Logique & Analyse 157 (1997), 9-29 A GENERAL TREATMENT FOR THE DEDUCTION THEOREM IN OPEN CA L CUL I * Arthur BUCHSBAUM and Tarcisio PEQUENO Abstract Aiming clear formulations o f the deduction theorem and precise account of its restrictions, a study concerning the rules introducing connectives related to its behavior is conducted. Special attention is devoted to operators, such as quantifiers and modalities, dealing with varying objects. Concepts and techniques able to cope with the subleties of tracing those objects are introduced. As a result, a classification of calculi in terms of robustness to applications of the deduction theorem is given. 1. In tro d u ctio n The material implication, no matter of having always being one of the most disputed connectives along all the history o f Logic, is almost universally present in the myriad o f logics nowadays available. In addition to this popularity, it plays a unique role in most o f the logics in which it occurs, because it has a direct connection to the very core of the particular relation of logical consequence that the logic intends to define. In fact, it works as a formulation fo r that notion, expressed in terms o f the internal logical formalism. The relation between this internal notion o f entailment it represents and the notion of logical consequence the logical calculus as a whole express is usually formulated through the so called deduction theorem, which establishes the ways and conditions for their interplay. In general, the way the implication is introduced in a particular calculus is related to the way the rules for quantifiers and other connectives alike, such as modalities, are introduced. In spite of being such a pervasive connective and a so important one, it can be verified that frequently it is not treated in a way so careful as it deserves, bringing about imprecision in the enunciation of rules and misappli- * Work supported by the LOG1A project (Protem - CNIN). 10 A R T H U R BUCHSBAUM A ND TARCISIO PEQUENO cations of theorems related to its definition. The aim of the authors in this paper is precisely the one of clearing out the issue, by making explicit all the elements and relations involved in the formulations of the rules for the introduction of some key connectives that have the power to interfere in the correct enunciation of the deduction theorem applicable in a particular calculus. Among these connectives there are quantifiers and other operators of the kind, such as modalities, whose occurrence in a formula may require some sort of tracing of the varying objects that they, whether explicitly or implicitly, may involve, and that may affect the behavior o f the terms occurring in the deduction theorem. This concept o f a varying object, which generalizes the one of a variable, together with two basic relations, named here as dependency and supporting, to be introduced later in the paper, are the key elements in our strategy to make explicit the mutual relations among the rules introducing those connectives and the way they affect the correct enunciation and applications of the deduction theorem. The treatment given here for the subject is conducted in the style of generalized logic, meaning the introducing of concepts, theorems and procedures with no special regards for particular systems but, instead of it, using them to characterize classes of logics with respect to properties related to these concepts. It turns out, from this strategy, the results being applicable to large classes of logics, whether known or yet to develop. The concepts here presented had been already applied by the authors in the formulations and generalized proofs of metatheorems for families of non classical calculi in [2] and in [3]. It can be observed by a careful examination of the classical logic books that two distinct choices for the introduction of implication and quantifiers have been made standard: st) T h e rule for introducing the implication has no restrictions, but there are constraints for introducing the universal quantifier and other operators of the kind. A calculus adopting this strategy is called closed in our context. It is more often used in calculi presented in natural deduction and sequent calculus style. Examples of closed calculi may be found in [1, 4, 5, 6, 9[. However, this closed option may be very cumbersome when used to calculi, presented in axiomatic style, having varying objects other than variables, such as in modal logics. 