smullyan analytic natural

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rHE J O U R N A L OF S Y M B O L I C LOGIC

Volume 30, Number 2, June 1965

ANALYTIC NATURAL DEDUCTION

R AY M O N D M . SMULLYAN*Introduction We consider some na tur al dediiction systems for quanti-

fication theory whose only quantificational rules involve eliminationofquantifiers. By imposing certain restrictions on the rules, we obtain asystem which we termAnaly t ic N atura l Deduct ion ;i t has the property tha tth e only formulas used in th e proof of a given formulaX are either sub-formulas of X , or negations of subformulas of X . By imposing furtherrestrictions, we obtain a canonical procedure which is bound to terminate,if th e formula being tested is valid. Th e procedure (ultimately in th e spirit

of Herbrand [11) can be thoug ht of a s a partial linearization of t he me thodof semantical tableaux [ Z ] , [3] . A further linearization leads to a variantof Gentzens system which we sh all stu dy in a sequel.

The completeness theorem for semantical tableaux rests essentially onKonigs lemm a on infinite graphs[4] . Our completeness theorem for na tur aldeduction uses as a counterpart to Konigs lemma, a lemma on infinitenest structures, as they are to be defined. These structures can be lookeda t as th e unde rlying com binatorial basis of a wide varie ty of na tu ra ldeduction systems.

I n 5 1 we study these nest structures in complete abstraction fromquan tification th eo ry ; th e results of th is section are of a pu rely combi-natorial nature. The applications to quantification theory are given in5 2.

1. Nest structures We let n be a finite 1-1 sequence a l , a2, . . . a n ,or a denumerable 1-1 sequence a l , a2, . . . an , . . . of elements calledpoin ts .For i 5 j by the interval [ae, a& we mean the sequence (ae, ai+l, . . . , a j ) .We allow the unit interval [a t ] , but we exclude the empty interval. Nowlet C be a finite or denumerable collection of these intervals, and letNbe the ordered pair (n, ) . We call N a nest structure iff the followingconditions holds.

( N 1 ) If two distinct intervals overlap, then one of them wholly contains theother, and properly extends it at both ends.

More precisely:(a) If two distinct interv als overlap, then one of th em wholly contain s

the other.

Received February 27, 1963.* This paper was presented a t th e Internation al Congress of Mathematics, Stock-

holm, 1962. Th e research was sponsored by t he In formatio n Research Division, Air

Force Office of Scientific Research, under Grant No. 433-63. We wish to thank thereferee for countless corrections and suggestions.

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124 R AY M O N D M . S M U L LYA N

b) N o point begins two distinct intervals, nor ends two distinct intervals.We call N a finite or inf ini te nest structure, depending respectively on

whether the sequence 7t is finite or infinite. We refer to the number of termsof n as the length of the nest structure N .We shall think of the sequence 7t as being written down vertically rather

than horizontally, and we shall display nest structures by drawing boxesto represent the intervals. In the following examples, (a) and (b) are neststructures, but (c), (d) and (e) are not (each one violates condition N 1 ) inone way or another)

a2 begins 2 distinct intervals) a5 ends 2 distinct intervals)

We shall call N a regzllar nest structure i f f the following condition also

( N z ) :For each positive integer i , if a6 ends a n interval , then ai+l does notholds.

be@ an interval.


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A N A LY TI C N AT U R AL D E D U C T I O N 125

In the above examples, (a) is a regular nest structure, b ut (b), thougha nest structure, is not regular. Regular nest structures are the only ones

which shall come u p in our stud y of quantification theo ry (cf.52). Thefollowing is our fund am ental lemm a.

LEMMA. or any infinite regular nest structure, infinitely many pointslieoutside a21 intervals.

PROOF. et N be infinite and regular. We calla point free if it liesoutside all intervals. We shall show: (i) N contains at least one free point;(ii) for any free point, there is ano ther free point under it .

(i) Consider the point a l . If it is free, then there is nothing more toprove. Otherwise, it begins some interval[ a l , as]. We assert tha t a i+l must

then be free. For ad+l cannot begin an interval, by the hypothesis ofregularity. Hence, any interval ab outaa+l would overlap [ a l , ad] in a mannerviolating condition N 1 . Thu s there exists at least one free point.

