n o , . . . ) 0,-1 +qL1}.

We let X be GMP-reduced (GSL-reduced) with respect to H if and only if S is a GMP-proof (GSL-proof) from H and there is no proper subsequence 8' of 8 which

12 WTLLIAM CRAIU

is a GMP-proof (GSL-proof) from H to the end sequent of S. Evidently. given any GMP-proof (GSL-proof) S from H one can find a t least one subsequence S' of S having the same end sequent as S such that S' is GMP-reduced (GSL-reduced) with respect to H . For example, one may form a desired S' by starting with the next to last term of S and, proceeding toward the first term, deleting all those terms, if any, whose deletion still leaves a GMP-proof (GSL-proof) from H .

A GMP (GSL) shall be in a sequence S if and only if there is a subsequence of S whose last term is the conclusion and whose other terms are the premises of the GMP (GSL).

Theorem 7. Let S = {O, -+ Q,, . . . , @, +a,> be a GMP-proof from H , and let (7), (8), (9), and (10) be the conditions listed 6elow.l) Then the condition that (0, + SL,, . . . , 0, - SL,,) is GMP-reduced with respect to H , the condition that (7) a d that H n (0, -' Q,, . . . , 0, -j Om} is GMP-minimal for (0, -+ Q",,}, and the condition that (7) , (8), (9), and (10) hold are

(7) S has no repetitions.

( 8 ) Each term of 8, except the last, is a premise of a GMP in S. (9) Each term of S is the conclusion of at most one GMP in S. (10) No conclusion of a GMP in 8 belongs to H . Proof. Let S = (0, + Q,, . . . , 0, - Q f j L ) be a GMP-proof from H . Assume

first that S is GMP-reduced with respect to H . Then evidently (7) and (8) hold. Since each major premise 0 --f 0 of a GMP is identical with its conclusion. it follows that

(1 1) Each major premise of a GMP in S has a nonempty antecedent. Since 0 -+ 0 is superfluous as a minor premise of a GMP when the major pre-

(12) Each minor premise of a GMP in S has a nonempty succedent.

mise has a nonempty antecedent, and since S is GMP-reduced it follows that

Hence

(13) No term of S is the major premise of more than one GMP in S.

Now suppose that for some k 5 m , Ok -+ Q k = 0 --f Q k is the conclusion of two distinct GMP in (0, 3 Q,, . . . , On -+ Qk). Let 0, + Qi and 0 += Q, be the re- spective major premise, i 5 j < k. Then, by (12), Oi + Qi and 0, + 52, are also distinct. Hence, by (13), either one of these can be deleted from S so that S is not GMP-reduced. Hence (9) holds. From (13) it also follows that if a conclusion of a GMP

I ) Conditions (7), (8), and (9) involve only 8 and thus are internal conditions on S, wherem (10) is external, involving also H . In general, a proof can be analyzed into a sequence of in- ferences in more than one way. If (7), (9), and (10) hold then the analysis into a sequence of GMP's is unique.

a) Thus if S satisfies (7), (8), and (9) and if J is the set of terms of 8 which are not the conclusion of a GMP in s, then S simultaneously establishes derivsbility of @, -+ Sam from J end nonderivability from proper subsets of J .


in S belonged to H , then the major premise of that GMP could be deleted from S so that S would not be GMP-reduced with respect to H . Hence (10) holds.

Conversely, assume now that (7), (8), (9), and (20) hold. By (7), one cannot delete the last term and still have a GMP-proof to 0, + 52,. Now consider any other term. By (8) it is a premise of some GMP in S. By (9) the conclusion of this GMP is not the conclusion of any other GMP in S. By (lo), the conclusion of this GMP is not in H . Hence the term cannot be deleted from S. Thus no term can be deleted from S , and hence S is GMP-reduced with respect to H .

Assume now again that S is GMP-reduced with respect to H . Let J = H n n (0, -+ Do, . . . ,Om -+ 9,) . If O,, $. 0, then evidently J = (0, --f Qn,} is GMP- minimal for (0, -> Q,}. Now assume that 0, = 0. By (10) and an argument similar to that which showed that (9) holds, no distinct sequents in J have the same succedent. Also, if Oi -+Qi is in J then 0 ->Qi is a term of S and hence is in 3, since otherwise 0, -+ Qi could be deleted from S. Hence, by theorem 2, J is GMP- minimal for i. Evidently 0, + 9, is in J u J . Now suppose there is a @ -+ Y in J u j such that not @ - .YyJvj 0, +9,. Then there is a @ +!Pin J u J such that Y $. 9, and such that if @’I + Y” $; @ -+ Y, then not @ -+ Y 5 ,,j @‘I -+ 4u“. Then deletion from S of CD + Y still leaves a GMP-proof from H to 0, --f 52,. Then S is not GMP-reduced with respect to H . It follows that @ --f Y &,j 0, + Q, for each CD -+ Y in J u J , and hence by theorem 5 that J is GMP-minimal for (0, -+ Qn,}.

Finally assume that (7) but that S is not GMP-reduced with respect to H . Then deletion of some @k -+ Q k from S still leaves a GMP-proof from H to 0, -+ 52,. If 0, + Qk is not in H , then it is the conclusion of a GMP in (0, -+ Q,, . . . , 0, -+ 52,). Let Oi --f Qj = Oi -+ 9 k be the major premise of this GMP. Since S has no repetitions, 0, =+ (3 and On: -+ Q k is the only conclusion of a GMP in S having Oj -+ Qi as one of its premises. Hence deletion of Oi -+ Qi along with Gk + Qk from S still leaves a GMP-proof from H to 0, + 52,. If Oi + 52, is not in H , we can repeat the argument. Thus eventually one obtains some Oi -+ 52, in H whose deletion, along with deletion of none or more later terms, from S still leaves a GMP- proof S’ from H to 0, --f Q,. Since S is without repetitions, the terms of S’ which are in H form a proper subset of the set H n (0, --f Q,, . . . , 0, -+ Q,} . Hence this set is not GMP-minimal for {Om 4 a,}. This shows that (7) together with the condition that H n (0, -+ Q,, . . . , 0, --z SZ,} is GMP-minimal for (0, -+ 52,) implies that (0, -+ Q,, . . . , 0, -+ Q,,J is GMP-reduced with respect to H . Q.E.D.

