Our question is: can we embed minimal negation in implicative logics weaker than I→? Previous results show how to define minimal negation in the positive fragment of the logic of relevance R and in contractionless intuitionistic logic. Is it possible to endow weaker positive logics with minimal negation? This paper prooves that minimal negation can be embedded in even such a weak system as Anderson and Belnap’s minimal positive logic

Mendez, J.

Robles, G.

Salto, F.

English

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REPORTS ON MATHEMATICAL LOGIC36 (2002), 117–130Jose´ M. ME´NDEZ, Francisco SALTO, Gemma ROBLESANDERSON AND BELNAP’S MINIMAL POSITIVELOGIC WITH MINIMAL NEGATION1. IntroductionWe understand the locution “minimal negation” as historically definedby Johansson in [3], where negative theorems are instances of positive in-tuitionistic theorems with A ! f abbreviating ¬A and f an arbitrarypropositional falsity constant. In particular, any logic with axioms for cus-tomary properties of conjunction and disjunction is endowed with minimalnegation if it contains as theorems:(i) A! ¬¬A(ii) (A! B)! (¬B ! ¬A)(iii) (A! ¬B)! (B ! ¬A)(iv) (A! ¬A)! ¬A(v) (A! B)! ((A! ¬B)! ¬A)(vi) (A! ¬B)! ((A! B)! ¬A)all of which are instances of positive implicative theorems of I! (intu-itionistic implicational logic).Received 10 April 2001, revised 13 February 2002118 JOSE´ M. ME´NDEZ, FRANCISCO SALTO, GEMMA ROBLESOur question is: can we embed minimal negation in implicative logicsweaker than I!? [4] shows how to define minimal negation in the positivefragment of the logic of relevance R and [5] in contractionless intuitionisticlogic. Is it possible to endow weaker positive logics with minimal negation?In this paper our aim is to prove that minimal negation can be embeddedin even such a weak system as Anderson and Belnap’s minimal positivelogic.Whenever implicative resources to instantiate minimal negation areabsent, we can still impose additional constraints on f to obtain minimalnegation. Interestingly, this strategy allows the semantical isolation ofdi↵erent principles of negation. Moreover, finegrained varieties of negationweaker than minimal are naturally considered in this setting, which o↵ersa kind of microscopical companion to [2]. Together with the modelizingof positive and minimal negation, these facts illustrate – we think- theconceptual import and motivation of the proofs.The structure of the paper is as follows. In §2, Anderson and Belnap’sminimal positive logic ML+ is presented. §§3-5 recall semantical consis-tency and completeness proofs for ML+ with respect to Routley-Meyertype relational semantics. §§6,7,8 define the logic ML+, f (ML with thefalsity constant f added to the sentential language). In §§9,10,11, we definethe logic MLm (ML+ with weak double negation and weak contraposi-tion, but without reductio). In §§12,13,14 the logic MLmr (MLm withthe reductio axiom). The paper ends in §15 with three notes on the resultsobtained.2. Anderson and Belnap’s Minimal Positive Logic:The logic ML+Anderson and Belnap’s minimal implicative logic is (see 8.11 of [1],labelled T!  