(ii) Suppose at is free. We m ust find a free point under U . Z . If u6+1 is free,there is nothing m ore to prove. Otherwise ai+ l is contained in some interv al;this interval cannot contain a6 (since ad is free), so ag+l must begin thisinterval. Let [at+1, aj be this interval. We assert that aj+l must be free.For suppose some intervalI contains q + l . The first point of I cannot beaj+l (since aj ends an interval, andN is regular). The first point of I alsocannot be a ny point in the interval[ai 1, j ] , or again condition N1 wouldbe violated. Therefore, th e first pointof I must be at, or some point abo veat,which is contrary to the hypothesis th atut is free. This concludes the proof.

We shall subsequently employ the following terminology. Consider anynest structure N ; et L be its length. Let i+ 1 5 L . Either ui+1 begins someinterval or it does not. If the latter, and if ui does not end any interval,then we call ai+l the sole successor of ai. If at+l does begin some interval[ai+1, q ] , hen we call u.2 a junction point, and we refer toai+l as the firstsuccessor of at, and aj+l as the second successor of ai (the lat te r, of course,on ly if aj+ l exis ts - .e. only if j + l S L ) . Thinking pictorially in termsof boxes, a junction point is one which comes just before th e to p of som ebox. If we cross th e top of th e box, we hit th e first successor of th e po int ;if we leap ov er t he box , we hi t t h e second successor of t he point.If pointb is a successor of a (either sole, first or second) then we shall say thata isconnected to b. We define a path as a ny finite or denum erable sequence ofpoints, beginning with the first free point ofN , and such that each termof the sequence is connected to the next. What properly results from theproof of our lemm a is th at any infinite regular nest struct,ure contains a ninfin ite path .1 In term s of our no tion of successors, ifa non-junction point

1 This is reminiscent of the Konig lemma that every infinite tree in which eachpoint branches to only finitely many points, must contain at least one infinite path.In a sequel to this paper, we shall show how our lemma on nest structures relatesto th e Konig lemma.


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126 RAYMOND M . SMULLYAN

is free, so is it s sole successor.If a junction point (of a regular nest structu re)is free, so is its second successor. Thus, we can tak e th e infinite path by

star ting with the first free point, an d going down th e sequenceTC, leapingover all boxes th at come our way. We note th at thisis the only infinite pa th ,for if we ever enter a box (i.e. tak e thefirst successor of a poi nt), the n wecan never get out again, and all boxes are finite. We shall refer to thispath as the principal path of N. It is also the path obtained by alwaystaking the second successor, when we areat a junction point.

I n prepara tion for the nex t section, we need a few more notions pertinentto nest structures. Let j be any positive integer less than or equal to thelength of N, and let i I . We shall say that at is covered at stage 1, if aiis contained in at least one box [ a k , a ~ ] hich terminates before the pointa5 - .e., which is such that < . If at is not covered at stage j then wesay th a t at is alive or free at stage j . We note that any point at is auto-matically alive at stage i (even if the unit interval [at] is present). Alsoany point at which is a free point of N is alive at stage j , for every j 2 i(prov iding, of course, th a tj is 5 to th e length of N). Finally, we observetha t the s ta tement at is connected to a, is equivalent to the statem enta6 is the last point beforeaj which is alive at stage j .

I n 52, th e poin ts of o ur nest structure s will be sentences (closed wffs)of quan tification theory , or mo re precisely,occurrences of sentenc es on a line(not contained w ithin a n occurrence of a larger formu la on tha t line). Inthis context, we shall indeed refer to our points as lines.

2. A system of quantification theory We shall consider quanti-fication theory as based on th e primitives (negation),A (conjunction),v (disjunction), (ma terial implication), V (universal quantification),3 (existential quantification). For an y formulaF , we use F synonymouslywith NF. We use individual variables XI 2, . . . , xn . . . and individualconstants a l , a2, . . ., a,, . . . These constants (sometimes called para-meters) differ from the variables in that they are not to be quantifiedupon .e. Vat ) does not occur as a pa rt of an y well formed formula.B y a sentence we mean a closed (well formed) formula; if no constantsare present, then we caII it a p u r e sentence. We use F, G as meta-linguistic variables ranging over all (well formed) form ulas;X, Y, 2 as metalinguistic variables ranging over allsentences; x y, a s m e t a -linguistic variables ranging over all individual variablesxl, 2, 3, . . . anda, b, c as metalinguistic variables ranging over our parametersa l , a2, as, . . By F: we mean the result of substitutinga for every freeoccurrence of x in F . [In our system of quantification theory , we never needsub stitu te a variable fora variable, but rath er onlya constant for a variable.

