Post - The Two-Valued Iterative Systems of Mathematical Logic

 

 AN  A N N A L S OF M A T H E M A T IC S ST STU U D IES IE S   NUMBER 5

 

 Th  T h e T w o - V a l u e d It Iter erat ativ ive e Sys Syste tems ms  Of M Mat athe hemat matic ical al Logic BY 

EMIL L. POST

PRINCETON  P R I NC N C E T ON O N U N I V E R S I T Y PR PR E S S LONDON: HUMPHREY MILFORD   OXFORD UNIVERSITY PRESS

1941

 

Copyri Cop yright ght 1 94 1  PRINCETON

UNIVERSITY

PRESS

P r i n t e d   i n   u .s . a .

Lithoprinted by Edwards Brothers, Inc., Lithoprinters   Ann Arbor, Mich ichig igan an,, 1941

 

Dedicated to CASSIUS J. KEYSER in one of whose whose pe da dago go gi gica ca l devices the author belatedly recognizes  the true source of his trutht truthtable able method.

 

INTRODUCTION

In its original form the present paper was prese nted to the America n Mathematical Soci Society ety, , April 2 k,  1920 , as a c com om  pan ion piece to the w ri te r ’ s disser dissertation, tation, hen cefo rth referre d to as Elementary Propositions [23  ]1. Tho ugh submi tted for pub lication the following year, at least a revision of the present pap er was editori ally sugges suggested. ted. Except for the finis hing toup ch ches es, , including therei n introduc tion and footnotes , the present version was effected in the years 1929 - 1932 . Though, Thou gh, as suggested by its its titl title^a e^a the main contribution of the present present paper ma y be considered t to o be the comp complete lete solution of a certain problem in classica classical l mathema mathematics, tics, the origin of the paper and., as far as the writer can see, its main interest lies in the study of two-va lued propos itional calculi calculi. . In the first of the four parts parts into into whi ch Elementary Propositions was div ided the proposi tional calculus of Principia Math emati ca [37 ]> base d on the primitives negation and disjunction, was studied; and, as a preli min ary to t that hat stu study, dy, it was shown th that at if what we term ed truth-tab trut h-tables les were assigned to ~ p and pvq in accor accordance dance with their heuristic heuristic meanings, meanings, then not only was a truth-table there by formally determined for every expres sion in the ca lculu s, but every (two-valued) (two-valued) truth-t able was thus obtained. The second part of that paper attempted to generalize the first part by a l lowing for an arbitrary finite number of primitive truth-func tions, of arbitrary finite numbers of arguments, to each of which a truth-table truth-table was arbitrarily assig assigned. ned. It was then Immediately obvious obvio us tha that t while each expression in such a calculu calculus s ag ain had a corresponding truth-table, the set of such truth-tables migjht Numerals In brackets refer to the bibliogra phy concluding the present paper. paper. 0 However, footnotes V 7  and ^8 (§2 5 ) were also writt en withi n the period stated. Ofi This rather ref refers ers to theroriginal title of the pre present sent ver  si sion, on, "Determination of all itera tivel y close d two- valu ed systems of fu nc ti on s. " ,

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INTRODUCTION

2

now be but a proper subset of the set of all possible truthtables. tabl es. Two problems thus suggested themselve themselves. s. First First, , what truth-tables could be assigned to the primitive functions so that the set of all possible truth-tables would result; second ly, allowing for arbitrary initial truth-tables, what are the various possible sets of truth-tables that can thus result. Ad de d interest was lent to the first pr obl em by N i c o d ’ s ac hie ve ment [21  ] in basi ng a propositiona l calculus on but one primi  tive truth-function in accordance wit h the pioneering work of Sheffer [2 7 ]^. Whi le the second pro blem was at f first irst viewe d as an aid to the solution of the first first, , its intrinsic interest soon became apparent, and made it our mai n pro problem. blem. A brief summary of the result of our solution of this main k pro blem appears in Elem entar y Propositions . In this connection it should be noted that the operation of substitution, which is the only operation used in building up the variou various s expre expressions ssions of a propositional calculus by means of variables and primitive functions, has its counterpart for the corresponding truth-tables. The problem can then be considered entirely in terms of truthtables. table s. The above summary i is s then at the moment best restated as it appears in the printed abstract of the present paper [2 2 ] ” there are 66  different systems generated by primitive tables table s with no more than three argu argument ments, s, and 8  infinite famil ......

ies of systems which require tables of four or more arguments.” Such a generated system of truth-tables clearly has the proper ty of bei ng closed under the operatio n of substit substitution, ution, or, or, as we shall say say, , itera tive ly close closed. d. It is the n significant, as was noted in our abstract, that —— the converse also is true .- 5  In On the the other ha hand, nd, such a basis for a logic of propositions as given by Tarski [3 0 ] is outside the scope of our inv esti ga tion; since, in addition to the primitive truth-function equiva lence len ce, , it emplo employs ys quanti ficat ion of propositio nal variables. -------- -----------------------------------

^

-----

See [2 3 ], §5, (pp. (p p. 17 3- ^) .

