A formal explication of the search for explanations. The adaptive - - PowerPoint PPT Presentation

▶
a formal explication of the search for explanations the
SMART_READER_LITE
LIVE PREVIEW

A formal explication of the search for explanations. The adaptive - - PowerPoint PPT Presentation

Dec 20, 2023 267 likes •1.47k views

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

slide-1
SLIDE 1

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 http://logica.ugent.be/hans

MBR’09 December 17–20 2009, Campinas

slide-2
SLIDE 2

Outline

1

Searching for Explanations Abduction? Logic–Based Approaches to Abduction Aim of this Talk

2

The Deductive Frame Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts

3

On Defeasible Inference

4

Enter Adaptive Logics Multiple Abduction Processes General Characterization Proof Theory Examples

5

Conclusion

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 2 / 42

slide-3
SLIDE 3

Outline

1

Searching for Explanations Abduction? Logic–Based Approaches to Abduction Aim of this Talk

2

The Deductive Frame Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts

3

On Defeasible Inference

4

Enter Adaptive Logics Multiple Abduction Processes General Characterization Proof Theory Examples

5

Conclusion

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 3 / 42

slide-4
SLIDE 4

Abduction?

Searching for Explanations

Backwards reasoning: from the phenomena to be explained to possible explanations.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 4 / 42

slide-5
SLIDE 5

Abduction?

Searching for Explanations

Backwards reasoning: from the phenomena to be explained to possible explanations. ⇒ Abductive inferences A ⊃ B, B ⊢ A (Affirming the Consequent)

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 4 / 42

slide-6
SLIDE 6

Abduction?

Searching for Explanations

Backwards reasoning: from the phenomena to be explained to possible explanations. ⇒ Abductive inferences A ⊃ B, B ⊢ A (Affirming the Consequent)

Example

A physician in search of the right diagnosis for a patient’s symptoms, a technician trying to find out why a machine broke down, a scientist trying to find an explanation for an empirical phenomenon contradicting some predictions derived from an accepted theory,...

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 4 / 42

slide-7
SLIDE 7

Outline

1

Searching for Explanations Abduction? Logic–Based Approaches to Abduction Aim of this Talk

2

The Deductive Frame Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts

3

On Defeasible Inference

4

Enter Adaptive Logics Multiple Abduction Processes General Characterization Proof Theory Examples

5

Conclusion

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 5 / 42

slide-8
SLIDE 8

Logic–Based Approaches to Abduction

Affirming the Consequent (AC) is not deductively valid !!!

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 6 / 42

slide-9
SLIDE 9

Logic–Based Approaches to Abduction

Affirming the Consequent (AC) is not deductively valid !!!

Backwards Deduction plus Additional Conditions

A number of conditions is specified that enable one to decide whether

  • r not a particular abductive inference is sound.
  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 6 / 42

slide-10
SLIDE 10

Logic–Based Approaches to Abduction

Affirming the Consequent (AC) is not deductively valid !!!

Backwards Deduction plus Additional Conditions

A number of conditions is specified that enable one to decide whether

  • r not a particular abductive inference is sound.

Example

Given the background theory Γ, A is an explanation for B iff Γ ∪ {A} ⊢ B Γ ¬A Γ B; A B ...

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 6 / 42

slide-11
SLIDE 11

Logic–Based Approaches to Abduction

Affirming the Consequent (AC) is not deductively valid !!!

Backwards Deduction plus Additional Conditions

A number of conditions is specified that enable one to decide whether

  • r not a particular abductive inference is sound.

= a realistic explication of abductive reasoning

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 7 / 42

slide-12
SLIDE 12

Logic–Based Approaches to Abduction

Affirming the Consequent (AC) is not deductively valid !!!

Backwards Deduction plus Additional Conditions

A number of conditions is specified that enable one to decide whether

  • r not a particular abductive inference is sound.

= a realistic explication of abductive reasoning

FOR

Focus is on abductive consequence, not on abductive reasoning

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 7 / 42

slide-13
SLIDE 13

Logic–Based Approaches to Abduction

Affirming the Consequent (AC) is not deductively valid !!!

Backwards Deduction plus Additional Conditions

A number of conditions is specified that enable one to decide whether

  • r not a particular abductive inference is sound.

= a realistic explication of abductive reasoning

FOR

Focus is on abductive consequence, not on abductive reasoning ⇒ Search procedures instead of a proof theory e.g. Tableau methods (Aliseda–Llera 2006, Mayer&Pirri 1993)

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 7 / 42

slide-14
SLIDE 14

Logic–Based Approaches to Abduction

The Adaptive Logics Programme

The inference rule AC is modeled as a defeasible inference rule (a default rule).

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 8 / 42

slide-15
SLIDE 15

Logic–Based Approaches to Abduction

The Adaptive Logics Programme

The inference rule AC is modeled as a defeasible inference rule (a default rule). ⇒ Abductive consequences are accepted only in case and for as long as certain conditions are satisfied. ⇒ Only the unproblematic applications of AC are retained.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 8 / 42

slide-16
SLIDE 16

Logic–Based Approaches to Abduction

The Adaptive Logics Programme

The inference rule AC is modeled as a defeasible inference rule (a default rule). ⇒ Abductive consequences are accepted only in case and for as long as certain conditions are satisfied. ⇒ Only the unproblematic applications of AC are retained. ⇒ Abductive reasoning steps are combined with deductive reasoning steps.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 8 / 42

slide-17
SLIDE 17

Logic–Based Approaches to Abduction

The Adaptive Logics Programme

The inference rule AC is modeled as a defeasible inference rule (a default rule). ⇒ Abductive consequences are accepted only in case and for as long as certain conditions are satisfied. ⇒ Only the unproblematic applications of AC are retained. ⇒ Abductive reasoning steps are combined with deductive reasoning steps.