We are th us not troub Ied with a n y possibility of collision of qu antifier s].For purely typographical reasons we sometimes abbreviateF:, b y F r. The


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A N A LY T I C N AT U R A L D E D U C T I O N 129

Next we consider thed i r e c t rules for the logical connectives.

X

X+

Y

- ( X v Y )-XN Y

V 1)

- (X Y )X

N Y

3 1 )

Next the discharge rules

V 2 x v Y

32)

d X

cont.Y

l ont.N Y

X 3 Y

+xcont.

Y

Before considering the rules for the qua ntifiers, we remark th at we couldta ke fewer logical connectivesas primitive and get along with fewer rules


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130 R AY M O N D M. S M U L LYA N

.g., if we tak e jus t - v as primitive, we need use only rules P ) , -),V l ) , v 2 ) .

The rules for the quantifiers are all direct. They are

[a is a n y parameter]

[a is any parameter]

_ _ _ _ provided a is a new para m eter (i.e. does not o ccurFZ in an y earlier line).

32)

- V X ) F, provided a is a new parameter.

-FE

By a natural deduction we shall mean any nest stru cture- inite or infinite-such that each line and box (interval) has been introduced in accordancewith t he abov e rules. Th e points (lines) of ou r nat ur al deductions are(occurrences) of sentences (with or witho ut 4 . e note th at (in our presentsystem) the first line of a natural deduction can never be discharged; alsono finite deduction has a box (interval) which includes the last line. Also

th e first line must bea premiss, and it cannot contain an y parameters. B y aclosed natural deduction we shall meana finite na tur al deduction which isin a contradictory state and which contains no premiss aliveat the laststage except for the first line. Bya refutation of a (pure) sentence X weshall mean a closed natural deduction whose first line is the premiss2 / X .By a proof of X we shall mean a refutation of X .

DISCUSSION.ule V1 is substantially Quines rule U . I . (universal instan-tiation) cf. [ 5 ] . Rule 31 can be looked at as a trivial variant of U . I .[In fact, it could be replaced b y th e rule to inferVx)-F from - 3 x )F . ]Rule 3 is substantially Quines rule E . I . (existential instantiation), and

V2 is a trivial variant of 3 2 . Th e intuitive idea be hind th e use of rule32 isthis. If ( 3 x ) F is true, then at least one element satisfiesF - et a be anysuch element. If we subsequently derive another sentence 3x)G, we cannotlegitimately say let a be any element satisfyingG, for we have alreadycommitted a to being th e name of some element satisfyingF , and we donot know that there is an element satisfying bothF and G. This is thereason for the restrictive clause in Rule3 2 ; analogous considerations applyto Rule V Z .Actually we can liberalize these two rules by replacing theclause providing a is new by providing a has not been previouslyintroduced b y Rule 3 or Rule V Z ,and does not occur in F . Under thisliberalization, proofs can sometimes be shortened (cf. second examplebelow).


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It would appear that our present natural deduction system- call it(N) s sound in the sense th at every refutable sentence is unsatisfiable(and hence every provable sentence is valid). We discuss this further in

As an ex ample of a na tura l deduction, th e following is a proof of th e 3.

sentence Vx) [Px Qx] [ Vx)P x Vx)Qx]

2/-[ V4 [px Qxl [ Vx) Px Vx)QxllV W x Q4- Vx) x Vx)Qx]

Vx)Px

- V4 Q X-Qai

PalPal 3 Qal

Qal

We wish to consider another examp le to illustrate a point. T he followingis a proof of the sentence 3y)[Py Vx)Px]

2/-[(3Y) [PY Vx)PxIl-[Pal 3 Vx)Px]

Pal- Vx)Px-Pa2-[Pa2 3 Vx)Px] (from line 1

Pa2

If we liberalize Rule V2 as indicated in our previous discussion, we can

obtain the following shorter proof

2 / - ( 3 Y W Y V4PxI-[Pal Vx)Px]

Pal- Vx)Px-Pal

ANALYTICN ATU RA L D ED U CTIO N . We shall call a natural deductionanalytic if it is constructed in accordance with the following restrictionR)on premiss introduction.


