Logique & Analyse 161-162-163 (1998), 107-120
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I I n t ro d u ct io n The paraconsistent logics was studied long before the concept of paraconsistency was formulated. One of such logics is a minimal logic suggested by Johansson [ I ] which is obtained from the intuitionistic logic by omitting the axiom scheme — considered the system L E, which is obtained by adjoining the Peirce law 1 D ((q) ( c (p p ) Dq ) to the minimal logic. The reference at this work can be found in [3]. Curry [3] considered the system L M fo r minimal logic and D the system 1 1 / )LE. along with other L—systems for negation, which are usually Ireferred n as Curry logics. He characterizes the properties of negation in L M and LE as simple and classical refutability, respectively. h i s Our studies also lead t o th e system L E , b u t it arises here in an u n p u b unexpected way, from the investigation o f the constructivity concept suglgested i sin 141. h The e important feature of this approach is that the constructive dsense o f a formula is determined by its outer connective and related to w o system. This allows to avoid technical difficulties concerned some formal rwith inductive k definitions, as in case o f realizability semantics, f o r in [stance. The 2 constructivity o f the whole system is defined via a simple condition, i.e., a condition imposed on the sets of formulae. A ]second order , Ksystem isr constructive if every theorem is constructively true relative to the isystem, pand exactly constructive if the sets o f its theorems and construcktively true e formulae coincide. As was shown in [5], the constructivity concept of [4] agrees with traditional approaches, for any constructive system, we can construct a realizability semantics. In this paper, we study the class E K o f formulae constructive in any exactly constructive system. It turns out that E K is a paraconsistent logic, which is equal to the intersection of the maximal extensions of the minimal logic. Using this fact we establish the maximality property fo r EK. The first example o f maximal paraconsistent logic P' was suggested by Sette 16]. Adjoining to P' a classical tautology, which is not provable in P the A similar property holds f o r E K . Adjoining to E K a 1 , classical g i v elogic. s new classical tautology gives the classical logic, and adjoining a new maxi-
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mal negative tautology (see Sec.2) gives the maximal negative logic. Note that EK has no other nontrivial extensions. Further, we describe an adequate algebraic semantics f o r E K , which easily implies the equality E K = Le. L e denotes the logic which corresponds to the deductive system LE, i.e., the set of theorems of LE. And we can see that the class of all nontrivial extensions o f the minimal logic is divided into three intervals: the well-known intermediate logics lying between the intuitionistic and the classical logics, the negative logics, i.e. logics with degenerate negation when all negated formulae are true, lying between the negative and the maximal negative logics (see Sec.2), and the proper paraconsistent logics, which all lie between the minimal logic and Le. In conclusion, we present a natural four-valued semantics fo r the logic Le.
2. P re limin a rie s In the present article, we consider only logics and deductive systems in the propositional language {A , v , D , 1 }, where I is the constant "contradiction". The negation is assumed to be a definable symbol, = = cpD I . As usually, by a logic we mean a set of formulae closed under substitution and modus ponens, and the deductive system is a collection of axioms and inference rules. An arbitrary set of formulae can be treated as a deductive system with empty set of inference rules. We say that a set of formulae has the structural property if it is closed under substitution. If L is a deductive system, Thm(L) denotes its set of theorems. We call a deductive system L formal if the set Thm(L) is recursively enumerable; the system L is consistent if I e Th m(L ) and weakly consistent if there is a formula (p with cp a Thm(L). Finally, we say that the deductive system L is inconsistent if L Fcp for all formulae (p. We adopt the following notation for propositional logics: I T is a positive logic; L j is a minimal or Johansson logic; Le is a Curry logic of classical refutability; L i is an intuitionistic logic; L k is a classical logic; Ln is a negative logic. Hilbert style deductive systems for logics Lp, L j, Le, Li, L k, and Ln are denoted LP, U . LE, LI, LK, L N respectively.
