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FACULTY OF ARTS AND PHILOSOPHY A formal explication of the search for explanations. The adaptive logics approach to abductive reasoning. Hans Lycke Centre for Logic and Philosophy of Science Ghent University Hans.Lycke@Ugent.be

  1. Aim of this Talk The Overall Aim To present a general approach towards the explication of abductive reasoning based on the Adaptive Logics Programme. Three Steps The deductive frame = To spell out the relation between abduction and deduction. On defeasible inference = To characterize the abductive inference rule in general. Enter adaptive logics = To characterize some adaptive logics for abductive reasoning. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 10 / 42

  2. Outline Searching for Explanations 1 Abduction? Logic–Based Approaches to Abduction Aim of this Talk The Deductive Frame 2 Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts On Defeasible Inference 3 Enter Adaptive Logics 4 Multiple Abduction Processes General Characterization Proof Theory Examples 5 Conclusion H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 11 / 42

  3. Abduction vs Deduction Intertwinement Abductive reasoning validates some arguments that are not deductively valid. IN CASU Applications of Affirming the Consequent ( AC ). H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 12 / 42

  4. Abduction vs Deduction Intertwinement Abductive reasoning validates some arguments that are not deductively valid. IN CASU Applications of Affirming the Consequent ( AC ). Abductive reasoning is constrained by deductive reasoning. FOR Abductive consequences of a premise set might have to be withdrawn in view of its deductive consequences. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 12 / 42

  5. Abduction vs Deduction Intertwinement Abductive reasoning validates some arguments that are not deductively valid. IN CASU Applications of Affirming the Consequent ( AC ). Abductive reasoning is constrained by deductive reasoning. FOR Abductive consequences of a premise set might have to be withdrawn in view of its deductive consequences. ⇒ Abductive inference steps are applied against a deductive background! H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 12 / 42

  6. Outline Searching for Explanations 1 Abduction? Logic–Based Approaches to Abduction Aim of this Talk The Deductive Frame 2 Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts On Defeasible Inference 3 Enter Adaptive Logics 4 Multiple Abduction Processes General Characterization Proof Theory Examples 5 Conclusion H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 13 / 42

  7. A Modal Frame In most logic–based approaches, deductive reasoning is captured by means of classical logic . e.g. Aliseda–Llera 2006, Meheus&Batens 2006,... H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 14 / 42

  8. A Modal Frame In most logic–based approaches, deductive reasoning is captured by means of classical logic . e.g. Aliseda–Llera 2006, Meheus&Batens 2006,... I will opt for the modal logic RBK ! H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 14 / 42

  9. A Modal Frame In most logic–based approaches, deductive reasoning is captured by means of classical logic . e.g. Aliseda–Llera 2006, Meheus&Batens 2006,... I will opt for the modal logic RBK ! Language Schema of RBK language letters logical symbols set of formulas L S ¬ , ∧ , ∨ , ⊃ W L M W M S ¬ , ∧ , ∨ , ⊃ , � n , � e , ♦ n , ♦ e � n expresses nomological necessity . � e expresses empirical necessity . H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 14 / 42

  10. A Modal Frame In most logic–based approaches, deductive reasoning is captured by means of classical logic . e.g. Aliseda–Llera 2006, Meheus&Batens 2006,... I will opt for the modal logic RBK ! Proof Theory of RBK = the axiom system of CL , extended by AM1n � n ( A ⊃ B ) ⊃ ( � n A ⊃ � n B ) AM1e � e ( A ⊃ B ) ⊃ ( � e A ⊃ � e B ) AM2n � n A ⊃ A AM2e � e A ⊃ A NECn From ⊢ A to ⊢ � n A NECe From ⊢ A to ⊢ � e A � n A ⊃ � n � n A AM3 � n A ⊃ � e A AM4 ♦ n A = df ¬ � n ¬ A ♦ e A = df ¬ � e ¬ A H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 15 / 42

  11. Outline Searching for Explanations 1 Abduction? Logic–Based Approaches to Abduction Aim of this Talk The Deductive Frame 2 Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts On Defeasible Inference 3 Enter Adaptive Logics 4 Multiple Abduction Processes General Characterization Proof Theory Examples 5 Conclusion H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 16 / 42