Theorem 7 has no obvious anatogues for GSL. Let S, be the sequence {{a,} -+ {a1),

the first three terms of S, , and let a,, . . . , a3 be distinct. Then H , is GSL-minimal for {{a,} -+ { o ( ~ ) } but S, is not GSL-reduced with respect to H , , since either {a,} --f {a2} or {al} + {as} can be deleted. Let S, be the rather long sequence given below, let H , consist of the fifth term {a,} -+ {a2} of 8, and of those nine terms of 8, which are not the conclusion of a GSL in S,, and let a,, a:, . . . , a, be distinct. Then one can verify that S, is GSL-reduced with respect to H , . However, H , is

-

-+ {.2>, {.2> + {a3} 2 {a,) -+ {.%} 7 {a,} -+ (a3) > {a,> -f (.,>> 9 let H , consist of

44 WILLIAM CRAIG

not GSL-minimal for {{ao, a;, a;} --f { M . ~ } } since S2 is also a GSL-proof from H , n - {{a,,} -+ {a2}} to the end sequent. Moreover {ao} -+ {a,} is the conclusion of two distinct GSL in S, and belongs to H , , so that the analogues of (9) and (10) also fail. Also note that the analogue of (13) fails, since {a,} + (017) is the major premise of the GSL in S, whose minor premise is (a;} --f {a2} and of the GSL in S, whose minor premise is {a:} -+ {a,}. S, is the sequence ((a,,} + {a:}, {ao} + {a:’}, i.2 + {%>, {a:’> - {.2> (M.0) -+ {.2> > {a,> + (4) 9 {a;} + {a;} I (47 + {a;: {a:> -> {a;} t {a:’} - > {a+:) 9 (a,,} - {a;}, { N O ) -+ (4‘) 9 { & 2 , 4 , 4 ’ } {aJ, (010) + {az} 9

(a3 > a;} + {ad} > {%, 4 , a:’} -> {a4)) *

Although arbitrary GSL-reduced CSL-proofs do not always have properties analogous to ( 7 ) , . . . , (13), there is under suitable conditions a t least one GSL-proof having these and related desirable properties. We now state this more precisely, improving theorem 1 (b) and also going somewhat beyond 0 5 of [S].

A sequent 0 --f yl shall be ( H , @)-restricted if and only if @ 2 0 and there is no proper subset 0’ of (1) such that HI=@’ + Y. A GSL shall be ( H , @)-restricted if and only if the conclusion and each minor premise are ( H , @)-restricted. A sequence S shall be (H, @)-restricted if and only if each GSL in S is ( H , @)-restricted.

Theorem 8. Assume that HI= 0 + 9 , that there is no proper subset H’ of H such that H’l= 0 -+ 9, that thew is 770 proper subset 0‘ of 0 such that H I= 0‘ --f 9, and that if HI= 0 -+ 0 thcn Q = 0 . Then there is a GSL-proof S” from H to 0 -+ 9 which satisfies the following c0nditions.l)

(7)” S“ has no repetitions. (8)” Each term of S”, except the last, is a premise of a GSL in 8”. (9)” Given any Y, there is at most one GSL in S” whose conclusion has Y as succed-

(10)” No conclusioiz. of a GSL in S” i s a logical axiom or belongs to H . (11)” @ n - 0 + 0 for each major premise r f ,

(12)” Y + 0 for each minor premise @ + y/ of a GSL in S”.

(13)” N o sequent is the major premise of more than one GSL in S”.

(14)” Each major premise of a GSL in S” belongs to H .

(15)” No major premise of a GSL in S” is also a minor premise or the conclusion

( 1 6)’’ No major premise of a GSL in 8“ is a logical axiom.

(17)” Either S” = (0 3 0) or logical axioms occur in Sft as minor premises only.

(18)” If a occurs in a logical axiom orcurrilzg i n X”, then a E 0. (10)” If neither (1) + Y nor (53’ + Y‘ is the conclusion of a GSL in X“, if both

(20)” S” i s ( I ] , @)-restricted.

l) As the proof will show, conditions (ls)”, . . ., (19)” can be derived from the others.

ent.

Y of a GSL in St.

of a GSL in S”.

occzw in S”, and if @ =$ @ I , then Y $. Y’.


Proof. Let H and 0 -+ 52 be as stated, let J = (0 3 {a}: a E O}, and let K = H u J . Then Kl=0-+52. Then, by theorem 1, either K l ~ ~ ~ 0 - + 5 2 or KkMpO+O. Now if KbMPO-+O, then H I = O - + B so that Q=O.Hence , in either case, K loMp 0 -+ Q. There is no proper subset H' of H such that HI u J 1~~~ 0 -+ Q , since otherwise H' I= 0 + Q . Also there is no proper subset J' of J such that H v J' la^=^ 0 3 SZ, since otherwise H I= 0' -+ Q for the proper subset 9' = ( a : 0 --f {a} € J'} of 0. Hence K = H v J is GMP-minimal for (0 -+ Q}. Moreover, H A J = 0 , since if 0 -+ {a} were in H and in J then H I= 0 n - {a} -+ SZ .