2 n introduction of the universal quantifier and other analogous operators is unconditional. This strategy is more often adopted fo r axiomatic formulad tions. From here on this kind of calculi are called open. Examples may be ) found in [7,8,10]. T h It is important to notice that these alternative strategies are not incompateible and have been adopted as variants for the definition o f the same logic, i n t r o d THE DEDUCTION THEOREM IN OPEN CALCULI I l classical logic, for instance. However, the resulting calculi are not strictly equivalent. They are equivalent just in the sense of having the same logical theorems but not in the stronger sense of implementing the same deduction relation. For instance, in an open formalization for the classical logic we have p(x) V x p(x), but the same does not hold for a closed formalization. Some well known formulations o f the deduction theorem, in the context of open calculi, presented in axiomatic style, that can be found in the literature, present some undesirable features, such as: • explicit use of the concept of demonstration, instead of an idea of a higher level dealing with syntactic consequence; • la ck of an adequate tracing to accompany the use o f varying objects in rules o f generalization, making it difficult to further applications of the deduction theorem, in a context, after the first time it has been applied. Furthermore, we consider it essential, for a deeper understanding of this matter, to conduct a survey, followed by a careful study, of the basic properties o f the consequence relations involved. We have discovered in this study two relevant relations of consequence, with different tracing systems for varying objects, here called "dependence" and "supporting". Under certain special conditions, to be precisely stated later, these relations can be proved to be equivalent, enabling the use of the most convenient properties of both of them. Below we will give two examples of formulations of the deduction theorem, commonly found in the literature, that suffer from the above-mentioned ills: • " Fo r the predicate calculus (or the full number-theoretic formal system), if I', A H- B with the free variables held constant for the last assumption formula A, then A B . " According to [7, pg. 971. • "Assume that F, A B where, in the deduction, no application of Gen to a wf which depends upon A has as its quantified variable a free variable of A. Then F I— A D B." According to [8, pg. 631. In this study we have found formulations for the deduction theorem that overcome these problems in a generalization that covers a broad spectrum of logics. 12 A R T H U R BUCHSBAUM A ND TARCISIO PEQUENO 2. V a ria tio n , Dependency and Supporting From here on, we will consider C an axiomatic calculus, a,O,y formulas in C, and F , be valid if the cited signals are used with primes or subscripts. to i9 Pre-Definition 2.1: For each application o f a rule of inference r in C, we c o l l consider as previously known the varying objects o f this application in C. e c t i If r is a rule in C, whose applications do not have varying objects, we say o n s that r is a constant rule in C; otherwise, we say that r is a varying rule in o C. We say that o is a varying object in C if there is an application of a rule f in C such that o is a varying object of this application. We also consider as f o previously known when a varying object o is free in a given formula a. The r m following additional conditions are to be fulfilled: •uthel number of varying objects of each application of a rule in C is finite; s varying object of an application o f a rule is not free in the conse•aeach i quence of this application. n Example C 2.2: In practice, we find the following varying objects: •; variables used in universal quantification; t o f the rule of universal h • x is a varying