^ Provided we exclude the null system of functio functions, ns, whi ch vacuously satisfies the condition of iterative closedness, or else allow a null set of generators, thus adding one to the numb er of genera ted sys system tems. s. Whil e this omissio n is easily supplied (but see footnote 51  (§2 5 )), more serious is our ex clu sion of functio ns of no variables, i.e i.e., ., constants. In this omissio n we follo w the usual form of a proposition al calc calculus ulus, , e.g., e.g ., such as that of Princ ipia Ma th em at ic a. Actually, t here would be little difficulty in modifying our count of systems in

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INTRODUCTION

3

terras of generated systems and logic, we may be said to have da te m i n e d all the non-equivalen non-equivalent t sub-langu sub-languages ages of the language of the complete two- value d propo sitio nal cal calculu culus. s. Since in itself a truth-table is but a two-valued function, two-valued referring t o n, the in range of arguments possible values tion, tio te terms rmsof ofvalues ite ratively cl osed and systems we are led of to func the title of this paper ? 1 or, or, to use the te rmino logy of Wiener, ^ we have determined all iterative fields of two-valued, functions. The first version of the present present paper was prese nted in terms of truth-tables, and. foll owed the me tho d of dis cover y via the concept of gen era ted system.^ That ev ery c cl losed, sed, sys tem of truth-tables could be generated by a finite set of tables only appeared after all such generated sys systems tems had been fo found und. . Sinc Since e It was impossible to present all the calculations and tabula tions, In addition to arguments, inherent in this procedure, that tha t first v ersion may be said t to o have bee n bu but t a repo report rt o n the complete solutio solution. n. The present versio n centers aro und the co n cept of closed system, and merely pauses in each instance to verify that the closed system in question can be generated by a the tw o-val ued p pt? t?ob oble lem m to all ow for the two constants there oc curring, curr ing, since those constants pla y essentia lly the same r61e as two of the four two two-va -value lued, d, func tions of one variabl e. On the other hand, generalizing our membership-functions of classes, (§2 ), to al low fo r arb itrar y constants wo ul d fun damen tally alter the scope of the present paper, as well as break down the iso morph ism with the t wo-valued problem. problem. Our count count of sys system tems s wou ld have be en considerably simpli fied if we failed to distinguish, for example, between functions of one variable and functions functions of two var variables iables whose values are independent of the value of one of those va ri ab le s. That such distinctions distinctio ns are real can be illustrated in a variet y of wa ways; ys; and whi le the s impli fication of the count count, , if desired, can be immediately effected, not so the meticulous recovery of the lost distinctions. See footnote 2a.

6 See O l ] , p. 7, for the de fi ni ti on of it er at iv e fie ld, [ij-o], p. 1 5 9 , for the cla class ss of Op er at io ns ’generated by a sin single gle op eration. eratio n. Where Wie ne r analyses the opera tion of substitutio substitution, n, which Is at the bottom of these concepts, into at least semiatomic parts, we have found It more convenient to use a more unifo rm def definit inition. ion. On the other hand hand, , Wiene r makes no refer ence to a specific class of variables to be employed, thus mak ing the first of the above definitions ambiguous, the second misleading. ^ ' A brie f outline of this met hod might be give given. n. Sin Since, ce, ne g lecting the particular variables heading a truthtruth-table table, , there are but four truth-tables of one argument, sixteen of two, while

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INTRODUCTION finite set set of generators.

Its synthetic development unif ies the

argumentative portions of the original development, and avoids all of the l at te rTs rTs tabulations, and most of it its s calculations , Q so that it is complet e in itsel itself. f. Furthe rmore, foll owi ng the suggesti on of th the e referee of the origi nal version version, , the truthtable has been replaced by logical expressions in the Jevons no tation. tati on. To preserve the preci sion of the truth-table development these expressions are interpreted not as truth-functions of pro positions but as what we call membership functions of classes. While the advantages are not all wi th the Jevons notation, it its s greater flexibility perhaps justifies the change.^ To avoid the unclarity of the first version of the present paper, we have devoted the first two sections of the present the tables of one or two arguments generated by a set of tables similarly restricted can be found by using only such ta table bles, s, mere syst systemati ematic c calculation and tabulation served t to o yi eld all the distinc t sets of such tabl es that could be s so o ge gene nera rate ted. d.. . On the other hand hand, , special methods a dapte d to the individual case cases s were needed to obtain a formula for an arbitrary table in each of the corresponding sy syste stems. ms. The next step was a systematic procedure which, again by special arguments, yielded formulae for all tables of more than two arguments which, take n singl singly, y, would generate systems not identical with one of the previously found system systems. s. By restricting these tables t to o t those hose of three ar guments, gume nts, all systems that could be g enerat ed by one table of three arguments, but not by tables of one and two arguments, were found., whence it was an easy matter to find all systems gener ated by arbitr ary sets of tables of no more tha n three argum arguments. ents. One repetition of this weeding-out process then so cleared the fie ld that it was p ossib le to f ollo w the proced ure for ’ thi rd order’systems in the case of n-th order systems with arbitrary n > 3 , and. thus com ple te the solu tion. On the ha hand, nd, information the origin al other solutio n ismuch los lost. t.additional But see foot note 50 ,yielded (§2 5 ). by y  Par ticu larl y for those systems satis fyin g the ’[A:a [A:a] ] con d i t i o n’, an d thus ad mit tin g for th eir memb ers the ’ nor mal [A [A:a :a] ] e x pa n s io n ’ , (see §8). It sh ou ld be me nt io ne d that i n 1920-21 Professor Oswald Veblen urged the writer to replace the truthtable solution by on one e which represented logic logical al functions by algebraic polynomials modulo 2  (the arithmetical forms of B. A. Bernstein [2 ], whose importance has since be en empha sized by Stone [2 8 ] In connecti on wit h his researches on Boo lean alg e bras. bra s. For their connection with Vebl en se see e [3 2 ], P. 9 ). How ever, it then seemed to the writer, and still seems so, that while the derivation of what we have called t the he alternating sys tems (§15) wo ul d be g rea tly s impli fied thereby, all of the rest of the solution would become correspondingly more complicated.