There is a sequence S = (0, -+ Q,, . . . , 0, -+ Q,) without repetitions which is a GMP-proof from K to 0 + Q. By theorem 7 and its proof, S is GMP-reduced with respect to K and satisfies conditions (8) through (13), with K in place of H in (10). By theorem 6, (0, -+ Q,, . . . ,Om -+ Q,%} = K v K . Now consider any j 2 m. By theorem 2, there is a unique Dj such that Qi + 9, is in K. Let Ki = {@ + !P: @ + -+Qf}, Hi = H n K j , and J i = J n K i . If O j $: 0 let @ - @ . , a n d i f 0 . = 0 l e t 3 - 7 7 O ~ = { a : O - + { a } E J i } . LetS '=(@;+Q, , ...,

Consider any j 5 m such that Oj = 0 . We want to show that 0; -+ Q . is ( H , 0)- restricted. Clearly, 0; 2 0. Now suppose there is a proper subset 0t of 0; 2 0 such that H I= 0; + Qi . Let J" = (0 -+ {a} : a E 0;). Then H u J" (=0 -+ 52$. Then, by theorem 1, either H u J" bMp 0 -+ 0 or H v J" l~~~ 0 -+ Qj. Suppose that H w J" IF^^ 0 -+ 0. Then H I= 0; + 0 and hence H I= 07 -+ 52, which contradicts our assumption that there is no proper subset 0" of 0 such that HI= 0" -+ Q. Suppose now that H w J" IcMp 0 -+ Qi. Then, for some H" 5 H , H" v J" is GMP-minimal for (0 -+ Qj}. Since H n J = 0 and J" is a proper subset of J , 2 J , K j = Hi u Ji is not a subset of H" u J" K. This contradicts the assertion near the end of the proof of theorem 5 according to which K , is the unique subset of K which is GMP-minimal for (0 -+ Q,} . Hence in no case is there a proper subset 0: of 0; such that H I= 0; + Q;. Hence 0; + 52 is ( H , @)-restricted.

Next, we want to show that S' = (06 3 Go, . . . , 0; -+ 9,) is a GSL-proof from H. Consider any j (= m. We distinguish four cases.

Case (i): O j $. 0. Then 0; -+Q7 = Oi + Q j is in H. Case (ii): Oi = 0 and Of -+ Qj is in H . Since H 2 K , Qi = Oj = 0. Hence

@ -+ Y LK Qj + Qi fails for any CD -+ !Pother than Q j + O f . Therefore 0; = 0 , and hence 0; -+Q, = Of -+Qj is in H .

Case (iii): O j = 0 and Oj +Qn, is in J . ThenOi+Qi=O +{a} for some a € 0. Since J C - K , Qj = Oj = 0 . Hence @ -+ Y sK Qi -+ Qj fails for any @ -+ Y other than 0 -+ {a}. Therefore 9; = {a}, and 0; 3 Q j is the logical axiom {a} + {a}.

Case (iv): Oj = 0 and Of -> Qi is neither in H nor in J . Then 0, -+ Q? is the conclusion of a GMP in 8. By (9), there is only one such GMP. Then for some positive

I

o:, -+ Qn,). 1)

l) or ZKL.sr induces a partial ordering of {@; + Q,, . . . ,OL --f D,} which is extended to a total ordering in Srr below.

46 WILLIAM CRAIG

integer q ( j ) and for some mapping h of the pairs <j, 0)) . . . , (j, q ( j ) + 1) into

@ h ( j , O ) --f Q h ( j , O ) , * . . Y @ h ( j , q ( j ) ) + @ h ( j , q ( j ) )

( 0 , . . . , j - 1)

are the minor premises and q ( j ) + l ) --f Q h ( j , q ( j ) + l ) is the major premise of the GMP. Now consider any a in 0;. Then a E 0 and 0 .+ {a} d K @ j + a, = @h( j ,q ( j )+ l ) +

+Qh(j ,n( j )+l) . Since @ h ( j , q ( j ) + l ) = Q h ( j , o ) u' + ~ . % ( ~ , ~ ( j ) ) , there is Some k I; q ( j ) such that 0 --f {a} & @ h ( j , k) + Q h ( j , k ) and hence a is in @ A ( j , k,. Conversely, consider any k s q ( j ) and any a in @ i ( j , k ) . Then a t 0 and 0 +{a} & @ h ( j , k ) +Qh(j,k). Then also0 -+ {a} 59 @ h ( j , q ( j ) + l ) --f Q n ( j , q ( j ) + l ) = @ j + QiSince, by (1% Q h ( j , k ) =k + 0 and hence Q h ( j , k ) A q ( j ) + l ) $: 0. Then a is in 0;. Thus 0; = &(j, 0) u - * . w

v @i, ( j ,g ( j ) ) . Hence O i ( j , o ) + O h ( j , o ) , . . . , Oi( j ,g( j ) ) fjh(j,p(j))form the minor prenzises and f&, q(j)+i) -+ Q h ( j , e(j)+l) the major premise of a GSL whose conclusion is

Thus S' is a GSL proof from H to 0; -+ Qvt. Now Q,,, = Q , 0; 2 0 , and there is no proper subset 0' of 0 such that H I= 0' + Q. Hence 0; = 0.

Consider any j 5 m such that case (iv) holds. We now distinguish two subcases. Case (iv) (a): @ h ( j , q ( j ) + l ) $ 0 . Since 0; 2 0 , @i(j,q(j)+l) + Q h ( j , g ( j ) + l ) =I= 0; +sL'_i. Since @ h ( j , q ( j ) + l ) -+ Qh(j,,(j)+l) is in H and K = H u J is GMP-minimal for K , 63; i Qi is not in H . Since moreover H n J = 0, Qi = Q h ( j , q ( j ) + I ) differs from each 0 -+ {a} in J and hence contains no a E 0 . Since 0; 5; 0 , therefore ni n 0; = 0 and hence not @;r Qi. A fortiori, 0; + Qj is not a logical axiom. Case (iv) (b): Oh( j ,q ( j )+ l ) 5 0. Suppose first that a € @ h ( j , q ( j ) + i ) . Then 0 --f {a} 3~ @h(j, q ( j ) + i ) --f

.+ Qh( j , q ( j ) t . l ) = Qj --f Qj and hence a E 0;. Conversely, suppose that a E 0;. Then 0 + {a} iK 0, + Qj = @ h ( j , p ( j ) + l ) -+ ~ & ( ~ , ~ ( j ) + ~ ) . Since Oh(j,g(j)+l) 2 0 and since, if ,!I E 0 and d> -+ {,!I} € K then @ = 0 , it follows that a E O h ( j , q ( j ) + l ) . Thus 0; =

= @ h ( j , q ( j ) + l ) ; By (11), @ h ( j , q ( j ) + l ) 0. @ h ( j , q ( i ) + l ) = @ h ( j , g ( j ) + l ) . Therefore, 0; -tQf = @ h ( j % # ( j ) + l ) + Qh(j,q(j)+l)

Assume that case (iv) (b) holds for j = m. Then 0 -+ Q is in H and we take (0 -+ Q) as our desired sequence.