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We recall that the logic L p is defined in the language (v, A , D), and the axiomatics for L j is obtained by extending the traditional axiom schemes for L p to the language v , A , D, 1 ). Other listed above logics are related as follows: Le = L j + 1(tp Lk = Li +
p
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) ,
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Below we give a few definitions and facts concerning the algebraic semantics for propositional logics. The detailed information can be found in [7, 8]. Let A be an algebra for the language I ). An arbitrary map V : {p is o called an A—valuation. Each A—valuation extends naturally to the set of all , p formulae. A formula (p is true in A, or is an identity of A, and we write A1 ( p , if 1/(tp) ,, .Obviously, I f o r the set L A = t p lA ( p ) o f formulae is a logic, which we call a. . na logic y o fAA. A logic o f the class o f algebras i s the intersection of logics of algebras in J-C, — } v— a l u a t› i Lo J (n= VA f l The I algebra A is a model for the logic L if L C L A . I f also L = LA, we .say f that A is a characteristic model for L. L r A I o Proposition 2.1.[7, Ch ill, Sec.3] Every logic has a characteristic model. A m J t The lattice A = (A,A,v,I w i t h the greatest element I is called implicative ifh forCany a)) E A there exists a supremum V[xia A x A l l implicative } f o rm a variety, and the lo g ic o f this variety is L p [8 , Theolattices e rem X.2.1 . l• s e By a .1—algebra we mean an implicative lattice, treated in the language t I) with in t e rp re t e d as an arbitrary element of the lattice. The o minimal lo g ic L j corresponds to the variety o f j—algebras 18, Th e o f X1.2.21. Equivalently, we can define j—algebras as implicative lattices rem p the negation satisfying the property a D b = b D — with lent lra . definitions T h e are e related q u as i vfollows: a - -loa ,p ao D I , I =s i t i o
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A Heyting a lg e b ra is a j—algebra wit h th e least element I T h e intuitionistic logic L i is the logic o f the variety o f Heyting algebras 18, Theorem XI.8.21. A negative j—algebra is a j—algebra with I = I. Obviously, the negative j—algebras are distinguished in the variety o f »algebras via the identity p. Therefore, the negative logic Ln is the logic of the variety of negative j--algebras. We call a Peirce algebra an implicative lattice satisfying the identity ((p D q) D p) D p. Let X and 2x be a set and its power-set. The set-algebra ( 2x,u,n,D,X), where Y D Z = ( X \ Y )u Z, for any Y,Z, E 2x, and u and n are ordinary union and intersection of sets, is a Peirce algebra. Moreover, we have the following Proposition 2.2. [8, Ex. 111.4] Let A be a Peirce algebra. Fo r a suitable set X, the algebra A is iso mo rp h ica lly embedded t o the set-algebra (2x,u,n,D,X). Let r = (10, I 1,A,v,D, I) be a two-element Peirce algebra. Proposition 2.3. L 2 l'= Lp + I ((/) D q) D p) D p l. Proof I t is clear that L p + I ((p D q) D p) D p ) C L2P. To prove the inverse inclusion we show that there is only one subdirectly indecomposable Peirce algebra, 2 . As is known, every variety is generated by its subdirectly indecomposable algebras. Let A be a Peirce algebra with more that two elements. We show that for any a E A there exists a filter F„ 0 ( I ) on A with a E F„. Take an 1 a E A. There is also a b E A with I 0 b 0 a. I f b a , then a E F (b), where F (b) = {Mx b l is a filter generated by b. Assuming b a we consider an element a D b. Using Proposition 2.2 we infer that a aD b a nWe d have thus constructed a collection ( F that ( a E A ) o f D ,la b f ni lF at = eI ),r s aEA n I o A , u c h i s . e . ,
M A X I M A L PARACONSISTENT EXTENSION OF JOHANSSON LOGIC