object of the application generalization; V . v a e • hidden variables used in introducing connectives associated with modals ities such as necessity, skeptical plausibility [2, Chapter 5], etc; such a variable may be indicated by the sign itself introduced by the rule; m a e • " L I " is the varying object of the rule Da c o n Definition 2.3: Let Z = b e a demonstration in C. We say that a v is relevant to a i fulfilled: fe i n Z •E n i . / and a 1•i(t ai s 3 a ijIi u s t i f , 1 -f ii.o ein. dC .and , athere exists a hypothesis f3 such that -1 E sinn k ) (nk) i i s . . . , p ) ) jZ 3 is Ir e l e, 2.4: Definition We say that a demonstration Z in C depends on a colleco f ua fc a n tion of h tvarying i sobjects if V contains the collection of varying objects o ssoo vt 11 of tapplications o f rules in Z having a hypothesis in which o is free such p p l i c a t i ta nn a o o ip r et 1 n3 fe m oi iifn k s i ee tu n ; dhe ie Z THE DEDUCTION THEOREM IN OPEN CALCULI 1 3 that there is a formula, justified as a premise in Z , whereon o is free too, relevant to this hypothesis in Z . If there is a demonstration in C of a from such that it depends on Y , we say that a depends on CV f ro m Fi n C, and we note this by F— a. I f o l ,F . — „ , oa b y a . I f Y 0 , we say that Z is an unvarying n ) a n d demonstration in C. n I , on 0 fro m F in C, we say that a is an unvarying consew If a depends e quence of F in C. a l s o nTheoremo2.5: A formula t ea depends on Y fro m F in C if, and only if, at least one of the following conditions is fulfilled: • a is an axiom of C; •a E a i , . . . , a • there is an application "If such a l , that , r o f a rule in C such that —a and, for every varying object o of this application, o E Y and for every a, (1 is I ' C I", such that o is not free in i n), if o is free in a „ then there and a,. If CV = 0 , we can replace the third clause by the following condition: , • there exists an application of a rule a , , . . . , a in 0 C , s u c h t h a t — a , and, for every varying object o of this application C andafor every a (1 i n ) , if o is free in a „ then there exists F' C l 0 " that o is not free in I ' and I ' — a such i . Example 2.6: In an axiomatic open calculus with the rules o f universal . " generalization and of necessity, we have the following examples of depenF dence: x, y • p (x y) V x V y p(x,y); • px x,E1 L Vx px. 14 A R T H U R BUCHSBAUM AND TARCISIO PEQUENO Theorem 2.7 : The following properties are valid for the relation" C . if there is a demonstration i n C of a from F whose collection of If varying objects of applications of rules of C in i s lt, then F — a; cv (ii) i f — a, then — a; (iii) i f I" — a, then there is a collection it of varying objects such that cv — a; (iv) — , 0 — a; li (v) i f F — a and cir C V', then f — a; sC C cir c' Y (vi) i f — then ir — a; lC a and F C I', C ror y (vii) i f F — ' a, then there is V C elf such that lr' is finite and f — a; C ,C If c i f (viii) i f — a, then there is PC F such that ir is finite and F' — a; C C (ix) i f — a and, for each o E o i s n o t then f ar e ; e i n c il r — a, (x) for each o E 14f , there e xI"' i—s at and s a then I a ' . C i F r s —u c c h t W h a t o cv THE DEDUCTION THEOREM IN OPEN CA LCULI 1 Theorem 2.8: The following assertions are not valid for the relation cv • if 5 Li C ' an, I al,•••,an) u then • if * f o r all i e {I a n d for a ll . free in a / lE 1 ,and ,F'. . . a, p 1 , i t hf e o , ( ncv 4 then F l t h r a e r n d o e , Proof i s x Let C be ea calculus whose axioms are given by the schemas " a -÷ a y 13" i s a, a -> and "Vx ta s a (At)" , and whose rules o f inference are a n d I 1 ' that the first is a constant rule and the second is a varying , such 3 rule Vxa C in which F the varying object of each application is the corresponding quantified variable. s u 0 , c tv —Ky We have h , however it is not true y t h Q c Vz(Ry y Sz) a ( Q(y,z) R A — that 1Vy tQ(y,z), 0 V z o ( yR y V counter-example , S z for )the ,first proposition. f r io ,m w h s i z c h w e n ) v h oa e , t a f cv —s• r Q ( 16 A R T H U R Likewise, we have BUCHSBAUM A ND TARCISIO PEQUENO — Rv c - , nevertheless Y