Assume henceforth that case (iv) (b) does not hold for j = m . Consider any j 2 m such that Q, = 0. Then, by theorem 6, either j = m or 0, -> Qj is the major premise of a GMP in S whose conclusion is 0, += Q n L . Hence if j + m , then

m and such that case (iv) (b) doe8 not hold. We form S" by listing, in ascending order of subscript, those 0; + Q, such that j E P. Thus A'' results from S' by none or more deletions. The last term of 8'' is 0; -+ Q, = 0 3 Q, since m E P . Since each term thus deleted from 8' also occurs earlier in S', it follows that 8'' also is a GSL-proof from H to 0 3 Q.

0; + Qj.

H .

0; == 0 . c 0. 1 =

Let P be the set of those j such that j

Consider any j m . By theorem 6, in case (i), (ii), or (iii), no term in (0, -> Qo, -+Qjnj-l) has Q, as succedent, and hence (3; += Qi differs from each of . . . ,

MODUS PONENS AND DERIVATIOP FROM HORN FORMULAS 47

the sequents 0; + Qo, . . ., Oi-i -+Qj-l. I n case (iv) (a), Oh(j,p(j)+l) --f Qh(j,q(j)+l)

is the only term in (0, + Q,, . . ., --f SZi-i) having Q, as succedent. Since O&(j)+l) --f fLh(j,,(j)+l, 0; --f Q,, therefore 0; + Q j differs from each of the sequents 0; --> Go, . . . , Oj-l -+ QjWl. In case (iv) (b), S” contains no corresponding term. Hence S” is a sequence without repetitions. Hence S” satisfies (7)”.

Consider any j E P such that 0; -+ Q, is the conclusion of a GSL in 8”. An argument similar to the one just given shows that case (iv) (a) holds and that the major premise of the GSL is @L(j,g(j)+l) .+ Q3(j,p(j)+l). Hence S” satisfies (11)” and (14)”. Now consider any k (= ria such that Ok -+ Qnn: is a minor premise of the GSL. Since case (iv) (a) holds, O j = 0 and hence 0; 2 0. Since 0; 5 @;, therefore 06 2 0. Hence Qk * 0 as we saw earlier. Hence 8“ satisfies (12)”. An argument similar to the one above shows that if Qt = Qk then either 0; = 0: or else 0; 0 , in which case the fact that 0; 2 0 precludes 0; -+ Ql from being a minor premise of the GSL. It follows that the minor premises of the GSL are uniquely determined. Now the only sequents in S” having Q j as succedent are 0; -+ Qj and p(j)+l , + + SZh(j,p(j)+l). Hence 8“ satisfies (9)” and (13)”. By theorem 6, for some Is m , 0 + Q k = + 9,. Then 0; .+ Ql is a term of S” and, as we saw earlier, is ( H , 0)- restricted. Then 0; = 0; as we saw a moment ago. Hence 0; + Q, is ( H , 0)- restricted. Moreover, since case (iv) (a) holds, 0, = 0 and hence, as we saw earlier, 0; --f Qj is ( H , @)-restricted. Hence the conclusion and each minor premise of the GSL are ( H , @)-restricted. Hence S” satisfies (20)”. As we also saw earlier, since case (iv) (a) holds, 0; -+ Qj is not in H and 0; -+ Q, is not a logical axiom. Hence S” sat,isfies (10)”.

S” does not in general satisfy (8)”. However deletion of those terms 0; -+ Qi other than the last which are not premises of a GSL in S“ yields a sequence satisfying (8)” without affecting any of the conditions (7)”, (9)”, . . ., (14)”, and (20)Il. Now if {a} + {a} E H , then HI/= 0 + SZ for the proper subset H’ = H n - {{a} + {a}} of H , contrary to one of our assumptions. Hence (14)” implies (16)”. Similarly, (14)”, (S)”, and (10)’’ imply (17)”. Likewise, (14)”, (S)”, and (20)” imply (IS)”. (20)” and (11)” imply (15)”. Now consider any distinct @ -+ !P and @’ --f y/‘ which occur in 8” but are not the conclusion of a GSL in 8“. If they are both in H , then Y + !€” since H u J is GMP-minimal for (0 + Q}. If they are both logical axioms, then evidently Y + !P‘. If, say, @ -+ Y is in H then Yn 0 = 0 since otherwise either HI= 0‘ 3 Q for the proper eubset 0’ = 0 n - !P of 0 or else H n - {@ -+ !P} (= 0 -+ 9. Then by (IS)”, if @’ -+ !P I is a logical axiom, then Y $. Y’. Hence S” satisfies (19)”. Q.E.D.

$ 2. Derivation from Horn formulas

Using theorem 1 and lemma 1 (for Q = 0) of $ 1, we shall first show that for certain parts of the propositional calculus, intuitionistic and classical implications coincide. Using these results, HERBRAND’S theorem, and the fact that the axioms for equality can be written as HORN sentences without the propositional constant f ,

48 WILLIAM CRAIG

we shall then show for certain sets of formulas of first-order predicate calculus with equality that the two kinds of implication also coincide.