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and F„ I f o r all a E A. The latter means that A is subdirectly decomposable. The proposition is proved. Let 2 , (I0,1 ),v,A,D,0,1 b e a two-element Heyting algebra, which is a characteristic model fo r the classical logic, L2 L k . And let 2' , 00,1 I, I ) be a two-element j-algebra. Proposition 2.4.17, Ch.V, Sec. 3, Ex.1J The logic L j has exactly two maximal extensions, L k and L2'. Pro o f Consider an arbitrary extension L o f L j and its characteristic model A, L , LA. I f — the 1 l universe A = of a Isubalgebra isomorphic to 2'. Consequently, L A C L2'. I f -LA= —11 A LAA IC L k. , t h e n The proposition is proved. A f o r e, v e r Proposition 2.5. L2' , L j + I q ) D p) D pl. yt h e 1n Proof Obviously, a n d ((p D q) D p) D p are identities of 2', and so h at e E L2' D L j + — s u1 p , ( ( p A l The ,b a inverse inclusion can be stated similar to Proposition 2.3. D g e The proposition is proved. t a )h b r e D sa Wep call L) T a maximal negative logic and denote it Lmn. eA D t ,l 3. E xpa c t ly) constructive systems aI . ,A In the introduction, we have just said a few words about the constructivity Ii concept of ILI]. For the sake of space we will not reproduce the system of s A notions given in Hi. We give only definitions of constructive and exactly ii constructive systems, which are equivalent to propositional versions of respective ss notions in VII and are reworded in a form the most convenient for o purposes. our m o Definition / . Let L be a formal system in the propositional language rv, A , D, 1 T h e system L is called exactly constructive (constructive), or p h i c t o 2
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shorter, ek-system (k-system) if and only if for any formulae (p following hold: o a n d conditions ( p i , (i) tL Fh (q)e (ii) L (çoo ( L ço 0 A ( , cP1) (iii) o0 1oL)r (cPoL ç o il) ( I ,c< pc= 1p> ) ; o As ( we) can see from Definition J. every classical tautology constructed via connectives i (m pp l i ev, A s , D is provable in an arbitrary ek—system, and we then have the following Lo Fa Proposition 3.1. Any ek—system L admits all axioms and inference rules ( p 1 ) . of nthe system L i and the Peirce law ((p p ) D p. d L only note that the reductio ad absurdum ((p D tp) D (((p D— We —l( i p ) D 1p D1 ( 1 ,1 p) Proposition 3.2. Any consistent ek—system admits a ll axioms and infer1 ) ; ence rules of the system LK. ) i D s Proof( Let L be a consistent ek—system. We check that formulae — a D a n d —1 c p D (p, i.e., (ço D 1 ) D (tp D qi) and (((p D 1 ) D D 1 (p ( D p are provable (p, in L. D a D r By assumption, L k I . Assuming L F q) we have L k D 1 by Defit nition I -, and hence, L ( p D 1 ) D ((p D N o w , let L ( p . Then 0 (p D tif and also ((p D 1 ) D ((p D tif) are provable in L. Thus, we i formulae ) ((p D 1 ) D Op D 0). have LF a D We consider the formula (((p D 1 ) D 1 ) D tp. I f cp is provable in L , we l immediately D have L 1- (((p D 1 ) D 1 ) D (p. Assume L k (p . Successively c applying Definition I (iii) we then have L c ) a and finally, L H (((p D 1 ) D 1 ) D (p. is ) The proposition is proved. oe D . 1 , L k (o c p D 3.3. Any weaklyDconsistent ek—system L, which is not con1f Proposition ) sistent, admits 1 , all axioms and inference rules of the deductive system L MN fto r the maximal negative logic. h e We need only check that L c p D 1 , but this is the case, because 1 is provable t in L by assumption. r a n s i t
MA X I MA L PARACONSISTENT EXTENSION OF JOHANSSON LOGIC 1 1 3
Proposition 3.4. I f L is a consistent ek—system, it does not have the structural property. Proof By Proposition 3.2, L H p y p . In view of L being exactly constructive, either L H p, or L — flicts of L. which proves the proposition. 1 / 3 with . Athenconsistency y w a y , t As we h can e see from the proposition, the set of theorems of a given exactly constructive s t r u csystem t u is rnot generally a logic. However, for several important classes of exactly constructive systems sets of theorems provable in all a l systems of the class are actually logics, i.e., they are closed under substitup and r modus o p e Wer denote byt Tcon. tion ponens. h eke class of all consistent ek— tsystems, y T„„ o n T not finally, .ce kconsistent, e t- For h ea class T of formal systems, let L(7) == tpI(p E Th m(L ) for all L E TI. kc ld a e ns o t e s ts h e co Theorem l f a I.s The s following equalities hold: o a 1. L(T,„„ l f e a l l l 2. k) L (=r „ ) = Lmn, e w 3. kLe ( T— L k , sa e ykk s t e m l Pro ) ys o f .1. For an arbitrary consistent ek—system L , define a mapping c :L{po' o 1n s Treating 3k i s (0,1 l as a two-element implicative lattice and using Definition I, we t 1n ,e•can • • nshow that the