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(3) KFKI-75-24. EASILY COMPREHENSIBLE MATHEMATICAL LOGIC AND ITS MODEL THEORY H . Andréka INELOR Systems Engineering Institute, Budapest, Hungary. T. Gergely Central Research Institute for Physics, Hungarian Academy of Sciences, Budapest. I . Németi Computer Center of Institute of Industrial Economy & Organisation of the Ministry of Heavy Industries, Budapest, Hungary.. ISBN 963 371 023 5.

(4) KIVONAT. A tanulmány újszerű közelítésmódban tárgyalja a matematikai- logikát. Rész letesen foglalkozik a nullad-, első- és másodrendű nyelvekkel és logikákkal. E logikák bizonyitáselméletének és modellelméletének is részletes leírását adja. Több alapvető tételnek pl. a teljességi tételnek uj, egyszerűbb bi zonyítása található benne. A tanulmány különösen hasznos anyagot nyújt a számítástudomány azon szakemberei számára, akik magasszintü programozási nyelvek, valamint automatikus tételbizonyitó programrendszerek, kifejleszté sén tevékenykednek.. ABSTRACT Present study discusses mathematical logic in a new approach. The zero-, firet- and second-order languages and logics are discussed in details. Their proof theory and model theory are also described. New and easily comprehen sible proofs are given for the basic theorem e.g. completeness theorem. It provides useful material to those engaged in the field of computing sciences especially in the development of high-level programming languages and auto matic theorem proving program systems.. АННОТАЦИЯ Настоящая работа посвящена изложению математической логики с новой точки зрения. Подробно излагаются языки и логики 0-го, I-го и 2-го порядка, вместе с их теорией доказательств и теорией моделей. Приводятся новые и легко до ступные доказательства основных теорем, так например теоремы полноты. Работа представляет интерес для тех специалистов науки вычислений, которые работа ют в области создания языков программирования высокого уровня и программ для автоматического доказательства..

(5) I, INTRODUCTION. In the present paper dealing with purely mathematical. in. vestigation of both classical first-order language and logic, the intuitive mathematical language will also be used, for clearer comprehension, as meta-language. In other words, the intuitive interpretation of ooncrete mathematical objects appearing in the mathematical con struction of the logic under investigation is described using the English terminology of mathematics. As it is well known, language and logic, are closely con nected with human thinking, more precisely with the acti vities of problem solving, knowledge aquisition, etc. Mathematical logic on the other hand is the mathematical study of these activities as they are connected to lan guage and logic.. Let us first consider the intuitive, everyday interpreta tion of language. Language is the means of cognition of objective reality. In that the world manifestly is diver sified and complex, thus a set of possible worlds becomes existent relative to the subjective mind during the course of cognition. Let us now investigate the role of language in the cogni tive process. Language must be able to describe the possible у.

(6) - 2. worlds in some extent, hence it must be based on a finite set of symbols and rules /grammar/ that underline the formation of expressions suitable for the description of the different properties of some or other worlds. Consequently a language must contain the following components: Language = ^syntax, bole. L= < F ,M. set of possible worlds, validity^.In sym-. ,K>.. Syntax oonsists of a set of strings of symbols /texts/ and the system of rules working on these texts. Validity is a binary relation characterizing the connection between the syntacti cally oorreot expressions and the possible worlds. The set of possible worlds together with the validity relation form the semantics of the language. It follows from the role of language that the modelling of semantics is primary over that syntax. It is for this reason that the so called Model Theory is of primary importance. In order to be able to make valid statements about worlds and in order to conclude from the world models we must switoh over to logic from language, Logio is comprised of the pair \ language, calculus)' , Caloulus is a system of rules of inference that permits reasoning about the worlds,. %. Mathematical logic is characterized by the fact that within its framework the possible worlds are represented by some mathemat ical objects. These mathematical objects are called models, /They are the mathematical models of the possible worlds,/.

(7) 3 -. So, in mathematical logic, the set of possible worlds is re presented by the set of models. The individual brandies of mathematical logic differ from each other in the mathemati cal objects used for the representation of the worlds. E.g. classical mathematical logic defines the models as structures. The mathematical theory of semantics is called Model Theory. The construction of a calculus to a language belongs to the so called Model Theory, and the refinement of already exist ing calculuses belongs to the so called Proof Theory. In this work wo restrict ourselves to the investigation of classical logic.. II. BASIC DEFINITIONS AND NOTATIONS. In this work we strongly rely on the tools «und concepts of naive set theory. Now we introduce the notations : iff. is a shorthand for "if and only if".. w.r.t. is a shorthand for "with respect to". Ж. marks the end of proofs. marks the end of other units e.g. definitions, examples. that is. denotes the set.

(8) k. -. Two sequences. <űbL. n. all. CL^ =. b ji. A А П. ,. -. and. ^. are identical iff for. ^n. -. An n-ary relation. is a setof sequences /of length n/ i.e.. n-ary relation on. the sets A. ,A. Aj. an. is a subset of И 4 X-. n. X /4,? .. A function is a set of pairs. Let the function f be suoh that. j- Q.. is the smallest set for which -fma In of f у ;A. The range of f(Rgf). ХЬ/. Similarly the do-. /А X. j. Q. D. o. X g> •. is onto if Rgf s B,. AD. f from A to В is a set of sequenoes of. i.e.. П ~/~у[. Q A x .. Х А. Thus an n-ary function is an. X. П-h í. .£>. i.e.. A i X-.--. .V A n d. \ccil. (...(] \,. cL^y --- ' v. \<ч. h/ \ — > £ > .. .J. Z. ary relation. This notation. is justified by the fact that to malte fine. .. the smallest set for which. An n-ary function length. X. means that Do f = A and Rgf ^ B #. —> JA. A. -f :. (bo f) is. А В. X. associative we de-. A d X / \ 3 )X. . V ' \ ; ,,. cz±, ol^ \jci3У;. ... , y. a.A. Mow we show how to use functions and relations, -j- •. If. ^. В. CLrb /. b y. / \. then €. ^ ( Cl± , . . .. В. mean« that. f .. we state that the elements. г where. f (AfTi ) ~. T Q /\$_Х-- У. C i L}..■, CZn. are in relation with. then this means that. x d £,. • CZ/Z/" ^ ^ •. Cf* \ c Z j _ i - . - j CD/ b A ^f- D. then the relation r does not hold. between the elements. } (.Zn. bhe function. f. c l ± / ... , If. is not defined on. DoJ. a.. then we say that.

(9) 5. A function is a special relation. The property which distinquishes a function from an arbitrary relation is the uniqueness of its values, that is A-. \. Let. r. j. implies that. Ь. is a funotion iff. *\С1/. ^. '. be an n-ary relation /or an /7-4. — ary function/. Then. the relation /or function, respect ive 1у/ 'T"/") П s triction of. r. to. P). is the re-. B. \. Now we introduce the conoept of algorithms.. II. 1. Concept of algorithms. ¥e shall use the intuitive concept of algorithm. To state our theorems it is enough to postulate some basic properties of algorithms. These postulates are satisfied by any intuitive concept of algorithm as well as by any mathematical concept of algorithm defined so far. It is left to the reader to de cide which concept of algorithm to choose e.g, computer prog ram, Turing machine, Markov algorithm, recursive functions.. We postulate that there is a finite set of symbols say. Z. by. which each algorithm can be described. That is any algorithm is defined by a text /finite string of symbols/ from. Z.. Let. Z* stand for the set of all finite strings of symbols from Z. /Thus an algorithm - or the description of that algorithm is on element of Z N /,.

(10) -. 6. Each algorithm ALG has a meaning: the execution of ALG, ,From this meaning we consider only the partial function ALG from a subset of ALg (x ). в. Z. Y. into. Z , For any X, Y f ALG6 Z' the statement. means that ALG is an algorithm and if we execute. ALG for the input. X. the execution stops in a finite number. of steps and the result is Do. A L G. then. Y,. The notations. Щ ALG. are taken in the strict sense, i,e, if. ALG. and. D0. f\L (s. is defined on it.. But for convenience purposes, just in the case of algorithms we sometime abuse the notation f we write. ALG :. is included in. tA. —. Z *,. A. - A. *even if. b. Do ALG-. We suppose that the set of numerals. ~X~. *7. Z and by a slight sloppiness we write cO ^ >0 ,. We further postulate that if we have a straightforward defi nition of a symbol-manipulation procedure then it is an algo rithm, and if we combine two or more algorithms in an intui tively straightforward way than the new object is again an / algorithm.. An algorithm ENU enumerates a set _^ and. iff. • An algorithm DEC deoides. Li о I !л ^ (j. elements of. B<Z Z. Ъ. where. and for any. Proposition 11,1,. Let. /\fj \. and <£. В c Z ,. aL. J \^ '. b o W U = со BA. Z. iff. and. Ibor L ) t C " / . .. are two distinguished. DAC* (А ) -. c f-f. Д t fc).. From any decision algorithm. /for В/ we can always construct an enumerating algorithm but not vica versa.. Now wo introduce the concept of language..

(11) - 7 -. II#2# Concept of language. A language. L. is a triple: \=z-. tences/ ,. where F is the syntax /set of sen. z7. M, F ^. is the semantics where. M. is the. class of models /class of possible worlds/ and is the validity relation, i.e, and. őt 6. C%. Let. ^. Y. Y 'f. then. Zj. cYY-. F , c. symbols Л of A. F jJ. F. 1 ). iff. F-- ( j /. C. is a model of. iff for all. In this case we also say that. Let. {j. is. f. we also say that (.. (f. Yj. is a sentence. ^ G. I i3 a model then di /=■cf means that. valid /true/ in If. if. F ^ MX F. d. t. (y9 . 1. (~j? €, ]>u. ^. is a model of. is a consequence of £. ■. all models of. ^. /i n. are also models. Y. .. G- í~. is a tautology of L /in symbols. set of models of ' j The symbol. hf- ^j. is exactly. h :(j /. iff the. M#. means that it is not the case that ( //. / A logic is a pair <. \ L, CAL. /. where. L. is a language. and CAL is'a calculus for L # An algorithm is a calculus for L.. An algorithm is a calculus for L CAL. #. enumerates a set of consequences of. CAL is complete for L iff it enumerates the set of. all tautologies of L. given of. if to any given. ipf”j. CAL. CAL is adequate for L. iff to any. enumerates the set of all consequences.