  12. Representing Abductive Reasoning Contexts Abductive Reasoning Contexts Situations in which people search for possible explanations for some puzzling (empirical) phenomena. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 17 / 42

  13. Representing Abductive Reasoning Contexts Abductive Reasoning Contexts Situations in which people search for possible explanations for some puzzling (empirical) phenomena. As RBK is a modal logic, it is able to capture some intensional elements of abductive reasoning contexts. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 17 / 42

  14. Representing Abductive Reasoning Contexts Abductive Reasoning Contexts Situations in which people search for possible explanations for some puzzling (empirical) phenomena. As RBK is a modal logic, it is able to capture some intensional elements of abductive reasoning contexts. Premise sets are taken to express abductive reasoning contexts: H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 17 / 42

  15. Representing Abductive Reasoning Contexts Abductive Reasoning Contexts Situations in which people search for possible explanations for some puzzling (empirical) phenomena. As RBK is a modal logic, it is able to capture some intensional elements of abductive reasoning contexts. Premise sets are taken to express abductive reasoning contexts: ◮ W N = { � n A | A ∈ W} Nomological Facts ◮ W E = { � e A | A ∈ S ∪ S ¬ } Empirical Facts H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 17 / 42

  16. Representing Abductive Reasoning Contexts Abductive Reasoning Contexts Situations in which people search for possible explanations for some puzzling (empirical) phenomena. As RBK is a modal logic, it is able to capture some intensional elements of abductive reasoning contexts. Premise sets are taken to express abductive reasoning contexts: ◮ W N = { � n A | A ∈ W} Nomological Facts ◮ W E = { � e A | A ∈ S ∪ S ¬ } Empirical Facts = The background knowledge ⇒ Necessities express contextual certainty ! H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 17 / 42

  17. Representing Abductive Reasoning Contexts Abductive Reasoning Contexts Situations in which people search for possible explanations for some puzzling (empirical) phenomena. As RBK is a modal logic, it is able to capture some intensional elements of abductive reasoning contexts. Premise sets are taken to express abductive reasoning contexts: ◮ W N = { � n A | A ∈ W} Nomological Facts ◮ W E = { � e A | A ∈ S ∪ S ¬ } Empirical Facts = The background knowledge ⇒ Necessities express contextual certainty ! ◮ W P = { A | A ∈ S ∪ S ¬ } Puzzling Facts H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 17 / 42

  18. Representing Abductive Reasoning Contexts Abductive Reasoning Contexts Situations in which people search for possible explanations for some puzzling (empirical) phenomena. As RBK is a modal logic, it is able to capture some intensional elements of abductive reasoning contexts. Premise sets are taken to express abductive reasoning contexts: ◮ W N = { � n A | A ∈ W} Nomological Facts ◮ W E = { � e A | A ∈ S ∪ S ¬ } Empirical Facts = The background knowledge ⇒ Necessities express contextual certainty ! ◮ W P = { A | A ∈ S ∪ S ¬ } Puzzling Facts = The explananda H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 17 / 42

  19. On Defeasible Inference AC in a Modal Environment The applications of AC that qualify for conditional acceptance are limited to those satisfying the following schema: AC m � n ( A ⊃ B ) , B , ∆ ⊢ A H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 18 / 42

  20. On Defeasible Inference AC in a Modal Environment The applications of AC that qualify for conditional acceptance are limited to those satisfying the following schema: AC m � n ( A ⊃ B ) , B , ∆ ⊢ A A can only be considered an explanation for B in case there is a statement expressing the nomological dependency of B upon A . ⇒ Relation with Hempel’s account of explanation. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 19 / 42

  21. On Defeasible Inference AC in a Modal Environment The applications of AC that qualify for conditional acceptance are limited to those satisfying the following schema: AC m � n ( A ⊃ B ) , B , ∆ ⊢ A A can only be considered an explanation for B in case there is a statement expressing the nomological dependency of B upon A . ⇒ Relation with Hempel’s account of explanation. The explanandum B may not be part of the background knowledge! OTHERWISE It wouldn’t be in need of an explanation. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 20 / 42