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INTRODUCTION

5

vers ion essentially to the st statement atement of our problem. The next three sections develop those aspects of the technique of the Jevons Jevo ns notation which are required in the solution of th the e prob  le lem. m. The last four sections of Pa Part rt I form an integral part of that solution, but are included among the preliminaries to expedite the presentation of that solution. Pan; II is devoted to the complete solution of what we have called our ma in problem. The various closed syst systems ems of memb er ship functions of classes are pres ented eith er individually, or or, , in the case of the infinite families of closed systems, with aid of a parameter. In general we ma y say that each closed system is given by a condition on it its s members. These conditions are such that mere Inspection is sufficient to tell of a given ex pressio n whet her the correspond corresponding ing function satisf satisfies ies that con dition. dition . The y are easily retranslat ed in terms of truth-tables. The first of the two sections comprising Part III unifies the set of all closed systems of membership functions, or truthtables, table s, wit h respect to the relatio n of inclusio inclusion. n. By anal ysing this relation in terms of immediate inclusion, we are led to the inclusion diagram, which gives a graphical summary of the solu tion of our mai n problem problem. . It should be no ted t that hat even this re vise d version of the section was writ ten before the chri christen stening, ing, and recent development, of the lattice concept . 10   The last sec tion gives our solution of the other of the two problems posed above. above . This prob lem was also solved in the first ver sion of the pape r under the added condition th that at the generat generators ors be indepen dent dent, , and the n further specialized In such a wa y th that at the infin ite number of independent sets of generators reduced to   es 36 sential ly different set sets. s. Actually, in the l light ight of the solu tion of our main problem problem, , the solution of the secondary problem without the condition of independence Is shown in the present version to be very s sim impl ple. e. A different and far easier sol solution ution than in the first version is here obtained of the problem of In dependent generators, generators, but wi th corresponding added complications in the deri vatio n of the specializ ed resul result. t. That the same 36 essentially different different set sets s are t hen obtained is but a check on both solutions. 10

See fo r example, [5 ].

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6

IN TRODUCTION

The writer is aware of but two papers directly touching on our two problems problems. . Both papers restrict themselves to second order functions, i.e., functions of but two variables, not only as generators but as functions genera generated. ted. There bei ng but six teen such functions, the doubly infinite character of our prob lem thus beco mes c ompl etel y finite, a an nd. ma nag ea bly so so. . In 1925 , E. Zylinski |>2] specifically determined the sets of second or der functions generate d by single second order functions. In a recent Investigation, W. Wemick [3 6 ] dis cus ses t he sets of second order function functions s generated by arbitrary sets sets of second or der functio functions, ns, and specifically determ determines ines what we wou ld call call the various sets of second order functions which are sets of in dependent generators of the sixteen second order functions, and hence, indeed, indeed, of all functions. Since our general solution of the problem of independent independent generators can but be g iven by condi tions on those generators, while our specialization rules out certain sets even consisting of second order functions, these results ma y be said but to overlap our own. own. We might note that ha d Zylinski seen our Elementary Proposi tions, publi shed four years ear earlie lier, r, he would have found the questio n ending his pa per answered in the affirmative, the conjecture immediately preced ing disproved. In the second part of Elementary Proposition Propositions s will be found the follow ing paragraph. "We "We thus s see ee that complete s systems ystems are equival ent to the s yst em of ’ Pr in ci pi a 1  not only in th the e truth table development but also postulationally. As other system systems s are in a sense degenerate forms of complete systems, we can con clude that nonew ne wlogi logical systems int rod uce d”. generaliza Perhap s the essentially logics cs yielde d by are our postulational tion blotted out our earlier idea that these subsystems of the complete system wo ul d constitute a so sort rt of logical playgro und for the construction o of f what are now called propositional cal cu culi li. . This earlier vis ion was r enewed in the observatio n that such a system as as that generated by Imp Implicat lication, ion, if admitting a pos tul atio nal develop ment ma ki ng I It t ( postu lationally ) ’ cl os ed 1, wo ul d not be ’ complete ly cl os ed 1 — to use concepts of the t hird part of Elemen tary Prop osi tio ns. Actually, quite a num ber of these systems should admit a postulational development in the ordin ary se sens nse. e. And, ind indeed eed, , thro ugh developments of Lukasie-