Ry —VyVz(Ry Y Sz) C { { it is not true that V y Q(y,z), Vy Q(y,z) —› Ry) vVz (Rv S z ) , from V C y Q which we have a counter-example for the second proposition. ( y , Definition 2.9: We say za demonstration i n C is supported by a collection Y o f varying objects i)f Y contains the collection o f varying objects o f applications o f rules in, s u c h that, for each conclusion o f such applications, there exists a premise relevant to it in I f there exists a demonstraV tion in C o f a f ro m yF such that it is supported b y Y , we say that ( a is supported b y l f from F in C, and we note this by F — a . I f Q 0 0 ( { o y l , demonstration in C. I f a s supported by 0 from we , . . say that i s a stable F . ,in C, we say that a isza stable consequence of F in C. ) o Theorem 2.10: I f i s a —collection of varying objects in C, a is supported n by 11 from F in C if, and > only if, at least one of the following clauses is ful} filled: R •a a is an axiom of C; Y •n a E ,•••, a ) •dthere exists an application o f a rule in C such that 1 na n I a , l such that o Y , then a l , . " a w wapplication , e . Ifa = 0 ,—we can replace the third clause above by the following condition: l ce a a •sthere exists an application ' o f a rule in C such that n o œ a c n F lf n — v a „ and, if there is a varying object o of this — o d c t application, then — —a a , e i n ] F f . — t a h b e y r THE DEDUCTION THEOREM IN OPEN CA LCULI 1 7 Example 2.11: In an open axiomatic calculus with the rules of generalization and of necessity, we have the following example of supporting: E1, y • Opx E Vy Opx. Theorem 2.12: The following properties are valid for the relation" _ C (i) i f there exists a demonstration i n C of a from F whose collection of varying objects of applications of rules of C in 3) i s V , t h e n — a; c v (ii) i f F — a, then F — a; ) i f F — a, then there is a collection V of varying objects such that — a; (iv) — a iff (v) i f ov a; a and V C V', then — a; (vi) i f F e a and F C P, then F' — a; lf (vii) i f a, then there exists V' C 'V such that V' is finite and F — a; C v (viii) i f a, then there exists IF' C I" such that F' is finite and — o c v — C a ; 18 A R T H U R BUCHSBAUM AND TARCISIO PEQUENO 1 C W (ix) i f a l , . . . then , F f 3 . a n The theorem below describes a way of expansion for the relation ,generic calculus. 1 a lTheorem 2.13: if F a a ,l . . n .,then . ., . F,aF1 l i , 0. . . . , 1 l u l i — a cur l 4 /Lemma 2.14: I f l — a , then r 3 , , . Proof . I f a is an axiom of C or a E F, there .is nothing to prove. , Let us suppose then there is an application a a , o f a r u l ne fulfilling o fthe conditions of C theorem 2.10. 1 By induction hypothesis, we have If clr F — a„. Given a varying object o of this application such thatc o I f , we have — — a a n l n d 2.7, to , theorem a h a esufficient n c econdition for concluding that F C a , — a , w . h i c h i . s , . c a3. S c o r Calculi d i p e cia l Axiomatic n , g F Definition 3.1: A calculus C is said to be partial stable if the following conditions are valid: • each varying rule o f C is unary, its domain is the collection o f all formulas in C, and each of its applications has exactly one varying object; v c il— c THE DEDUCTION THEOREM IN OPEN CA LCULI 1 9 • f o r each application — of a varying rule in C, if its varying object is not 0 free in a', then a' — a ; a , .., o f a constant rule in C, if a,' l a . are respectively conclusions of applications of a varying rule over a • f o r each application a„, a using the same varying object, then a ; , . . . , a , Lemma 3.2: I f C is partial stable and F — a , then, for each application -• C o Proof it is similar to the proof of lemma 3.1 I. f a v Theorem 3.3: If C is partial stable, then F — a c a iff r y Proof it is similar to the proof of theorem 3.12. i n Theorem g 3.4: I f C is partial stable, then — " has the following additional property: r u 0 0 * l — a ,P — e c 0 i i t a n 1 n o, ] 0 C •, i f * f .1o r ,every i E „I 1,...,n j and for every j E ( I,...,p), if o, is 1 . i free in a • f j• . • . i ,and l t tap h e s nl v e x i a s t s r F y ' i C n F 0 — c a . 