Méndez Rodríguez, José Manuel

Robles Vázquez, Gemma

THEORIA An International Journal for Theory History and Foundations of Science

English

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Two versions of minimal intuitionism with the CAP. A note Gemma ROBLES and José M. MÉNDEZ Received: 2004.09.20 Final Version: 2005.04.02  BIBLID [0495-4548 (2005) 20: 53; pp. 183-190]  ABSTRACT. Two versions of minimal intuitionism are defined restricting Contraction. Both are defined by means of a falsity constant F. The first one follows the historical trend, the second is the result of imposing special constraints on F. Relational ternary semantics are provided. Keywords: Intuitionistic logic, Contraction Axiom, Converse Ackermann Property, Constructive Falsity.  1. Motivation      As it is known, minimal intuitionism can be viewed as a definitional extension of posi-tive intuitionistic logic. The idea is to add to the positive language a propositional falsi-fity constant F along with the definition ¬A =df A → F. Then, given the intuitionistic positive theorems,                a.  [A → (B → C)] → [B → (A → C)]      b.  A → [(A → B) → B]     c.  (A → B) → [(B → C) → (A → C)]     d.  [A → (A → B)] → (A → B)     e.  [A → (B → C)] → [(A → B) → (B → C)]     we have, for example (and to limit ourselves to conditional-negation theorems) the following minimal intuitionistic theses:     a’. (A → ¬B) → (B → ¬A)     b’. A → ¬¬A     c’. (A → B) → (¬B → ¬A)     d’. (A → ¬A) → ¬A     e’. (A → ¬B) → [(A → B) → ¬A]  Theoria 53: 183-190, 2005. Gemma ROBLES and José M. MÉNDEZ 184     Minimal intuitionism departs (from this syntactical point of view) from full in-tuitionistic logic in the absence of, e. g.,     f.  A → (¬A → B) or     g.  (A ∧ ¬A) → B though, we note,     h.  A → (¬A → ¬B) or     i.  (A ∧ ¬A) → ¬B are present.  On the other hand, let  us focus on the CAP. A positive logic with a falsity con-stant F  has the Converse Ackermann Property (CAP) if all the formulas of the form (A → B) → C are unprovable whenever C contains neither → nor F. The CAP can intuitively be interpreted as the non-derivability of necessitive propositions from non-necessitive ones (A is necessitive if A is of the form NB, that is, if A is equivalent to (B → B) → B (Anderson and Belnap 1975, §4.3)).     The question about which systems do possess the CAP is first proposed in Ander-son and Belnap (1975, §8.12). In Méndez (1987) it is answered for implicative and positive logics. Syntactically speaking, the solution consists in restricting contraction ((d) above) and assertion ((b) above) to the case in which B is an implicative formula (A is implicative iff A is of the form B → C). Thus, logics with the CAP are contractionless logics. Actually, they are the natural bridge between strict contractionless logics and logics with contraction.     But what about negation in these logics? That is, which kind(s) of negation(s) is (are) compatible with the CAP? We briefly note the results we are aware of. In Mé-ndez (1988) a sort of semiclassical negation, in Kamide (2002), Kamide (2003) a so-called “strong negation” is added to the positive logics of Méndez (1987). Now, the aim of this paper is to define minimal negation in the positive intuitionistic logic with the CAP of Méndez (1987).     As the  title makes clear, we consider two possibilities. The first one follows the his-torical trend commented above: we definitionally extend positive intuitionistic logic with the CAP, I0+, with F. This gives us the logic  I0m1. Now let us try to motivate the second one. As (a)-(e) above are not provable in I0+, (a’)-(e’) are not provable either. Nevertheless, (a’)-(e’) are, of course, not only minimally acceptable but also desirable indeed. Moreover, they are CAP-compatible. In consequence, we shall show how to impose additional constraints on F to make these formulas valid, which give us the second possibility. This gives us the logic  I0m2.  