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INTRODUCTION

7

wicz, Tars wicz, Tarski, ki, Be ma y s , L6sni L6sniewski, ewski, Mihallescu, and Wajsberg, four of these systems ma y be said to have bee n s so o treate treated. d. In pa r ticular, we may refer to developments in terms of equivalence only ([14], [ 3 3 h   and [16]),1 [16]),11 1 whi ch generates the syst em we sym bol ize 2 , dual (§1 5 [18], ), equivalence n [1 7 ], and. equiv alence and Lits each pair and of negatio which functions generates L 1 , (§1 5 ) , 12   im pl ic at io n ([34], [10] [10] ), wh ic h gen er at es F^° , (§2 2 ), and. implication and conjunction [10] and perhaps equiva lenc e a nd di sj un ct io n [1 9 3> among other others, s, w hic h generate the im  p o r t a n t C 2 , ( §1 9 ).1^ We might add that where an o rdin ary pos tulational development is not possible, a complete logic might be given for a system in other wa y s .1^ Whi le our mai n proble m and its soluti solution, on, ret retra rans nsla late ted, d, in terms of truth-tables, fits hand in glove with developments of two-valued propositional calculi bas ed on a finite number of primitive trut truth-fun h-functions, ctions, difficulties in formulating a conc concept ept In connection with these developments, and also the two immediately succeeding, see [24],

1o

The re vie we rs1 obser vatio n that not all enunciations in terms of these primitives capable of proof in ordinary logic can be dedu ced here makes it q questionable uestionable whether we can say that the postulates are for system L - . Thou gh here it ma y be ju just st a case of incompleteness, in other instances it ma y be that not the two-valued logic is under consideration. ^ We ma y say that as syst em C- is the con cern of the com plete two-valued calcu calculus lus of prop propositions ositions, , system 0  Is the concern of the po si ti ve l og ic 1, ([9 ], p. 68). In thi s co nn ec  ti on C,, C,, the dual of C , Is dis tin gui she d by the fact that under tne functions functions whose of classes it consists of all such values,interpretation for given classes, are inde pendent of the particular universal cla class ss containing t the he given classes. class es. Retur ning to the postul ation al developments, we must admit that the intent of the authors is not to develop a pos tulational development development for a give n system via certain prim i tives, but merely to study the inter-relations of those primi tives amon g themselves themselves. . Thus Thus, , in [10] a again, gain, we fin d a study of expressions built up by implication, conjunctio conjunction, n, an d negation, while either of t the he first two primitives with the last generate the complete two-valued system. 1^ Thus, in the duals of the first class of systems a 1cont ra dict ory lo gi c1 is possib le (se (see e [1 [11] 1]), ), while in all cases it ma y be possible to set up a complete logic solely by means of rules of derivation (see [2 0 ]). ]).

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8

IOTRODUCTION

of identity of truth-functions, as against mere equivalence, pre vents us from giving even a definitive formulation of the prob lem of determining all all iteratively closed systems of truth-f unc tions . That our sol ution of the cor respond ing pro blem for truthtables tabl es ma y here be of little help is indic ated by the fo llowing specialization of the gene general ral proble problem. m. In the propositional calculus of Principia Mathematica let two truth-functions be con sidered identical identical when an d only whe n their express expressions ions in ter terms ms of the primitives ~ p and p v q are identi identical. cal. The concept of an iterati vely closed system of (~,v (~,v) ) truth-functi ons i is s then precise. But whil e to each closed system of (~ (~,v ,v) ) truthfunctions there corresponds the closed system of corresponding truth-tables, the problem of determining all such closed systems of truth-functions is obviously entirely independent of the cor responding problem for truth-tabl truth-tables, es, being a problem in symbol structure only. only. A s olution of this prob lem in the same sense as that of the present pap er is hardly to be expect expected, ed, since the card ina l nu mb er of (x >xi2 i2 >.. >... . by X 1,X2,...,XQ respectively. And. the exp res sio n f (X1 ,X2 , ... ,X ,Xn n ), wh ic h not on ly rep rese nts this function but indicates the manner in which It is to be ob tained, will be referred to as an Iterative process depending on th e va ri ab le s

x^ * *x x j2 > • • •*x jm

811(3 yi el di ng the fu nc ti on in

on See footnote 5 (Introduction). 1o

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§1 . IT ERA TIV E CLO SED NES S A N D GE NER ATI ON question. We now define F to be iterat ively closed, wit h re spect to V ifreplacing if replacing the arguments of a func tion in P by variables in V or functions inP in P al always ways results in a func  ti on in F, i.e i.e., ., i if f wh en ev er f ( x ^ >x -^2 > • • ,x ,xin in ) a fac ti on in P on dis tinc t var iab les x* ,x. ,. ..x ..x. . , an d X,, 1n In F, X g ,.. ,...XQ .XQ represent variables in V or-M functions then • the f unc tio n yielde d b y the iterati ve pro cess f (X 1 ,X2, .. .Xn .Xn ) Is also in F. A system of functions iteratively cl closed osed wit h respect to V wil l also be r ref efer erre red, d, to as an iterati vely closed p  i system over V. A functi on of n distinct independent variables will be said to be of order n. We then Immediately verify tha that t an iteratively closed system F either consists wholly of first order functions or else pos sesses functions of every finite order. For if F does not co consist nsist who lly of first order functions It will possess some funct ion f ( x ^ x ^ , .. ) of order m .> 1 . Then, by the closedness definition, the first order function f ( x . , x . , ...x. ) be lo ng s to F, and. if th e n -t h or de r fun c"1 y 1 J1 ti on g(x. ,x ,x. . ,...x. ) is in F F, , and x . is disti nct J1 32  Jn Jn+ Jn+1 1 f r o m x . , x . ,. • .x • ,f [ g( x . ,x . ,.. ,x . ) , x . ... .x . 1 wil l Jl J 2  Jn J1 3 2 ’ Jn Jn+1 Jn+1 be an (n+1 )-th )-th ord er fun cti on in F. The resu result lt follows by mathematical induction. Our proof actual ly s shows hows that i if f a closed system F does not consist wholly of first order functions, then it possesses at least one function of each finite set of variables belonging

to

V.