In the propositional case, elaboration of the argument would show that one can move, so to speak, from the formulas po, . . ., pp whose conjunction forms the antecedent to the consequent y in a way which reflects a partial ordering of the kind considered in 0 1 and induced by {po, . . . , rp,.}. Relatively few properties of the propositional connectives are used. I n the quantificational case, the propositional aspects of the derivation can be madc part of a linear movement.

We begin with three propositional calculi. I n each, an atomic propositional formula is either a propositional variable or one of the two propositional constants t or f . A propositional formula shall be formed in the usual way from atomic formulas using the connectives 13, =, A , and v . The formulas a. A a1 A a2 2 a3 v ar and [[ao A all A olz] 13 [a3 v 4 , for example, shall be the same. We let ao, al, az and a3, a4 be the terms of the conjunction p = a. A a1 A az and of the disjunction y = a3 v a4 respectively, and p the antecedent and y the consequent of p 2 y . We let cp y and b y if and only if rp 2 y or y respectively is provable in the classical propositional calculus. and shall be defined similarly for the intuitionistic and the minimal propositional calculus respectively. The latter can be obtained, for example, from the calculus described in Exercise 26.19 of CHURCH [Z] by ad- joining the constant t and the two axioms t 3 [ f 3 f ] and [ f 13 f ] I> t .

The basic propositional HORN formulas shall be those of the form yo A . . A yr 3 6 such that y o , . . . , yr , 6 are atomic propositional formulas. Now let x = yo A . . A y r 2 6 be a basic propositional HORN formula. Then 17~ and ( ~ x are each equivalent to the condition that either (i) some y k is 6 , k 5 r , or (ii) 6 is t , or (iii) some yk is f , k 5 r . Also, t- is equivalent to the condition that either (i) or (ii). Now let x’ = y; A . - - A yr , ~3 S’ be a basic propositional HORN formula, and assume that neither kx nor b-x’. Then each of the three conditions 1 % ~ = X I ,

hx 7 x’, and l ~ l x = x’ is equivalent to the condition that { y o , . . . , y7} w {t} = = { y o , . . . , y:.} u {t} and 6 = 8’. Also, 8 = 6‘ is equivalent to the condition that (6) w { f } = (6’) w { f } . We are thus led to say that a propositional formula x and a sequent @ - !?’ correspond to each other if and only if x is a basic propositional HORN formula yo A - A yr 2 6 such that 6 is not t and yk is not f , k 2 r , {yo, . . . , y,.} w {t} = CD w { t } , and (6) w {f} = Y u { f } . It follows that if x and x’ correspond to the same sequent, then

Each of the rules GSL, GMP, TA is valid not only classically, as was remarked prior to theorem 1, but also for the minimal calculus and hence intuitionistically. More precisely, if xo, . . . , x9, correspond to the minor premises of a GSL, x ~ + ~ t o the major premise, and x ~ + ~ to the conclusion, then xo A - . A xn A xn+l Ixxn+2. Similarly for TA. TS is similarly valid intuitionistically and hence classically, but not! for the minimal calculus. Thus theorem 1 (a) gives rise to a theorem about IT, and theorems 1 (b) and 1 (c) to theorems about and 13. I n each case, a somewhat sharper theorem can be obtained which is also of interest classically. For this purpose, one formulates a rule corresponding to GSL, GMP, TA, and TS respectively and concerned with propositional formulas corresponding to sequents. I n

M,x

= x’.

MODUS PONENS AND DERIVATION FSOM EORN FORMULAS 49

general these rules must be supplemented by others to take care of commutativity of A, etc. We omit details.

Theorem 9. Let xo , . . . , x g be basic propositional HORN formulas, and x , 1, , . . . , A, conj tm ctions of atomic propositional formulas.

(a) If xo A . . A xq x 3 1, v - . v A,, then xo A - . - A xq x 3 li for some i 5 r and hence xo A - . A x g t i x 3 A,v . - v A,.

(1) ) If p 5 q , f is the consequent of xp+i, . . . , xg, not xo A - . A xq x 2 f , and ~ o ~ ~ ~ ~ ~ ~ q ~ x ~ l o v ~ ~ ~ v A , , then X , A . . . A X ~ ~ X ~ A ~ for some i 5 r and hence xo A . - ' A xP lxx 2 A, v . - v 1,.l)

Proof. 2, Let x o , . . ., xn, x , A,, . . . , A, be as stated. We first consider the case where each xi corresponds to some sequent and each Ai either is f or does not contain f , j 2 q , i r . Let H be the set of those sequents @ + Y which correspond either to some x, , i 5 q , or to some t 2 y such that y is a term of x and different from t , or to some Ai 2 f such that Ai is not f , i I; r .

Assume that xo A - + . A xq (7 x 3 1, v - . v 1,. Then HI=0 -+ 0 and hence, by theorem 1 (c), H l c ~ ~ l ~ 0 -+ 0. Then by lemma 1, J l~~~ 0 -+ 0 for some J 2 H such that no distinct sequents in J have the same succedent. Hence J contains a t most one sequent of the form @ -+ 0 and hence a t most one sequent corresponding to some 1, 2 f , i 5 r . If J contains no sequent corresponding to some Ai 2 f , then xo A . . . A xq A x 17 f by theorem 1 (c), hence xo A . - . A xQ 17 x 2 A( for any i r , and hence xo A . . - A x4 k x 2 1, v . + v 1,. Now suppose that J contains a sequent corresponding to ili 2 f and let Ai = ya ,o A - . A yi, llzl. Then xo A . A xg A x 11 yi,r for each j 5 mi. Then, by theorem 1 (a), xo A . . . A xq A x for each variable y i , j , and evidently xo A - a . A xu A x yi,j if y i , j is t , j 5 m i . Hence xo A . . A xg A x Ai , therefore xo A . . - A xq x 3 ili, and therefore again xo A - . A xq

yi,

x 2 A, v - - - v 1,. For the case considered this proves (a).