  22. On Defeasible Inference AC in a Modal Environment The applications of AC that qualify for conditional acceptance are limited to those satisfying the following schema: AC m � n ( A ⊃ B ) , B , ∆ ⊢ A A can only be considered an explanation for B in case there is a statement expressing the nomological dependency of B upon A . ⇒ Relation with Hempel’s account of explanation. The explanandum B may not be part of the background knowledge! OTHERWISE It wouldn’t be in need of an explanation. Certain additional conditions have to be fulfilled before AC m may be applied. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 21 / 42

  23. On Defeasible Inference Additional Conditions? Some are equal to those stated by the backwards deduction –approaches to abduction. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 22 / 42

  24. On Defeasible Inference Additional Conditions? Some are equal to those stated by the backwards deduction –approaches to abduction. Some can only be presumed in a defeasible way! H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 22 / 42

  25. On Defeasible Inference Additional Conditions? Some are equal to those stated by the backwards deduction –approaches to abduction. Some can only be presumed in a defeasible way! ⇒ The formulas expressing those conditions are obtained by means of defeasible inference rules, such as ⊢ ¬ � n ( A ⊃ B ) (NNN) ⊢ ¬ � e A (NEN) H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 22 / 42

  26. On Defeasible Inference Additional Conditions? Some are equal to those stated by the backwards deduction –approaches to abduction. Some can only be presumed in a defeasible way! ⇒ The formulas expressing those conditions are obtained by means of defeasible inference rules, such as ⊢ ¬ � n ( A ⊃ B ) (NNN) ⊢ ¬ � e A (NEN) ⇒ These defeasible inference rules are prior to AC m . H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 22 / 42

  27. On Defeasible Inference Additional Conditions? Some are equal to those stated by the backwards deduction –approaches to abduction. Some can only be presumed in a defeasible way! ⇒ The formulas expressing those conditions are obtained by means of defeasible inference rules, such as ⊢ ¬ � n ( A ⊃ B ) (NNN) ⊢ ¬ � e A (NEN) ⇒ These defeasible inference rules are prior to AC m . ⇒ Abduction processes are layered processes! ⇒ The adaptive logics needed are prioritized adaptive logics . H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 22 / 42

  28. Outline Searching for Explanations 1 Abduction? Logic–Based Approaches to Abduction Aim of this Talk The Deductive Frame 2 Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts On Defeasible Inference 3 Enter Adaptive Logics 4 Multiple Abduction Processes General Characterization Proof Theory Examples 5 Conclusion H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 23 / 42

  29. Multiple Abduction Processes H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 24 / 42

  30. Multiple Abduction Processes Practical Abduction In case of multiple possible explanations, only the disjunction of all possible explanations is derivable. ⇒ The logic AbL p H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 24 / 42

  31. Multiple Abduction Processes Practical Abduction In case of multiple possible explanations, only the disjunction of all possible explanations is derivable. ⇒ The logic AbL p Theoretical Abduction In case of multiple possible explanations, all possible explanations are derivable. ⇒ The logic AbL t H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 24 / 42

  32. Multiple Abduction Processes Practical Abduction In case of multiple possible explanations, only the disjunction of all possible explanations is derivable. ⇒ The logic AbL p Theoretical Abduction In case of multiple possible explanations, all possible explanations are derivable. ⇒ The logic AbL t Prioritized Abduction In case of multiple possible explanations, only the most plausible explanations are derivable. ⇒ The logic AbL pt H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 24 / 42

  33. Multiple Abduction Processes Earlier Attempts H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 25 / 42

  34. Multiple Abduction Processes Earlier Attempts J. Meheus et al. Ampliative Adaptive logics and the foundation of logic-based approaches to abduction. In: L. Magnani, N. Nersessian and C. Pizzi. Logical and Computational Aspects of Model-Based Reasoning , Kluwer, Dordrecht, 2002, pp. 39–71. Some extra–logical features are incorporated. BUT ⇒ No formal logic is provided. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 25 / 42

  35. Multiple Abduction Processes Earlier Attempts J. Meheus et al. Ampliative Adaptive logics and the foundation of logic-based approaches to abduction. In: L. Magnani, N. Nersessian and C. Pizzi. Logical and Computational Aspects of Model-Based Reasoning , Kluwer, Dordrecht, 2002, pp. 39–71. Some extra–logical features are incorporated. BUT ⇒ No formal logic is provided. J. Meheus and D. Batens. A formal logic for abductive reasoning. Logic Journal of the IGPL , vol. 14, 2006, pp. 221–236. Only abductive inferences at the predicate level. BUT Only practical abduction could be characterized. BUT ⇒ Abductive reasoning is captured in a limited way. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 25 / 42