W ):A1. A! AA2. (A! B)! ((B ! C)! (A! C))A3. (B ! C)! ((A! B)! (A! C))ANDERSON AND BELNAP’S MINIMAL POSITIVE LOGIC WITH MINIMAL NEGATION 119with the rule Modus ponens (if ` A and ` A! B, then ` B).To this logic ML! we add the axioms:A4. (A ^B)! A (A ^B)! BA5. ((A! B) ^ (A! C))! (A! (B ^ C))A6. A! (A _B) B ! (A _B)A7. ((A! C) ^ (B ! C))! ((A _B)! C)A8. (A ^ (B _ C))! ((A ^B) _ (A ^ C))and the rule Adjunction (if ` A and ` B, then ` A ^B) thus definingthe logic ML+. We note that ML+ is BR+ plus C5 of [1] (p.294), alsocalled (T  W )+.The following formulas (useful in the proof of the completeness theo-rem) are derivable:T1. (A ^B)! (B ^A)T2. ((A _B) ^ (C ^D))! ((A ^ C) _ (B ^D))T3. ((A! C) _ (B ! D))! ((A ^B)! (C _D))T4. ((A! C) ^ (B ! D))! ((A ^B)! (C ^D))T5. ((A!C)^(B!D))!((A_B)!(C_D))3. Semantics for ML+AML+ model is a quadruple < O,K,R, ✏>where K is a set, O a non-empty subset of K and R a ternary relation on K subject to the followingdefinitions and postulates for all a, b, c, d 2 K with quantifiers ranging overK:d1 a  b =def 9x(x 2 O and Rxab)d2 R2abcd =def 9x(Rabx and Rxcd)P1 a  aP2 a  b and Rbcd) RacdP3 R2abcd) 9x(Racx and Rbxd)P4 R2abcd) 9x(Rbcx and Raxd)120 JOSE´ M. ME´NDEZ, FRANCISCO SALTO, GEMMA ROBLESFinally, ✏ is a binary valuation relation between elements from K andsentences ofML+ satisfying the following conditions for all sentence lettersp, formulas A,B and a, b, c 2 K:(i) a ✏ p and a  b) b ✏ p(ii) a ✏ A _B i↵ a ✏ A or a ✏ B(iii) a ✏ A ^B i↵ a ✏ A and a ✏ B(iv) a ✏ A! B i↵ for all b, c 2 K, Rabc and b ✏ A) c ✏ BA is valid in ML+ [✏ML+ A] i↵ a ✏ A for all a 2 O in all models.4. Semantic consistency of ML+We prove:Lemma 4.1. a  b and a ✏ A) b ✏ AProof. Induction on the length of A (Use P2 in the case of the condi-tional). ⌅Lemma 4.2. ✏ML+ A ! B i↵ for all a 2 K in all models, a ✏ A )a ✏ BProof. Use P1, d1 and Lemma 4.1. ⌅Now we can prove:Theorem 4.1. (Semantic consistency of ML+) If `ML+ A, then✏ML+ A.Proof. Since all axioms of ML+ are conditional formulas, we can useLemma 4.2. to immediately render A1, A4-A8 and the rules. A2 and A3are proved with, respectively, P3 and P4. ⌅5. Completeness of ML+We begin by recalling some definitions. A theory is a set of formulasof ML+ closed under adjunction and provable entailment; a theory a isANDERSON AND BELNAP’S MINIMAL POSITIVE LOGIC WITH MINIMAL NEGATION 121prime if whenever A _ B 2 a, then A 2 a or B 2 a; finally, a is regular ifall theorems of ML+ belong to a.Now we define the ML+ canonical model. Let KT be the set of alltheories and RT be defined on KT as follows: for all formulas A,B anda, b, c 2 KT , RTabc just in case if A ! B 2 a and A 2 b, then B 2c. Further, let KC be the set of all prime theories, OC the set of allregular prime theories and RC the restriction of RT to KC . Finally, let✏C be defined as follows: for any w↵ A and a 2 KC , a ✏C A i↵ A 2 a.Then, theML+ canonical model is the quadruple < OC ,KC ;RC ,✏C>. Inwhat follows in this section we sketch a proof of the completeness theorem,beginning with some previous Lemmas.Lemma 5.1. Let A be a w↵, a 2 KT and A /2 a. Then, A /2 x forsome x 2 KC such that a ✓ x.Proof. Define from a a maximal theory x without A. If x is notprime, then for some w↵s B,C,B_C 2 x,B /2 x,C /2 x. Put [B, x] = {E :9D(D 2 x and ` (B ^D)! E}. Define [C, x] similarly. Well, both [B, x]and [C, x] are theories strictly including x. As x is maximal, A 2 [B, x]and A 2 [C, x], whence it is easy to prove that A 2 x, which is impossible.