Now assume in addition that p 5 q , f is the consequent of . . . , xu, and not xo A . r , such that the above set J contains a sequent corresponding to Ai 2 f and there is no i , p < j q , such that the above set J contains a sequent corresponding to xj. Let Ai = yi,o A . - . A yi, m r .

Since not x0 A - . A xq A x for each variable y i , i , j j mi. Arguing as above we can conclude that xo A - . - A x p 1% x II) 1, v . - . v 1,. For the case considered this proves (b).

Now consider the case where xi does not correspond to a sequent, where xo, . . . , xj-l, x,+~, . . . , xq correspond to sequents, and where q =+ 0 . Assume that xo A - . - A xq lc x 3 1, v . . a v A,. Since l ~ x ~ , therefore xo A . + - A xj-l A A . . A xq la IT x 2 A, v . . v A, and hence xo A * . - A ~ j - ~ A X J + ~ A A xq kx 2 A,v . . v A,.

9 A xg 17 x 2 f . Then there is exactly one l i , i

f , theorem 1 (c) implies that x0 A - . . A x p A x IF yi,

l) For the case where p = q and r = 0, an analogous theorem holds for the implicational

2, G. KREISEL, in a letter, gave an essentially similar proof, and also conjectured that the

4 Ztschr. f . math. Logik

calculi of HILBERT-BERNAYS [9], pp. 422439.

result would generalize in certain directions to the predicate celculue.

50 WILLIAM CRAIG

But also xo A xo A . . . A xj-, A xjil A * - - A xq. This proves (a) for this case.Ifq=O,useof somecr2a inplace o f X o A , . . . ~ x j - 1 ~ ~ j + l ~ . . . ~ ~ 4 p r o v e s (a) similarly. By induction, (a) holds for any basic propositional HORN formulas x O > . ' ' 9 XQ' For (b) the proof is similar. One similarly extends (a) and (b) from the case considered above to the case where Ri may contain f in addition to other terms, i 5 r . Q.E.D.

The proof shows that if x is t and case (b) holds, then a rule corresponding to GMP together with certain rules for A and v is sufficient for deriving a,, v . . . v 1, from xo A . . . A xp. Somewhat similar remarks apply to the other cases.

To each of the three propositional calculi there corresponds a first-order predicate calculus with equality. Each of these has the same formation rules. For each n we allow symbols for functions mapping n-tuples of individuals into individuals. The individual terms and the atomic formulas shall be formed in the usual way. Thus an atomic formula is either one of the two constants t or f , or contains the symbol N

for equality or contains another predicate symbol. A formula shall be formed in the usual way from atomic formulas using 3 , V, 3, =, A, and v.

For all three calculi, provability shall be related to provability in the corresponding propositional calculus in the same way. More precisely, to define provability we first change the axioms and rules for the corresponding propositional calculus to take into account the new notion of formula, and then add in each of the three cases the same axioms and rules for quantifiers and the same axioms for equality (see, e.g., KLEENE [lo], p. 82 and $ 73). We let y be tautologous if and only if y is obtained by substitution of atomic formulas for propositional variables from a propositional formula g~ such that (c y ~ . Henceforth we let ~i y and b y if and only if p 2 y and y are respectively provable in the classical predicate calculus with equality. Also we henceforth let and be defined similarly for the other two predicate calculi with equality.

A basic HORN formula shall be any formula of the form yo A . * . A y r 3 6 such that y o , . . , , y r , S are atomic formulas. A prefix shall be any finite, possibly empty, string of quantifiers. A prefix shall be universal if and only if each quantifier in it is universal. A HORN formula shall be of the form ( P ) [xo A . A x,] such that (P) is a prefix and each x1 is a basic HORN formula.

Theorem 10. Let ( P ) , (Q) be prefixes, xo, . . ., x q basic HORN formulas, and x , R o , . . . , I r conjunctions of atomic formulas.

(a) If (&) i s universal and if ( P ) [xo A . . A xq] 17 (Q) [ x 2 A, v v 4 1 , then (P) [xo A . . A x41 IT (Q) [x 3 I$] for some i 5 r and hence ( P ) [x0 A - * - A x4] IT

(b) If p 5 q , f is the consequent of x p + l , . . ., xQ, not (P) [XO A . . . A X q ] IF f , and

A xq

I-, (Q) [x 3 A0 v ' . * v A,].

(PI [xo A - a + A xpl IF (Q) [Ao v . - * v 4 1 , lhen (P) [xo A . - . A xpl 15 (Q) 4 for some i s r a n d h e n c e ( P ) [ ~ o ~ ~ ~ ~ ~ ~ x p ] ~ ( Q ) [ A o ~ . ~ . ~ A ~ ] .

be universal, let 6 = x IJ A, v - * v AT, and assume that (P) [xo A . Proof. To prove (a), let (P), (Q), xo, . . .,xg, x , I , , . . ., I , be as stated, let (&)

A xq] IT (&) 5 .


There are axioms (P),i0, . . ., ( P ) * L ~ for equality such that (P),i0 A - . . A (P)8is A A ( P ) [x, A . . . A x4] 3 (Q) t is provable in the classical predicate calculus without axioms for equality. Each ( P ) is a universal prefix and each ti a basic HORN formula, j 5 s. Let r ] = (P),i, A - . A ( P ) 8 ~ 8 A (P) [xo A . . + A xn]. Without loss of generality we can assume that no variable occurs in r ] 2 t both bound and free and that no variable in (&) occurs in r ] . We now apply HERBRAND’S theorem or GENTZEN’S