(12) -. Л logic. * L, СAI,. complete. /о т. is complete. (PeZ. adequate/ If CAL is. /о т. adequate/ for L. Note, that if CAL is a calculus. for L and to a given for any. 8. F j also. CAL enumerates the set. (j } j ’ h '- -. Ц у'. X FZ. F. , then. .. Л language bas implications iff there is a sentential connec tive —. such that for any. CJ? ,. G. h. } (У С G. ;. M. /. Proposition II. 2. plete for. L.. Let. L have implications and CAL be com. Then an adequate caloulus CAL*. for. L. can be. obtained from CAL in the following way: C A L ': F a nd. X UJ — >. such that for any. J~~. П < LCŐ ijy. CAL Proof#. Ш .. (V.n. =. u f. ■ C A L. Cm). 9G_ t~. ■= (. otln c r w i s e. The proof is immediate from the definitions. THE MODELS OF A T-TYPE LANGUAGE ( /^f ^ ). In our approach every language has a fixed type define the concept of type. Throughout this article. is a fixed symbol.. t. Now we. y / ).

(13) - 9 -. Definition 111,1 , A type is a pair of funotions. 1./. such that. b. 4>. , b. is disjoint from. > €I. **•/ < c O / о. Do t. L. R.O -Í'Q CO \{01 ] 2./Ras. 3#/ Dot. Do t. =r. Dot. n. II. is called the set of relation symbols, and II. is called the set of function symbols». If r is a relation symbol, then t ( r ) is its arity, and simi larly for function symbols, /Note that the purpose of <(COJ (//><£ b. will be seen at the de. finition of t— type models /see Definition 111,2,/,. Remarks where by. In this paper we restrict ourselves to such types t,. R ^ - L '!. ßq í. I. R c j i - “й и з ,. ^Cj. í. ChfőL/. This restriction can be easily replaced when. Ord. is the class of ordinal num. bers. Though, these generalized languages are not classical in the strict sense. Such generalized languages were investigated in. 0 41.. .. I. In this paper we restrict ourselves to the case where. t and t. are countable, moreover we suppose the existence of algorithms II. ALGT' is. and. ALGT^ whioh enumerate. M J Z T 1:. for whioh. such. ALGT. t' and t. that for any </T,/?>£. (m) = K r / П У''. R. respectively- That there is an. YYl <. similarly for ALGT .. A t-type model is a set on which fixed funotions corresponding. //.

(14) Io. to the function symbols, and fixed relations corresponding to the relation symbols from. t, are defined.. Definition III.2. A t-type model is a funotion c%. such that с} { ( с. set, and for all relation symbols Л / т /. is a. i ( r ) -. tion symbols f,. Ő. т9 Л С Т )Э А .. Л /\± в. / \. a. /that is. ary relation on А/, and for all funo ->/f and if t /f/ = 0. t ( f :. then. Э й р е A. That is the zero—ary funotion symbols are the constant sym bols /and the corresponding elements in ? are the constants/.. Now we M. define the d a s s of t-type models ~L. which. is Ä t-type model. Ő. Note that. < /c 0 , о У Э 1. 3tCo)-j)/ from. M. l. :. \\. serves to exclude the empty model /for M^#. Notation: We denote models by German capitals, and to denote the values of a model. as of funotion we use the following notation:. őt ( o ) ^ X é. /the corresponding Roman capitals/. L/L, (7") =. ő t (f). -. t%f. for all. nr. O o i. for all^. £ t)o. Í. ,. .. If it is not stated otherwise we suppose that a type II к >. is fixed and we call an object. if it is a t-type model.. a. a model.

(15) 11 -. Sometime« we «bell consider functions a« special relations.. Definition III.3, o&'. is a submodel of the model. Öt Ъ. Sr. 9. c / C о9. u ff fo r a ll. o r <т,п-Л>е1. Sr- = öl-y- n nb. The definition implies that B ^ A ,. the relations and func. tions are simply restricted to B, and since В has to be of \. -type. t. we have, that В is closed w.r.t. the functions of. Theorem III.4. Let. be a model. The set of all submodels of. tially ordered by. г г. ,. (X. is par. and this ordering has a smallest. element.. Proof: C 's. being a partial order follows from its definition. The. intersection of an arbitrary set of submodels of a sumbodel of. őt. is again. . This follows from the fact that all sub—. models contain the constants and we always have at least one constant of. . Now take the intersection of all submodels. This is also a submodel and of course it is the small*. est one.. Having defined the models we turn to define the languages. t L- , ^L. and2 /_. treated in this work 0. . To this end we.

(16) 12 -. define their syntax and validity e.g#0-f~. t. -t. All these. /=. Jo. three languages have the same class of models M'# That is. L. i. =. < 0F *,. У. '. e. ,-L.. c .. XV. THE T-TYPE ZERO-ORDER LANGUAGE. IV. 1. The syntax of the language:. The syntax of the languages investigated in this work have an analogous structure to that of the natural languages; -. first we construct the set of nouns. These will be oalled terras /Т/.. -. from the nouns we construct the prime sentences /Р/ by re lation symbols.. -. from the prime sentences we construct the sentences /F/ by the sentential connectives. // are symbols.. Remember, that the elements of. Definition IV.1. t. a »/ The set of t-type zero order terms. 1. is the smallest. set /of strings of symbols/ for which:. L) if i Сc) = О. -élnem c.e0 ~J--L. и) if (t" (f)= П and. d. 2“ ,...,. i. )dnen. Note, that all terms are constructed from constant symbols. %f\c..

(17) 13. by using function symbols. Any ©lement 7 a noun, since in a. n. y. M. of. /. is really. ■6 to the term. there corresponds. L. a certain denotation, actually it denotes an element of A,. b./ The set of t-type zero-order prime sentences. о^ ■ = c . /. (-ct/-,T„) :■£'(T)= > ?. { V. < W. / 0 <. 7;,.,. / \. ^ ^. p. The set of t-type zero-order sentenoes / о r~ / : is the \. smallest set /of strings/ for whioh the following three conditions hold:. P C? p * ?. c. -. G. U~). L. LLL. c f </? e. о. p -t. VP'. 4ben 7. <7 4 —'. G- o b. b. О ( f. Aу / -)t. Note, that the zero order sentential connectives are the symbols. 7. (n o b. ). n n c {. Since to the set /of pairs/ whioh enumerates. .b \. t. (C. *. jn C /) -. there is an algorithm ALGT. t , to the set of terms. D. T. there is an. algorithm ALGTER0 whioh enumerates all the terms. That ----- — A ~t* is a l g t e r :-:(ű —>0 J and it is onto. The construction of ALGTERo from ALGT. is left to the reader.. Analogously we oan define an algorithm ALGFo -£ ates the set o F ,. whioh enumer-. We shall introduce abbreviations to our language in the.

(18) Ik. -. usual way, In order to make sentences more readable. The sym bols V /or/, _> /implies/. and < ^. /iff*/ are abbreviations. defined as follows: (< f. V yO. for. \ - r < f ). for (~l ОУ \ y Ljy ). (lf> ^ y ) (. S. f. f. o. r. У 7 — 7-. (u> -?. Another abbreviation which we shall adopt is to live out un necessary parentheses.. IV,2. The semantics of the language. о. L. ~b. We are now ready to build a "bridge” between the sentences /. f-. /. model. and the models M^. The denotation of a term 2 4 7 in a < Х е н ь. i.. a t.. Z. • We shall express the fact that a. sentence / is valid /true/ in a model. (Ж. tion. The symbol. by the special nota-. 4 is an abbreviation for the relation. Now we define the relation I—. -0 ^. (—. i I. 'Ь =. i^. -L. Л7 X о. / and. the denota. f f. a function. tion of the terms,. Definition IV. 2. Let. c. be о constant symbol i.e.. symbol /for which. t' (f)=. tion symbol /for which t /( r)=m/.. Í. (c ) =. Q. and r. a rela.

(19) 15. ./. ä ckc. с. (. 3 í f ( Z L, . . , T n ). b./ •ЛУоТ. t=. lff \. ... £Vi. h - T Íf ,. (ií)fy. Л С. Х„. ) Z ±, - - ■. bcrlh fftl=<P. iff. A. f. Ő i T l ) ■■- , Л. '/. L'^ z r ^ /. апс/ c Y Í=. iff. Example ГУ«3». f ä{ ! а l ' X. { f f ,. l>. The model с А ^ Л Г. ‘ A. "C. =. fx .r ^. /is defined as. }. I&-JL. I. a,, й г >. ,. < « / /a. I. ,. >. cÁ СХл,. Jtf (. А а.X. Л СУ. Чг. о!. <^ S 07 1. Л, с. а. Now с А. (Л С / =. 1 =. °±. 'f ( с ,f. f. сХ. > С/А. Гс;. Л ffс с). Л г _ .f. С^. For the Illustration of this examplesee Fig. 1.. Fig# 1.. {с ). e.

(20) 16 -. -. Note, that we cannot talk about. a^. since no noun refers to. it. Remember, that nouns are constructed from oonstants ■7-t according to the definition of 0 \ .. Now, we are ready to define the t-type zero-order language. [} , о. ■ -i. Lb Í < o1=± M. ./. 0. -i. Note, that. is usually denoted by. D. .. Using the terminology of Section 2 the semantics of <. и. / /=>. and its syntax is. € ,. 7. is. 0. Preparing Exercises XV»if. IV.if.l.. IV.if,2.. Ót. ConstruotVmodel . ^ for any C, j. Show that if ^ ^. cP. but J P. (. in which if. ~ТГ. then. (ó t. •. ~6 0 Г. ^ 1. .. is suoh that for any. or there is a. ф ÓÍL. С^-ф~С^. ТГ€. J l. for which. p & 0 P. , then there is a model. either 7T^= ~l. such that . ^. J C ^ J L .. IV. if. 3.. Let. /" and. b. ooouring in. Show that if there exists a model. JSA ' P о /2 г } .Let. P. —. then for any ^. ^. tences oocuring in a of Г. the set of primsentenoes. IÓ G c P. I. c%t=r. also. auch that all primesen— ,. ZÓÉE Q Ó. occur in an element. as well. Prove the proposition of Exercise. IV.if. 3. for this case too.. 7.