Now If

x ±i ,x^2 , .. ,XjLn , an and. d. x j - | ’**X J*n are 8I1^

two sets of n distinct vari able s in V, then, if f(x^ ,x^ ,.. ...x. is in F, f (x. , x . ,.. ,.. .x . ) w i l l al so b e in F , 1 a n S -Ln ji 32  Jn conversely. conve rsely. It easily follows tha that t the totalit y of functions in F on the first set set of n variables is transfo rmed in 1-1 fash ion into the total ity of functions in F on the second secondset set of n variables by any 1-1 replacement of the variables in the first set by the variables In the second se set. t. We likewise see that a closed F whic h doe does s cons consist ist wholly of first order functio functions ns  p i

Or iterat ively closed syste system, m, or close d sys system tem. .

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I. PRELIMINARIES possesses at least least one functi on o of f each sin single gle variable in V; and the totality of functions in F on one variable is trans formed in 1 1-1 fashion into the totali ty of functions in F on any other variable by replacing the first variable by the second.. We ma y therefore say that the functions in F on any one set of n variab les are me rel y repe repeat ated ed, , over all other set sets s of n variables belon ging to V. This redundan cy can be overcome as follows. Let V be simpl y orde ordere red, d, in the series ,x2 ,x ^, . ...., .... , an and. d. let le t J   be composed of those functions in F whose arg ume nts are (x^ ),(x^ ,xQ ), (x^ (x^ ,x2 ,x^ ), etc. Th en J   posesses all of the essentially different functions to be found in F pp wi th ou t the l a t t e r ’ s dup lic ati ons . IF wi ll be call called ed, , the cont contra ract cted ed, , close d system corre spon ding to F. Note that wh en F consists whol ly of first order functions functions, , ? possesses only func tions of x ^ ; otherwise, ^ posses ses at least one fu nct ion of (x1,x2, ... ...x xn ) for every positive integral n. We shall often implicitly use the concept of a contracted, closed system for the purpose of exhibiting the functions possessed, by a closed system. On the othe r hand., the replace ment process, wh ich is the basis of our concept of iterative closedness, demands the very duplication of the ’ same fu nc ti on s’over di ffer ent sets of vari abl es fo fou und nd, , in cl clos osed ed, , systems, an d del delet eted ed, , the ref rom in fo rm in g con contra tracte cted, d, closed sys system tems. s. Our derivations will therefore explicitly refer to closed, systems as originally defined..2^ Ulti mate econ omy wou ld be achi achiev eved ed, , by form ing the set of all ’ abs trac t fu nc ti on s ' cor res pon din g to fun cti ons in F, two functions bein g said to correspond t to o t the he sam same e abstract func  tion if they can be made Identical by a 1-1 replacement of ar guments. The resu lting ’ abstrac t closed syste system* m* corr espo ndin g to F Is, Is, however, not as prac tica l as 7. It wo ul d carr y us too far afi eld to relate these ideas to the concept f (x 1,xp ,.. ,... xn ) of Princ Principia ipia Mathe mati ca. 2^ The requirement that V be a denume rably infinite clas class s is In agreement with the usual form of a propositional calculus. Note that were W any Infinite clas class s of variables, G an iter ative ly closed system over W, the then, n, if each funct ion in G Is on a finite numbe r of arguments, a system F Iterati vely closed wit h respec respect t to a denumer ably infinite clas class s V could be con stru cted such that the abstrac t clos ed systems systems, , in the sense of thelast the last footnote, correspo ndin g to G and F are iden identical. tical. If then V be inf infinite inite, , it ma y as well be denumerab ly infin infinite. ite. V finite, however, l leads eads to somet hing new. Note that if F Is iter ativ ely clo closed, wi th respect to V, the n if V 1 is an exis tent subclass of V, F f the class of functio ns in F wi th ar-

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51. ITERATIVE CLOSEDNESS AND GENERATION