extended Hauptsatz, modifying the version given as theorem 50 on page 460 in KLEENE [lo] by considering in place of each sequent of formulas the corresponding conditional and by noting that thinning is not needed (see GENTZEN [6], p.409, note 6). By the above provability of r ] IJ (Q) 6 and thus of q 2 6 and by this modified theorem 50, there is a sequence r ] , 3 6, . . . , qt 2 t such that qt = r] , r ] , 3 t is tautologous, and each qrn I> t yields 3 t by one of the four rules corresponding t o V-introduction, 3-introduction, Contraction, and Interchange respectively in the antecedent of a sequent, rn < t . It follows that each term of the conjunction r] , is obtained from one of the formulas i , , . . . , i s , xo A + - A xq by substitution of individual terms for individual variables and hence is a conjunction of one or more basic HORN formulas. Since qo 2 [ x 3 A, v . - v A,] is tautologous and by theorem 9 (a), i t follows that r ] , 17 x 2 Ai for some i 5 r . Moreover, each of the four rules just mentioned is also valid for the illtuitionistic calculus and if qm 2 6 yields r]m+l 3 6 by one of these four rules, then qrn 3 [ x IJ A?] yields qm+l 2 [ x 3 Ail by the same rule, m < t . Hence hr] 2 [x 2 At] and thus also l l r ] 2 (Q) [ x ~ 1 4 1 by our choice of (Q). Since l ~ ( P ) ~ i ~ , i 2 s , also Iy(P) [xo A * * A xq] 2 (Q) [ x 3 Ail and hence also k ( P ) [x, A - - - A xq] 3 (Q) [ x 3 A, v . - v A,]. This proves (a).

To prove (b), let ( P ) , ( Q ) , xo, . . . , xa, A,, . . ., A, be as stated, let p 5 q , let f be the consequent of x ~ + ~ , . . . , xq, and assume that not ( P ) [x, A - + . A xq] f and that ( P ) [x, A - - - A xq] 10 (Q) [A, v 9 . v A,]. There are axioms (P) ,L, , . . . , (P),i8 for equality such that (P),L, A 9 A (QL, A ( P ) [x, A - - . A xJ c (Q) [A, v . + v AT] is provable in the classical predicate calculus without axioms for equality. Then by the modified theorem 50 of [lo], there is a sequence r] , 3 to, . . . , qt 2 tt such that qt = (P),i0 A - - A ( P ) , L ~ A ( P ) [x, A . - A xn], f t = (&) [A, v . v A,], r], 2 E,, is tautologous, and each qrn2 Ern yields r]rn+l IJ trn+l by one of the eight rules corresponding to V-introduction, 3-introduction, Contraction, and Interchange respectively in the antecedent or in the succedent of a sequent, m < t . It follows that each term of the conjunction q0 and of the disjunction 5, is obtained from one of the formulas L,, . . . , L,, xo A - - - A xn or from A, v . . . v A, respectively by substitution of individual terms for individual variables. Hence each term of the conjunction ?lo is a conjunction of basic HORN formulas and each term of the disjunction 6, a disjunction A; v - . - v A; of r conjunctions of atomic formulas. Also, r] , =I f is not tautologous, since otherwise the sequence r ] , 2 f , . . . , qt 2 f would provide a proof of q t 3 f in the classical predicate calculus without axioms for equality so that (P) [xo A . A xq] f . Now let r],!, be obtained from r ] , by deleting from each term xh A - - A x p A xp+l A . - . A x: obtained from xo A . . A xp A xp+l A - A xq (by substitution for individual variables) the terms . . . , x i . Then, by theorem 9(b) for the case where x is the constant t , there is a term A: of one of the disjunctions

I f

4*

52 WILLIAM CRAIG

A6 v . . v 2; which are the terms of the disjunction to such that q; A:. To each term xi A - . - A ~1 of q, there corresponds in an obvious way a term (P)”[$ A - - A x’&] of qm such that (P)” is a final segment of (P) , m 5 t . We let v, be the result of replacing in qm each such term (P)” [x: A . A xyl, by (P)” [xt A . - . A xg], m 5 t . Similarly to the term 1; v - . . v 1; of 6 containing ,Ii as term there corresponds in an obvious way a term (Q)” [A’,’ v . - . v A:’] of 6, such that (Q)” is a final segment of ( Q ) . m 2 t . We let 5, = (&)”A” for each m 5 t . Evidently, v, is qh and 5, is A:. Also, v, is (P),L,, A . - . A ( P ) 8 ~ 8 A ( P ) [x, A - - A x p ] and tl. is (&)Ai. Also, if q, z, 6, yields tlv,+l 3 t,ra+l by one of the four rules concerning the antecedent, then v, 3 5, yields v,+~ 2 clr,+l by the corresponding rule, if q, 3 6, yields q7m+l z, tm+l by the rule corresponding to Contraction or to Interchange in the succedent of a sequent, then v, 3 C, is v,+~ 2 5m+l, and if q, 3 6, yields q,+l 2 by the rule corresponding to V-introduction or to 3-introduction in the succedent of a sequent, then v, 3 i, either is v,+* 3 5m+l or yields v , + ~ 2 by the same rule. Each of these rules is also valid for the minimal calculus. Since I , M Y , I> T o , therefore I=vt 3 ct. Since l%(P),~j, 7 * < = 8 , also b ( P ) [xo A A x p ] z, and therefore ~ M ( P ) [x, A . . r> xp] 3 (Q) [A,v - - . v A,]. This proves (b). Q.E.D.

As a special ease of theorem 10 (a) we note that for any HORN formula x, if I:x 3 f then 17~ 3 f , so that if a HORN formula is inconsistent classically it is also inconsistent intuitionistically.