Advantages

(Some of) the conditions of BD can be incorporated. A nice proof theory for abductive reasoning is provided.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 8 / 42

slide-18
SLIDE 18

Logic–Based Approaches to Abduction

The Adaptive Logics Programme

The inference rule AC is modeled as a defeasible inference rule (a default rule). ⇒ Abductive consequences are accepted only in case and for as long as certain conditions are satisfied. ⇒ Only the unproblematic applications of AC are retained. ⇒ Abductive reasoning steps are combined with deductive reasoning steps.

Advantages

(Some of) the conditions of BD can be incorporated. A nice proof theory for abductive reasoning is provided. ⇒ The adaptive logics approach provides a more realistic explication of the application of abductive inferences in human reasoning!

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 8 / 42

slide-19
SLIDE 19

Outline

1

Searching for Explanations Abduction? Logic–Based Approaches to Abduction Aim of this Talk

2

The Deductive Frame Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts

3

On Defeasible Inference

4

Enter Adaptive Logics Multiple Abduction Processes General Characterization Proof Theory Examples

5

Conclusion

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 9 / 42

slide-20
SLIDE 20

Aim of this Talk

The Overall Aim

To present a general approach towards the explication of abductive reasoning based on the Adaptive Logics Programme.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 10 / 42

slide-21
SLIDE 21

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.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 10 / 42

slide-22
SLIDE 22

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.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 10 / 42

slide-23
SLIDE 23

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

slide-24
SLIDE 24

Outline

1

Searching for Explanations Abduction? Logic–Based Approaches to Abduction Aim of this Talk

2

The Deductive Frame Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts

3

On Defeasible Inference

4

Enter Adaptive Logics 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

slide-25
SLIDE 25

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

slide-26
SLIDE 26

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

slide-27
SLIDE 27

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

slide-28
SLIDE 28

Outline

1

Searching for Explanations Abduction? Logic–Based Approaches to Abduction Aim of this Talk

2

The Deductive Frame Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts

3

On Defeasible Inference

4

Enter Adaptive Logics 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

slide-29
SLIDE 29

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

slide-30
SLIDE 30

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

slide-31
SLIDE 31

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 LM S ¬, ∧, ∨, ⊃, n, e, ♦n, ♦e WM n expresses nomological necessity. e expresses empirical necessity.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 14 / 42

slide-32
SLIDE 32

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) ⊃ (nA ⊃ nB) AM1e e(A ⊃ B) ⊃ (eA ⊃ eB) AM2n nA ⊃ A AM2e eA ⊃ A NECn From ⊢ A to ⊢ nA NECe From ⊢ A to ⊢ eA AM3 nA ⊃ nnA AM4 nA ⊃ eA ♦nA =df ¬n¬A ♦eA =df ¬e¬A

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 15 / 42

slide-33
SLIDE 33

Outline

1

Searching for Explanations Abduction? Logic–Based Approaches to Abduction Aim of this Talk

2

The Deductive Frame Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts

3

On Defeasible Inference

4

Enter Adaptive Logics 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

slide-34
SLIDE 34

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

slide-35
SLIDE 35

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

slide-36
SLIDE 36

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

slide-37
SLIDE 37

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:

◮ WN = {nA | A ∈ W}

Nomological Facts

◮ WE = {eA | A ∈ S ∪ S¬}

Empirical Facts

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 17 / 42

slide-38
SLIDE 38

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:

◮ WN = {nA | A ∈ W}

Nomological Facts

◮ WE = {eA | 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

slide-39
SLIDE 39

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:

◮ WN = {nA | A ∈ W}

Nomological Facts

◮ WE = {eA | A ∈ S ∪ S¬}

Empirical Facts

= The background knowledge ⇒ Necessities express contextual certainty!

◮ WP = {A | A ∈ S ∪ S¬}

Puzzling Facts

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 17 / 42

slide-40
SLIDE 40

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:

◮ WN = {nA | A ∈ W}

Nomological Facts

◮ WE = {eA | A ∈ S ∪ S¬}

Empirical Facts

= The background knowledge ⇒ Necessities express contextual certainty!