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132 R AYM OND M . S M ULLYAN

R) Aside from the first line, no premiss may be introduced except underthe following conditions:

a) If X v Y s alive (at stage n , where n is th e length of t he ded uctiona t ha nd ), and if neither X nor Y is alive, then 2 / X may be introducedas a premiss (i.e. adjoined as linen+ 1).

(b) If -(X Y ) s alive, but neither X, Y re alive, then 2/X maybe introduced as a premiss.

(c) If X r Y s alive, but neither X nor Y re, then we may introduce2/X as a premiss.

It is obvious that in ananalytic deduction, every formula which ap pearsmust be a subformula or the negation of a subform ula of th e first line.Our aim is now t o show tha t every valid pure sentence is provable by some

analytic natural deduction.A UNI FYI NG NOTAT I ON. It will save us considerable repetitio n of es-

sentially similar argu me nts in the metat heo ry, if we use th e unified notatio nin [ 6 ] .

We use a to d eno te an y sentence of one of t h e formsX Y , X v Y ) ,- X Y), -X. By a1 we respectively mean X, X, , , n d b y a2we respectively mean Y, Y, -Y, . E.g., if a = -(X Y ) , hena1 = X and a2 = -Y]. n each case, a is truth-functionally equivalentto th e conjunction of a1 and a2. We might refer to sentences a as of con-junctive type (this includes as special cases sentences which are literallyconjunctions). By a sentence t of disjunctive type we mean a sentenceof one of the forms X v Y, (X Y), Y ; y j31 we respectivelymean X, X, -X and by t32 we respectively mean Y , Y, . n eachcase /I s truth-functionally equivalent to thedisjunction of /?2. By asen ten ce of universal type - enoted by y we mean a sentence of o ne ofthe two forms ( V x ) F,~ ( 3 x ) F ;or any parameter a , b y y a ) we respectivelymean FZ, -FZ. By a sentence of existential type denoted by 6, weme an a sentence of on e of t h e two forms( 3 x )F, - ( V x ) F ; b y d a) we re-spectively mean FZ, -FZ.

To sum marize, we are usinga , p y , 6 as metalinguistic variablesranging ove r sentences which are respectively of co njun ctive , disju nctiv e,universal, existential types. Every sentence, except for an atomic sentenceor it s negation, is uniquely of one of these four types.

Our rules for natural deduction - other than premiss introduction -can now be more succinctly formulated as follows

Rule A - [Direct sentential] -I

El

a


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Rule B - [Discharge] P

YRule C4

6Rule D - providing a is new.

d a)

For analytic natural deduction, the three cases for premiss introduction(other tha n the first line) are subsumed under the rule.

R) If is alive a t stagen, but neither P I ,, are so alive, then 2/181 maybe taken as a premiss.

A CANONICAL PROOF PROCEDURE. We now wish to give a canonical proofprocedure having the property that any analytic deduction, whose firstline is 2/-X, and which is constructed in accordance with the procedure,is bound to terminate in a proof ofX, roviding X s valid. To technicallyfacilitate the description of t he pro cedu re, we introduce th e colon as a newformal symbol, and we define anauxiliary sentence to be an expression ofthe form y : a n . [We could interpret ( V x ) F : a ,as saying that F holds for

everyx

other than81,

.an,

and we could interpret4 3 x ) F : a n

s sayingtha t N F holds for every x other than a1, . . ., an. Actually it is not reallynecessary to consider the question of the interp reta tion of these auxiliarysentences; they are basically but technical devices to force our deductionsto have a purely syntactical closure property which we need]. We shalluse y : 0 synonymously w ith y , and y :n ) or y : a n , n = 1, 2, . . .

Let A )be our system of analy tic natura l deduction. We let(A1)be thesystem obtained from A ) by replacing Rule C by the following.

RuleC1 - y : ~ n = 0, 1, 2, .)

Y n+ 1)y : n + 1

This is really a collection of four rules (where nown is a 9ositive integer).

( V 4 F ~ ( 3 x ) F ( V x )F : a n ~ ( 3 x ) F :,

F ( a d NF(a1 ) F (an+ l ) N F ( a n + l )( V x ) F : a l N ( 3 x ) F : a l ( V x ) F : a , + l N ( 3 x ) F : a n + l

Suppose D1 is a natural deduction in the system(A1) .Som e of t he lines

will, in general, not be sentences of quan tification th eor y, bu t rath er theseauxiliary sentences. However,if we delete these aux iliary lines, the re sulting


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134 R AY M O N D M. S M U L LYA N

nest structure D that remains is obviously a natural deduction in thesystem A ) .