13

We give an independent definition of the system of functions gener ated by a given existe existent nt finite se set t of function functions. s. Var ia  bles and values are to be in V and. v as hereto fore. Ther e is no loss of generality in writing the functions that are to genera te the system in the for m f (x1 ,xg, .. . x ^ ), i = 1,2 1,2,.. ,.. .v, n^ ^ 0. 0. We t hen define the sys tem of function s ge gene nera rate ted, d, by these generators over the given denumerably infinite class of varia bles V as the class of all functions that can be a ass ssig igne ned, d, to the system by the use of the follow ing cr criteri iterion. on. If X 1, X 2 ,.. «X «Xni ni> > 1 = 1,2,. 1,2,. . .v, rep rese nt var ia ble s in V or fu nc  tions in the generated system, then the function represented by f^(X.j ,Xg , .. . X ^ ) also belo ngs to the gen era ted syste system. m. We shall say also that every function th thus us obtainable can be g enerated by the give n generators. Note th that at in the initi initial al application s of this inductive defi nitio n X 1 ,X2, .. . X ^ can only be variables in V. It follows from this definition that each function in the generated system can be expressed in fini finite te for m in term terms s of the genera tors of the system system, , and vari ables in V. The se expr express ession ions, s, or operations as the y may be cal called led, , can be clas classi sifi fied ed, , accord ing to the ir ra n k . It is conv enie nt to re ref fisr to va ria ble s In V as operations of rank rank z zer ero. o. The n i if f the highest rank of the oper  atio ns I 1 ,X2 ,.. ,.. .Xni is p p, , t he op er at io n ,X2 , .. . X ^ ) wi ll be s sa aid, to be of rank p+ 1. We also indu ctiv ely define the comp onen ts of f^(X^ .X „,. „,.. . •V to be X 1 ,X2 , .. . X ^ and. the ir components, a variab le in V be ing -u unde nderstood ood, to hav e no components. Clearly, an oper ati on of rank p has at least one compo nent of eac h rank less th an p. Thus, wi th f 1 (x^ ,x2 ) an d f 1 !f2 !f2 (x 1 ,x2 ),f1 [ t ^ X y X ^ ) , x Q ] | is a n operation of rank three; and among its components are the opera tio ns f 1 [f2 (x ^, x1 ),x2 ],f2 (x ^, x1 ),x^ ),x^ of rank s t two, wo, one, z ero, respectively. respec tively. Assoc iate d wi th the g gen ener erat ated ed, , system of functions is thus a system of operation operations. s. Each operati on of ran rank k greater than zero in the latter system defines a unique function in the former. On the othe r hand., for most of the gen era ted systems f 2 (x^,x2 (x^,x2 )  as gen era tor s,

guments in V * , the n F f is iterati vely closed wi th respect to V 1. This resu result lt easily transforms the solution of the principa l prob lem of the present paper paper, , give n for infinite V, into solu tions of the same prob lem for finite V !s. In this connection see also the next footnote.

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I.

PRELIMINARIES

that we shal shall l stud study, y, each function in the former sy stem is yie l ded by an infinite numbe r of different operations in the lat latter. ter. By means of these operations we readily verify that if f (x. x.^ ^

• • •x in )

is a fu nc ti on in a ge ne ra te d sys tem on di s

t i n ct v a r i a b l e s • • *x j[n > 8X1(1  re P pe pese serr rrt t variables in V or functions in the sys system tem, , then the functio n re pr es en te d b y f (X.j ,X2 , .. .X^)  also belong s to the syste system. m. That is is, , ev ery gen genera erated ted, , sys tem is it era tiv ely cl os ed . Clearly, if each function in a given set of generators belongs to a closed system F, then the system of functions generated by t those hose gen erators is conta ined in F. Hence, if in addition, each func tio n in F can be generat ed by those generato generators, rs, the genera ted system is F. There is no a-pri ori reas reason, on, however, for an arb itr ary closed system admitting of generation by a finite set of genera tors. We define the order of a finite set of generators as the highes t orde r of the several functio ns in the s set et. . If, If, then then, , a clo closed, system can be gene rate d by a finit e set set of generators, we define the order of the system as the lowest order that a finit e s set et of generat ors of the system can have. The phrase, 'closed, system of finite' order 1  is then equivalent to the phrase 'generated, system 1  provided the number of generators is underpii stood, to be finite. Since functi ons of one varia ble can only generate functions of one variable, it follows that a closed system of the first order consists wholly of functions of the first orde order. r. On the other hand hand, , a closed system of finite order greater than one must possess at least one function of order grea ter th an on one, e, and. hen ce p osses ses functio ns of e very order. The concept of generated system, however, continues to be vali d if an infinite numb er o of f generators is allow allowed. ed. This less les s stringen stringent t conce concept pt lea leads ds to a correspondin gly wid er con cept of 'closed, syste m of finite order'. The foll owin g obs er vatio n assumes this wid er co conce ncept. pt. Clea Clearly, rly, a closed s system ystem over V of finite order n is determined by it its s n-th and lower orde r functions. It follow s that if V» consists of say the first n variables in V, then there is thus determi ned a 1 - 1 correspondence bet ween the closed systems over V of order les less s than or equal to n and the closed systems over V' in the sense of the preced ing footno footnote. te. See also footnote 39  (§1 0 ). Where, as in th the e proble m of t the he present paper paper, , there are but a finite number of functions on a given finite set of arguments, there is no diffe rence bet we en the two concepts of 'closed sed, sys tem of ord de er n ' .