(21) 17. "6 V. SOME PROPERTIES OF THE ZERO-ORDER LANGUAGE Recall the definition of submodel. /see Definition. III,3#/# Now we investigate some properties o f 0 L. which. are related to the notion of submodel.. Theorem V,1, Let. ^. then. £. iff for any. ^. сX o'. t~'. ~>. != -. iz. Proof, The proof goes by induction on the length of the sentence (y9,. For prime sentences the theorem holds by the definition of and. }J = ^. o f y 'C. *. Suppose, that the theorem holds for the sentences у. and. Now, the theorem holds for / , \ 1 as well by the defini tion of 6%/=;. tj7Л /L /see. holds for / Ljy. Definition IV, 2,c,/# The theorem _. as well by the definition of. К/. /see Defi. nition IV,2,c,/,. A Notation:. Q. Now we consider set 0. T. <{ c. (%. :. l". ( О. \. as a function which is defined on the. That is the funotion. function. = О. án(CxÁ rom. ( / Í. is an extension of the. the set of constant symbols to the. set of all nouns /or terms/. This is illustrated on Fig, 2.. ■-.

(22) - 18 -. Recall that. is the range of the funotion. w. f.. Theorem V,2, Let. 'а ё '. be the smallest submodel of (. t noM. With other words, the smallest submodel oonsists of exaot~ ly those elements which have nouns referring to them /see Definition XV.l#/ #. Proof, For any. а. С. у. ё. /. (. we have. In the same time ^ /. A <•'J ( A. .. / is closed under the functions of. /and thus forms a submodel of —г ' A the definition of 0 /. (. ( /# This follows from. A Notation;. Let. now iff for any. ,. < J t i = k ..

(23) 19 -. Lemma V. 3# Let ''П*0 í é , V “ or there is a. be euch that for any X G N e i t h e r /О "Ó "for whioh and j o [JJ]. Then there Is a model <4^ suoh that. c O t/= .. 0 П .. Proof, Let. be as stated. Now, we oonstruot a model. ^. <#.•. <X0 = A = I. . X. -. f. = {< x L/ Now, \ implies that. (c <. ..., > П т. sinoeX ~. dfo. ••^ 4 , ) €. HÜ. Implies that 7 J ^ “—. -ф. №. A. У ( /у-/ ~ b € j *. OU. ybrany foerOm.. and ^7 ^ "чг.. 4. Definition V, k 0. Fl is. The. a tautology:. -. !=. fo. ift. -. d l a. ..................................... =. <y>. ^. for all. c. X. .. See also Seotion 2,. Definition V,5,. л 0=. vt Cfi. *=. ; iff. < X. iff there is an cgI X. a =-. «éi. 6. for. ck ß= foe.. for whioh. M *. ,n. i. Theorem V, Let. X V fo / У 1/. 6. / Compactness {. X LP l C foU D. C ff. о. j. theorem of 0 Z- /.. then there is an /7<"6c9 for whioh. /=7 Ц ? < С<П. Proof, 1./ One direction is trivial, namely if \= = \ / L П. (nO. b= c<n. as well,, by Definition V,5. then. '.

(24) 2o. 2,/ Let us suppose that J= existence of an. H<U). <#• hold. We have to prove the. V. for which. V ^ ri c^n 1. as well*. To this end suppose the oontrary, i.e, suppose that for all. П^иУ,. fc b. V/■ t - P iy . From this hypothesis we have to n. derive. V. t/c •. ÚUJ. For all n, let. I 1. t k be. ő. such that. the existence of a model. ő í u. CY. ^ l ^ w e. have to show. for which с5У t< c o. Now we construct a set T T now we define Т. . ' У. J li. by an easy indue t ion J ]J J ~. for all. Let. 0^. / ^. (^JJ^&nd. = ° { j i Го. ^. ф ПР. Suppose that JL ^ is already defined so that for infinitely many n ,. Now, either there are infinitely many/ n. for which. (b ° [jL J ]iO T*. if this is not the oase, then. there are infinitely many n for which j _-j /С In the first case and in the seoond case. T. has a model. ő l u. by Lemma V # 3#. We show that. Ö. Í. for all. yC. Let. be such that all prime sentences oocuring in. к. "deoided” by 7 7 6 ^. ^. Ус. /= 7 ^. LaO .. O i^ ^ th a t. is, if. or. Now, there is an only finitely many. JL. ocoures in 0 ^ then. are either. .. n. i. such that. '-. because there is. n £= i, but by the construction of -J L. k. f or.

(25) 21. infinitely тешу. n. If. t n? ¥ L. n>i,. then. Lú. tion of. ő. к.. we have. beoause < # V. С&Сд/,= ~ .Г ТП JL^. C 7L. \/c P. .. By the oonstruo/7 ^ implies. '(P,. and ао. (у с. сд-_. Now, we investigate the connection between the syntactical structure of a formula and its validity* Notioe that any model defines a function. cyUjo.i] on. tences in the following wayt. the set of prime sen. flz ( j c ) - P. if f. t f f f= ^. 7l ~. Definition V* 7. ,. For any. ;0. f i. P. --- 5»/. 0j. we define its extension. by postulating!. С) Á m é Let. A. (jo. sßl (. p o t). f i (. ^. 1Ге 0P. be already defined, then. and. (c t ). if. t. ^ m jn. /\. *. 7 Y ). f A { u>) -A ( ^ ). i -JE. Notat ions be arbitrary*. Let &. ( У. iff for all. -. /otherwise. ßb ( P ). ~= О. P). У. P i. P. ( P>). ~. L. / •. Lemma V*8* Let. P i. Рг (J t)=. and d .. h. be suoh that for all Then for any. .p. e. J E C 0 'P ,. cA p. (fA f'/=. JC cA f. Proof* The proof is immediate from the fact that the definition of.

(26) - 22 г. £. is similar to the definition of. Lemma V »9» For any. ^. there exists an. - h :o. Ж е вР^. A (. jz)^_ l. Ж. such that for all. Ж. c/f cfiC. Proof: The lemma follows from lemma V# 3# Take / { j z .* А с ж ) = 1 3. •. (ж )~. Corollary У Д о » Let. 2. £. Then, the following are equivalent. „^. ( C). There is a model. (u i). There is a. Ő Í. ßb ■’0 P. —. for which 4 o ,. ŐC. for whioh. d ]. ". A. (. 2. ). -. d. A Corollary V*ll, Let IQ. Then, the following are equivalent /. *=i о 2. ~b (it). For all. V. f i b! d: ] О. 7. -fib,P. --- ^. Á Note, that by the above corollaries and lemmas we estab— lished a strong connection between the language ö Z_. and. the so called propositional /or sentential/ language* About the propositional language see. L2 3 .. Definition V,12, \T _ T-1Í Let 2 <= 0 F . Г) ^. I. is defined as the set of prime formulas ocouring in the. elements of. Д. ExercisesV*13*. ( .. (.

(27) 23. -t From now on. У,13»1». Prove Lemmas 2, 3 and corollaries 1 , 2 for the case. A : P ^ ^ -jo, 4. V.13.2.. instead of. Note, there are exactly. l f P\. P j. fit •. P —^ fiО/ Л.\. many different func-. fii.. P P0~fi be finitely satisfiable, Show that for. every finite. V, 13»^». 0 /d ]. is finite, then there are only. finitely many different functions. V» 13» 3» ■ LetAi. Ч. -t. r<=0P. Show that if. t ions. -L. Ръ / _ 'P. TcA there is a fib. ‘jo/ 1 tj for. whioh*. $ъ (Г’') ~ J- .. Let P. .t 0P be finite and let H be an infinite, set. of functions and. such that for every a. Ab :. Ae H there is a. for whioh. Show that there is an inf inite such that for any. -Ac © ; =. P qИ. and a. l .. A(r)-i. A 1с И1 , A <9. /Hint: use exeroise V, 3»2», and conolude the infini tely many elements of H will oontain exactly one function. A. In the following exercises. ;T v. — *. Г л. 'Ус€ьэis an increasing sequenoe. of finite sets of formulas. That is. Pa is finite and г^_г+. for all ^ АА ,<ЬЭ. Furthermore we suppose that^. № satisfiable that is to each. o . 4. is finitely. c<U) there is a h such that fit СГс)=d.

(28) у в 13*5» ~'fb. :. Let P >^ t. S. (0 Á j ( i L). V. 13.6.. be arbitrary. Show that there is a. A^U). suoh that. 1. for infinitely many -. ( I~ S ) ~. 4-. .. <c<?iJbe arbitrary, and let. Let. there io a. such that. ÁbA. -fib A. AA t ALo. be the function. constructed in exercise V.13#5# Show that there. A .-.a P i *for which the conditions of exercise <£-// ___. exists a. V.13.5. hold, i.e. many A j. V.13*7.. L-t/ ^J АьА. (. ) -. í j. k jr j- i. there is a А / 2. suoh that. ± .. Show that there is an increasing sequence / that is if j. 7 <AA then. f or every. V.13•S .. and for infinitely. Let <. c A. </. A,. ca>. for which А -. A;i /. / for which. d. Á. \. Á l-. A ^. f c. h. /-. cAA .. be an increasing sequence of functions. А Г М. Show that if A. for every. — CJ A. • then. jt c oo.. -A. (ССой UP.i.. ^ ~. Á. -. /Hint: use the definition of -fl(£) and the fact that for any. V,13.9». h £ u rc there is a n ^ C Osuch that te-ILICcú. /. Prove the compactness theorem by using Corollary V #l#l e and exeroises V #13#7 # V,13#8 # /Hint: suppose that for all which. UL. is an. Sc. n<U) there is an. 7(^1/ <y^/|oonclude that for all for which. ck *=An7^7 Take. ^. n. for there. ^ yV. is clearly finitely satisfiable, Construct a function h. forü. using exercises V.13.7# and V #13 .8 #.