It is now important to note that if D is any deduction, either in thesystem A ) r the system A I ) , f finite leng th n, which is in a contradictorystate, but not closed, then itis possible to use a discharge rule- moreoverin such a manner that the subjoined lineis not a repetition of an earlierline alive at stage n+1.2 To see this, consider the last premiss alive atstage n ; et k be th e num ber of t he line where it appe ars (clearly1 < k 5 n ) .Now, by the restriction rule R) on premiss introduction, the premiss inquestion must be of the formj31, where j3 is alive at stage k - 1 , but ,Bzisnot alive at stage k - 1 . If we then discharge 81 and subjoin j32, we get norepe tition of a line aliveat stage n+ 1 . [I t is obvious th at th e only lines aliveat stage n+l are those alive at stage k-1 together with the n+lth lineitself].

When we employ the premiss introduction rule (in accordance withrestriction R))we say that we use j3 t o introduce /?I When we employ adischarge rule, then we say that weuse ,B to subjoin ,62. When we employa direct rule X / Y ,we say tha t we use X to adjoin Y . We note that in ouroriginal system A ) ,a sentence y could be used to adjoin infinitely manysentences. I n our present system A I ) , can be used t o adjoinat most twosentences. Thus, e.g., we cannotuse ( V x ) Fto obtain F ( a z ) rathe r we use

Vx)F to obtain ( V x ) F : a l ,which in turn may be used to obtain F(a2) .More generally, we can use a sentence of th e formy : n n only two possibleways. We shall say th at a line on a path P is used on path P iff it has beenused to obtain some line (perforce below it) on pathP. We say tha t a lineon path P is fulfilled (on path P ) iff one of t h e following condition s hold s:

i) it is a sentence of t he form a , and both a l , a2 appear on P.(ii) it is of th e form j3,and either j31 or Bzappears on P.

(iii) it is of t h e form y , and both y ( 1 ) and y : appears on P.(iv) it is of th e formy :n, for n a positive integer, and either a,+l does not

appear in an y line on pathP , or else both y n+ 1 )and y :n+ 1 appear

on path P.(v) it is of t h e form 6, and for some positive integern, 6 n) ppears on P.Suppose a natural deductionD (in the system A l ) is infinite. Then, by

our fund amen tal lemma on infinite regular nest structures, infinitely m an ylines lie outs ide all box es; these lines are th e elements of t h e so-calledprincipal path. If a deduction D is finite, we also speak of th e principalpath of D ; th e elements of th is pa th are those lines which are alive a t th elast stage of th e deduction. WhetherD is finite or infinite, we shall sayt h a t D is fulfilled iff th e following conditions both hold : 1 ) Every line

2 The additional fact about avoiding repetition is not needed in this paper, butwill be used in a sequel.


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ANALYTIC NATURAL DEDUCTION 135

on the principal path of D is fulfilled (on the principal path); (2) D is notclosed.

If an infinite analytic deductionD in the system A 1 ) is constructed atrandom, it is not at all necessary that D is fulfilled. The purpose of acanonical procedure is to guaran tee t h at (i) every infinite deduc tion con-structed according t o th e proc edure is fulfilled; (ii) every finite deductionconstructed according to the procedure which cannot be extended furtherwithout violating the procedure is either fulfilled or closed.

Many canonical procedures could be given. One such is as follows. Startthe deduction with any premiss. Now suppose the deduction has beencarried through to stage n. If, as of this stage, th e ded uction is eitherclosed or fulfilled, then we stop.If the deduction is neither closed nor

fulfilled, then our next act is determined by the following conditions.(1) If the deduction is in a contradictory state, then we must use a

discharge rule (which is indeed possible, as we have shown). And we usethe earliest line which can be used to effect the discharge. [This line isperforce of type j an d unfu lfilled a s of stagen, b u t fulfilled as of stage

(2) If the deduction is not in a contradictory state, then we pick theearliest unfulfilled line.If it is of t he form a , the n we adjoin th e first of th epair a l , a2 which is not alive a t stagen. If it is a sentencej3 then we adjointhe premiss 2/j31. If it is of t he form y : n , the n we adjoin the first of t hepair y l z ) , y : n which is not a live.If it is of the form 6, then we take the firstinteger wz such tha t am occurs in no line alive a t stagen, and we adjoin 6 m).