◮ WP = {A | A ∈ S ∪ S¬}

Puzzling Facts

= The explananda

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 17 / 42

slide-41
SLIDE 41

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: ACm n(A ⊃ B), B, ∆ ⊢ A

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 18 / 42

slide-42
SLIDE 42

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: ACm 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

slide-43
SLIDE 43

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: ACm 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

slide-44
SLIDE 44

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: ACm 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 ACm may be applied.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 21 / 42

slide-45
SLIDE 45

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

slide-46
SLIDE 46

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

slide-47
SLIDE 47

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 (NNN) ⊢ ¬n(A ⊃ B) (NEN) ⊢ ¬eA

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 22 / 42

slide-48
SLIDE 48

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 (NNN) ⊢ ¬n(A ⊃ B) (NEN) ⊢ ¬eA ⇒ These defeasible inference rules are prior to ACm.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 22 / 42

slide-49
SLIDE 49

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 (NNN) ⊢ ¬n(A ⊃ B) (NEN) ⊢ ¬eA ⇒ These defeasible inference rules are prior to ACm. ⇒ 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

slide-50
SLIDE 50

Outline

1

Searching for Explanations Abduction? Logic–Based Approaches to Abduction Aim of this Talk

2

The Deductive Frame Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts

3

On Defeasible Inference

4

Enter Adaptive Logics 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

slide-51
SLIDE 51

Multiple Abduction Processes

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 24 / 42

slide-52
SLIDE 52

Multiple Abduction Processes

Practical Abduction

In case of multiple possible explanations, only the disjunction of all possible explanations is derivable. ⇒ The logic AbLp

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 24 / 42

slide-53
SLIDE 53

Multiple Abduction Processes

Practical Abduction

In case of multiple possible explanations, only the disjunction of all possible explanations is derivable. ⇒ The logic AbLp

Theoretical Abduction

In case of multiple possible explanations, all possible explanations are derivable. ⇒ The logic AbLt

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 24 / 42

slide-54
SLIDE 54

Multiple Abduction Processes

Practical Abduction

In case of multiple possible explanations, only the disjunction of all possible explanations is derivable. ⇒ The logic AbLp

Theoretical Abduction

In case of multiple possible explanations, all possible explanations are derivable. ⇒ The logic AbLt

Prioritized Abduction

In case of multiple possible explanations, only the most plausible explanations are derivable. ⇒ The logic AbLpt

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 24 / 42

slide-55
SLIDE 55

Multiple Abduction Processes

Earlier Attempts

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 25 / 42

slide-56
SLIDE 56

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.

BUT

Some extra–logical features are incorporated. ⇒ No formal logic is provided.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 25 / 42

slide-57
SLIDE 57

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.

BUT

Some extra–logical features are incorporated. ⇒ 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.

BUT

Only abductive inferences at the predicate level.

BUT

Only practical abduction could be characterized. ⇒ Abductive reasoning is captured in a limited way.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 25 / 42

slide-58
SLIDE 58

Outline

1

Searching for Explanations Abduction? Logic–Based Approaches to Abduction Aim of this Talk

2

The Deductive Frame Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts

3

On Defeasible Inference

4

Enter Adaptive Logics 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

slide-59
SLIDE 59

General Characterization

Prioritized Adaptive Logics

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 27 / 42

slide-60
SLIDE 60

General Characterization

Prioritized Adaptive Logics

1. A Lower Limit Logic (LLL)

The LLL determines the inference rules that can be applied unrestrictedly.

2. A Set of Abnormalities (Ω = Ω0 > Ω1 > ... > Ωn)

Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences

◮

A ∨ B∈Ω A , unless B cannot be interpreted as false. 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.

3. An Adaptive Strategy

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

slide-61
SLIDE 61

General Characterization

Prioritized Adaptive Logics

1. A Lower Limit Logic (LLL)

The LLL determines the inference rules that can be applied unrestrictedly.

2. A Set of Abnormalities (Ω = Ω0 > Ω1 > ... > Ωn)

Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences

◮

A ∨ B∈Ω A , unless B cannot be interpreted as false. 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.

3. An Adaptive Strategy

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

slide-62
SLIDE 62

General Characterization

Prioritized Adaptive Logics

1. A Lower Limit Logic (LLL)

The LLL determines the inference rules that can be applied unrestrictedly.

2. A Set of Abnormalities (Ω = Ω0 > Ω1 > ... > Ωn)

Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences

◮

A ∨ B∈Ω A , unless B cannot be interpreted as false. 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.

3. An Adaptive Strategy

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

slide-63
SLIDE 63

General Characterization

Prioritized Adaptive Logics

1. A Lower Limit Logic (LLL)

The LLL determines the inference rules that can be applied unrestrictedly.

2. A Set of Abnormalities (Ω = Ω0 > Ω1 > ... > Ωn)

Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences

◮

A ∨ B∈Ω A , unless B cannot be interpreted as false. 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.

3. An Adaptive Strategy

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

slide-64
SLIDE 64

General Characterization

Prioritized Adaptive Logics

1. A Lower Limit Logic (LLL)

The LLL determines the inference rules that can be applied unrestrictedly.

2. A Set of Abnormalities (Ω = Ω0 > Ω1 > ... > Ωn)

Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences

◮

A ∨ B∈Ω A , unless B cannot be interpreted as false. 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.

3. An Adaptive Strategy

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

slide-65
SLIDE 65

General Characterization

Prioritized Adaptive Logics

1. A Lower Limit Logic (LLL)

The LLL determines the inference rules that can be applied unrestrictedly.

2. A Set of Abnormalities (Ω = Ω0 > Ω1 > ... > Ωn)

Elements of Ω are interpreted as false as much as possible The result: some conditionally derived consequences

◮

A ∨ B∈Ω A , unless B cannot be interpreted as false. 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.