Two versions of minimal intuitionism with the CAP. A note  185     The logics we present here have not been, to our knowledge, defined in the litera-ture. Nevertheless, it is not hard to derive them from some published work, as we show below. 2. The logics I0m1,  I0m2Positive intuitionistic logic with the CAP, I0+, can be axiomatized as follows (see Mén-dez (1987)).     Axioms:     A1.  A → A     A2.  A → (B → A)     A3.  (A → B) → [(B → C) → (A → C)]     A4.  A → [[A → (B → C)] → (B → C)]     A5.  [A → [A → (B → C)]] → [A → (B → C)]     A6.  (A ∧ B) → A  /  (A ∧ B) → B     A7.  [(A → B) ∧ (A → C)] → [A → (B ∧ C)]     A8.  A → (A ∨ B)  /  B → (A ∨ B)     A9.  [(A → C) ∧ (B → C)] → [(A ∨ B) → C]     A10. [A ∧ (B ∨ C)] → [(A ∧ B) ∨ C]     Rules:     Modus ponens (MP): if  A → B and  A, then  B.     Adjunction (Ad): if  A and  B, then  A ∧ B.  Now, I0m1 is a definitional extension of  I0+ with the propositional falsity constant F along with the definition ¬A =df A → F. For instance, the following theorems be-long to I0m1:   Gemma ROBLES and José M. MÉNDEZ 186     T1.  ¬F     T2.  (A → B) → (¬B → ¬A)     T3.  ¬B → [(A → B) → ¬A]     T4.  F → ¬A     T5.  ¬A → (A → ¬B)     T6.  A → (¬A → ¬B)     T7.  (¬A ∧ ¬B) ↔ ¬(A ∨ B)     T8.  (¬A ∨ ¬B) → ¬(A ∧ B)     T9.  ¬(A ∧ B) → (A → ¬B)     T10. (¬A ∨ ¬B) → (A → ¬B)     T11. (A ∨ ¬B) → (¬A → ¬B)     T12. (A ∧ ¬A) → ¬B     T13. [(A ∨ ¬B) ∧ ¬A] → ¬B  On the other hand, I0m2 is the result of adding to I0m1 the axioms:     A11. A → [(A → F) → F] i.e., A → ¬¬A, and     A12. [A → (A → F)] → (A → F) i.e., (A → ¬A) → ¬A.  In addition to T1-T13 of I0m1, we have, for example, the following theorems:     T14. (A → ¬B) → (B → ¬A)     T15. (A → B) → [(A → ¬B) → ¬A]     T16. (A → ¬B) → [(A → B) → ¬A]  Two versions of minimal intuitionism with the CAP. A note  187     T17. (A ∧ B) → ¬(A → ¬B)     T18. ¬(A ∧ ¬A)     T19. ¬¬¬A → ¬A     T20. ¬¬(A ∨ ¬A) 3. Converse Ackermann Property It is proved in Salto, Méndez and Robles (2001) that LC0m has the CAP. I0m2  is (syn-tactically) included in LC0m. Therefore, I0m1 and I0m2 have the CAP. 4. Semantics for I0+Given a pair <K, R> where K is a non-empty set and R a ternary relation on K, let us define the binary relation ≤, the quaternary relation R2 and the five element relation R³ by, for every a, b, c, d ∈ K,     d1.  a ≤ b  iff (∃x∈K) Rxab     d2.  R²abcd  iff (∃x∈K)(Rabx and Rxcd)     d3.  R3abcde  iff (∃x∃y∈K)(Rabx and Rxcy and Ryde)  An I0+ model is a triple <K, R, > where K is a non-empty set, R is a ternary rela-tion on K satisfying the following conditions for every a, b, c, d ∈ K:     P1.  a ≤ a     P2.  a ≤ b  and  Rbcd ⇒ Racd     P3.  R²abcd ⇒ (∃x∈K)(Racx and Rxbd)     P4.  R²abcd ⇒ R²bacd     P5.  R²abcd ⇒ R³abbcd     P6.  Rabc ⇒ a ≤ c     Finally,  is a valuation relation from K to the sentences of I0+ satisfying the fol-lowing conditions for all formulas p, A, B and a point a in K:   Gemma ROBLES and José M. MÉNDEZ 188     i.    a  p  and  a ≤ b ⇒ b  p     ii.   a  A ∨ B  iff  a  A  or  a  B     iii.  a  A ∧ B  iff  a  A  and  a  B     iv.  a  A → B  iff for all b, c, e ∈ K, (Rabc and b A) ⇒ c  B  A is valid in I0+ iff a  A  for all a ∈ K in all models.  Remark: the postulates     P7.  R²abcd ⇒ (∃x∈K)(Rbcx  and Raxd)     P8.  Rabc  b ≤ c are derivable.  It is not difficult to prove along the lines of [3] that a formula A is valid iff A is a I0+ theorem. 5. Semantics for I0m1 A I0m1 model is a quadruple <K, S, R,  > where S is a non-empty subset of K and <K, R,  > is a I0+ model such that the relation  satisfies in addition the clauses     v.   a ≤ b  and  a  F ⇒ b  F     vi.  a  F  iff  a ∉ S  A formula A is I0m1 valid  iff a  A for all a ∈ K in all models. Semantic consistency (semantic soundness of I0m1 relative to the semantics of I0m1 models) is immediately derived from that of I0+. (Note that being S non-empty, F is not valid). 