20 A R T H U R then I" BUCHSBAUM A ND TARCISIO PEQUENO 0 Proof: It is similar to the proof of theorem 3.13. Corollary 3.5: I f C is partial stable, then the following additional properties are valid for the relation" „ . C • 0 0 0 0 if F — a P l 0 ,..•,r — a C aP — l a ,... ' ,F 1 a • i1f * f o r every i E t n ) and for every/ E p } , if o, is , . . free . in a , a 1j 0 0 , and F' — a , t h e C 0 t then n- P. h e x i e s t s Proof: it suffices to use theorem 3.3, the ninth proposition of theorem n F and ) . Ptheorem 3.4. I ' ' C 3.6: A calculus C is said to be partial strong if the following Definition clauses are F satisfied: s —ua a ; c 0 (ii) 0 h a t > h [ application P i'• • o f a constant rule of C, (iii) f o raeach 3 t 0 1ao—) ; — > On1 , c i " s n • o t f • THE DEDUCTION THEOREM IN OPEN CALCULI 2 1 Theorem 3.7: The following propositions are equivalent: (i) C is a partial strong calculus; (ii) f o r any F and a, where F is a collection of formulas in C and a is a 0 0 formula in C, F U (a ) — )3 implies F — a —) (3. Proof: it is similar to the proof of theorem 3.16. Definition 3.8: A calculus is said to be partial strong stable if it is partial strong and partial stable. Theorem 3.9: If C is a partial strong stable calculus, then F U { a l 0 implies F — a —› 0 . 0 c Proof: it suffices to use theorems 3.3 and 3.7. Definition 3.10: A partial stable calculus C is said to be stable if it has the following additional property: • f o r each application o f a varying rule in C, where o is its varying a object, if p' and a' are respectively conclusions of applications of a varying rule in C over p and a using the same varying object, then 0 —a Lemma 3.11: I f C is stable and F — a, then, for each application — cv a avarying ' o frule ain C, such that its varying object is not free in F, F — a '. Proof: If a is an axiom of C, then — a, and so — a ', and therefore If a E F, then the varying object of the application considered is not free 0 in a, and so, as C is stable, a — a ' , and therefore — a '. I f there is an application of a constant rule a ' cv — a n , w e h a v —,,a a in C such that 22 F A R T H U R BUCHSBAUM AND TARCISIO PEQUENO - a t ' t , of a w h l c e r bstable, it follows that a i e yl f t' Let us- suppose,athen, that there exists an application — o f a varying rule ; a h cv a ' , a e of C such a n thatdF— 0, and, if its varying object does not belong to v , then r sh e n 0 a. eLet o be thet varying object of —. I f o V , — a, hence — a ', and a c— C C C 'V a a m F therefore I' — ar ' . Consider [3' as consequence of 0 by the same rule in e a , C e cv r. which a ' ia consequence of a. If o E V, as g — g ' , and so, as r u e l 0 C s [3' — a', and therefore F e is stable, we have p i e n c clf w a iff Theorem 3.12: I f tC isi stable, then h — v i c e c Proof: l h the lemma 2.14, we have By — a implies — a , so it remains to y a prove the converse. r c c1 I — ' Let us suppose that a. o i n s Let b e a demonstration in C of a from F depending on 1 , g the first s a occurrence of a formula in j u s t i f i e d as a consequence of an application e c of — such that its varying object does not belong to 'V and o a varying rule q u n some premise is relevant e to f3' in L e t o be the varying object o f this s application. n e If o is not free inc [3' , then, as C is stable, we have [3' — [3, and hence, as q e u the considered occurrence of [3' precedes f3 in Z , we have F— f3' , and s e — c S' n c e o f a . THE DEDUCTION THEOREM I N OPEN CA LCULI therefore, by the transitivity o f " 2 3 c If o is free in ,63', then, as o E Y, there exists F' C F, such that o is not free in F' and I ' lemma 3.11, I" — 0 ', and hence, as C is stable and in accordance with — /3, and therefore C In any case, there is a demonstration Z Y.oReplacing the considered i n C o f occurrence of i n 32 by Zi3, we obtain, given Z ,[ a demonstration in 3 f r Co of am from F, in which the number of applications of Fvarying rules, whose varying objects do not belong to Y and whose hypotheses have premises relevant to them in the new demonstration, has s u p p o r t e d decreased one unit. Repeating the same process a finite number of times, b y we obtain a