  36. Outline Searching for Explanations 1 Abduction? Logic–Based Approaches to Abduction Aim of this Talk The Deductive Frame 2 Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts On Defeasible Inference 3 Enter Adaptive Logics 4 Multiple Abduction Processes General Characterization Proof Theory Examples 5 Conclusion H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 26 / 42

  37. General Characterization Prioritized Adaptive Logics H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 27 / 42

  38. General Characterization Prioritized Adaptive Logics A Lower Limit Logic ( LLL ) 1. The LLL determines the inference rules that can be applied unrestrictedly. A Set of Abnormalities ( Ω = Ω 0 > Ω 1 > ... > Ω n ) 2. Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences A ∨ B ∈ Ω , unless B cannot be interpreted as false. ◮ A Prioritized: Ω is a structurally ordered set of sets. Consequences obtained by falsifying abnormalities of a certain ◮ priority may necessitate the withdrawal of consequences obtained by falsifying abnormalities of a lower priority. An Adaptive Strategy 3. The adaptive strategy determines which of the conditionally derived formulas have to be withdrawn. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 27 / 42

  39. General Characterization Prioritized Adaptive Logics A Lower Limit Logic ( LLL ) 1. The LLL determines the inference rules that can be applied unrestrictedly. A Set of Abnormalities ( Ω = Ω 0 > Ω 1 > ... > Ω n ) 2. Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences A ∨ B ∈ Ω , unless B cannot be interpreted as false. ◮ A Prioritized: Ω is a structurally ordered set of sets. Consequences obtained by falsifying abnormalities of a certain ◮ priority may necessitate the withdrawal of consequences obtained by falsifying abnormalities of a lower priority. An Adaptive Strategy 3. The adaptive strategy determines which of the conditionally derived formulas have to be withdrawn. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 27 / 42

  40. General Characterization Prioritized Adaptive Logics A Lower Limit Logic ( LLL ) 1. The LLL determines the inference rules that can be applied unrestrictedly. A Set of Abnormalities ( Ω = Ω 0 > Ω 1 > ... > Ω n ) 2. Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences A ∨ B ∈ Ω , unless B cannot be interpreted as false. ◮ A Prioritized: Ω is a structurally ordered set of sets. Consequences obtained by falsifying abnormalities of a certain ◮ priority may necessitate the withdrawal of consequences obtained by falsifying abnormalities of a lower priority. An Adaptive Strategy 3. The adaptive strategy determines which of the conditionally derived formulas have to be withdrawn. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 27 / 42

  41. General Characterization Prioritized Adaptive Logics A Lower Limit Logic ( LLL ) 1. The LLL determines the inference rules that can be applied unrestrictedly. A Set of Abnormalities ( Ω = Ω 0 > Ω 1 > ... > Ω n ) 2. Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences A ∨ B ∈ Ω , unless B cannot be interpreted as false. ◮ A Prioritized: Ω is a structurally ordered set of sets. Consequences obtained by falsifying abnormalities of a certain ◮ priority may necessitate the withdrawal of consequences obtained by falsifying abnormalities of a lower priority. An Adaptive Strategy 3. The adaptive strategy determines which of the conditionally derived formulas have to be withdrawn. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 27 / 42

  42. General Characterization Prioritized Adaptive Logics A Lower Limit Logic ( LLL ) 1. The LLL determines the inference rules that can be applied unrestrictedly. A Set of Abnormalities ( Ω = Ω 0 > Ω 1 > ... > Ω n ) 2. Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences A ∨ B ∈ Ω , unless B cannot be interpreted as false. ◮ A Prioritized: Ω is a structurally ordered set of sets. Consequences obtained by falsifying abnormalities of a certain ◮ priority may necessitate the withdrawal of consequences obtained by falsifying abnormalities of a lower priority. An Adaptive Strategy 3. The adaptive strategy determines which of the conditionally derived formulas have to be withdrawn. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 27 / 42