3. An Adaptive Strategy

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

slide-66
SLIDE 66

General Characterization: Practical Abduction

The Adaptive Logic AbLp

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

slide-67
SLIDE 67

General Characterization: Practical Abduction

The Adaptive Logic AbLp

1. Lower Limit Logic (LLL) 2. Set of Abnormalities Ω = Ωbk > Ωp Ωbk = Ωp = 3. Adaptive Strategy

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

slide-68
SLIDE 68

General Characterization: Practical Abduction

The Adaptive Logic AbLp

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωp Ωbk = Ωp = 3. Adaptive Strategy

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

slide-69
SLIDE 69

General Characterization: Practical Abduction

The Adaptive Logic AbLp

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωp Ωbk = {xA | x ∈ {n, e} and A ∈ W} Ωp = 3. Adaptive Strategy

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

slide-70
SLIDE 70

General Characterization: Practical Abduction

The Adaptive Logic AbLp

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωp Ωbk = {xA | x ∈ {n, e} and A ∈ W} Ωp = {n(A ⊃ B) ∧ B ∧ ¬eB ∧ ¬A | ⊲ B ∈ S ∪ S¬, ⊲ A in Conjunctive Normal Form, and ⊲ B is not a subformula of A } 3. Adaptive Strategy

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

slide-71
SLIDE 71

General Characterization: Practical Abduction

The Adaptive Logic AbLp

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωp Ωbk = {xA | x ∈ {n, e} and A ∈ W} Ωp = {n(A ⊃ B) ∧ B ∧ ¬eB ∧ ¬A | ⊲ B ∈ S ∪ S¬, ⊲ A in Conjunctive Normal Form, and ⊲ B is not a subformula of A } 3. Adaptive Strategy = Reliability

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 28 / 42

slide-72
SLIDE 72

General Characterization: Theoretical Abduction

The Adaptive Logic AbLt

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 29 / 42

slide-73
SLIDE 73

General Characterization: Theoretical Abduction

The Adaptive Logic AbLt

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωt Ωbk = {xA | x ∈ {n, e} and A ∈ W} Ωt = 3. Adaptive Strategy = Reliability

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 29 / 42

slide-74
SLIDE 74

General Characterization: Theoretical Abduction

The Adaptive Logic AbLt

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωt Ωbk = {xA | x ∈ {n, e} and A ∈ W} Ωt = {n((A1 ∧ ... ∧ An) ⊃ B) ∧ ¬[An

1 ⊃ B]∧B ∧ ¬eB

∧¬(A1 ∧ ... ∧ An) | ⊲ A1, ..., An, B ∈ S ∪ S¬, ⊲ B is not a subformula of A1 ∧ ... ∧ An, and } 3. Adaptive Strategy = Reliability

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 29 / 42

slide-75
SLIDE 75

General Characterization: Theoretical Abduction

The Adaptive Logic AbLt

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωt Ωbk = {xA | x ∈ {n, e} and A ∈ W} Ωt = {n((A1 ∧ ... ∧ An) ⊃ B) ∧ ¬[An

1 ⊃ B] ∧ B ∧ ¬eB

∧¬(A1 ∧ ... ∧ An) | ⊲ A1, ..., An, B ∈ S ∪ S¬, ⊲ B is not a subformula of A1 ∧ ... ∧ An, and ⊲ ¬[An

1 ⊃ B] =df ¬n((A2 ∧ ... ∧ An) ⊃ B)

∧ ¬n((A1 ∧ A3 ∧ ... ∧ An) ⊃ B) ∧ ... ∧ ¬n((A1 ∧ ... ∧ An−1) ⊃ B) } 3. Adaptive Strategy = Reliability

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 29 / 42

slide-76
SLIDE 76

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

slide-77
SLIDE 77

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

slide-78
SLIDE 78

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

slide-79
SLIDE 79

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

slide-80
SLIDE 80

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

slide-81
SLIDE 81

General Characterization: Prioritized Abduction

The Adaptive Logic AbLpt

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 31 / 42

slide-82
SLIDE 82

General Characterization: Prioritized Abduction

The Adaptive Logic AbLpt

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωpt1 > Ωpp1 > Ωpt2 > Ωpp2 > ... > Ωt

Ωbk and Ωt as for theoretical abduction. Ωpti = Ωppi =

3. Adaptive Strategy = Reliability

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 31 / 42

slide-83
SLIDE 83

General Characterization: Prioritized Abduction

The Adaptive Logic AbLpt

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωpt1 > Ωpp1 > Ωpt2 > Ωpp2 > ... > Ωt

Ωbk and Ωt as for theoretical abduction. Ωpti = {n((A1 ∧ ... ∧ An) ⊃ B) ∧ n♦i

e((A1 ∧ ... ∧ An) ∧ B)∧B

∧¬eB ∧ ¬(A1 ∧ ... ∧ An) ∧ ¬[An

1 ⊃ B] |

⊲ For the most part as for theoretical abduction, except for } Ωppi =

3. Adaptive Strategy = Reliability

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 31 / 42

slide-84
SLIDE 84

General Characterization: Prioritized Abduction

The Adaptive Logic AbLpt

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωpt1 > Ωpp1 > Ωpt2 > Ωpp2 > ... > Ωt