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§2. TWO -VA LUS D SY STEMS OP FUNCT ION S

15

Continuing to to restrict restrict ou r attention to function functions s whose ar guments are in V and values in v, we define a conditi on im posed. up on such functions to be iter ative 2^ if when eve r f ( x ^ y x. ,. ..x ..x. . ) ), , on di st in ct va ri ab le s x. , ,x x. , ,.. .. .x. , sa ti sf ie s 1  2  in -*-1 12 11nn the condition, and. X 1,X2 ,...XQ repres ent varia bles in V, or functions satisfying the condition, then the function represen ted ed, , b y f ( X 1 ,X 2 , .. .X^)  also satisfie s the condition. Clearly, if each of several conditions Is iter ative, then the combined condition that a function satisfies each of those conditions is iterative. There are two important consequences of a conditi on bei ng iterative. iterative . On the on one e hand, the system of all functions s atis fying the condition i is s iterat ively cl close osed. d. On the oth other, er, if each function in a given set of generators satisfies the condi tion, then every function in the generated system satisfies the condition. condit ion. We shall in every case say that if each func tion in a closed system satisfies a certain iterative condition, then the system satisfies that con condition dition. . The discove ry of iterative conditions thus plays a very Important r8le in the determination of iteratively closed, and generated., systems of functions. §2

TWO-VAL UED SYSTEMS OF FUNCTIONS and THE RELAT ED L LOGIC OGIC OF CLASSE CLASSES S Variables , functions, an d s systems ystems of functions for whi ch the class of values v consists of but two memb ers wil l be s sa aid, to be two -v al ue d. The fol low ing dis cus sio n is res restr tric icte ted, d, to a single cla class ss v whos e two members are symbolized + and - re spectively. Our definition of the identity of functions on the same set of variables implies the adoption of the extensional point of view wi th res respect pect to function functions. s. Eve ry two- value d function function, , 2^ We have bor borro rowe wed, d, the wo rd "iterative" in these connecti ons from Wie ne r (s (see ee footnote 6) 6). . But our "iterativ "iterative e condition" i is s not the same concept as Wiener's "iterative characteristic" [41].

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I. PRELIMINARIES can therefore therefore be uniformly exhibited in finite form by a table such as

p,(B (B), ),

i.e. i.e., ,

B:b

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by

 

52

II. DERIVATION OF CLOSED SYSTEMS

0:1 :1. . f(A,B) wit h expansi on A+B:a b becomes a funct ion g(A,B) wit h expansi on A:a. The system therefore has the a-func tion g(A,B) g(A, B) wit h relev relevant ant variable A, irrelevant variable B, and hence possesses all a-functi ons reducible to functions in S 1 . On the other hand, the presence of p-functions In the system is consistent wi th all of its a-fu ncti ons be ing in S 1. Thus, if any variable in a logical sum funct ion be repl aced by a pfunc tlo n wi th expa nsio n 1 :o, the exp ansi on of the logical sum fun ctio n also reduces to 1:0, 1:0, and but a fSfS-fun functi ction on results results. . The various various possibiliti es are now easily s see een. n. We shal shall l continue the symbolism use d for the R syst systems ems, , and in addi tion use cr to symbolize an arb itr ary logical sum function, a an arbi trary functi on reducible to a logical sum functio function. n. Then the only closed subsystems of the complete logical sum system other than closed systems reducible to first order are the fol lowing. S 1 : [o^-j jcr],

S g : [ a ^ ,o ] , a],

S^ :

>o ] ,

S ^ : [ a ^ ,(3 ^ *ex]>

S ^ : [q^ ,*y ^ ,

S 6 :[ S 1 , ^ , ^ . , 0 ] .

The above deriva tion shows tha that t each of these 'lo 'logic gical al sum sys tems 1, can be gener ated by a suitable choice of generators from the functio ns P,(A), ^ ( A ) , f(A,B) wi th exp ans ion A+B:ab, g(A,B) g(A,B ) wit h expansion A:a. Hence they are all of the second order. The dual of the exp ans ion

A +B + . . .+I: ab.. .1   is the exp an

sio n A B ... I :a + b + ... +1 . Hen ce the dual of a logical sum fun c tion Is a 'lo 'logic gical al product function', i.e i.e., ., a func tion whi ch is the logical product of two or more distinct variables, while the dual of a function reducible to a logical sum function Is a functi on 'red 'reduci ucible ble to a logical product function function'. '. By the principle of duality we may then immediately write down the closed systems which have no other functions than those reduc ible to first order a,  (3, an d -y fun cti ons , an d to lo gi cal product functions, and whic h are not themselves reducible to fir st order. In tr od uc in g the sym bol s it an d Tt as dual s * of a and o,  we thu thus s have the follo wing si six x 'lo 'logic gical al product ----------------------------£0-------------------: Allowing a closed system t to o be a n improper subsystem of itself.

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§14. [g,P, [g,P,-y] -y]

SYSTE MS

53

sy st em s1 which, in order, are the duals of the above six logical sum systems, and. which are also of the second order. •^1 •

],

Pg * t^ i

^9

1 

  C^i

*

[0£^, £^,P^, P^,TC ],

3 1 

Pj ^ • t ^ T t

]i

• [0£-| ,"V i ,|j |j^ ,Tt Tt ] • §14

[a,p,-y]

SYSTEMS

By refe rring to §8 we see that the o nly functi ons sati sfy ing the [A:a [A:a] ] cond itio n are p an d -y-y-funct functions ions reduc ible to first ord order, er, and a-functions whi ch admit a normal [A [A:a :a] ] ex pansion. Since the [A:a [A:a] ] con diti on is Iterative, the syste m of [A:a :a] ] mfusymb nct ion s edis anclosed. W eed thus hav e the cl clos ose ed. [a,pall ,-y] [A syste oliz d desc rib as follows. A 1.