  43. General Characterization Prioritized Adaptive Logics A Lower Limit Logic ( LLL ) 1. The LLL determines the inference rules that can be applied unrestrictedly. A Set of Abnormalities ( Ω = Ω 0 > Ω 1 > ... > Ω n ) 2. Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences A ∨ B ∈ Ω , unless B cannot be interpreted as false. ◮ A Prioritized: Ω is a structurally ordered set of sets. Consequences obtained by falsifying abnormalities of a certain ◮ priority may necessitate the withdrawal of consequences obtained by falsifying abnormalities of a lower priority. An Adaptive Strategy 3. The adaptive strategy determines which of the conditionally derived formulas have to be withdrawn. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 27 / 42

  44. General Characterization: Practical Abduction The Adaptive Logic AbL p H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

  45. General Characterization: Practical Abduction The Adaptive Logic AbL p Lower Limit Logic ( LLL ) 1. Set of Abnormalities Ω = Ω bk > Ω p 2. Ω bk = Ω p = Adaptive Strategy 3. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

  46. General Characterization: Practical Abduction The Adaptive Logic AbL p Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω p 2. Ω bk = Ω p = Adaptive Strategy 3. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

  47. General Characterization: Practical Abduction The Adaptive Logic AbL p Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω p 2. Ω bk = { � x A | x ∈ { n , e } and A ∈ W} Ω p = Adaptive Strategy 3. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

  48. General Characterization: Practical Abduction The Adaptive Logic AbL p Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω p 2. Ω bk = { � x A | x ∈ { n , e } and A ∈ W} Ω p = { � n ( A ⊃ B ) ∧ B ∧ ¬ � e B ∧ ¬ A | B ∈ S ∪ S ¬ , ⊲ A in Conjunctive Normal Form , and ⊲ B is not a subformula of A ⊲ } Adaptive Strategy 3. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

  49. General Characterization: Practical Abduction The Adaptive Logic AbL p Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω p 2. Ω bk = { � x A | x ∈ { n , e } and A ∈ W} Ω p = { � n ( A ⊃ B ) ∧ B ∧ ¬ � e B ∧ ¬ A | B ∈ S ∪ S ¬ , ⊲ A in Conjunctive Normal Form , and ⊲ B is not a subformula of A ⊲ } Adaptive Strategy = Reliability 3. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

  50. General Characterization: Theoretical Abduction The Adaptive Logic AbL t H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 29 / 42

  51. General Characterization: Theoretical Abduction The Adaptive Logic AbL t Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω t 2. Ω bk = { � x A | x ∈ { n , e } and A ∈ W} Ω t = Adaptive Strategy = Reliability 3. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 29 / 42

  52. General Characterization: Theoretical Abduction The Adaptive Logic AbL t Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω t 2. Ω bk = { � x A | x ∈ { n , e } and A ∈ W} Ω t = { � n (( A 1 ∧ ... ∧ A n ) ⊃ B ) ∧ ¬ [ A n 1 ⊃ B ] ∧ B ∧ ¬ � e B ∧¬ ( A 1 ∧ ... ∧ A n ) | A 1 , ..., A n , B ∈ S ∪ S ¬ , ⊲ B is not a subformula of A 1 ∧ ... ∧ A n , and ⊲ } Adaptive Strategy = Reliability 3. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 29 / 42

  53. General Characterization: Theoretical Abduction The Adaptive Logic AbL t Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω t 2. Ω bk = { � x A | x ∈ { n , e } and A ∈ W} Ω t = { � n (( A 1 ∧ ... ∧ A n ) ⊃ B ) ∧ ¬ [ A n 1 ⊃ B ] ∧ B ∧ ¬ � e B ∧¬ ( A 1 ∧ ... ∧ A n ) | A 1 , ..., A n , B ∈ S ∪ S ¬ , ⊲ B is not a subformula of A 1 ∧ ... ∧ A n , and ⊲ ¬ [ A n 1 ⊃ B ] = df ¬ � n (( A 2 ∧ ... ∧ A n ) ⊃ B ) ∧ ⊲ ¬ � n (( A 1 ∧ A 3 ∧ ... ∧ A n ) ⊃ B ) ∧ ∧ ... ¬ � n (( A 1 ∧ ... ∧ A n − 1 ) ⊃ B ) } Adaptive Strategy = Reliability 3. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 29 / 42