⌅Lemma 5.2. Let RTabc, a, b 2 KT , c 2 KC . Then, RTxbc for somex 2 KC such that a ✓ x.Proof. Define from a a maximal theory x such that RTxbc. Suppose xis not prime and define [A, x] and [B, x] as in Lemma 5.1. Since both [A, x]and [B, x] are theories strictly including x, we deduce not RT [A, x]bc andnot RT [B, x]bc. But then, it is easily shown that c is not prime, which isimpossible. ⌅Lemma 5.3. Let RTabc, a, b 2 KT , c 2 KC . Then, RTaxc for somex 2 KC such that b ✓ x.Proof. Similar to that of Lemma 5.2. ⌅122 JOSE´ M. ME´NDEZ, FRANCISCO SALTO, GEMMA ROBLESLemma 5.4. If `ML+ A, there is some x 2 OC such that A /2 x.Proof. ML+ is the minimal regular theory such that A /2ML+. ByLemma 5.1, there is some x 2 KC such that ML+ ✓ x and A /2 x. Butx 2 OC . ⌅Lemma 5.5. Let a, b 2 KT . The set x = {B : 9A(A ! B 2 a andA 2 b)} is a theory such that RTabx.Proof. It is smoothly proven that x is closed under adjunction andprovable entailment. Obviously then, RTabx. ⌅Lemma 5.6. a C b i↵ a ✓ b.Proof. Suppose a C b. By d1, RCxab with x 2 OC . As A! A 2 x,if A 2 a, then A 2 b, i.e., a ✓ b. Suppose now a ✓ b. Granted that a is atheory, clearly RTML+ aa, and, so, RTML+ ab. By Lemma 5.2, there issome x 2 KC such that ML+ ✓ x and RCxab. As x 2 OC , a C b by d1.⌅Lemma 5.7. Let a, b, c 2 KT and d 2 KC . Moreover, let RT2abcd.Then, there is some x 2 KC such that RTacx and RT bxd.Proof. Suppose RT2abcd, that is, RTabx and RTxcd for some x 2 KT .We have to prove that there is some x 2 KC such that RTacx and RT bxd.Define [cf. Lemma 5.5] the theory z = {B : 9A(A ! B 2 a and A 2 c)}with RTacz. Deduce now RT bzd using A2. By Lemma 5.3, RT bxd withz ✓ x and x 2 KC . By RTacz and definitions, RTacx. ⌅Lemma 5.8. Let a, b, c 2 KT and d 2 KC . Further, assume RT2abcd.Then, there is some x 2 KC such that RT bcx and RTaxd.Proof. Proceed as in Lemma 5.7, but use A3. ⌅Lemma 5.9. The canonical postulates hold in the ML+ canonicalmodel.Proof. P1 is trivial by Lemma 5.4. P2 holds by Lemma 5.4 anddefinition of RC . Use Lemmas 5.7 and 5.8 to prove, respectively, P3 andP4. ⌅ANDERSON AND BELNAP’S MINIMAL POSITIVE LOGIC WITH MINIMAL NEGATION 123Lemma 5.10. The canonical ✏C is a valuation relation satisfying con-ditions (i)-(iv) of §3.Proof. (i) is trivial by Lemma 5.4 (ii) and (iii) are easy given ML+and properties of the members of KC . (iv) from left to right is immediate.So, interest focusses on clause (iv) from right to left. Assume a 2C A! B.We show that there are x, y 2 KC such that RCaxy, x ✏C A and x 2C B.Define first b = {C :` A! C}. Notice b is a theory and A 2 b [` A! A].Define now (cf. Lemma 5.5.) the theory c = {C : 9D(D ! C 2 a andD 2 b)} such that RTabc. We do have: B /2 c (if otherwise B 2 c, thenC ! B 2 a, C 2 