Ωbk and Ωt as for theoretical abduction. Ωpti = {n((A1 ∧ ... ∧ An) ⊃ B) ∧ n♦i

e((A1 ∧ ... ∧ An) ∧ B)∧B

∧¬eB ∧ ¬(A1 ∧ ... ∧ An) ∧ ¬[An

1 ⊃ B] |

⊲ For the most part as for theoretical abduction, except for ⊲ n ♦e...♦e | {z }((A1 ∧ ... ∧ An) ∧ B)

i times

} Ωppi =

3. Adaptive Strategy = Reliability

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 31 / 42

slide-85
SLIDE 85

General Characterization: Prioritized Abduction

The Adaptive Logic AbLpt

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωpt1 > Ωpp1 > Ωpt2 > Ωpp2 > ... > Ωt

Ωbk and Ωt as for theoretical abduction. Ωpti = {n((A1 ∧ ... ∧ An) ⊃ B) ∧ n♦i

e((A1 ∧ ... ∧ An) ∧ B)∧B

∧¬eB ∧ ¬(A1 ∧ ... ∧ An) ∧ ¬[An

1 ⊃ B] |

⊲ For the most part as for theoretical abduction, except for ⊲ n ♦e...♦e | {z }((A1 ∧ ... ∧ An) ∧ B)

i times

} Ωppi = {n((A1 ∧ ... ∧ An) ⊃ B) ∧ n♦i

e((C1 ∧ ... ∧ Cm) ∧ B) ∧ B

∧¬eB ∧ ¬(A1 ∧ ... ∧ An) | ⊲ 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

slide-86
SLIDE 86

General Characterization: Prioritized Abduction

The Adaptive Logic AbLpt

1. Lower Limit Logic (LLL) = the logic RBK 2. Set of Abnormalities Ω = Ωbk > Ωpt1 > Ωpp1 > Ωpt2 > Ωpp2 > ... > Ωt

Ωbk and Ωt as for theoretical abduction. Ωpti = {n((A1 ∧ ... ∧ An) ⊃ B) ∧ n♦i

e((A1 ∧ ... ∧ An) ∧ B)∧B

∧¬eB ∧ ¬(A1 ∧ ... ∧ An) ∧ ¬[An

1 ⊃ B] |

⊲ For the most part as for theoretical abduction, except for ⊲ n ♦e...♦e | {z }((A1 ∧ ... ∧ An) ∧ B)

i times

} Ωppi = {n((A1 ∧ ... ∧ An) ⊃ B) ∧ n♦i

e((C1 ∧ ... ∧ Cm) ∧ B) ∧ B

∧¬eB ∧ ¬(A1 ∧ ... ∧ An) | ⊲ For the most part as for theoretical abduction, except ⊲ that ¬[An

1 ⊃ B] is absent, and

⊲ that C1, ..., Cm ∈ {A1, ..., An}. }

3. Adaptive Strategy = Reliability

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 31 / 42

slide-87
SLIDE 87

Outline

1

Searching for Explanations Abduction? Logic–Based Approaches to Abduction Aim of this Talk

2

The Deductive Frame Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts

3

On Defeasible Inference

4

Enter Adaptive Logics 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

slide-88
SLIDE 88

Proof Theory (1)

General Features

An AbLx–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

slide-89
SLIDE 89

Proof Theory (1)

General Features

An AbLx–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

slide-90
SLIDE 90

Proof Theory (1)

General Features

An AbLx–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 AbLx 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

slide-91
SLIDE 91

Proof Theory (1)

General Features

An AbLx–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 AbLx are based on the same LLL, the deduction rules are

the same for all of them.

Marking Criterium

◮ As all AbLx 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

slide-92
SLIDE 92

Proof Theory (2)

Dab–Formulas

Dabx(∆) = (∆), with ∆ ⊂ Ωx

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 34 / 42

slide-93
SLIDE 93

Proof Theory (2)

Dab–Formulas

Dabx(∆) = (∆), with ∆ ⊂ Ωx

Deduction Rules

PREM If A ∈ Γ: . . . . . . A ∅ RU If A1, . . . , An ⊢RBK B: A1 ∆1 . . . . . . An ∆n B ∆1 ∪ . . . ∪ ∆n RC If A1, . . . , An ⊢RBK B ∨ Dabx(Θ) A1 ∆1 . . . . . . An ∆n B ∆1 ∪ . . . ∪ ∆n ∪ Θ

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 34 / 42

slide-94
SLIDE 94

Proof Theory (3)

Minimal Dabx–consequences

Dabx(∆) is a minimal Dabx–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 Ωx′ such that Ωx′ > Ωx, 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

slide-95
SLIDE 95

Proof Theory (3)

Minimal Dabx–consequences

Dabx(∆) is a minimal Dabx–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 Ωx′ such that Ωx′ > Ωx, and (3) there is no ∆′ ⊂ ∆ for which the same applies.

The Set of Unreliable Formulas of a Certain Priority

Ux

s (Γ) = ∆1 ∪ ∆2 ∪ ... for Dabx(∆1), Dabx(∆2),... the minimal

Dabx–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

slide-96
SLIDE 96

Proof Theory (3)

Minimal Dabx–consequences

Dabx(∆) is a minimal Dabx–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 Ωx′ such that Ωx′ > Ωx, and (3) there is no ∆′ ⊂ ∆ for which the same applies.