(29) -. 25. Complete the proof by oorollary 5.1.1. showing that there is an. for which. c f y k jtr V c L oo. c p. ^. ■. /. Preparing exerciae V«l^«. V»1*1«1,. Construct by exercise V, 13«2« and Corollary V«ll« an algorithm whioh decides the set of tautologies« Ú. That is contruot an algorithm whioh, to any gives the result 4. if. ^. £. .. о. and gives the result. ^. f. otherwise«. VI. THE T-TYPE ZERO ORDER LOGIC. Recall /from Section 2/ that a logic is a pair where. L. is a language and. for any sentence. ^. CAL. /. I. CAL. >. is an algorithm, such that. of the language. L. cAL. computes suoh. sentences of the language whioh are oonsequences of. When defining a calculus. CAL.. to a language. L. we shall al. ways take care of its soundness, i.e. we shall ensure that if. Cj--J. pute. is not a consequence of (j;. then. CAL would never com. as a consequence of. Ф. Def ini t ion VI.1. The algorithm ( / of L. is adequate w.r*t.. L if for any sentence. CAL enumerates the set of all consequences of. That is if from. CAL. L jy. is a consequence of. by CAL.. Lf. then. U. is obtainable. ,.

(30) -. 26. -. Definition VT. 2. The algorithm L. •. CAL is complete for. L. if all tautologies of. oan be obtained by CAL without giving any sentence. to. ^. CAL.. Now, we turn to give an adequate caloulus to the language 0. First we define an algorithm. CAL*which. c. [_. will be shown to be. / ^. Actually, we shall fchow more than a complete oalculus for, 0z_ just completeness o f 0. CAL* :. we shall show that 0. CAL. decides. / -é. the set of tautologies of. 0 L_. Definition VI. 3# Given an arbitrary first;. ó. 7^ the algorithm ^. computes the set of the functions. A = ■{. — -9 -f о/ A. ^ /Note that. H. is finite by Exercise V. 13*2. and. thus computable from second:. computes /Note that. third:. ^. /. for all functions rb. ~Á £. H. .. is computable./. computes the minimum of. -A C f). this minimum is the result of. Notationt. C A L. л. : &. £. CA L. H. and. .. Given an algorithm ALG, then the funotion computed by ALG is denoted by ALG. That is if ALG is applied an ^. then the result Is denoted by ALG. (< = £ ). •.

(31) 27. Theorem VI, 4, if. is a tautology then ^. o -h ~. О. j. <. otherwise. CA L. i,e, the algorithm. decides the. 0< E A i-. set of zero—order tautologies.. Proof: The theorem is immediate by oorollary V,ll, and exercise V #13#l# and V #13* 2, A Corollary VI,5» \. /Completeness theorem of zero-order logio/t a complete caloulus for 0. C A L. L—. Exercise VI,6, Construct an algorithm. fECAL-i. which enumerates the. set of zero order tautologies 9 that is: A. iff there is an. 4^. Pz. UJ. о. b. c. a. l. *1. =. Definition VI,7#. -t. 6 The algorithm. A C A L. computes the funotion. 0 such that for any о A C A L -t. rr-i r .. л ( i f. O '). A. °. -. A. b = Ljy'. iff for every. Instead of. l. LP у. ЦУ. L~ j. Ó t f. we write. ‘0 С. X JA >. .. 0* ( ( < ? —. Notation: is a consequence of. CAL.. implies. r ■.

(32) 28. -. -. T h e o r e m VT, 8, /Adequateness. ACAL. t h e o r e m of z e r o - o r d e r l o g i c /. t. is an adequate calculus to the zero— order i. logio, T h a t. is f o r a n y. ^. °. d АСА. //9°. ^. /. (yy и/)= A. •. Proof#. The proof follows from the definition VI.7#. >. A Note, that the important result is formulated in Theorem V I # 4# which postulates completeness and decidability of zero-order logic#. Exercises VI#9» Vi. 9.1.. Using. construct an algorithm. which for any given C e n u m e r a t e s — '— -t £ of y° . That is^ A C A L ^ • 0 А and any. j. !such that. ACAl^^fCfi/b) A. A. ~. LjyJ. A '. VI« 9# 3#. U. 0. for which. LO. °. £ (fi. /2 ) -. (js ~ j =. “С. from О. • -é , which. ACAL. /7. enumerates the set of all consequences. .. i n t r o d u c e s o m e c a l c u l u s e s w h i c h are c o m p l e t e. g e n e r a l i z e d sense, sets of f o r m u l a s .. i.e». Cf?. ACA. ~L А С А L n. Construct a calculus. to any finite. N o w we. such that for any. With other words:. Show the adequateness of. ^. A. £. there ie a n / Z f or which 0A C A lA ^. :. VI# 9« 2#. of. the set of consequences. there is an /7-. *. ^ /A C A L ^. c a l c u l u s e s w h i c h c a n treat. in some infinite.

(33) 29. We say that an algorithm ALG is the definition of J1 ALG enumerates which. i.e.. G. =. ALG ( h ). у?. S. - { c f. 0A. there is a. I. if UJ. for. ,. An algorithm which to any set. C. C. enumerates the set of. C О. the consequences of jC. from the definition of. is called a. G. generalized calculus.. Definition VI.Io. The algorithm. / ^ (SCAL is. defined on the set of algorithms. as follows t for any given algorithm ALG. oGCAL^( ALG)= ± ^ L i ) m a x { О. ^. CAL ^. f t. (V. AL■'. and it is undefined otherwise. Ф. The existence of. (SCAL.. as an algorithm follows from the. fact, that if /i/ holds then for some Thus <5. G cal. (c c J C A L. C .)/ A L G A ^ c. can successively compute these values for until an/?^. rt-d-jdj..-.. If such an. H<AcO. n. lD. is found for which. is found, then the result of'jG?. and the algorithm. G ^ A 1—. C 'ccJ. A A L.. holds. is. J_. stops. Otherwise it proceeds in. finitely, i.e. the result is undefined.. Theorem VI. 11. /Generalized completeness theorem of zero—order logic/: For any algorithm ALG. , /=■• VA£-£Oi.tf (GCAL6(ALG) n<cco. /. 1. iL.

(34) Эо -. Proof,. -- ---. ^^ , mG\/o-o AL <£га). х. Dy the Compactness Theorem, (Theorem V, o.y implies that. t=V A LGfO. for some. tc io. /ПРА> and. thus 6. GCAL -6. stops for this n. The other direction is trivial by the definition of the concept of calculus /see Section 2./.. Definition VI,12. The algorithm. A G CAL. is defined on the Cartasian pro-. duct of the set of algorithms and For any given ALG and. ^. as follows:. ^. J&et A 'LG ; UA> ^ A ^. be defined as. 0. = ( A ALG(d. —?. / У //TO A.<LT). /. ш. for all. h i A. A. For any given ALG and. ÄGCÄJ ( ALG,. U. C. o E. y)=. _____. GCAL. ,. ( A LG). Theorem VI.13» /Generalized adequatness theorem of zero—order logic./ For any algorithm ALG and. Ä IG ( O :. ^ «i. A. LfA Pc 0 F. cfy. ^. _____. i/f.A G C A L (ALG,. Proof, The theorem is immediate by the Compactness Theorem /see Theorem V.6./.. Á Note, we can also say that the generalized calculus ”proves11. (/. from the set of Hhypotheses". ( J. (. l-. L t..

(35) 31. Recognise, that an infinite set of hypotheses can only be given ’’effectively" by an algorithm /say ALG/. /Such a set of hypotheses is e.g, Peano's axiom system for arithmetic,/ Thus, a logic also involves a concept of "proof" or "deriva tion" /w,r,t, its language/.. In the textbooks on propositional logio the generalized adequateness theorem is better known then the generalized completeness theorem. For the reader familiar with the for mer but unfamiliar with the latter it is very easy to show that the generalized completeness theorem /Theorem VI.ll/ is an immediate consequence of the generalized adequateness theorem /Theorem VI.13#/#. E x e r c i s e s VI.. VI.l4.1.. Prove a Generalized completeness theorem from the Generalized adequateness theorem. /Hint: Ä L JS. Let. — ^ 0 P~ be arbitrary, and. A L & •' (+Э. = ~> И L & L A ). Now,. for all. t= V A L & cjlff. ( /Д. C f €. 0 f=. ^. /\L &. /By the Generalized Adequateness theorem, Theorem. ÄGCAL1j_AL(S (<fЛ ~7( ~f)~ Í i defme ß CAL l CA L 4 ^ & c A. VI. 13./, n o w. VI,l4.2.. ,. Define an algorithm. A. C/A. which for any. enumerates the set of all consequences of Hint: use. ALGF0i:. /. ^. 4. 0 f~. (- j. see Section TV,1,/.. / /. VI.lU. 3.. Show that. "1. С-AL-. / t. ^. is an adequate calculus for f.L.

(36) 32 -. VI,l4,4,. Define explicitely the concept of "proof1' in the case of. VI, l4, 5,. <0. AGAL. and. Define an algorithm. A C .A LD? which for any ALG,. Á & C -A. enumerates the set of all consequences of A L G. that. Сm ). ;. °{ A L G. /7 < i^s> ^. ,. ( n ) : П- <. Of course it is supposed ]. C p F. .. Hint: proceed similarly as in the oase of exercise 1 , however in this case a "diagonalisation" is neoessary, /This diagonalisation involves a simi lar technique to the simulation of parallel prog rams on a serial processor or to the organisation of time-shearing operating systems,/. VI, 14,6,. VII,. A A CA L ^ •. Prove the adequateness of. PROOF THEORY OF ZERO-ORDER LOGIC. If the reader is not interested in Proof Theory, this sec tion can be skipped. The following sections can be entirely understood without reading this one.. To find a calculus which is complete for a language is an exercise in Model Theory, However,. if a complete calculus. is already given, to find new ones which satisfy certain additional requirements /effectiveness,. naturalness etc,/. belongs to the area of Proof Theory, The tools of Proof Theory are purely syntactical; and this makes Proof Theory. ,.