b, therefore, ` A ! C, and, so, A ! B 2 a, i.e.,a ✏C A! B, contradicting the hypothesis). By Lemma 5.1, there is somey 2 KC such that c ✓ y and B /2 y. Applying definitions, RTaby. ByLemma 5.3, there is some x 2 KC including b and RCaxy with A 2 x[A 2 b]. ⌅Lemma 5.11. The ML+ canonical model is indeed a ML+ model.Proof. By Lemma 5.4, OC is non-empty. Obviously, RC is a ternaryrelation on KC . Then, Lemma 5.11 follows by Lemmas 5.9 and 5.10. ⌅Finally, we proveTheorem 5.1. (Completeness of ML+) If ✏ML+ A, then `ML+ A.Proof. By contraposition, if 0ML+ A, then 2ML+ A: Lemmas 5.4 and5.11. ⌅6. The logic ML+ fIn order to define the logic ML+ f , we add to the sentential languageof ML+ the propositional falsity constant f together with the definition:¬A =def A ! f . Note that, for example, the following are provable inML+ f :T6. (A! B)! (¬B ! ¬A)T7. ¬B ! ((A! B)! ¬A)124 JOSE´ M. ME´NDEZ, FRANCISCO SALTO, GEMMA ROBLEST8. ¬(A _B)! (¬A ^ ¬B)T9. (¬A _ ¬B)! ¬(A ^B)7. Semantics for ML+ fAnML+f model is a quintuple < O,K, S,R, ✏> where < O,K,R, ✏>is anML+ model and S is a subset of K such that S\O 6= Ø. The clause:(v) a ✏ f i↵ a /2 Sis satisfied in all models.A is valid in ML+ f [✏ML+f A] i↵ a ✏ A for all a 2 O in all models.We note that f is not valid: let a 2 S \ O. Then, a 2 f . But a 2 O.So, 2ML+f f .Theorem 7.1. (Semantic consistency of ML + f) If `ML+f A,then ✏ML+f A.Proof. Since analogues of Lemmas 4.1 and 4.2 are immediate, thetheorem follows by Theorem 4.1. ⌅8. Completeness of ML+ fWe define the ML+ f canonical model as the quintuple< OC ,KC , SC , RC ,✏C>where < OC ,KC , RC ,✏C> is the ML+ canonical model and SC is inter-preted as the set of all consistent theories. A theory a is consistent i↵f /2 a.Now we need to prove:Lemma 8.1. SC \OC is not empty.Proof. Given that 2ML+f f [see §7], we have, by theorem 7.1., 0ML+ff , i.e., f /2ML+ f . As ML+ f is a theory, Lemma 5.1. applies and thereis some x 2 KC such that ML + f ✓ x and f /2 x. Obviously, x 2 OC[since ML+ f ✓ x]. As f /2 x, x 2 SC . ⌅ANDERSON AND BELNAP’S MINIMAL POSITIVE LOGIC WITH MINIMAL NEGATION 125Lemma 8.2. Clause (v) holds in the canonical model.Proof. Lemma 8.1 and definition of SC . ⌅Lemma 8.3. TheML+f canonical model is in fact aML+f model.Proof. Lemmas 5.11, 8.1 and 8.2. ⌅Theorem 8.1. (Completeness of ML+ f) If ✏ML+f A, then`ML+f A.Proof. It su ces to note that an analogue of Lemma 5.4 holds forML+ f , since the theorem follows by Lemmas 5.4, 8.1 and 8.3. ⌅9. ML+ with minimal negation: The logic MLmMLm is defined when we add to ML+ f the axiom:A9. A! ((A! f)! f)We note that, in addition to T6-T9, the following are exemplary the-orems of MLm:T10. A! ¬¬AT11. (A! ¬B)! (B ! ¬A)T12. ¬¬¬A! ¬AT13. (¬A ^ ¬B)! ¬(A _B)Alternatively, MLm can be axiomatized withA9’. If ` A! (B ! f), then ` B ! (A! f)orA9”. (A! (B ! f))! (B ! (A! f))instead of A9, among other possibilities. Proof is left to the reader.10. Semantics for MLmAMLmmodel is anyML+f model with the addition of the postulate:P5. Rabc and c 2 S ) 9x(x 2 S and Rbax)A is valid in MLm (✏MLm A) i↵ a ✏ A for all a 2 O in all models.126 JOSE´ M. ME´NDEZ, FRANCISCO SALTO, GEMMA ROBLESTheorem 10.1. (Semantic consistency ofMLm) If `MLm A, then✏MLm A.Proof. Given Theorem 7.1, just the validity of A9 is at issue. Use P5.