This concludes the description of th e canonical procedure.It is a routinem atte r to check th at this procedure does have th e desired properties.

Suppose now that D is an infinite deduction constructed according tothe canonical procedure. Let S* be th e set of all sentences which occur(with or without a check mark) on the principal path. SinceD is fulfilled,then S* must satisfy the following conditions.

M o : No atomic sentence and its negation are both inS*.M I : If a&*, then a l , az are both in S*.M z : If , I&*, hen either j31 or P Zis in S*.M 3 : If &S*, then for at least one parame tera, 6 a)cS*.M i : For each positive integern, if y:ncS*, hen y ( n + l ) and y : n + 1 are

both in S*.

n 11.

Let S be th e se t of all elem ents ofS* which are not auxiliary sentences.The set S obviously obeys conditionsMo-M3 (reading S for S* . ndsince S* obeys condition M i , then by an obvious induction argument, theset S obeys the condition:

M 4 : If ye s , then for every positive integer n y ( n ) ~ S .


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136 RAYMOND M. SMULLYAN

Thus the set S of all non-aux iliary sentenc es on the p rincipal pat hof Dobeys conditions Mo-M4. Such a setS is called a model set in Hintikka [3 ]an d is denumerably satisfiable.3 Th us if the canonical deduction, star tingwith a given premiss 2 / X , runs on infinitely, then X is denumerablysatisfiable.

Suppose that we run a canonical deduction starting with2 / X , and thatwe reach a stage in which every line on the principal path is fulfilled, butthe deduction is not in a contradictory state. Leta, be the last parameterwhich occurs on the principal path. Then the setS of all non-auxiliarysentences on th e principal pa th satisfies conditions Mo-M3, and in placeof M4 it satisfies the condition.

M k :Such a set S is satisfiable in a finite domain of n elements (by an

argument analogous to that in footnote3).

We have thus shown tha tif X has no finite or denumerable model, thenthe canonical deduction starting with2 / X must terminate in a refutationof X. herefore, if a sentenceX s valid (or even denumerably valid) thenthe canonical deduction starting with2 /X ' must terminate in a refutationof X . herefore, if a sentence X is valid (or even denumerably valid)then the canonical deduction starting with2/X ' must terminate in aproof

of X. o, of course, every valid (pure ) senten ce is prov able b y some analyticdeduction, since the canonical deduction is analytic.We have , of course, given a canonical procedure for th e systemA l ) , nd

have shown that every valid pure sentence is provable by an analyticdeduction in A ). B ut this implies th at every pure valid sentence is provableby an analytic deduction in the system A) .

REMARKS ONCERNING THE PROCEDURE. The above procedure, thoughtheoretically adequate, is not a particularly good one from the point ofview of obtain ing sho rt proofs. The following procedure, tho ugh a bit moredifficult to justify, is considerably better as a working procedure.

Suppose at stagen the deduction is not in a contradictory state. In theprocedure we have given, we then always give priority to the earliest un-fulfilled line. It is bette r rath er t o give first priority to lines of the form6,second prio rity t o lines of t he forma hird priority t o lines of the formy:n. And w ithin each of these groups, we give prio rity t o the earliestunfulfilled line of t h a t group .

If yeS, then for every i 5 n , y ( i ) ~ S .

3 Briefly, th e argum ent is this. Take for the universe of discourse th e set of p ara-meters themselves. Assign to each n-a ry pred icate, th e set of all n-tuple s of para -

meters such that the predicate followed by the n-tuple lies in S. Under this inter-pretation , every sentence X of S is true, by an induction argument on th e number ofoccurrences of logical connectives and quant ifie rs in X ) .


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A few working examples will convince th e reade r of the s upe riority of thisprocedure. N eedles to say, it is still capable of ma ny practical improvemen ts.Such a study is a subject in itself, and we shall deal with this elsewhere.

3. Concluding remarks 1 ) We now see how ou r lemma on infinitenest structures can be used in place of the Konig graph lemma in provingth e completeness of quan tification theory. One can easily use th e com-pleteness of analytic na tural dedu ction to prove the comp leteness of th ebetter known Hilbert type formalizations, as well as the Gentzen formu-lations.