(37) 33 -. entirely different from Model Theory in oharaoter as well. The main reason for this is that in Proof Theory we need not care about the meaning of the syntactical entities we just oompair syntactical properties of different syntactical systems.. Now, г/e oonstruct some more effective algorithms /calculuses/ which are equivalent to. 0CAL. .. That is more effective algo. rithms which deoide the set of tautologies. After this it is a trivial matter to construct adequate calculuses from these complete ones similarly to the case when ed from. о. ^ ^. ACAL was. construct. •. One of these more effective calculuses is the so called Reso. RCAL. lution algorithm. . Since we do not need it later. we shall not define it precisely here, instead we illustrate it in the following example, /The precise definition can be found in any textbook on automatic theorem proving,/. Example VII,1,. RCAL. The algorithm. Ь. is defined as follows:. Let the sentence (-/^ be given to a,/. dRCA L. then:. Compute the disjunctive normal form of (_/. ; form a. matrix from this. e,g. Let the disjunctive normal form of (e x. Л У. CL I b . C. / /. A. <~i. D. 1 с) V 'P. ). \. ( a. A. ~ lh. /17. then its matrix is:. be с ). И г where.

(38) 1h. -. -. ------- --. 4. Cl.. 2. a 7a. 3. 1/C АуС 1c €. b An. 4. b,/. Resolve two rows of the matrix, to obtain a new row, and attach it to the matrix. E,g, in the above matrix resolving rows. A. and 2 - w© obtain th© row. ( C£/. /C ). and then we join this new row to the matrix as row 5 * ------------ r-. b /. a. 1C. 0 Repeat step 2 until either we get the empty row or a. c ,/. deadline is reached. If the empty row is reached then <:<-/>). R C A L. 0. 1 R -A L _. (. .in the case of a doadline ' у-) -- O _ This completes the definition of l. lt is not too difficult to see that R C A L £ the same result as 0 C A L ) that is О. Ö. t L A C R. '. ti Ш. gives. C A L 1. Ex ere isos Y U #2 , V Г Г. . 1 ,. Show that. 0. J\ C A /. ’. is a complete calculus for. . j I L. -1-1. 2 * 2 .. Construct an adequate calculus from. AC-AL. ’ow ve ere going to introduce some so celled p::ic:;n ;.*<• С О I<'. I■1; о *=•. c. Aj.

(39) 35 -. Notation: ^uo (j~. ). Stands for the set of all finite subsets of. о F~. Definition VII,3, An axiomatic calculus oonsists of: (£ ). an algorithm which deoides a set of tautologies called the set of axioms of the calculus,. (O t). Rules of inference. These are given in the form of an algorithm which decides a subset of. S M (0 -F- J X 0 t. This set is called the set of rules of inference.. Definition У П Л , Л derivation of , ■ - ■/. °^c ) - - -. from у. is a finite sequence. of sentences for which. cTn. Co) cTh= (tc j. for all. ify either. As С. П. is a finite. T. U/. U. \/|П/ c £ >. c. &. C</. %. U X c. or there. X. such that. is a rule of inference,. # If such a derivation exists we say that (.in symbols. 2- j. t. ^. (/. is derivable from. •. In the following when we want to distinquish between say c a l. and. ,. instead of. Now. we. C A l~ ^. /—. we write /. 1. and. f—. respectively. *. show an easy /but not very elegant/ example of an axio. matic calculus:. t.

(40) - зб -. ^. Definition VII,5,. The axiomatic calculus 0 A X C /j. (jL). is defined as follows:. the set of axioms: ' Ay t. (X j L). =. { t f & a F. ±. the rules of inference:. MP={<. }y y / >. The set of rules of inference. MP. ;. if/ / ' e / -tf. is called Modus Poneus,. jLЛ. Theorem VII»6, ! T ■ Ллхс, 1» an adequate calculus for. 0. 4. j. jAXO*. =. i. d. L_ . With other words. j=_. Proof# -L. The theorem is immediate from the adequateness of Actually. АСАL. and. A AC,. 0/\(2Á L.. ere different definitions of. the same algorithms,. A In the above definition we utilised the decidability of the set of zero-order tautologies. Now, we introduce an important axiomatic calculus. The im portance of this calculus is in the intuitively natural form of the definition of its set of axioms.. Definition VII,7« ~b. A\ V d. The axiomatic calculus 0. is defined as follows: 2.

(41) 37 -. (Pi). The set of rules of inference is. Remark:. AIP. t. The advantage of. is the fact that for any. X\ ^. we can decide by first sight whether. OX A=r.0A У 2. -6 (A?. or not. t. without any computation, /it is often said that in the every day practice a calculus of this sort is used in general./ Now we inve stigate this. о. AXCé. in details,. Lemma VII.8 ,. ' ' xc. Cf.. c. 1lc)Ä C a. ! ‘. t. h= (^J. i. . That is for any then also. 4. /. /. 7 6 оF. ^/ A. Proof. A. The proof is easy by checking that if and that if. P (/Л> У MP j. 1 '. if. then. X *2.. '~P -- 0. P. -b. k -. и. then. A. ? ,.

(42) 38. -. Note, that the lemma states that for. о. /^. is a oalculus. 0Л У С 2. /see Section 2,/#. I. From now on, in this section |-- stands for. -Vc ^. /-. i. Notation:* i means that it is not the case that. ^>1. /— —. ,. i- consistent if there is at least one. ^. Gr 0. ^. suoh that. /-7^. ^. •. Exercise VII»9» Let. LfZ G-. 0 Í~. consistent iff. We say that 2 _j. be arbitrary but fixed# Show that 27 Z-/ -. 7. Л. 7. ). У !. •. is ^maximal consistent11 iff. is consistent. but the only consistent set of sentences which includes is. is. TTj. itself#. Lemma VII#lo# /Lindenbaurn’s Theorem/: Any consistent set is contained in a maximal consistent set of sentences.. Proof, be consistent, and let. We shall form an increasing chain of consistent sets of sentences:.

(43) If* Д г nia«. U. j. i < f n. f. /L. у. Д. ^ V ^ / o t her. /7. <7. ^. ,. is consistent* Suppose not*. but in the derivation only finitely could be used, that is there is a finite. L1. such that. 0. for some. Г 4 У 5 n - if. many elements of ^. =. J 5 n ^, = X / n •. Now we claim that Thus. ie consistent then. 0 0. 0 0 -n. у 7 /1 Т у 7,.since. 0 .. is finite,. 0. This means that. is inoon~. 00n. sis tent which is a contradiction*. Ve. [7 now claim that /. For suppose. U. ^ f. (n. Then. is maximal consistent* *7. is consistent, for some. if^. is also consistent and thus. Á Corollary VII,11# For any axiomatic calculus, if contained in a maximal. P. is. then. for which Г. l~ F. CJ?. .. Proof# The proof is entirely analogous to the proof of Lindenbaum’s theorem.. Throughout this section we should have kept ourselves to the rule, that in Proof Theory all the arguments and statements should be syntactical, that is models. cP& M ior. A. should. not appear in the text. Eg. Completeness theorems should be.

(44) —. ho. proved by referring to an existing complete calculus, say / . Yet, in the following lemma we brake this rule, о. CAL~. and use. simply because this proof fits better to the. materials studied so far. However, a purely syntactical -t. proof /referring to 0 C A. instead of. / could be obI. tained with the same effort.. Lemma VII,12, /Deduction theorem/ t PL о. Let 1. 1. then. f. I. P Cl 'f. and. — a. П h- /. f. Proof, S u p p o s e that Thu s. there. Г. 4 ./. и. /. is a s e q u e n c e of s e n t e n c e s. Ax и. which either tuch that c£ - (/У/~ Def ine. A—. c Q ) and. / ( c. (p.. {cj_. /— >. This will. L et. us s u p p o s e. L. V. /. Otherwise This means. / X. ~~V* ( '"y. complete. that f o r a n y. r " /C7r t. On. for. Í, A. x. Jf. =- /6? .. that/ /. the proof j <. A. .. f o r all. 'б. since. , / V. At. }. c/J. d ; ) b y the a x i o m. /Afv;. d - ( a —>. t h ere are that. I— -id,. 7■ > ч c.,7•■/ ) 4-Ac J /. V /л V /// /. 'bus completes. c// C / b y the a x i o m (c. f, kl < 44 X. ai.i/l tliis is rn a x i o m. This. / /X. c ;. C'x. c/<; “ ^. ■-,<d.. or t h e r e a re. N o w we s h o w by i n d u c t i o n o n l <C. x. <p. the proof.. , -/. ■X x. £. such the t °4 - /^ 'been.use x- / .A " )) (у ’( '( t - <>г. > c/ .) í. О.

(45) kl. -. Lemma VII« 13, Let / ].C— Then. оh. ( jl ). ^. be maximal consistent.. cf?f \jS. G- 1. iff. P. g. (l l ). *. Cj? A ^ Y. ^. г. .. 4P ■. iff. Proof; First note that by the maximal consistency of. Г3. implies that ( l ) P I—. у7 I LJJ. iff. P. Cf . U. I—. OP /Л уУ. by the axioms. у. ) ) ,. C PA v7 < f/\ u ; — J Y. ;; > Ui. '. if i maximality of. and by the axiom. and so ((~~l U. Г. U u> I. —^ f. ^. ■. then JP U p 7 U?1 f—. i1. A— ^. ^. ■/У/Л 1. again by the. by the deduction theorem. 7 <3 Я. Lemma VTI.l^. Let 5. be arbitrary.. :. 0СНЯ. is consistent iff there is anC/£f*or dlt which. Proof: 1 ./. If. /^=Ai. then it is easy to see that. is con. S ‘t. sistent 2./. Let. be consistent. By Lindenbaum’s theorem there is. a maximal consistent Г. of 1. Now. 2-2.. We now construct a model. UPä <fc> P*U IJ 1. JC. ;Jl. By Lemmas V. 3« and VII. 13* there is an. G 0 IP cP. 4= --j i. )A.