equivalence t }L I L . cp e -m1/((p)= 4k —)n holds every formula g; in the language A , \ / , D I. Putting 1/(1) = O. we s s. yfor e have V( ( p ) = 17(q; D .1) = 1 if f 1/((p) = O. The latter means that L k cp , s t i.e.,t L c p D I . because L I and L is an ek-system. We have thus e i n provedmthe equality Th m(L ) cp11/(cp) = I ) with V treated as a 2—valuation. s g , w At 1 the same time, for any effective 2—valuation V. the set of formulae L h ==v7q)11/((p) = I f o r m s a consistent ek—system. It can be easily deduced i ( Definition p from 1. The effectiveness condition is needed, because ek—systems c 1 must be formal. h Obviously, ) = L k = L2= first a fI l equality ( L is proved. r Ii eN f i s aL n ep f f e c t ii v e 2 — a vn a l u a
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2. For L E ' 7 For - any 2 any of the form c p is inferable if L, by Proposition 3.3. These 1 „ . „formula „ facts - . e kand Definition I immediately imply the equivalence v a, l u a t cp i odn17(g)) e, f =i I n L 1e/ 2 for all tp. As above, every effective 2 ( 11 ) true, gives a weakly consistent ek-system L 1 == that -v v a note l=u a Lt4m i o)nn 1= L TVV - (n, c { 1p. , ) 1proving v a l the u second a = ) . vw h i c1 h equation. , t i o n iI 1 / i tss 1 a n wV The equality rn 3.f e m c tat i Lvi(Te n s e f o e habove econsideration. ti be d 2 'k - e on t i c a l n yThe ctheorem is proved. vl ) ay Ll uk a t i o n . e tn t h E K •==uL(Tek,)•Th e set of formulae E K is a logic as an interV hLDenote n ssection m osf two logics. The next section is devoted to the investigation o f ( ei this — ra logic. n 1 ui m m e ( ld i a t 4. T h e logic EK p ee )Let 1c L o n s =extensions / q ouf the minimal logic. Their intersection L oe 1axiomatizable, o =(e n n L c L ffrom IL pe j n Iff o i Ls a l s o o)if+0io.'bi yn= i t e l y rknown 1)f u j for extensions of the intuitionistic logic [9]. However, it is not hard sL E verify that it remains valid for minimal logic. By definition. E K is an ato + b ahIt s t intersection o f two finitely axiomatizable extensions o f L j: L mn = L j + ni1 t u naih ((p D q) D p) D p 1 and Lk, which can be represented as L j + p ytidvfei o p (p D q), ((p D q) D p) D p). We have thus proved the following qn yLef )o inLProposition 4.1. .f1 =ob Ip lLpp Ar E Ki = L j + t((p D q) D p) p , ' P ( no E jr i ap 1 +o. The problem of finding the most natural ' q axiomatics for EK remains open do p s ,(yet. D dij0o t ( i ie »s o q tn i .E / D ia }Jt l , r ovw h )i ) na e r bo ) ,ren 1 a i4 f ,
MA X I MA L PARACONS1STENT EXTENSION OF JOHANSSON LOGIC 1 1 5
Lemma 4.2. The formulae p ( p A -1 - / 13 areAclassical tautologies, which can be easily deduced in q These ) , ( formulae / ) the maximal negative logic using the scheme — ) q ) ' 1, ( p . a Lemma 4.3. E K does not contain —' —'p D p and ( p D a). n Proof I . Assuming E K p , we have L mn p D p. But Ldmn h e n c e LmnE- p, and the structural property implies that any formula is provable in Lmn, a contradiction. - 2. Arguing as above we obtain L mn p D (p D a), whence L mn H p 1 a. Substituting f o r p in the latter formula we again have L mn 1- a, a D contradiction. ( p y We list some interesting properties of EK. a ) Proposition 4.4. ( 1. E K is decidable; 2. E K is not a k—system; 1 3. A n y k—system L extending EK has a negation constructive in the PIp lowing sense: A - I f I, I- - 1 1 (q ) q Proof A L We have (p E E K iff (t) is true in 2 and in 2'. Thus, the condition c)c E 0E K , is effectively verified. E E K , but p b 2. tTheh disjunctive property fails for EK. Indeed, p and — i p are not in EK. e e n is an easy consequence of the definition of k—systems and the fact l 3. This c that EK contains the formula — o p isp proved. 1n (The p proposition A o ag ) . r we consider models for EK. Now, t I o Proposition 4.5. Let A = b e a j—lattice with E K C L A , E i.e., a —model for E l( The following properties are then true: K 1 . 1. A (treated as an implicative lattice) is a Peirce algebra; 2. A n interval [0,I1 53. [i 0s, I IAais a Boolean algebra; s u b a l g e b r a o f A i n t h
