Toward the use of the contraposition law in multi-adjoint lattices

../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Toward the use of the contraposition law in multi-adjoint lattices Nicol´ as Madrid UNIVERSITY OF M´ ALAGA SPAIN VRSR 2016 Malenovice 12.11.2016 ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Introduction In crisp logic, the connectives hold interesting relationships among them via tautologies. Some very well known examples are p∧q←→ ¬(¬p∨ ¬q)(the Morgan’s law) p→q←→ ¬q→¬p(the contraposition law) p→q←→ ¬p∨q(the material implication) ¬(¬p)←→ p(the double negation law) However, the satisfiability of such relationships in Fuzzy logic depends on how the connectives are interpreted. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Outline In this talk I focus on the double negation and on the Contraposition law by: showing limitations of residuated structures to deal with both laws at the same time, presenting a structure based on multi-adjoint structures where, somehow, both laws hold, and providing some theoretical results of the mentioned structure. Finally some future work are presented. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work The residuated lattice Let me begin by recalling the notion of residuated lattice. Definition A residuated lattice is a triple L=((L,≤),∗,→)such that: 1(L,≤)is a complete and bounded lattice with largest element 1 and least element 0. 2(L,∗,1)is a commutative monoid unit element 1. 3∗and →form an adjoint pair, i.e: z≤(y→x)iff y∗z≤xfor all x,y,z∈L. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Contraposition law in residuated lattices Definition A negation in a complete lattice (L,≤)is any antitonic mapping n∶L→Lsuch that n(0)=1 and n(1)=0. The contraposition law in a logic based on a residuated lattice ((L,≤),∗,→)and a negation nrequires that x→y=n(y)→n(x) for all x,y∈L. It is not hard to prove that if the equality above hold, the negation nmust coincide with: n(x)=x→0. for all x∈L. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Double negation law vs contraposition law On another hand, the Double negation law in a logic based on a residuated lattice ((L,≤),∗,→)and a negation nrequires that n(n(x)) =x for all x∈L; i.e., the negation nmust be involutive The problem with requiring both laws is that the negation defined by x→0 is seldom involutive, from a theoretical point of view, somehow the contraposition law is preferred, but from a practical point of view, somehow the double negation law is preferred. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Let’s be shortly informal. After checking the theory based on residuated lattices we can think that, in general, Contraposition Law and Double negation law do not hold in fuzzy environments but ... informally the do! If I kick powerfully the ball, then it goes far. If the ball did not go far, then I did not kick the ball powerfully. One option is to restrict ourselves in residuated lattices where both features hold; for instance in Łukasiewicz logic. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Adjoint Triples The definition The Multi-adjoint lattice structure is base on adjoint triples. Definition Let (P1,≤1),(P2,≤2),(P3,≤3)be three posets. The mappings &∶P1×P2→P3,↘∶P2×P3→P1, and ↗∶P1×P3→P2form an adjoint triple among P1,P2and P3whenever: x≤1y↘zif and only if x&y≤3zif and only if y≤2x↗z for all x∈P1,y∈P2and z∈P3. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Adjoint Triples Properties and one restriction Lemma If (&,↘,↗)is an adjoint triple w.r.t. P1,P2,P3, then 1&is order-preserving on both arguments, 2↘,↗are order-preserving on the first argument and order-reversing on the second argument. In this paper I consider a simplified structure of adjoint triples: (P1,≤1),(P2,≤2)and (P3,≤3)are equal to a lattice (L,≤). ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work The ↗n-adjoint triple Similarly, we define the ↗n-adjoint triple. Definition Let (&,↘,↗)be an adjoint triple defined in a lattice Lwith an involutive negation n. The ↗n-adjoint triple (&n ,↘n ,↗n)of (&,↘,↗)is given by the following operators: x↗ny=n(y)↗n(x)for all x,y∈L x&ny=n(y↘n(x)) for all x,y∈L x↘ny=n(n(y)&x)for all x,y∈L. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Hey! Are they adjoint triples? The definition above requires a proof to justify the use of the term ‘adjoint triple’ in it. Lemma Let (&,↘,↗)be an adjoint triple defined in a lattice L with an involutive negation n. Then (&n,↘n,↗n)and (&n ,↘n ,↗n)are adjoint triples as well. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work What about doing a reiterative construction? By composing twice the constructions of adjoint triples, there are, a priori, four possible new adjoint triples, namely: The ↘n-adjoint triple of (&n,↘n,↗n): ((&n)n,(↘n)n,(↗n)n) The ↗n-adjoint triple of (&n,↘n,↗n): ((&n)n ,(↘n)n ,(↗n)n) The ↗n-adjoint triple of (&n ,↘n ,↗n): ((&n)n ,(↘n)n ,(↗n)n) And the ↘n-adjoint triple of (&n ,↘n ,↗n): ((&n)n,(↘n)n,(↗n)n) ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work The main result Theorem Let (&,↘,↗)be an adjoint triple defined in a lattice L with an involutive negation n. Then the following equalities hold: 1x(&n)ny=x(&n)ny=x&y. 2x(&n)ny=x(&n)ny=y&x. 3x(↘n)ny=x↘y. 4x(↗n)ny=x↗y. 5x(↘n)ny=x↘y. 6x(↗n)ny=x↗y. 7x(↗n)ny=x↘ny. 8x(↘n)ny=x↗ny. 9x(↗n)ny=x↘ny. 10 x(↘n)ny=x↗ny. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work A closed framework for the contraposition law. Given an adjoint triple (&,↘,↗)defined on a lattice Lwith an involutive negation n, the multi-adjoint framework given by: (L,≤,(&,↘,↗),(&n,↘n,↗n),(&n ,↘n ,↗n)) is a closed framework where it is possible to apply the contraposition rule an unlimited (numerable) number of times. For instance, for every x,y∈Lwe have: x↗ny=n(y)(↗n)nn(x)=n(y)↘nn(x). That is: it is possible to apply the contraposition rule to the implication ↘nby using the implication ↗n ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work The case of a residuated lattice. Before to end the talk, I study the construction above from a residuated pair (&,→)and an involutive negation n. By commutativity of &, the multi-adjoint framework associated with the construction described above is: (L,≤,(&,→),(&n,↘n,↗n)) Moreover, the following equalities hold: x→y=n(y)↘nn(x), x↘ny=n(y)→n(x), x↗ny=n(y)↗nn(x) for all x,y∈L. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Conclusions In this talk I have described some drawbacks about constructing a framework based on residuated lattice to hold both: the contraposition law and the double negation law. recalled the notion of multi-adjoint lattices and constructed a simple multi-adjoint framework where we can apply the contraposition rule an unlimited number of times. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Future Work As future work I plan to study properties on the given framework for specific cases: G¨ odel connectives with negation n(x)=1−x, product t-norm and implication with negation n(x)=1−x As a long long term work, I would like to develop an algebras (formal logic systems) based on the multi-adjoint frameworks presented here. ../../ESTYLF2016/paloma.png Introduction Residuated approach Multi-adjoint approach Conclusions and Future Work Toward the use of the contraposition law in multi-adjoint lattices Nicol´ as Madrid UNIVERSITY OF M´ ALAGA SPAIN VRSR 2016 Malenovice 12.11.2016