The Set of Unreliable Formulas of a Certain Priority

Ux

s (Γ) = ∆1 ∪ ∆2 ∪ ... for Dabx(∆1), Dabx(∆2),... the minimal

Dabx–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, ∆ ∩ Ux

s (Γ) = ∅.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 35 / 42

slide-97
SLIDE 97

Proof Theory (3)

Minimal Dabx–consequences

Dabx(∆) is a minimal Dabx–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 Ωx′ such that Ωx′ > Ωx, and (3) there is no ∆′ ⊂ ∆ for which the same applies.

The Set of Unreliable Formulas of a Certain Priority

Ux

s (Γ) = ∆1 ∪ ∆2 ∪ ... for Dabx(∆1), Dabx(∆2),... the minimal

Dabx–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, ∆ ∩ Ux

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

slide-98
SLIDE 98

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

slide-99
SLIDE 99

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 ⊢AbLx 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

slide-100
SLIDE 100

Outline

1

Searching for Explanations Abduction? Logic–Based Approaches to Abduction Aim of this Talk

2

The Deductive Frame Abduction vs Deduction A Modal Frame Representing Abductive Reasoning Contexts

3

On Defeasible Inference

4

Enter Adaptive Logics 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

slide-101
SLIDE 101

Example: Practical Abduction

Definition

A, B =df n(A ⊃ B) ∧ B ∧ ¬eB ∧ ¬A

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 38 / 42

slide-102
SLIDE 102

Example: Practical Abduction

Definition

A, B =df n(A ⊃ B) ∧ B ∧ ¬eB ∧ ¬A

Example

1 n(p ⊃ q) –;PREM ∅ 2 n(r ⊃ q) –;PREM ∅ 3 q –;PREM ∅

Set of Unreliable Formulas

Ubk

3 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

Up

3(Γ) = ∅

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 38 / 42

slide-103
SLIDE 103

Example: Practical Abduction

Definition

A, B =df n(A ⊃ B) ∧ B ∧ ¬eB ∧ ¬A

Example

1 n(p ⊃ q) –;PREM ∅ 2 n(r ⊃ q) –;PREM ∅ 3 q –;PREM ∅ 4 ¬eq –;RC {eq} 5 p 1, 3, 4;RC {eq, p, q} 6 r 2, 3, 4;RC {eq, r, q}

Set of Unreliable Formulas

Ubk

6 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

Up

6(Γ) = ∅

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 38 / 42

slide-104
SLIDE 104

Example: Practical Abduction

Definition

A, B =df n(A ⊃ B) ∧ B ∧ ¬eB ∧ ¬A

Example

1 n(p ⊃ q) –;PREM ∅ 2 n(r ⊃ q) –;PREM ∅ 3 q –;PREM ∅ 4 ¬eq –;RC {eq} 5 p 1, 3, 4;RC {eq, p, q} 6 r 2, 3, 4;RC {eq, r, q} 7 p, q ∨ r ∧ ¬p, q 1,2,3,4;RU {eq} 8 r, q ∨ p ∧ ¬r, q 1,2,3,4;RU {eq}

Set of Unreliable Formulas

Ubk

8 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

Up

8(Γ) = {p, q, r ∧ ¬p, q, r, q, p ∧ ¬r, q}

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 38 / 42

slide-105
SLIDE 105

Example: Practical Abduction

Definition

A, B =df n(A ⊃ B) ∧ B ∧ ¬eB ∧ ¬A

Example

1 n(p ⊃ q) –;PREM ∅ 2 n(r ⊃ q) –;PREM ∅ 3 q –;PREM ∅ 4 ¬eq –;RC {eq} 5 p 1, 3, 4;RC {eq, p, q}

  • 6

r 2, 3, 4;RC {eq, r, q}

  • 7

p, q ∨ r ∧ ¬p, q 1,2,3,4;RU {eq} 8 r, q ∨ p ∧ ¬r, q 1,2,3,4;RU {eq}

Set of Unreliable Formulas

Ubk

8 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

Up

8(Γ) = {p, q, r ∧ ¬p, q, r, q, p ∧ ¬r, q}

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 38 / 42

slide-106
SLIDE 106

Example: Practical Abduction

Definition

A, B =df n(A ⊃ B) ∧ B ∧ ¬eB ∧ ¬A

Example

1 n(p ⊃ q) –;PREM ∅ 2 n(r ⊃ q) –;PREM ∅ 3 q –;PREM ∅ 4 ¬eq –;RC {eq} 5 p 1, 3, 4;RC {eq, p, q}

  • 6

r 2, 3, 4;RC {eq, r, q}

  • 7

p, q ∨ r ∧ ¬p, q 1,2,3,4;RU {eq} 8 r, q ∨ p ∧ ¬r, q 1,2,3,4;RU {eq} 9 p ∨ r 1,2,3,4;RC {eq, p ∨ r, q}

Set of Unreliable Formulas

Ubk

9 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

Up

9(Γ) = {p, q, r ∧ ¬p, q, r, q, p ∧ ¬r, q}

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 38 / 42

slide-107
SLIDE 107

Example: Theoretical Abduction

Definition

A1 ∧ ... ∧ An, B =df n((A1 ∧ ... ∧ An) ⊃ B) ∧ B ∧ ¬eB ∧ ¬A ∧ ¬[An

1 ⊃ B]