6. Semantics for I0m2 A I0m2 model is similar to a I0m1 model but with the addition of the postulates     P9.  (Rabc  and c ∈ S) ⇒ (∃x∈S) Rbax     P10. (Rabc  and c ∈ S) ⇒ (∃x∈K)(∃y∈S)(Rabx  and Rxby).  Two versions of minimal intuitionism with the CAP. A note  189  A formula A is I0m2 valid iff  a  A for all a ∈ K in all models. Semantic consis-tency is left to the reader (the validity of A11 is proved with P9; the validity of A12 with P10). 7. Completeness of  I0m1 and I0m2As noted in §3, I0m1 and I0m2 are sublogics of LC0m. In Salto, Méndez and Robles (2001) it is proved the completeness of LC0m with respect to an extension of the se-mantics here provided for I0m2. Now, it is not difficult to prove completeness theo-rems for I0m1 and I0m2 using appropriate restrictions of the theorems and lemmas there employed (just let aside any references in Salto, Méndez and Robles (2001) to P7-P10 in the case of  I0m1 and to P10 in the case of I0m2). Acknowledgements Work partially supported by grants BFF-2001-2066 Ministerio de Ciencia y Tecnología, España (Ministry of Science and Technology, Spain) and SA088/02 of the Junta de Castilla y León, España.  REFERENCES Anderson, A.R., and Belnap, N. (1975). Entailment: The Logic of Relevance and Necessity, vol. 1. Princeton University Press. Kamide, N. (2002). “A canonical model construction for substructural logics with strong negation”, Re-ports on Mathematical Logic, 36, 95-116. ——— (2003). “Normal substructural logics with strong negation”, Journal of Philosophical Logic, 32, 589-612. Méndez, J.M. (1987). “A Routley-Meyer semantics for Converse Ackermann Property”, Journal of Philoso-phical Logic, 16, 69-76. ——— (1988). “Converse Ackermann Property and semiclassical negation”, Studia Logica, 47, 159-168. Salto, F., J.M. Méndez, and G. Robles (2001). “Restricting the contraction axiom in Dummett’s LC: a sublogic of LC with the Converse Ackermann Property, the logic LC0”, Bulletin of the section of Logic, 30, 3, 139-146.    Gemma ROBLES is a PhD candidate for the Universidad de Salamanca. Her doctoral dissertation is ex-pected to be read at the beginning of 2005. Her publications include over twelve items on contrac-tionless logics and semantics for different concepts of negation in Bulletin of the Section of Logic, Re-ports on Mathematical Logic and Teorema. She works with Francisco Salto and José M. Méndez in the research project directed by the latter.  ADDRESS: Departamento de Filosofía y Lógica y Filosofía de la Ciencia. Campus Unamuno. Edificio FES. Universidad de Salamanca, 37007 Salamanca. E-mail: gemm@usal.es.    Gemma ROBLES and José M. MÉNDEZ 190 José M. MÉNDEZ is Professor of Logic in the Universidad de Salamanca. He has been working ever since the eighties on relevance logics and different implicative systems. His results are published in Journal of Philosophical Logic, Notre Dame Journal of Formal Logic, Journal of Symbolic Logic, Bulletin of the Sec-tion of Logic, Studia Logica, Logique et Analyse and other research journals and contributed and edited books. From the mid-nineties onward he has published several works with Francisco Salto and/or Gemma Robles on contractionless and akin logics. His research projects and interests cover many non-classical logics and semantics. ADDRESS: Departamento de Filosofía y Lógica y Filosofía de la Ciencia. Campus Unamuno. Edificio FES. Universidad de Salamanca, 37007 Salamanca. E-mail: sefus@usal. 

Two versions of minimal intuitionism with the CAP. A note

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Two versions of minimal intuitionism with the CAP. A note

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