⌅11. Completeness of MLmWe define the MLm canonical model exactly as the ML+ f canonicalmodel, but with this di↵erence: now a theory a is consistent i↵ the negationof a theorem does not belong to a.Lemma 11.1. f 2 a i↵ a is inconsistent.Proof. Suppose f 2 a. By A9, (f ! f) ! f 2 a. Thus, a isinconsistent. Suppose now a is inconsistent. Then, for a theorem A, A!f 2 a. By A9, (A! f)! f is a theorem. So, f 2 a. ⌅Next, we note that an analogue of Lemma 8.1 for MLm becomes im-mediate. We proveLemma 11.2. Let a, b, c 2 KT with c consistent and RTabc. Then,there is some x 2 SC such that c ✓ x and RT bax.Proof. Assume RTabc, with a, b, c 2 KT and c consistent. Define (cf.Lemma 5.5) the theory y = {B : 9A(A! B 2 b and A 2 a)}. We go intoproving that y is consistent. Suppose it is not. Then, (cf. Lemma 11.1)f 2 y. By definition of y, A ! f 2 b, A 2 a. By A7, (A ! f) ! f 2 a.Given RTabc, f 2 c, which is impossible c being consistent.So, we have a consistent theory y such that RT bay. Since f /2 y,Lemma 5.1 applies and we have some x 2 KC such that y ✓ x and f /2 x.So x is consistent, that is, x 2 SC . Given that y ✓ x and RT bay, weconclude RT bax. ⌅Lemma 11.3. The canonical P5, i.e., RCabc and c 2 SC ) 9x(x 2SC and RCbax), holds in the MLm canonical model.Proof. By Lemma 11.2. ⌅ANDERSON AND BELNAP’S MINIMAL POSITIVE LOGIC WITH MINIMAL NEGATION 127Lemma 11.4. Clause (v) holds in the MLm canonical model.Proof. Lemma 11.1. ⌅Lemma 11.5. The MLm canonical model is indeed a MLm model.Proof. Lemmas 8.3, 11.3 and 11.4. ⌅Finally we turn toTheorem 11.1. (Completeness of MLm) If ✏MLm A, then`MLm A.Proof. First we note that an analogue of Lemma 5.4 for MLm isimmediate. Then, theorem 11.1 follows by Lemmas 5.4 and 11.5. ⌅12. MLm with the reductio axiom: The logic MLmrWe add to MLm the axiom:A10. (A! (A! f))! (A! f)Note that, in addition to T6-T13, the following are provable inMLmr:T14. (A! B)! ((A! ¬B)! ¬A)T15. (A! ¬B)! ((A! B)! ¬A)T16.(A! ¬B)! ¬(A ^B)T17. (A! B)! ¬(A ^ ¬B)T18. ¬(A ^ ¬A)T19. ¬¬(A _ ¬A)MLmr can be axiomatized alternatively withA10’. (A! B)! ((A! (B ! f))! (A! f))that is, T14, orA10”. (A! (B ! f))! ((A! B)! (A! f))which is T15, orA10”’. (A! (B ! f))! ((A ^B)! f)i.e., T16, or128 JOSE´ M. ME´NDEZ, FRANCISCO SALTO, GEMMA ROBLESA10””. If ` A! B, then ` (A! ¬B)! ¬Aamong other possibilities instead of A10. Proofs are left to the reader.13. Semantics for MLmrModels forMLmr are defined as those forMLm but with the additionof the postulateP6. Rabc and c 2 S ) 9x9y(Rabx and Rxby and y 2 S)Note. The postulateP6’. Rabc and c 2 S ) 9x(x 2 S and Rcbx)used in [5] for A10 still would work here. We have preferred P6 becauseit is exactly what is needed in the semantic consistency and completenessproofs of the alternatives o↵ered in §12 (we leave to the reader the proofsof these facts).Theorem 13.1. (Semantic consistency of MLmr) If `MLmr A,then ✏MLmr A.Proof. Given Theorem 10.1, we have to prove that A10 is valid. UseP6. ⌅14. Completeness