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 39 / 42

slide-108
SLIDE 108

Example: Theoretical Abduction

Definition

A1 ∧ ... ∧ An, B =df n((A1 ∧ ... ∧ An) ⊃ B) ∧ B ∧ ¬eB ∧ ¬A ∧ ¬[An

1 ⊃ B]

Example

1 n(p ⊃ q) –;PREM ∅ 2 n(r ⊃ q) –;PREM ∅ 3 q –;PREM ∅

Set of Unreliable Formulas

Ubk

3 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

Ut

3(Γ) = ∅

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 39 / 42

slide-109
SLIDE 109

Example: Theoretical Abduction

Definition

A1 ∧ ... ∧ An, B =df n((A1 ∧ ... ∧ An) ⊃ B) ∧ B ∧ ¬eB ∧ ¬A ∧ ¬[An

1 ⊃ B]

Example

1 n(p ⊃ q) –;PREM ∅ 2 n(r ⊃ q) –;PREM ∅ 3 q –;PREM ∅ 4 p 1, 3;RC {eq, nq, p, q} 5 r 2, 3;RC {eq, nq, r, q}

Set of Unreliable Formulas

Ubk

5 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

Ut

5(Γ) = ∅

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 39 / 42

slide-110
SLIDE 110

Example: Theoretical Abduction

Definition

A1 ∧ ... ∧ An, B =df n((A1 ∧ ... ∧ An) ⊃ B) ∧ B ∧ ¬eB ∧ ¬A ∧ ¬[An

1 ⊃ B]

Example

1 n(p ⊃ q) –;PREM ∅ 2 n(r ⊃ q) –;PREM ∅ 3 q –;PREM ∅ 4 p 1, 3;RC {eq, nq, p, q}

  • 5

r 2, 3;RC {eq, nq, r, q}

  • 6

p, q ∨ r ∧ ¬p, q 1,2,3;RC {eq, nq, n(r ⊃ q), n(¬p ⊃ q)} 7 r, q ∨ p ∧ ¬r, q 1,2,3;RC {eq, nq, n(p ⊃ q), n(¬r ⊃ q)}

Set of Unreliable Formulas

Ubk

7 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

Ut

7(Γ) = {p, q, r ∧ ¬p, q, r, q, p ∧ ¬r, q}

?

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 39 / 42

slide-111
SLIDE 111

Example: Theoretical Abduction

Definition

A1 ∧ ... ∧ An, B =df n((A1 ∧ ... ∧ An) ⊃ B) ∧ B ∧ ¬eB ∧ ¬A ∧ ¬[An

1 ⊃ B]

Example

1 n(p ⊃ q) –;PREM ∅ 2 n(r ⊃ q) –;PREM ∅ 3 q –;PREM ∅ 4 p 1, 3;RC {eq, nq, p, q} 5 r 2, 3;RC {eq, nq, r, q} 6 p, q ∨ r ∧ ¬p, q 1,2,3;RC {eq, nq, n(r ⊃ q), n(¬p ⊃ q)} 7 r, q ∨ p ∧ ¬r, q 1,2,3;RC {eq, nq, n(p ⊃ q), n(¬r ⊃ q)}

Set of Unreliable Formulas

Ubk

7 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

Ut

7(Γ) = ∅

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 39 / 42

slide-112
SLIDE 112

Example: Prioritized Abduction

Definition

A, Bt ∈ Ωt, A, Bpti ∈ Ωpti and A, Bppi ∈ Ωppi

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 40 / 42

slide-113
SLIDE 113

Example: Prioritized Abduction

Definition

A, Bt ∈ Ωt, A, Bpti ∈ Ωpti and A, Bppi ∈ Ωppi

Example

1 n(p ⊃ q) –;PREM ∅ 2 n♦e(p ∧ q) –;PREM ∅ 3 n(r ⊃ q) –;PREM ∅ 4 n♦e♦e(r ∧ q) –;PREM ∅ 5 q –;PREM ∅

Set of Unreliable Formulas

Ubk

5 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 40 / 42

slide-114
SLIDE 114

Example: Prioritized Abduction

Definition

A, Bt ∈ Ωt, A, Bpti ∈ Ωpti and A, Bppi ∈ Ωppi

Example

1 n(p ⊃ q) –;PREM ∅ 2 n♦e(p ∧ q) –;PREM ∅ 3 n(r ⊃ q) –;PREM ∅ 4 n♦e♦e(r ∧ q) –;PREM ∅ 5 q –;PREM ∅ 6 p 1, 2, 5;RC {eq, nq, p, qpt1} 7 r 3, 4, 5;RC {eq, nq, r, qpt2}