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4. F o r any a E A, a 5-- 0, we have — l a I f =( ) 1I and 5. ; [0,11A t A, then A contains an element incomparable with a Proof I Th is is implied by ((p D q) D p) D p E EK. All other statements o f the proposition Can be easily deduced from the representation theorem for Peirce algebras (see Proposition 2.2). Consider, for example, the last statement. 5. There exists an element a under 0 by assumption. Take an implication 0 D a, which is obviously incomparable with O. The proposition is proved. Consider the lattice 4' = (I0,1,-1,a 1 , ), where -I • a 1 . - 1 - ( ) I , and the elements a and 0 are incomparable. It is a Peirce algebra. The operation — which, the properties listed in Proposition 4.5. It i x =as=is easily x seen, satisfies D is 0 routine to tcheck that p v ( u r n s quently, 4' —i t exemplifies a model for EK. In fact, 4' is a characteristic model for l ai EK,Dnwhicht (canaobe deduced from the following Da r ) j) — 4.6. Let a j—lattice A be a model for EK. Then either A is a i Proposition s l a t t i c e , model for Lmn, o r A is a model for Lk, o r L A C EK, i.e., A is a characa n teristic model for EK. i d e n t i t y Proof. Let Ab e a j—lattice and E K C L A . I f 0 1 = 1. then — o forl aany=a E fA1 by Proposition 4.5.3. The formula ((p D q) D p) D p is ob4 ' in A. Thus, . viously true A is a model for Lmn. C Assume o 0 1 n, then s[0,11 e A Ai sis then - a model for the classical logic. a Finally, assume n o n t r i v that i a 0l 0 1 and [0,11 algebra isomorphic A A . I B o o l ento 2, a whence, L A C L 2 = L k. Consider a filter F (0) generated by 0 and the corresponding quotient. Since — tn h i s A, the lattice AIF (0) satisfies = 1. Therefore, L (A / F (0)) l a ca 0al sg f eo a the n identity y e ,r b Ln. Moreover, ((p D a) D p) D p is an identity of A/F (0) as a quotient of a A E a . A. rcConsequently, L (A /F (0)) L m n , and so L A C L k n L mn = EK. o n ft a i n s I The proposition is proved. a [ 0 , 1 ] sCorollary u b A 4.7. 4' is a characteristic model for EK. = A was noted above, 4' is a model for EK. In view of 0 0 1, 4' is not a As model , for Lmn. Moreover, L4' 0 L k since [0,11 =4 EK by Proposition 4.6. , 0 4 ' . C o n s e q u e n t l y , L 4 '
M A X I M A L PARACONSISTENT EXTENSION OF JOHANSSON LOGIC 1 1 7
Remark. The lattice 4' is the simplest among characteristic models fo r EK. Propositions 4.5 and 4.6 easily imply that any characteristic model for E K must contain at least four elements. Indeed, unity differs from zero, there is a third element under zero, and there is a fourth element incomparable with zero. Now we are in a position to state the maximality property for EK. Theorem 2. Let tp be a formula, not in E K . There are three possible cases: I. E K + {(p} is inconsistent; 2. E K + (p i = LEM; 3. E K + ço l = Lk. Proof Assume that E K + ( istic I model. The inclusion L A C E K fails, since cp is not in EK. By Proposition D } 4.6, i s we then n ohave t either LA = L k or L A = Lmn. The theorem is proved. i n c o n s i s t e n t We 1 a Peirce-Johansson algebra (pf-algebra) if a call nA = d, (A,y,A,D,1) is a Peirce algebra and the constant I is interpreted as an arbiA trary element of A. These algebras provide an adequate algebraic semantics fori the logic sEK. i t s cProposition h a 4.8. rAn algebra a A = (A ,y,A ,D,I,1 ) is a model for EK if and c if tA is ae pi-algebra. r only Proof. I f A is model for EK, it is a j-algebra satisfying the Peirce law ((p q) D p) D p, i.e. a pi-algebra. Assume that A is an arbitrary pj-algebra. Its reduct to the language 1V,A,D) is a Peirce algebra, and we need only verify that i p y ( - r)) is an identity of A. We have - c I for any a E A. the second disjunctive in L j to ( — It/ Dterm is( equivalent a checkAthat qI )y ( lc/ D ( —,we may think o f I , 12, and c as subsets o f some set X, which is the r2.2, l bh of tunity e Athe r ealgebra. f o We r ethen , have ) t ib ( ( X \f M Yf I ) Ai b ) cD c =e( b As I ) D e = ( X \ ( 1 9 A 1 ) ) y e . sD u ) tc o And finally, I y (X \ (b A I ) ) y , X as was to be proved. I f o r a n y b ,