  54. General Characterization: Prioritized Abduction How to Represent Priorities? By integrating the knowledge of priorities in the background knowledge. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 30 / 42

  55. General Characterization: Prioritized Abduction How to Represent Priorities? By integrating the knowledge of priorities in the background knowledge. ⇒ If � n ( A ⊃ B ) then � n ♦ e ... ♦ e ( A ∧ B ) expresses that H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 30 / 42

  56. General Characterization: Prioritized Abduction How to Represent Priorities? By integrating the knowledge of priorities in the background knowledge. ⇒ If � n ( A ⊃ B ) then � n ♦ e ... ♦ e ( A ∧ B ) expresses that A is a possible explanation of B , and H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 30 / 42

  57. General Characterization: Prioritized Abduction How to Represent Priorities? By integrating the knowledge of priorities in the background knowledge. ⇒ If � n ( A ⊃ B ) then � n ♦ e ... ♦ e ( A ∧ B ) expresses that A is a possible explanation of B , and the lesser ♦ e ’s, the more plausible A is as an explanation of B . H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 30 / 42

  58. General Characterization: Prioritized Abduction How to Represent Priorities? By integrating the knowledge of priorities in the background knowledge. ⇒ If � n ( A ⊃ B ) then � n ♦ e ... ♦ e ( A ∧ B ) expresses that A is a possible explanation of B , and the lesser ♦ e ’s, the more plausible A is as an explanation of B . How to Make Use of Priorities? There are multiple possibilities! HERE in a straightforward way. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 30 / 42

  59. General Characterization: Prioritized Abduction The Adaptive Logic AbL pt H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 31 / 42

  60. General Characterization: Prioritized Abduction The Adaptive Logic AbL pt Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω pt 1 > Ω pp 1 > Ω pt 2 > Ω pp 2 > ... > Ω t 2. Ω bk and Ω t as for theoretical abduction. Ω pt i = Ω pp i = 3. Adaptive Strategy = Reliability H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 31 / 42

  61. General Characterization: Prioritized Abduction The Adaptive Logic AbL pt Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω pt 1 > Ω pp 1 > Ω pt 2 > Ω pp 2 > ... > Ω t 2. Ω bk and Ω t as for theoretical abduction. Ω pt i = { � n (( A 1 ∧ ... ∧ A n ) ⊃ B ) ∧ � n ♦ i e (( A 1 ∧ ... ∧ A n ) ∧ B ) ∧ B ∧¬ � e B ∧ ¬ ( A 1 ∧ ... ∧ A n ) ∧ ¬ [ A n 1 ⊃ B ] | For the most part as for theoretical abduction, except for ⊲ } Ω pp i = 3. Adaptive Strategy = Reliability H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 31 / 42

  62. General Characterization: Prioritized Abduction The Adaptive Logic AbL pt Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω pt 1 > Ω pp 1 > Ω pt 2 > Ω pp 2 > ... > Ω t 2. Ω bk and Ω t as for theoretical abduction. Ω pt i = { � n (( A 1 ∧ ... ∧ A n ) ⊃ B ) ∧ � n ♦ i e (( A 1 ∧ ... ∧ A n ) ∧ B ) ∧ B ∧¬ � e B ∧ ¬ ( A 1 ∧ ... ∧ A n ) ∧ ¬ [ A n 1 ⊃ B ] | For the most part as for theoretical abduction, except for ⊲ | {z } (( A 1 ∧ ... ∧ A n ) ∧ B ) � n ♦ e ... ♦ e ⊲ i times } Ω pp i = 3. Adaptive Strategy = Reliability H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 31 / 42

  63. General Characterization: Prioritized Abduction The Adaptive Logic AbL pt Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω pt 1 > Ω pp 1 > Ω pt 2 > Ω pp 2 > ... > Ω t 2. Ω bk and Ω t as for theoretical abduction. Ω pt i = { � n (( A 1 ∧ ... ∧ A n ) ⊃ B ) ∧ � n ♦ i e (( A 1 ∧ ... ∧ A n ) ∧ B ) ∧ B ∧¬ � e B ∧ ¬ ( A 1 ∧ ... ∧ A n ) ∧ ¬ [ A n 1 ⊃ B ] | For the most part as for theoretical abduction, except for ⊲ | {z } (( A 1 ∧ ... ∧ A n ) ∧ B ) � n ♦ e ... ♦ e ⊲ i times } Ω pp i = { � n (( A 1 ∧ ... ∧ A n ) ⊃ B ) ∧ � n ♦ i e (( C 1 ∧ ... ∧ C m ) ∧ B ) ∧ B ∧¬ � e B ∧ ¬ ( A 1 ∧ ... ∧ A n ) | For the most part as for theoretical abduction, except ⊲ } 3. Adaptive Strategy = Reliability H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 31 / 42