Set of Unreliable Formulas

Ubk

7 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 40 / 42

slide-115
SLIDE 115

Example: Prioritized Abduction

Definition

A, Bt ∈ Ωt, A, Bpti ∈ Ωpti and A, Bppi ∈ Ωppi

Example

1 n(p ⊃ q) –;PREM ∅ 2 n♦e(p ∧ q) –;PREM ∅ 3 n(r ⊃ q) –;PREM ∅ 4 n♦e♦e(r ∧ q) –;PREM ∅ 5 q –;PREM ∅ 6 p 1, 2, 5;RC {eq, nq, p, qpt1} 7 r 3, 4, 5;RC {eq, nq, r, qpt2} 8 p, qpt1 ∨ r ∧ ¬p, qpp2 1,2,3,4,5;RC {eq, nq} 9 r, qpt2 ∨ p ∧ ¬r, qpp1 1,2,3,4,5;RC {eq, nq}

Set of Unreliable Formulas

Ubk

9 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 40 / 42

slide-116
SLIDE 116

Example: Prioritized Abduction

Definition

A, Bt ∈ Ωt, A, Bpti ∈ Ωpti and A, Bppi ∈ Ωppi

Example

1 n(p ⊃ q) –;PREM ∅ 2 n♦e(p ∧ q) –;PREM ∅ 3 n(r ⊃ q) –;PREM ∅ 4 n♦e♦e(r ∧ q) –;PREM ∅ 5 q –;PREM ∅ 6 p 1, 2, 5;RC {eq, nq, p, qpt1} 7 r 3, 4, 5;RC {eq, nq, r, qpt2} 8 p, qpt1 ∨ r ∧ ¬p, qpp2 1,2,3,4,5;RC {eq, nq} 9 r, qpt2 ∨ p ∧ ¬r, qpp1 1,2,3,4,5;RC {eq, nq} 10 r ∧ ¬p, qpp2 8;RC {eq, nq, p, qpt1} 11 r, qpt2 9;RC {eq, nq, p ∧ ¬r, qpp1}

Set of Unreliable Formulas

Ubk

11 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

Upt2

11 (Γ) = {r, qpt2}

Upp2

11 (Γ) = {r ∧ ¬p, qpp2}

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 40 / 42

slide-117
SLIDE 117

Example: Prioritized Abduction

Definition

A, Bt ∈ Ωt, A, Bpti ∈ Ωpti and A, Bppi ∈ Ωppi

Example

1 n(p ⊃ q) –;PREM ∅ 2 n♦e(p ∧ q) –;PREM ∅ 3 n(r ⊃ q) –;PREM ∅ 4 n♦e♦e(r ∧ q) –;PREM ∅ 5 q –;PREM ∅ 6 p 1, 2, 5;RC {eq, nq, p, qpt1} 7 r 3, 4, 5;RC {eq, nq, r, qpt2}

  • 8

p, qpt1 ∨ r ∧ ¬p, qpp2 1,2,3,4,5;RC {eq, nq} 9 r, qpt2 ∨ p ∧ ¬r, qpp1 1,2,3,4,5;RC {eq, nq} 10 r ∧ ¬p, qpp2 8;RC {eq, nq, p, qpt1} 11 r, qpt2 9;RC {eq, nq, p ∧ ¬r, qpp1}

Set of Unreliable Formulas

Ubk

11 (Γ) = {n(p ⊃ q), n(r ⊃ q)}

Upt2

11 (Γ) = {r, qpt2}

Upp2

11 (Γ) = {r ∧ ¬p, qpp2}

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 40 / 42

slide-118
SLIDE 118

Further Research

Finetuning of

the adaptive logics for prioritized abduction.

Development of

adaptive logics for abduction based on inconsistent background knowledge, and of adaptive logics that combine abduction with induction.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 41 / 42

slide-119
SLIDE 119

References

ALISEDA–LLERA, A. Abductive Reasoning. Logical Investigations into Discovery and Explanation, vol. 330 of Synthese Library. Kluwer, Dordrecht, 2006. BATENS, D. A universal logic approach to adaptive logics. Logica Universalis 1 (2007), 221–242. BATENS, D., MEHEUS, J., PROVIJN, D., AND VERHOEVEN, L. Some adaptive logics for

  • diagnosis. Logic and Logical Philosophy 11–12 (2003), 39–65.

HEMPEL, C.G., AND OPPENHEIM, P. Studies in the logic of explanation. Philosophy of Science 15 (1948), 135–175. MAYER, M. C., AND PIRRI, F. First–order abduction via tableau and sequent calculi. Logic Journal of the IGPL 1 (1993), 99–117. MCILRAITH, S. Logic–based abductive inference. Tech. Rep. Number KSL–98–19, Knowledge Systems Laboratory, July 1998. MEHEUS, J., AND BATENS, D. A formal logic for abductive reasoning. Logic Journal of the IGPL 14 (2006), 221–236. MEHEUS, J., VERHOEVEN, L., VAN DYCK, M., AND PROVIJN, D. Ampliative adaptive logics and the foundation of logic–based approaches to abduction. In L. Magnani,

  • N. J. Nersessian, and C. Pizzi, Eds. Logical and Computational Aspects of

Model-Based Reasoning, Kluwer, Dordrecht, 2002, pp. 39–71. PAUL, G. AI approaches to abduction. In D. Gabbay, and R. Kruse, Eds. Abductive Reasoning and Uncertainty Management Systems, vol. 4 of Handbook of Defeasable Reasoning and Uncertainty Management Systems, Kluwer, Dordrecht, 2000, pp. 35–98.

  • H. Lycke (Ghent University)

The adaptive logics approach to abductive reasoning MBR’09, Campinas 42 / 42