of MLmrThe MLmr canonical model is defined similarly as the correspondingone for MLm. Hence, an analogue of Lemma 8.1 for MLmr is immediate.Next we prove,Lemma 14.1. Let a, b, c 2 KT c being consistent and RTabc. Then,there is some x 2 KC and y 2 SC such that RTabx and RTxby.Proof. Suppose RTabc, a, b, c 2 KT and c a consistent theory. Define(cf. Lemma 5.5) the theory u = {B : 9A(A! B 2 a and A 2 b)} such thatRTabu. Define now the theory w = {B : 9A(A! B 2 u and A 2 b)} suchthat RTubw. We prove first that w is consistent. Suppose it is not. Then,ANDERSON AND BELNAP’S MINIMAL POSITIVE LOGIC WITH MINIMAL NEGATION 129f 2 w (cf. Lemma 11.1.). By definition of w, B ! f 2 u (B 2 b). Bydefinition of u, A! (B ! f) 2 a (A 2 b). Then, by T16, (A^B)! f 2 a.But, given that A^B 2 b (A,B 2 b) and RTabc, f 2 c, which is impossiblec being consistent. Therefore, we have u,w 2 KT (w being consistent).Lemma 5.1 applies and in consequence there is some y in SC such thatw ✓ y and RTuby. Now, by Lemma 5.2 there is some x in KC such thatu ✓ x and RTxby. As RTabu (and u ✓ x), RTabx as required. ⌅Lemma 14.2. The canonical P6 holds in theMLmr canonical model.Proof. By Lemma 14.1. ⌅Lemma 14.3. TheMLmr canonical model is in fact aMLmr model.Proof. Lemmas 11.5 and 14.2. ⌅Theorem 14.1. (Completeness of MLmr) If ✏MLmr A, then`MLmr A.Proof. Lemma 5.4 for MLmr and Lemma 14.3. ⌅15. Additional remarks1. Both MLm and MLmr can also be defined with a negation con-nective instead of the falsity constant f , by means of the general strategyfollowed in [5].2. MLm and MLmr are di↵erent logics. We refer to the results of [5].There, the logic I +  C [positive intuitionistic logic without contraction]was defined and next extended with a minimal negation [Im   C] andreductio [Imr   C]. These extensions are defined from I + C exactly asMLm and MLmr were defined from ML+. Well, in that paper it wasproved that Im   C and Imr   C are di↵erent logics, the first includedin the second. Therefore, MLm and MLmr are di↵erent logics, MLmrincluding MLm.3. Can a genuine intuitionistic negation, that is, a minimal negationplus130 JOSE´ M. ME´NDEZ, FRANCISCO SALTO, GEMMA ROBLES¬A! (A! B)be introduced inML+? The answer is a rmative, but we leave the matterfor another paper.We thank the contribution of two RML referees.References[1] A.R. Anderson and N.D. Belnap, Entailment I. Princeton, 1975.[2] M. Dunn, Generalized ortho-negation, in H. Wansing (ed.) Negation. A notion inFocus. Berlin, 1996, 3–26.[3] I. Johansson, Der Minimalkalku¨l, ein reduzierter intuitionistischer Formalismus,Compositio Mathematica 5 (1936), 119-136.[4] J.M. Me´ndez, Constructive R, Bulletin of the Section of Logic 16 (1987), 167–175.[5] J.M. Me´ndez and F. Salto, Intuitionistic propositional logic without contraction butwith reductio, Studia Logica 66 (2000) 301–310.Universidad de Salamanca37007 Salamanca, Espana

Anderson And Belnap's Minimal Positive Logic With Minimal Negation

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Anderson And Belnap's Minimal Positive Logic With Minimal Negation

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