(46) k2. -. Wo prove by induction that also. iff. (j '?^. (. ^. f. Dy definition and Lernma VII. 13* the statement holds for prime formulas. Lot us suppose that the statement holds for у . Then the statement holds for because!. Us (ф [ ^. since. 7{ j j. and iff. is maximal consistent. The statement. Л У. also holds for. /. ^. (~ff y7jУ <7L. у/ Л. since. 1. because. Í. is maximal consistent.. Corollary VII. 1*5. iff. LF. is inconsistent. { i. Theorem VII,16, /Completeness theorem of. é , /:. ^ЛУС. iff. Proof. 1 •/. If /—. 2./. Suppose. <уУ. then clearly. !=■. t = y -. ^ .. by corollary VTI.15,. 7 (y7 /—. is inconsistent and thus. (ljs Л 7. by Lemma VII.12.. '(^ Л. ..>. /у /and now by the axiom. we oonclude. <a>. Corollary VII,17, /Adequateness theorem of 1. 1. СуУ /=г (_yy. Á X C. ^. that is for any V. *. /. 0/. t 7~.

(47) Proof. The corollary follows from the definition of. MP. that is. from the inverse of the Deduction theorem /Theorem VII.12./,. Exercises VII,18 » VII»18»1». Prove Corollary VII,15.. VII»18»2». Prove by a purely syntactical argument that for any. г. °. Г. i - consistent iff there is о A : T —* {o, Í. for which. VII»18» 3». ?. A. ( Z. ). - 1. Prove /by a purely syntactical argument/ that ./4XC, -t iff о. /. Note, that this is the completeness theorem of -j0. A \C X. in a purely syntactical /Proof Theore—. tical/ form..

(48) 8,. THE T-TYPE FIRST ORDER LANGUAGE. 8.1 The models of the t-type first-order language are the t—type models, defined in Section 3# The -t-type first-order language is ^ where '. м. Ч. с. Л. ;. 4. *. L~ F. ТГ. £. -t. ^. is a t-type model. с Л. ^ as well as ,F=. . It у. and. У. will be defined in the following, 1. 8.2 The syntax of the language. rr*. / A'. /1. Let V be an infinite set of symbols which we shall call the set of varables. Of oourse, V should be dijoint from. Do i'U. Dol" U-iBv , í , r A é. Definition 8,1, a/. h. The set of t— type first-order terms « 4 1 ’ i smallest set /of strings/ for which ~ \ f <DD _ -j-i s ^ if b = /7 and С1} Z , Z h é ^ / ) then. -é. T. is the. 1 ^. and 77п 7;. b/. The set of t-type first-order *prime-formulas ........ ..... . ■■. p ä {t ( where °/. b ' ( Л ). 1. r4l.w1 h) :re =■ /7 .. The set of t— type first-order — formulae e— jf 1. ^. 7,. is the smallest set /of strings/ for whioh the. following four conditions hold^. -7 4 , X.

(49) -. ^. 41 G fF. “T. (/>;. '€ < F. The symbol. >4 II I гЧ.. A ihen 7 y e , / : t FуL ^ Ihen~7~ 7 1 / / '7F<.1' hheei Jv- y? e i. iS c/?é‘. V. (7,). T). ~a.. /• « ’. -. --t. Is called quantifier.. :J. The symbol Vvy- /for all v/ is an abbreviation for Ve use also the abbreviations. ~7 А. A. .. defined in Section 4,1,. We suppose that the set V is suoh that there exists an algo rithm is an. ALG V- a?-> У which enumerates the set V. Thus, - t which enumerates the set algorithm ALGA, A. there. A. From now on. r\ Pi: and. f. .A. A. stand for. A. ?. JL. and , A i. respectively. A. 8 .3 The validity relation ± k Given a model. a. we shall use functions к to render concrete. values from A to the variables of V. These functions к are called assignement functions.. Definition 8.2. Let Then. and. И A. ^. : V. -. — 5*. A. is the natural extension of к to. ; T ^ — A>A. that is: k. CALA). =. /' г. с П. for all ■ 7. F. •V 7 7. 'V 'é. f. (&. )/. - /^. (. 7. -6.

(50) - 46 -. öt,^. N o w we d e f i n e by r e c u r s i o n We w r i t e W i n s t e a d. of ^. о. (L)(Xh='T(zLr../ (U)(ßt t=. cfCÁ'J.. J. ÖL 3 Ö\ 'Á^ 'a~d-j/1 , ...; ^. iff. IfLkJ. iff. (iu)őih= (m a X)LA7. c ö í Cß ' l. iff. <=7. f=z-. f. (J s L k ]. there exists a. iff. that. is valid in. that is iff -é. Note, that^. (\fk ,. ^. ^ jh -. /1. Now,. 1=. Ö Í. /=. C-. L j ? C k ] \for. all k.. iff. C Ö .H. ) t Qt. h~. for both. for all. t. F -. .. and. l .. О. О <LL / j. C. ö té M. é. t=. /v' 4 /. iff for all. <-jj. ( f í ö. 1 =. which justifies our habit. ;X. we also say that. l t = Ö. (-f. Y A. [0 F. >I L -‘. , f=i-. а7. holds. If ( -. e. é. /= =. of using the same symbol. Л. if. and. \. all. /. öt. Let. Ö t. such. Ál. ( K / ) -k (\x . /Jf or. Á. (x/£ L. L=rLf7[_. a n. /. (tVJcÖ/= Э ф (J/L. We say that. k(zh)>. £. is a model of. ( i.. If. h = Y ,. Ö. .. then У. is a tautology. An "occurence" of v in formula of the form. Сl. is bound if it is inside of a subAn occurence of a variable is. d '& W .. free if it is not bound.. A sentenoe is a formula, without free variables.. t. d. Sd. /. 4 J. Remark: if (У С. r-t. /. F. öP t p. 6. Í. F :4. contains no free variables. is a sentenoe, then iff. (. 9. /. lp. Í. k. j. fox’ some k.. / 7..

(51) b. With this, the first-order t-type language 1. i. L -< F l M \h. is defined.. Returning to the terminology of Section 2. the semantics of / t is L. and its syntax is. t. P.. 4. Remark:. i In a certain sense, the important part of^ ._ is the language / -Y j /. But mainly for technical reasons we shall treat. L. ■ь. -1. instead of this.. Exercises 8.3. i. 8,3*1 Let. /. and. Show that a ,/ C?t. A r J. means that there is an element. x (~P. X£r / А. which satisfies. ^. , /More pre. cisely, given the ©ssignement function к / z (x ) -C there is an A such that if and.. Ь ./ c. k. >7-. L/. /-. Y. t=. ■ < /!. (t [ ). rxy-. d ь. V. then. •>. means that for all X é satisfies. X é A. the elements. b{. (. //?. means, that there is an. h= 3 x V y < f. element. X^Ca. A 1]. the element x o . f ő í. for all. such that for all. X/^. satisfy. d./ Shox* the same statements for similar examples e.g.. Y y Yu r z. i.

(52) - 48 -. 8,3,2, Show that for any X /. X. and. XX. Vx (f. <•/. Preparing exorcise 8,4, ,0/ , /v I 8.4,1, Let. UL ^. i. I. tand. let. /' <(п-ы, • ' / ^ j Je y- y-. bo an. equivalence relation on A, such that for all. / p. //. \. C and for all \. X?. У. (X. I. -(Xy ^ )/А..a -r(X/XX(/°X/-7УХ/УХ Let. ß. =. and for all. /. A> • ' <. \ p f. b I. CL / 6 C'Y^. £ .• Ct_ g Д. or. /7 > X. < p. , h. - i >. X. ;7 /xV£,., Ул )í7-there аге such that \ X 1; / X? c > X f we defined the model X r и\which is the. Now,. €. I. é. ^. / - X :. same as. : -y // X. y. -s^. except that the relation r is "changed to identity", that is if r^a,b/ holds in СУ ( then a and b are considered identical in. ,X . i. Show that for any. ?6. A. a • CA. T. Xr X. 9. THE T-TYPE FIRST-ORDER LANGUAGE WITH EQUALITY The t-type first-order language with equality is f L ,. d ~x. ~ \ F j.А У / = X J >, 'L } ' ' ' is the same as in the previous sections,. .where ^. j. А/ <0 .;. and /'. are defined in the following. f- d. Г i X1. ;/ , i. Í ; the equality symbol. t F " ' /— where f. is obtained from t by including. as a new binary relation symbol, i..

(53) - 49 V. ■ L. /. ) 1. b. Thus,. 1~. T. ~L ± —. is an extension of —. 1 -. ^. í. U{<=L,3.>] 1. /I. -L. P. by including иequations” -/•. into the set of prime formulas /for any. C-. l b € jL- j. 7. / .. (—. To define validity //=—-. /it is enough to define it for the new. prime formulas:. л и - ’I. A —. cx. [_ f t. 7 iff. ^. (~c±) = b. e. ). for any. The remaining part of the definition of. h=-. b. is the same os for $>. 7-j 7b. A set of formulas. is algorithmically axiomatisable if. there exists an enumerable axiom system for ^. and it is. called decidably axiomatisable if there is a decidable axiom system for it^more precisely:. Definition 9*1» r-i a./ A set 2 j С. г is algorithmically axiomatisable in^. A ÄL G:UJ-^Z for. there is an algorithm. i t. iff. which to any. b. { AAL& (n)r ri<lő'] f= to . and to any о/ £ /A A LG (Vi J ' СО } 1 b #/ A set. ^. A. ,F tL.. an algorithm. Л. ■. ± D. is decidably axiomatisable iff there is b. dalg •1 D — ^. Do /JALG=X h band 'o'é. j. for which iff. - -1M A. i. ' ^. Theorem 9*2, j. The set of tautologies of. is algorithmically axioniatisobl. £. in. , moreover it is deoidably axiomatisable in i ~; i. '. as well*.