  64. General Characterization: Prioritized Abduction The Adaptive Logic AbL pt Lower Limit Logic ( LLL ) = the logic RBK 1. Set of Abnormalities Ω = Ω bk > Ω pt 1 > Ω pp 1 > Ω pt 2 > Ω pp 2 > ... > Ω t 2. Ω bk and Ω t as for theoretical abduction. Ω pt i = { � n (( A 1 ∧ ... ∧ A n ) ⊃ B ) ∧ � n ♦ i e (( A 1 ∧ ... ∧ A n ) ∧ B ) ∧ B ∧¬ � e B ∧ ¬ ( A 1 ∧ ... ∧ A n ) ∧ ¬ [ A n 1 ⊃ B ] | For the most part as for theoretical abduction, except for ⊲ | {z } (( A 1 ∧ ... ∧ A n ) ∧ B ) � n ♦ e ... ♦ e ⊲ i times } Ω pp i = { � n (( A 1 ∧ ... ∧ A n ) ⊃ B ) ∧ � n ♦ i e (( C 1 ∧ ... ∧ C m ) ∧ B ) ∧ B ∧¬ � e B ∧ ¬ ( A 1 ∧ ... ∧ A n ) | For the most part as for theoretical abduction, except ⊲ that ¬ [ A n 1 ⊃ B ] is absent, and ⊲ that C 1 , ..., C m ∈ { A 1 , ..., A n } . ⊲ } 3. Adaptive Strategy = Reliability H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 31 / 42

  65. Outline Searching for Explanations 1 Abduction? Logic–Based Approaches to Abduction Aim of this Talk The Deductive Frame 2 Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts On Defeasible Inference 3 Enter Adaptive Logics 4 Multiple Abduction Processes General Characterization Proof Theory Examples 5 Conclusion H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 32 / 42

  66. Proof Theory (1) General Features An AbL x –proof is a succession of stages, each consisting of a sequence of lines. ◮ Adding a line to a proof is to move on to a next stage. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 33 / 42

  67. Proof Theory (1) General Features An AbL x –proof is a succession of stages, each consisting of a sequence of lines. ◮ Adding a line to a proof is to move on to a next stage. Each line of a proof consists of 4 elements: ◮ a line number, ◮ a formula, ◮ a justification, and ◮ an adaptive condition (= a set of abnormalities) H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 33 / 42

  68. Proof Theory (1) General Features An AbL x –proof is a succession of stages, each consisting of a sequence of lines. ◮ Adding a line to a proof is to move on to a next stage. Each line of a proof consists of 4 elements: ◮ a line number, ◮ a formula, ◮ a justification, and ◮ an adaptive condition (= a set of abnormalities) Deduction Rules ◮ As all AbL x are based on the same LLL , the deduction rules are the same for all of them. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 33 / 42

  69. Proof Theory (1) General Features An AbL x –proof is a succession of stages, each consisting of a sequence of lines. ◮ Adding a line to a proof is to move on to a next stage. Each line of a proof consists of 4 elements: ◮ a line number, ◮ a formula, ◮ a justification, and ◮ an adaptive condition (= a set of abnormalities) Deduction Rules ◮ As all AbL x are based on the same LLL , the deduction rules are the same for all of them. Marking Criterium ◮ As all AbL x are based on the same adaptive strategy, the marking criterium is the same for all of them. ◮ Dynamic proofs H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 33 / 42

  70. Proof Theory (2) Dab –Formulas Dab x (∆) = � (∆) , with ∆ ⊂ Ω x H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 34 / 42