demonstration in C o f a fro m F supported by Y , o r rather, 'V F — a. Theorem 3.13: I f C is stable, then erty: — " has the following additional prop- o F — v • i f * f o r every i E {I then o t and i I' s f ov rC e e i n a j , t h e — c a ' } and for every j E 1,...,p), if o, E 'V and 24 A R T H U R BUCHSBAUM A ND TARCISIO PEQUENO Proof Let • Z p be respectively demonstrations in C of a ported l , . . .by , a11, and let CS be a demonstration of 0 from I a by lp fOf r o m F 0 r. i, .. .. C ,s. in ,uafrom p .p ,Let o y be the first occurrence of a formula in '3.) justified as a consequence n ) s u p p o r t e d of ) . an application — 7'of a varying rule, such that its varying object does not C belong to Y and some element of F is relevant to y ' in Z . As o are demonstrations supported by Y, we have the considered occurrence of n y' appears in CS, and hence, considering o the varying object of the appliccation, we get o E 0 a 1 t a e , . Let . . , a and o C be defined by j n n = a/ a Itt is easy to verify that there exists / E a finite I ' , such that F' c F, 0 is not i n i f { free g in I ' and, for every 5 e 8. Therefore, by the construction of E- I cv ( I , F r" u a C 1 . ; l , . u the considered occurrence of y' precedes yin , we have , As . . — w . , eaa . p ol a , } clf b—n r'' du — y', and therefore, byplemma a 3.I I, u C — y. ty 1 n o a' For a F d— 5, and hence, once again due to the every 8 E u C, we have i,i n o nas transitivity o f " — " , F — y. dOr irather, there exists a demonstration ann o s o of y from F supported by Y. Replacing ddC in the considered occurrence of y i f in eht b y y , we have a new demonstration Z ' in C of 13 from F, in which s r f the number of applications of varying rules, whose varying objects do not me n e r belong to 11 and each hypothesis has some premise relevant to it in 3.Y, has on o e decreased e one unit. Repeating the same process a finite number of times, nc t o if [3 from F supported by Y , o r rather, we obtain a demonstration in C see f n t,i r a r bn e ) ayu e ) ttC i i h. n oe a nt i -r I THE DEDUCTION THEOREM IN OPEN CALCULI 2 5 Corollary 3.14: I f C is stable, then the following additional properties are valid for the relation " If I r • i f Ir 1C a l, . . C . , f P a ' l„ a l , . then . . , a lt t C u_ p _ ) U V fp aL • i f * f loJr every i E { n } and for every j E 1 ,...,p i f o , 1a n d i y o .4 i ./ i a.sn . dr7 ,0 f then c a; r p e l Proof it esuffices to use theorem 3.12, the ninth proposition o f theorem 2.12 and theorem 3.13. i n Definitiona 3.15: A partial strong calculus C is said to be strong if it has the followingf additional property: , t • f o r each application SI ' h • e where i f is the collection of varying ' ' '(a O n Ph and no element of V is free in a. _ o efof the application n objects a x v , aa r y i n g The — u i3.16: rTheorem l e following propositions are equivalent: s (i) C is a strong calculus; o O f t Cn , s ) C a F p s , u c h t P 26 A R T H U R BUCHSBAUM A ND TARCISIO PEQUENO clr u t a l — (ii) f o r any I' and a, if j cs for every o E c 3 if V, o then F — i s c a – of› (i) implies (ii): n o t Proof f r e e 0 suppose . Let us that C is a strong calculus, F u {a } — t3 and, fo r each n o E o is not free in a. i a If 13 is an axiom of C, then — 1 according to clause (ii) of 3, cv definition 3.6, — a –> t3, and therefore a if n If E I", then F — 0 , and d io hence, according to clause (ii) of definition h 3.6, e n If [3 a , then, according to clause (i) of definition 3.6, — a –) a , and c cif e therefore F — a -+ , If there is an application P ' •'—' nf o f a rule of C such that 3 u (a ) u (a l — p„, we have, by induction hypothesis, c — a –› O i n f application that does not to 'V, then, according to theorem 2.10, . Ibelong f t h e r [3, and hence oncee again by clause (ii) of definition 3.6, — a –) ) 3, i s and therefore F— a • /3 . If every varying object of this application a belongs to if, then, asvC isa strong, we conclude that I" r y i n g Proof of (ii) implies (i): o b Let us suppose (ii). j e c t As a — a, we haveo — a –>a . C0 Cf 0 As 1 0,a1 — c 0, wet get 0 — a –› 13. h i s cv T , a " 3 THE DEDUCTION