(2)** We wish to briefly discuss some related natu ral d educ tion systems.Th e system N ) which we first introdu ced is of li ttle interes t in itself. Fo rsuppose we introduce a premiss. Unlessit be of the form 2/ @1 where pis already alive, it can never be discharged. Supposeit is of th e form 2/,91,with p already alive. If is already alive, the introduction of 2 / p 1 isstrategically pointless, and will only create a redundancy; ifpz is alive,then 2 / p 1 cannot be discharged without incurring a repetition ofpz. Thusany closed deduction in N ) ,which contains no redundancy, is automaticallyanalytic.

Of m ore intere st is the following extension N ) * of N ) . The rules ofN ) * re th e rulesP , A , B, C , D of N ) ogethe r with th e following discharge

rule

Rule E -

2/ X

cont.

X

In the system N ) * , he first line of a de duction is allowed to b e dischargedand th e deduction then continued. By aproof of (apure sentence) X n N ) *

we shall mean a finite deduction in N ) *whose last line is X, nd whichcontains no p rem ise s al ive a t th e last stage.

The system N ) * (which obviously does not obey the subformula prin-ciple ven in th e weaker form allowing negations of subfo rmulas) com esmuch closer to the usual natural deduction systems than does our analyticsystem A ) . ndeed it is trivial to o btain Quine s rules of cond itionalizationand truth-functional implication as derived rules in( N ) * .Alternatively,we could start with the system - call it ( M ) - whose rules are Quine srules of premiss introduction, truth-fu nctional implication a nd prem iss

* *Added February 28, 1964.


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138 R AY M O N D M . S M U L LY A N

introduction, combined with our quantificational rulesC , D, and easilyestablish th e rules of ( N ) * s d erived rules.4

If D s a proof of X n the system ( A ) or indeed in ( N ) ) ,hen one appli-cation of Rule E converts D into a proof of X n the system ( N ) * .So itis trivial that every sentence provable in A ) or in ( N ) )s provable in ( N ) * .

An argument justifying thesoundness of th e sys tem ( N ) * and hence alsoof th e sys tem s( N ) nd A ) ) s briefly a s follows. The crucial point to observeis that if S is a se t of sentences which is (simu ltaneously) satisfiable, andif 6eS and if a is a parameter which occurs neither in6 nor any elementof S , then SU{d(a)}is again satisfiable. Therefore RuleD preserves soundnessof na tu ra l deductions in th e following sense: Call a finite deduc tionD soundif for every inte rp reta tion of all predica tes alive inD (i.e., which occur inlines alive at the last stage of D n a non-empty universe U , there existsa n interp retatio n of th e live param eters ofD such tha t if all live premissesare true und er th e interpretation,so ar e a ll live lines ofD. Th at the remainingrules P , A , B, C, E also preserve sound ness is completely trivial. Therefore(by induction) every finite natural deduction (in( N ) ) s soun d. This easilyimplies that every provable sentence is valid (sinceit contains no para-meters, and no premisses are alive a t th e last stag e of a proof).

The soundness of ( N ) * , ogether with the completeness of A ) , mpliesth at every sentence provable in( N ) s also provable in ( A ) .Of course, thisproof is purely non-finitary. A finitary proof of t he equivalence of t h e

systems ( N ) * nd A ) s virtually eq uivalent to a finitar y proof of Gentzen sHaupsatz ndeed it could be obtained as a consequence of th e H aup -satz, or co nstructed indepe nden tly along th e lines of Gentzen s proof.

REFERENCES

[ l ] J . HERBRAND, echerches sur la Thdorie de la ddmonstration, Warsaw1930.

[2] E. W. BETH, Semantic entai lment and formal derivabil i ty, Mededelingen derKoninklifjke Akademie van Wetenschappen, Afd. letterkunde, n.s. vol. 18,no. 13 1955), pp. 309-342.

[3] K. J. J . HINTIKKA, orm and content in. quan tif icat io n theory, Acta Philoso-phica Fennica no. 8, Helsinki 1955, pp. 7-55.

4 Pedagogically we have found th e systems M , (N)*, o be particularly suitable asan introduction to Quantification Theory. We are not restricted to analytic proofs;indeed the systems are technically quite versatile. The restriction in Rule D in allour systems) is of a relatively simple so rt ; there is no worry abou t alphabetic variance,flagging of variables, nor collision of quantifiers. The soundness is relatively easy toestablish since we do not have generalization rules working in conjunction with

instantial rules, though it is not difficult to establish certain generalization rules asderived rules of the system).


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smullyan analytic natural

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