  71. Proof Theory (2) Dab –Formulas Dab x (∆) = � (∆) , with ∆ ⊂ Ω x Deduction Rules PREM If A ∈ Γ : . . . . . . A ∅ RU If A 1 , . . . , A n ⊢ RBK B : A 1 ∆ 1 . . . . . . A n ∆ n ∆ 1 ∪ . . . ∪ ∆ n B If A 1 , . . . , A n ⊢ RBK B ∨ Dab x (Θ) RC A 1 ∆ 1 . . . . . . A n ∆ n B ∆ 1 ∪ . . . ∪ ∆ n ∪ Θ H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 34 / 42

  72. Proof Theory (3) Minimal Dab x –consequences Dab x (∆) is a minimal Dab x –consequence of Γ at stage s of a proof, iff (1) it occurs on an unmarked line at stage s , (2) all members of its adaptive condition belong to a Ω x ′ such that Ω x ′ > Ω x , and (3) there is no ∆ ′ ⊂ ∆ for which the same applies. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 35 / 42

  73. Proof Theory (3) Minimal Dab x –consequences Dab x (∆) is a minimal Dab x –consequence of Γ at stage s of a proof, iff (1) it occurs on an unmarked line at stage s , (2) all members of its adaptive condition belong to a Ω x ′ such that Ω x ′ > Ω x , and (3) there is no ∆ ′ ⊂ ∆ for which the same applies. The Set of Unreliable Formulas of a Certain Priority U x s (Γ) = ∆ 1 ∪ ∆ 2 ∪ ... for Dab x (∆ 1 ) , Dab x (∆ 2 ) ,... the minimal Dab x –consequences of Γ at stage s of the proof. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 35 / 42

  74. Proof Theory (3) Minimal Dab x –consequences Dab x (∆) is a minimal Dab x –consequence of Γ at stage s of a proof, iff (1) it occurs on an unmarked line at stage s , (2) all members of its adaptive condition belong to a Ω x ′ such that Ω x ′ > Ω x , and (3) there is no ∆ ′ ⊂ ∆ for which the same applies. The Set of Unreliable Formulas of a Certain Priority U x s (Γ) = ∆ 1 ∪ ∆ 2 ∪ ... for Dab x (∆ 1 ) , Dab x (∆ 2 ) ,... the minimal Dab x –consequences of Γ at stage s of the proof. Marking Definition Line i is marked at stage s of the proof iff, where ∆ is its condition, ∆ ∩ U x s (Γ) � = ∅ . H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 35 / 42

  75. Proof Theory (3) Minimal Dab x –consequences Dab x (∆) is a minimal Dab x –consequence of Γ at stage s of a proof, iff (1) it occurs on an unmarked line at stage s , (2) all members of its adaptive condition belong to a Ω x ′ such that Ω x ′ > Ω x , and (3) there is no ∆ ′ ⊂ ∆ for which the same applies. The Set of Unreliable Formulas of a Certain Priority U x s (Γ) = ∆ 1 ∪ ∆ 2 ∪ ... for Dab x (∆ 1 ) , Dab x (∆ 2 ) ,... the minimal Dab x –consequences of Γ at stage s of the proof. Marking Definition Line i is marked at stage s of the proof iff, where ∆ is its condition, ∆ ∩ U x s (Γ) � = ∅ . Marking Proceeds Stepwise First for the highest priority level, and afterwards for the lower ones. H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 35 / 42

  76. Proof Theory (4) Derivability A is derived from Γ at stage s of a proof iff A is the second element of an unmarked line at stage s . H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 36 / 42

  77. Proof Theory (4) Derivability A is derived from Γ at stage s of a proof iff A is the second element of an unmarked line at stage s . Final Derivability A is finally derived from Γ on a line i of a proof at stage s iff (i) A is the second element of line i , (ii) line i is not marked at stage s , and (iii) every extension of the proof in which line i is marked may be further extended in such a way that line i is unmarked. Γ ⊢ AbL x A iff A is finally derived on a line of a proof from Γ . H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 36 / 42

  78. Outline Searching for Explanations 1 Abduction? Logic–Based Approaches to Abduction Aim of this Talk The Deductive Frame 2 Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts On Defeasible Inference 3 Enter Adaptive Logics 4 Multiple Abduction Processes General Characterization Proof Theory Examples 5 Conclusion H. Lycke (Ghent University) The adaptive logics approach to abductive reasoning MBR’09, Campinas 37 / 42

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