THEOREM IN OPEN CA LCULI 2 7 .0t b e an application of a rule o f C whose collection , ••• of varying objects is V , and a a formula o f C where no element o f clt is , free. 0 We have { a -0 1 hence, as 1 0 and /.3, we have {a - 0 1, . . . , a 1 ,therefore . .0 ,„ ,Oa I} a -› [ 3 ,...,a-*P — n n 1 ), . . . , a ,a} - ) O Definition 3.17: A calculus is said to be strong stable if it is strong and n stable. Finally, let 0 C is a strong stable calculus, Theorem 3.18: I f u ( a) — p, then 1 - for every o E If, o is not free in a, Proof: it suffices to use theorems 3.16 and 3.12. Definition 3.19: We note by C[F] the calculus obtained from C with the addition of F as a postulate. If 1" is a unitary conjunct of the form I a) , then we also note C H by g a t Theorem 3.20: The following assertions are valid for C[Fl: • u • F' u F — a iff 0 C[F] 0 a; a; C[r ] c cil If • i f u Fa a; — C[F] i c • if F' f a a, then there exists c C[F] f , W D 1 F'r • i f F' u — t a, then a ; s u C[F] c h h e t h a n t r ' u a; 28 A R T H U R • if • • • • • • if if if if if if a , C[1] BUCHSBAUM A ND TARCISIO PEQUENO then there exists W D l r such that r• u F a ; C is partial stable, then C[11 is partial stable; C is partial strong, then C[ r] is partial strong; C is partial strong stable, then C[11 is partial strong stable; C is stable, then C[11 is stable; C is strong, then C[F] is strong; C is strong stable, then C[11 is strong stable. Theorem 3.21: C[11 has the following properties fo r the introduction o f implication: • i f C is partial strong and F , u a ) C[F] • i f C is partial strong stable and I ' u ( a} u ( a) • i f C is strong and C[11 )3, then 0 a C[11 0 –0, /3, then I"' C[11 C [ 1 ] for every o E l i , o is not free in a, then I"' C[F] u { C[1] } • i f C is strong stable and 1 for a every o E 11 then a , o i s 4 1 3 . C[11 n o t f r e e 4. Conclusions i n a , fo r the deduction theorem fo r a We have found optimized formulations broad class o f open axiomatic calculi, which overcome all the problems that we pointed out in the beginning, within every spectrum of possible restrictions in their deductive functioning —from the partial stable and partial strong to the strong stable calculi. The weakest formulation o f the deduction theorem belongs among the partial strong stable calculi. An example of a calculus o f this type can be seen in [2], pg. 133, which is a translation to a first order language o f the Logic of Skeptical Deduction, defined in the same work, in Chapter 5. This calculus was essential for the proof of completeness of an axiomatic calculus with respect to the semantics of this logic. THE DEDUCTION THEOREM IN OPEN CA LCULI 2 9 The strongest formulation of the same theorem belongs among the strong stable calculi, which constitute the great majority o f open axiomatic calculi, concerning the material implication, found in the literature. Our initial motivation was the search fo r a conceptual basis fo r an abstract proof of completeness regarding generic calculi, which was done in [2], pgs. 72-88. A concise exposition of this proof will be the subject of a future paper. Departamento de Informatica e de Estatistica, Florianópolis/SC — Brazil, Email: [email protected] Laboratério de Inteligência Artificial, Fortaleza/GE — Brazil, Email: [email protected] REFERENCES 111 [2] 131 [4] [51 [6] [7] [8] [ 9 ][10] Bell, J. L. & Machover, M., A Course in Mathematical Logic, NorthHolland Publishing Company, 1977. Buchsbaum, Arthur, L6gicas da Inconsistência e da Incompletude: Sem:antica e Axiomdtica, PhD Thesis, Pontificia Universidade Cat6lica do Rio de Janeiro, 1995. Buchsbaum, Arthur & Pequeno, Tarcisio, A Logic for Careful Nonmonotonic Reasoning, to appear. van Dalen, D., Logic and Structure, Springer-Verlag, 1989. Ebbinghaus, H. D. & Hum, J. & Thomas, W., Mathematical Logic, Springer-Verlag, 1984. Enderton, Herbert B., A Mathematical Introduction to Logic, Aca demic Press, 1972. Kleene, Stephen Cole, Introduction to Metamathematics, North-Holland Publishing Company, 1971. Mendelson, Elliot, Introduction to Mathematical Logic, D. Van Nostrand Company, 1979. 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