(54) - 5o. Proof Let. у. I У. /. X. x. у ..v. А. I ^í/. ■у ^ 7 6. C o n s i d e r the f o l l o w i n g set of a x ioms 1 ./ e q u i v alence:. X A. a-) b.) o.). 2.. X. (У éУ A и. л. X J r K-. !. Ч. --- >. 1. X. / congruence /for functions/. y- i - ^. A.... for all 3.. X. A. Ay„. ?/Х> > é. )—. ^ ^/Xi / •/ ^. „. / lq. A. x. A - ^ A ■. A X, A y , 7) " ^ ( A(Av^,. v AX J^ <frr. ^. - t '' '. / for relations:. for all. ^. nrOi r/%]. > £. Now, it is not too difficult to prove that I (-L ) , /= ^ iff Д х whioh just means that. i. logies of. l. A X i. is an axiom system for the tauto-. /and of course it is decidable/. See also pre. paring exercise 8.4. Note, that if t is finite, then. /\X j. is also finite!. According to the above theorem the first-order language with equality. ]_ and the first-order logio with equality can be / ' considered as a special case of the language without equality / ~t A 1- and the corresponding logic. More precisely the logic with. equality is just the logic without equality together with a deoidable set of axioms. So there is no point in studying.

(55) 51 // separa tely. Thus it is enoußh to study,/ and to add 4 Y/later if wo wont to apply the results to , L ) j /. I. л. l .. Proving properties of. 1. directly would serve nothing but to obscure the essence. of the proofs by the presence of. /\x L. . All the basic properties. and constructs con be reached in a more natural and straight/ /. forward liraу by studying "pure" first-order language i.e. instead of a "mixed" version of it e,g,. /. L~. ■. 1 /. For this reason in the following we restrict ourselves to the I ± 1-. investigation of. J1 1. 1. ,. Z— .4. using. , /the results can always be applied to. AXL A. lo. THE T-TYPE SECOND-ORDER LANGUAGE^ ж jL~ The t-type second-order language is i(/ / - F 4, <4 -/ ,' , 1 1 ," n1 '. A. I. =. / 7/. the class of models t ions,. ^F. and. /. F. M L is. where. the same as in the previous sec-. are defined in the following.. а/че / ^ lo,l The syntax of the language —f is defined as follows: The syntax. F. First wo fix infinite disjoint sets /of symbols/ which are disjoint from any other used set /for example they are disjoint from. F )О F. V. v, •, , \. /, the set of first-order variables. f. ,.. the set of n-ary f unction-variables f or each. / ?ч 7<-'. the set of n-ary relation-variables„for each / К /<.

(56) - 52 -. Definition lo.l. a/ The second-order terras J ■t %. is the smallest set for which. V -Q. F. / /I). И. or. /?. l f (. then. F. T. jj (. ■*-). П. F. Z. 4 1. F J. and. -r-b F h Fz о 7. C.//. z. 1. b/ The second-order prime formulas;. f = { rfe..../k) 4 1“' l. c/ The. ' V - ^ e 3~ ^. semnc*- crc/er -fordulást. /=. <4) (n). 4. Ciii). if. M. if. the smallest. -(er wrliiclv. F. ( «ffef ßiict w. U 1 neuo. x& (. then. F e. f. Zl. (ifA y )e. then. , The validity relation Given a model. xf~t. (-fc'(r>n orreV^J ] .. J. 7. 4. zF b. еF. ■fc. we define the set of second order assign. ment functions as: 3.. К. -. {. к : Do k^ t)éiű U (VFUVf)U\/.and ’. if creV. . if and if That is the functions. k^K. then. кMe A. then Г€. VJJ "then. k(r) €=. nf\. render concrete elements to the. first-order variable, concrete functions to the function vari ables and concrete relations to the relation variables.. Now we proceed analogously to the definition of first-order validity.. j-.

(57) 53. Definition lo,2.. First given. we define the extension. К. dz. of. to. h?. ■ 't. the set г <^/ k (ny- ) = /г г^гг/. for any. ér F ( y ) ,. k(. r. {. (. t. -/Л. ( г п )),. t f f é V ,. Z;J'> ' ''. .. "^ “v. ^. ( Z a j , - , lecte,. 1 6 ^. //. 2 Now we define by recursion -t stead of 0 / =. We write /=-. (-PL'teJ.. in—. rKk(x4)r n 4,(znf>^kcT ) , i f те Z. )uL h -. TY^. ,-■,. Y )/. ík. 7. iff. <W. r -, Z e Y to X Y Y. to o t d d c p E1t 7 ifT c (to ^ ^ ( f a ¥ ) /-77 7 iff (E i h /■ 4. for any. !. Z. 2: é? LIS t. /7<Z07. a n d. iff there exists a. & fw )-= k (b u ) and. >7. i f <*г,. 7. VnFUV*jUV. A. *R. tm. 6-77 A - Y 7 ^ 7. -•. te. (2 ^. suoh that. 9b\^^W é(ypyR y\. 6^ /==•y7Z- 7e'].. Now, the definition of second-order t-type language is complete. The free variables and sentences are defined analogously as in the first-order case. Note, that using the same symbol t^for the second-order, firstorder and zero-order validity relation is completely Justified by the fact, that, denoting the i-th order relation b l. P. j. , -Linen j f =. 7- (. n. P. y. v. n. i h. У the seme relation.. ,. y. ^. )Y. That is, they are practically.

(58) - 54 -. C-. F~. W i t h o t h e r words,. d. ^. A. A d;. /=. =. ". / ^ is s i m p l y the " r e s t r i c t i o n of^ T h u s I— the r e s t r i c t i o n of 9 L Now we. 7:. Я ^ Y dX'A d A d. 1. t. andp h =. £. i. 4. J. t o Q id. and. L{i,. to ,J~ L. /. i n t r o d u c e the f o l l o w i n g a b b r e v i a t i o n f o r. é toot. AL. I'" A. is a n a b b r e v i a t i o n f o r. /for any d ' d. /A. 1. vrh ruVn)UV)/.. /7<Гёи. Exercises lo,3. -■6. lo. 3« 1« Let A d. Show that. 0. and. F. ./. ёЯ A= IfLP. :% -+ A. F. \y. means that there is a function. which satisfies. /. cisely given -. such that if. / i n ( Y /. More pre-. there is a function. /7 Л. k '(f)-. and. y. AC. other variables;then. M. ■ 9 ' * -. - A ('W ). for all the. ч?Аk. J^*J3*j2*_ Construct the parallel of* exercise 8*3*1* f o r and. 1 / /U \ / ‘ -■г. ё. (. ( /?<ёÍaJ. у— ~б. 1о.3.3- Show that / --- /ö. \ n F u. f--Ь...... d / — /. d.. -Г. ^. i d ). u. \. •. - t. 7 .. ~k. r. ,. and. t=*dn ^/лЯ A"/ *"■ л> J^ “d A ( x йA { . «<7. f. ". y^. /Hint: show that if б/ d. T a- V ten also for. iff. ( /F F. X , /~ / ■X. Cl' f .. 7. <9. '. A. -/. A 7 . etc./.. then also. ip. Y <-Y^ /-. and. similarly for (/-’б /•.

(59) 55. - i. lo .3 .b .. Let. Z7'. Cf?j L j'. Show that for any. { ~j' where ° lip/к3. f = ( Lf. and. X<773 //. /--- ( ^. Г "é / '. -. 'V C//yS is a text which is obtained from the. texts o/^ (./’ and. by concatenating them, (e.g. if. c x j - lo h , t j ? =. and в"я. ~. /Ъ> -~ (Íj l/ £/. then. JUL " ' j-. 10.3.5. Show that for. Л^ч’. A. a n y lf ié r. f =. X. Oi,/. X. 9. L) " <иг. *. £ г ... \. is the. Г*. iff. and. lo. 3.7. Let. /. and ‘{ х. set of free varables of. H j'é. X. lo« 3« 6 « Show that if. then. ( f. ^. j~. and any. ÖL. iff. '. f L. >7<?X5 X. 7 x (A7. be quanti-. and. '^. fier-free. Show that a) l f <£ ^ ъ} Л. then also. f = 3. <y/. \. h * d x u /. does not imply. ^7". з7 X. c.) all conditions are necessary to prove. implies. lo. 3«9« Let Let. zj X ^ 4. (Jy. (JlLLf/Z j. sT~. CJi. , 3. iß. X. not. occur e. in. LX. be the formula obtained from. Qy. Show that for any. V^juj. СЯ.). nor vicg versa.. replacing each occurance of. di. Lid t. Show that neither. -я. lo« 3*8» Let o/jr. .. ^. Lf. and '/■é. Lj. 1. by. by the term. C. (fiehl' iff- uL f—. '{(s 7.

(60) - 56 -. /Hint: а/ If there is an XG then to this. /4. /\. for which all Ij. A. also. satisf ie. Will satisfy. (/.' </ #. as. /by any choice of f./.. (j_. b/ If there is no Xé7 /4 fy. (J.'/. G. A , А СX ). = /у. would satis. Ip 6' , A. then there is a function./ 7-A. to any X. . SOME. f°r which all. A. such that. does not satisfy. Y. /.. PROPERTIES OF THE LANGUAGES INTRODUCED SO FAR. 11. Now we introduce the following notation; t. Let g. I. G F". / (l-. J. Cg. ’. n</oo. l\'/O'' L 1 /n (A '. 7'<r 7. end. -t. denotes the formula obtained from. each free occurrence of. in. nG". by. l/. by replacing and of oourse re. C. naming bound variables to avoid collisions. More precisely; 'L h 'j V ' / r. ^. C. /. is defined as above /there is no collision/, ОТ / A ). 7 ^ 7 —r. 7A. c/. .7 A ICC’*/'?]. J =". if ЪЭ' does not occure in / ~? - . . \ r /_. 7 cZ. :-3w. C. ). then. ^ ) [ n /т] =(g^)A ((//. otherwise, let. rJ. be a totally new variable,. ("TV L/ X - ir / z J =J ' 77* Similarly, ■/. Cf?[ ?p. /7/ / •' •/. L. h J). 7. a. /7.7. denotes the formula obtained from. by replacing each free occurrence of. and each free occurrence of 7/ ^. In у. renaming bound variables if necessary/.. 'Zp. by. I^. in /. by. 'ZG. y. /and of course.