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Corollary 4.9. The logic E K coincides with the Curry logic Le = L j + ((p D q) D p) p I [3]. It is clear, that the Peirce—Johansson algebras are distinguished in the variety of j—algebras via the identity ((p D q) D p) D q. Now, we make an interesting observation on the structure o f the class Jhn of all nontrivial extensions of the Johansson logic L I Let In t == L I , E Jhn, ( p D q) E L I be the class of all intermediate logics; let Neg == L I L E Jhn, p E L I be the class of all negative logics, i.e. Neg consists of logics with a trivial negation. Finally, let Par == ih n ( I n t u Neg) be the class of all proper paraconsistent logics. Obviously, the class Jhn is a disjoint union of the classes Int. Neg, and Par. It is well known that L E Int if and only if Li C L C Lk. Moreover, the following assertion holds. Proposition 4.10. Let L E Jhn. Then the following equivalences are true: (i) L E Neg if and only if Ln C L C Lmn; (ii) L E Par if and only if Lj C L C Le. Proof (i) I f L E Neg, then L n is contained in I, by definition. A t the same time, L can not be extended to Lk, consequently, L C Lmn. (ii) Let L E Par, and let A be a characteristic model for L. In A, I , hence (1 , 1 ) is a nontrivial Boolean algebra, which is a subalgebra of A. and so L C L k. Further, L OE In t, therefore, the quotient AIE ( I ) is nontrivial and has the greatest element I , hence it contains the two-element subalgebra isomorphic to 2'. The latter means that L C L mn . Thus, L C Le L k n Lmn. The proposition is proved. As we can see from the last assertion, the class Jhn decomposes into three disjoint intervals, one o f which, Par, contains logics that are really paraconsistent. Two other intervals demonstrate degenerate cases of paraconsistency. The class In t consists of consistent logics, i i s the least element in the models fo r such logics, and the class Neg consists o f logics with identically true negation, I is the greatest element in their models. In conclusion, we say a few words on the semantics for Le. The characteristic model 4' shows that Le is a finite-valued logic, and the number of truth values is four. We give a representation of the algebra 4' which makes
MA X I MA L PARACONSISTENT EXTENSION OF JOHANSSON LOGIC 1 1 9
the truth values of Le more sensible. From this point on, we use boldface notation for elements of 4': 1, 0, -1, and a. Consider the Peirce algebra 2' x 2P, whose elements are pairs of classical truth values, 0 and 1, and operations v, A, and D are classical on each of the components. The operation ---, algebra a = = '6 isomorphic a Dto 4'. Thus, the truth values of 4' correspond to the pairs: ( 0 1, —›1 (I )I ) 0 —› (0,1), -1 --> (0,0), and a —› (1,0). In Le, every statement by the pair of parameters, which are classical, in t u is thus r ncharacterized s some sense. 2 P x However, the question what means fo r a statement o f the logic L e to 2 P have a truth value (a,f3), a, 13 E 0 , 1 ] , remains open. A t first glance we imay understand n t this pair o of truth values, for example, as follows. The first a component is a propositional characteristic o f the statement, " tru e " o r p "false", and the second components is the measure o f definiteness, with iwhich the statement has its propositional truth value. Thus, the unity 1 = — is the definite truth, and the contradiction I = (0,1) is a definitely (1,1) false statement. In Z, we have — well 1 ( 1 with , 0 ) our=intuition. Indeed, it is natural to consider as definitely false the truth leads to a contradiction. But this interpre( 1statement , 0 ) that indefinite D tation of truth values for Le is not satisfactory. For example, it is question( 0 , 1 ) able to assume that the negation of indefinitely false statement is definitely = true, however — ( 0 , 1 ) , way the equality a v I = (1,0) y (0 , 1 ). (1,1). 1 w ( 0 , 0h) i c = h Of course, this is only a preliminary remarks to the semantic study o f Ia g, rextensions e paraconsistent o f Johansson logic. However, the semantics for 1 ) eLe described s above contains a key allowing to construct natural formal i n semantics for a wide class of logics in Par, which will be considered in the Z . subsequent works. I t i s Institute of mathematics, Koptyuga 4 a l Novosibirsk, 630090, Russia s o e-mail:
[email protected] i m p REFERENCES o s s i b l e t[1] I . Johansson. Der Minimalkalkill, ein reduzierter intuitionistischer Foro malismus. Compositio Mathematika, 4, 1937, 119-136. e[2] S.A.x Kripke. The system LE. unpublished. p a n i n t h
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