Al l functi ons satis fyin g the

[A:a [A:a] ]

condition.

Now suppose that a funct ion belon ging to a closed [a [a,P ,P/y /y] ] syste m does not satis fy the [Asa [Asa] ] condition. Then, as seen in §8, ther e wil l be a te rm in the f irst ex pr ess ion a an nd. a t er m in the second expression of the complete expansion of this function with respect to which the arguments of the function fall into at most three groups as follows: te rm in the' first exp re ss io n : cap ital ter m in the seco nd exp ress ion : capit al

small small

sma small, ll, capi capital tal. .

Clearly, the third, group cannot be empty, or the two terms would not be dist distinct. inct. Now replace the variables in the first group by p^ p^(A (A), ), in the second, by ^ ( A ) , in the third by A. The two terms become a and A respectively, henc e the complete exp ans ion a:A a:A, , an d the fun cti on 61 (A ). The gi ven clo closed, sed, sys tem woul d therefore possess the function a> either its a-functions are all reducible to first order functions and logical sum functions, or they are all reducible to first order functions and logical product functions.

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§15. AL TE RN AT IN G SYS TEM S

55

As the ft an d *y*y-func function tions s in the syste m must be redu cible t to o first order, it follows that the sy stem can then only be an 0 0, , R,

S,

or P system system. . We can ther efor e conclud e that the onl y systems are 0 Q, R 1Q , R ^ , S^, S^, P 6 , a nd A 1 .

§15

ALTERNATING SYSTEMS A functi on will be said t to o satisfy the alternating condi tion if replacing an ar argument gument of the funct ion by its negative either leaves the func tion unchanged, or changes It Into its n e g a t i v e . 1+1 B y §4  the arguments of the first kind are the ir relevant variables of the function function, , those of the second ki nd its relevant variables variables. . The set of relevant variables of an alte rnati ng func tion will be called its relevant subs et. Clear ly ly, , If any number of argumen arguments ts of an alternating function are changed Into their negatives the function is left unchanged, or is turned into its negative, according as an even, or odd, num ber of variables in its relevant subset have been changed Into their negatives. It follows tha that t the altern ating condi tion is it era ti ve . Fo r if f(M,N, ...W ...W) ) is an iter ative proc ess built up out of alternating functions and depending on variables A, B, ...I ...I, , changi ng an y one of these vari ables into its neg ati ve leaves M, N, .. .W uncha nged or turn ed Into Its negative, an d hence leaves f(M, f(M,N, N, ...W ...W) ) unch ange d or turne d into its its neg a tive. From the definition of a n alternating function we se see e that a compl complete ete expansion correspond corresponds s to an alternating function whe n and only when interchanging any capital and corresponding small letter throughout throughout the expansion either leaves the expans ion un changed, chan ged, or has the effect of intercha nging the two expressions of the the expa expansion. nsion. Since in the first case the letter in ques-

For truth-functions a typica l exa mple of an 1alte rnati ng function* Is p - q q. . Thus Thus, , rei nterpr eted as closed systems of truth-functi truth -functions, ons, L below may be said to be generated by equivalence, L_ by the dua dual, l, wh ic h is also the negative, of equiva lence, Li' by, fo r example, equ ival ence an d negatio n.

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56

II. DERIVATION OF CLOSED SYSTEMS

tion corresponds to an irrelevant irrelevant v ariable of the functio function, n, In the second to a relevant variable, It follows that two terms of the complete complete expansion of an alternating function which differ in the capitalization of but a single letter are in the same or different expressions of the expansion according as the letter corresponds to an irrelevant or relevant variable of the func tion. tio n. By successive appli catio n of this spec special ial re result sult we ob tai n the follo wing general re su lt . Two terms of the complete expansion of an alternating function which differ in th the e capi talization of any number of letters are in the same or differ ent expressions of the expansion according as an even or odd number of those letters correspond to variables in the relevant subset of the function. In particular, particular, a term of the complete expan sion of an alt erna ting fun cti on f(A,B f(A,B, , ...I ...I) ) is In the sam same e expression as AB...I, or In the opposite expression expression, , accordi ng a as s a n even or odd numbe r of the small letters in the the term correspon d to va variables riables in th the e relevant sub se set t of f( A, B, .. ... .1 ). An alterna ting functio n f(A,B,...I) is therefore deter  mined ined. . by its relevant subse subset, t, an d the ex pres sion of its complete expansio n in whic h AB AB. . ..I appear appears. s. Corre sponding to any s set et of variab les A,B, ...I ...I, , and subset of it chos en as the relevant subset, there are thus but two alternating functions, one with AB...I in the first expression of it its s complete expansio expansion, n, the other wit h AB...I, in the second expres expression. sion. Dual terms differ in the capitalization of all their let te ters rs. . Hence Hence, , whe n the relevant sub subset set of an altern ating func tion has an even numbe r of members, dual terms of the complete expan sion of th the e functi on are alway always s in th the e same exp expression ression, , when odd, in different expressions. Alternating functions with odd relevant subsets are therefore selfself-dual, dual, and conversel conversely. y. By considering the dual terms AB...I and ab...i we furthe r se see e that alternating functions with even relevant subsets are p or ^-fu nctio ns, those wi th odd. relevant s subub-

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§15. AL TE RN AT IN G SY STE MS sets set s

a

or

57