(61) - 57 -. Pof inltion 11.1.. r. ~. 4. '. C ß -=---- С',''. У. iff. (V. iff. 0°. <%. )( olh: cp L jy. c ? s ic /. ,; U. c Y A - XC. ‘k. (j-. Л formula is called quantifier-free iff it contains no quanti« fier, i,e. it does not contain the symbol -/. Definition 11,2, The formula -. is in prenex normal form, iff. z/x± V7. where. I. --- '7. A. (j/. is quantifier-free.. Theorem 11,3* To every first-order formula (. there exists a. ЕЕ С/. which is in prenex normal form.. Proof X* —. The proof goes by induction on the length of (. and is based. /. on the following equivalences;. 7 E x < /. =;. Ex. 1 У x (f ■ 3 i éé VV í. Let. not occurs in. XV\. 9" Л Ч' '. c f. X 7 A у. ■. ß iи/Cx/zl) Л У X. Then:/г/X t/y)A 37 s (V x A ) A A. 7. ytxhü) Л Х =. = ("V a. ц /-. Ч/ ЛСГ:. These equivalences can easily be seen to hold by the definition s. of. f. To complete the proof see exercise 8.3*2,.

(62) -. 58. Now, the above equivalences con be used mechanically to obtain a "prenex normal form” of a formula. This gives birth to the following algorithm.. Definition 11,4, -------. The algorithm for any. PREN ;. G. -i. -(-. F x - -> J=c. P R E h J (y > ). =-. u>. and. is defined as: Р&ЕЛУ X C f). is in prenex normal form.. The existence of the algorithm PREN follows from the proof of the above theorem.. Notation: и ...З х п=. 3x± З х л. Theorem 11.5*. •■• .X and similarly for the symbol. /Skolem^s normal form Theorem/,. ...xn where. Xi r .., Уп / ^ . . . ^ п е / а п й ^ 1 n. do not occure in. Ц. у*).. in I f n. е / Р. for all. /. <r/г. and. ■ '. Proof It is enough to show that ' Э Х Г К ^ , „ Х = *>/ -7-» ^ , 'si/ for any 1 £ P /note that ) i might contain quantifiers/. If we have shown this, the proof can be completed by an easy induetion. The statement to be proved folloi^s from the definition of in the following way:. ■ '.

(63) - 59 -. He give the proof here for 7v V ^ X the case of J X , If. a ( '/r. - :J X V. a. ,i f (J. each. ^. k (L j)tf\. is trivial.. X. then there exists an element will satisfy. C { /-—' :~ / x. V. A 'V. cX Cy J. then. X f. .. Thus,. X X. X. С X ) ^/iwill. k. C. there is no. Á ( x ) <XL / \. / f к. C X )]. (x ). for which. is ’'good”. This defines a function which renders to. ). the corresponding ’’bad’1 Mi).. к ( X) ( X(f. Thus,. V ij. as well, i,e,. Otherwise, if all. Л. Lj. f oi' which any satisfy Я. ...Ущ. • the generalisation for. ^ f X x X. i cf /. У. / 1. (X) J. ■. Definition 11,6, We soy that a type t has enough function symbols iff t contains infinitely many function symbols of each arity, ±,e, for any n, *J_ оf there are infinitely many symbolsy^ , ^ n j ■ such that for all / v it'. ■{. I. v/. c). From now on we suppose that the type. / г .. t lias enough function symbols.. Corollary .11,7, If the type t has enough function symbols then to any formula /1. ( ' L. there are /\71. < I,. ^ ), -. /,, U ' ■//? / (. ,/ /.'. i. iff /- ■. such that ••••\/;X '. • ' X . t ; L 'k г. ' Xn. ' {/. l. ГгооГ From the above theorem ,.it,. 1 /, ••;. „lb,-/. iff / ■ l X r “ •//;> /vr- •. !<n: if wo choose 'to each, function variable Lon symbol ‘ • a :,. |Mm. /•. from. О П :uro I11. II. V>/. •/ f. !t a correspond •. such that г ( /. then I>v the deflnl I ion e.f. and ’1.

(64) бо. /. \':fl ■ ■ ■ jn v/Xi-- X. Ь. :J\i■■■ Xi (j ' I. X ; ' "7. lftl V ”. I Ifi. [It !. {"(y4). Definition 11,3» Л formula СД. is an existential formula if. = Х. у. , .. •.X n. У. I. where C/7 is quantifier-free.. Corollary 11,9* If the type t has enough function symbols, then there exists 7 -t т-t an algorithm £ such that for any t у t /. X/• ’7. /=-■•F V/ сД ,. iff and. (Д) is an existential formula.. Proof From Definition. ll.k ,. .. V R E N M. By the above corollary there exists an. F Y f (7/0 •: /XXЛ' (''(/').'. existential formula Í. Ct. Recall, that Д < ■ / Í ~~ 1. is the set of t-type first-order sentence. Theorem 11,lo. If the type t has enough function symbols, then there is an a lg or it Inn. ~ 7 l~ /V . , •О /-С Г К • .о X. i. h - ( f. 7. v. /(')’■ ". F H V. у. '<3. /. such that for any. x/;x. , v;. <: ). Y X CO. Froof Jet. £\7(7/-0=-:/\;г...х. и.. .where. j. (]. is- quantifier—free. t ■.

(65) ~. V© give th© proof for. 61 -. Jxy without. loss of generality.. Let / •'(./ ' and so A~ :7л (// . О/ л ,(_ For every U /v / by Theorem 3.**. there is a smallest submodel of £. \y). Since /— У у. there is a {' I. V. L t: о. A( 74 ^ /. T. els о. A-’ 'J X i J / Thus by Theorem 5*2.. for which iTjr /=■'- (jy [_ X ^ T X .. and rfer. Су / we have (S. and so, since. J. UZ. X /Z C j . /. by Theo-. rem 5 *1/. Therefore. В [-"x■. W /У//. 7. гé In the other direction: '•. '. 7 4 x / r 7. trivially implies. 7. 7 7 ; /г.;. Now, define is an algorithm. Z - /. k. ^. С Ш. 7. /. Т Ш. :/ Xх C.// / h j IS. inoe. FA' 7. is also an algorithm.. Note, that the above theorem /Theorem 11.lo./ has nothing to do with Ilerbrand's Theorem 13.15, because. (l). Herbrandfs Theorem. is a purely syntactical /Proof Theoretical/ statement while the above theorem is strongly semantical; rem states the existence of an. Z xZVcck. П. <. (Z. Ilerbrand's theo-. (C l). /for which. CJ. iff. ( L f , the above / у ) theorem states while. ^XП nothing of this sort.. Note tion: \. For any Z. ] ]. iff for any. 1 V.. is a "1 * ■''. such that. (. . У. V...". (. Corollary 11,.11. bet the type t have enough function symbols.. C/O A /. there.

(66) 62. Let. <( I(. 7-1t:r .. .. f. ■'L. ((. t - V. be the set of algorithms. There exists an algorithm ■’ /. )■. ?» ./ (_) / /. t-. {< ? € ,;. 3 ^ the algorithm. such that for any ó s é. ("CC. has the propertys. 57:. /-. 2. y ) ^ - d. iff. j. Proof We clef ine 5 /,/’(( f] /7 ). ". (4^7. i. V. (/. otherwise. ' /. decision algorithm. iff there is. 4.. 71 (7')(7. /. (. 1. CO. for which. The existence of the. follows from the fact that any (/. 77. con. tains only finitely many terms.. Remark: Dy the above corollary the algorithm rithms / K l >Г rithm /. Г'. f\. is similar to the algo-. I. and /; / / , To any first order formula. ^. (J). the algo-. I. correlates a deoidable set Ch of zero-order formu, , \ / „ las such that / •-/ iff / V y;More precisely to any <~/7 i j ■) -. /. computes the definition of (.// algorithm//. ~ 'f(p. This definition of. which decides the set. C p.. ( /’. is an. Recognise, that the. existence of a decision algorithm is a stronger result than the existence of an enumerating algorithm. That is from г / easy to construct a. 2 1 ^. such that. ZJ.'jl. у. it is. enumerates the set. (j. Now, wc are ready to construct a complete calculus for the first. order language. To any first-order sentence ^ / by / wc ecu / v compute the definition of 7 Í ) and by the generalised comp.lci.o/ ness theorem of zero-order logic we can prove. !. Vo./-. !-. с/-, / '.

(67) -6з ~ 12. . THE. T-TYTE FIHST-ORDER LOGIC /. -{ Now, we define an algorithm. C*/\ /. which proves the. first-order tautologies by reducing a first-order sentence C. to a set of zero-order sentences. For any sentence the corresponding zero-order set is *) the definition. Z -tz R . ( U \ l l ).. /7 g t \J. \. of which is the algorithm г7 / f ^ ) •. In the following we define a calculus. ( 4/ for the first-. order language by using the generalised calculus , Zz (PA L of the zero—order logic and the algorithm . t „- / in the following way: for any ( / ’ ( - . ' z. P. E~. ^ ;. ?. -J). л е. jCAl1 (if) ~ 0& C A lP (?E, i'f9 ) ) ■ Notice, that the use of //. instead of ZE means that we. ,. do not use the result of Corollary 11.11 in the construction of complete first-order calculuses since (‘ 1 ’ {/. is. complete w.r.t. enunerable sets of formulas as well. To make the following definition simpler we do not use 17 J ' ■ 1 / ♦. there the algorithm. Dcf ini ti on 12.1 . To any. ( /'• '. ‘'. such t!mt for any. ,/■. a:. (/.•>,v. we define the algorithm 1/. I. (j. -. ’ •’. l. R. v •’. 'á 11. ' / " / 11 '. .. Í. /Note, that by the terminology of the remark following Corollary 11.11. the algorithm ,/4/ A ,, , / No;-/, we define the algorithm g '/ /. f о 11 о ws :. 4 /’( < < / 1 / . -4 . •• j. ' -. • ) • ■ ! ). v' :. '.