The condition that (Q) is universal cannot be omitted from (a). Nor can 1, v . . . v ?,, be replaced in (b) by x 3 A, v . v A,. This can be seen by the following example, partly due to FRED CALVIN. Let xo = Gxyx A F x y 3 Hz and x1 = G.syy 2 Pxy . Then xo A x1 Gxyx 3 H z , [Pxy 3 f ] A x1 Gxyy 2 f , hence [Fxy 2 f ] A x1 Gxyy 2 H z , and therefore xo A x1 [Gxyx 3 Hz] v [Gzyy 3 Hz] . However, not xo A x1 3 w[Gxyw z, Hz] . This can be shown using that part of the modified theorem 60 of [lo] which con- cerns 17. If there is an intuitionistic proof of xo A x1 I> 3 w [Gxyw z, H z ] then there also is a proof with a final segment vo 3 to, . . . , vt 3 Tt such that vt = xo A xl, ct = 3 w [Gxyw z, Hz] , v, 3 To is quantifier-free, (T Y, 2 T o , and each v, 2 5, yields v,+~ 2 [ m + l by thinning or one of the eight rules described earlier. Then v, is a conjunction whose terms are xo and x l , and 5, is a disjunction A, v . - - v 1, such that each A; results by substitution of an individual term for w in [Gxyw 2 H z ] . Now a given cut-free intuitionistic proof of v, 3 A, v . . v 1, can be transformed into a (cut-free) intuitionistic proof of vo 2 Ai for some i 5 r , since, roughly speak- ing, no axiom of the given proof contains v in the succedent, and any v-introduction in the succedent preceding the use of another rule can be permuted with it. How- ever, xo /! x1 17 l., fails for each i 2 r , hence k v o 2 A, fails for each i 5 r , and hence kv, 2 [, fails. Hence, not xo A x1

3 w [Gxyw 3 Hz] , since F x y A xo

3w[Gxyw 3 H z ] .

Our argument shows in particular that not xo A x1 JT [Gxyx 2 Hz] v [Gxyy 3 Hz] , whereas xo A x1 I F [G‘xyx 3 Hz] v [Gxyy 2 Hz] as we saw earlier. This indicates one direction in which theorem 9 fails to generalize.


Remarks on page 31 of [ll] yield that case of theorem 10 (a) where ( P ) is universal, r = 1 , and f is absent.l) In addition, they indicate for the quantifier-free part of the derivation in that case how one can move, so to speak, from (P) [xo A - . - A x4] to (&) [ x 3 A. v . . v 4 1 . Similar results for the general case are implicit in our proof. Moreover, by the methods of [3] they can be extended, classically and intuitionistically, to the quantificational level, since all L-rules on pages 252-253 of [3] other than V-importation and Matrix change are intuitionistically valid, since V-importation is not used here, and since Matrix change i s used only to an extent which is intuitionistically valid. To take care of the properties of equality one may use either axioms as above or, perhaps preferably, rules.

The purely classical content of theorem 10 (a) overlaps considerably with theorem 1 on page 66 of [12], while the additional purely classical content of theorem 10 (b) appears to be a new result.2) That part of it which is obtained by letting p = q was discovered by F. GALVIN. A theorem yielding both MCKINSEY’S theorem and the classical parts of theorems 10 (a) and 10 (b) will be given in [4]. The purely intuitionistic content of theorem 10 is related to a theorem by GODEL (see, e.g. [lo], p. 486).

References [l] P. BERNAYS, Betrachtungen zum Sequenz-Kalkiil. 1-44 Contributions to Logic and Method-

ology in Honor of J. M. Bochenski. North-Holland Publishing Co., Amsterdam 1965. [2] A. CHURCH, Introduction to Mathematical Logic, Vol. I. Princeton University Press, 1957,

p. 378. [3] W. CRAIG, Linear Reasoning. A New Form of the Herbrand-Gentzen Theorem. Journal

[4] W. CRAIG and F. CALVIN, A Satisfiability Theorem for Theories Preserved under Direct

153 G . GENTZEN, tfber die Existenz unabhilngiger Axiomensysteme zu unendlichen Satz-

[6] G. GENTZEN, Untersuchungen uber das logische SchlieOen. Mathematische Zeitschr. 39

[7] P. HERTZ, Reichen die ublichen syllogistischen Regeln fur das SchlieBen in der positiven Logik elementarer Sltze aus? Annalen der Philosophie und philosophischen Kritik 7 (1928),

[S] P. HERTZ, tfber Axiomensysteme fur beliebige Satzsysteme. Mathematische Annalen 101

Symb. Log. 22 (1957), 250-268.

Product (Abstract). Notices, American Mathematical Society, 1966, 624-625.

systemen. Mathematische Annalen 101 (1932), 329-350.

(1934), 176-210, 405-431.

272 -277.

(1929), 457-514.

1) In addition to rules (i) and (ii) given there, one also needs a rule for deriving a conjunction from all its terms. Otherwise, given F x 2 G,x, F x 3 G,x, G,x A G,x 2 Hx, and Fx one can derive G , x and G, x but not G,x A G,x and hence not H x .

‘2) I am grateful to A. OBERSCHELP both for calling MCKINSEY’S theorem to my attention and for rai singthe problem whether in case (b) one can delete from the conjunction in the antecedent the “negative” terms xP+,, . . . , xn. Note that %on-negativeness” of the consequent is not by itself sufficient, even in the propositional case. From theorem 9 (b) one cannot omit the condition that the other terms xo,. . ., x p of the conjunction are basic propositional HORN formulas. For example, let xo be a v /?, where a and /? are distinct, and let xl, x , and 1 be p + /, t , and a respectively. Then xo A x1 17 x 3 I and not xo A x1 x 2 1. x 3 f . Yet not x4,

54 WILLIAM CRAIG

[9] D. HILBERT and P. BERNAYS, Grundlagen der Mathematik, Vol. 2. Julius Springer, Berlin

[lo] S. C. KLEENE, Introduction to Metamathematics. Van Nostrand, Princeton, N. J., 1952,

[ll] G. KREISEL and W. TAIT, Finite Definability of Number-Theoretic Functions and Para-

[I21 J. C. C. MCKINSEY, The Decision Problem for Some Classes of Sentences without Quantifiers.

1939, p. 497.

p. 650.

metric Completeness of Equational Calculi. This Zeitschr. 7 (1961), 28-38.

Journal Symb. Log. 8 (1943), 61-76.

(Eingegangen am 17. Februar 1966)

Modus Ponens and Derivation from Horn Formulas

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