presentation clmps july 2011

8/6/2019 Presentation CLMPS July 2011

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Three Complications in Modeling Abduction in Science

Tjerk Gauderis

Centre for Logic and Philosophy of ScienceGhent University

[email protected]

CLMPS 2011

Nancy · July 26th, 2011

Tjerk Gauderis (UGent) Three Complications CLMPS11 1 / 28

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Outline

1 Modeling Abduction by means of Adaptive LogicsAbductionThe Idea behind Adaptive Logics for AbductionTwo Types of Factual Abduction

2 The Logic MLAs

3 Three Complications for Modelling Abduction in ScienceInsufficient Arguments

AnomaliesHierarchical Background Knowledge


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Goal

We want to create adaptive logics which enable us to model both:

deductive stepsdefeasible steps according to the Peircean Schema of factualabduction

The surprising fact, C is observed;

But if A were true, C would be a matter of course,Hence, there is reason to suspect that A is true.

which we will formalize as:

(∀α)(A(α) ⊃ B (α))B (β)

A(β)

We presuppose that A(α) and B (α) share no predicates.


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Outline


2 The Logic MLAs




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1 (∀x )(Px ⊃ Rx ) PREM2 Ra PREM


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1 (∀x )(Px ⊃ Rx ) PREM2 Ra PREM3 Pa


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1 (∀x )(Px ⊃ Rx ) PREM2 Ra PREM3 Pa ∨ ¬Pa Tautology


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Pa is a possible explanation for Ra unless ¬Pa is the case


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4 Qb ∨¬Qb Tautology


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4 Qb ∨¬Qb Tautology5 Pa ∨

(∀x )(Px ⊃ Rx ) ∧ Ra ∧ ¬Pa

1;RU


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(∀x )(Px ⊃ Rx ) ∧ Ra ∧ ¬Pa

1;RU

approximates the idea of “unlessPa

is not possible as an explanation forRa

”


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(∀x )(Px ⊃ Rx ) ∧ Ra ∧ ¬Pa

1;RU

approximates the idea of “unless Pa is not possible as an explanation for Ra”

Advantage: every time we derive a formula of the form

A(β) ∨

∀α

A(α) ⊃ B (α)

∧

B (β) ∧ ¬A(β)

we can consider A(β) as a hypothesis (according the peircean schema)


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(∀x )(Px ⊃ Rx ) ∧ Ra ∧ ¬Pa

1;RU



A(β) ∨

∀α

A(α) ⊃ B (α)

∧

B (β) ∧ ¬A(β)


5 Pa 1,2; RC

(∀x )(Px ⊃ Rx ) ∧ Ra ∧ ¬Pa


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(∀x )(Px ⊃ Rx ) ∧ Ra ∧ ¬Pa

1;RU



A(β) ∨

∀α

A(α) ⊃ B (α)

∧

B (β) ∧ ¬A(β)


5 Pa 1,2; RC

(∀x )(Px ⊃ Rx ) ∧ Ra ∧ ¬Pa

idea from [Meheus(2010)], which is behind all adaptive logics for abduction


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Outline


2 The Logic MLAs




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Multiple explanatory hypotheses

Example

Γ =

(∀x )(Px ⊃ Rx ),(∀x )(Qx ⊃ Rx ),Ra

→ We have to decide which consequences we want to derive defeasibly.


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Example

Γ =

(∀x )(Px ⊃ Rx ),(∀x )(Qx ⊃ Rx ),Ra

→ We have to decide which consequences we want to derive defeasibly.Γ Pa ∨ Qa ?

Γ Pa and Γ Qa ?

Γ Pa ∧ Qa ?


M l i l l h h

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Example

Γ =

(∀x )(Px ⊃ Rx ),(∀x )(Qx ⊃ Rx ),Ra

→ We have to decide which consequences we want to derive defeasibly.Γ Pa ∨ Qa ?

Γ Pa and Γ Qa ?

Γ Pa ∧ Qa ? ⇒ NOT

We can already rule out the last option:

not sensible, both hypotheses are sufficient explanations

if hypotheses are contradictory ⇒ Logical Explosion


T T f F l Abd i

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Two Types of Factual Abduction

Depending on the kind of reasoning we try to model:

Γ Pa ∨ Qa

Γ Pa

Γ Qa

Γ Pa

Γ Qa

Γ Pa ∨ Qa


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T T f F t l Abd ti



Two Types of Factual Abduction

Depending on the kind of reasoning we try to model:

Practical Abduction

Γ Pa ∨ Qa

Γ Pa

Γ Qa

reasoning aimed at acting onthe conclusions (derivedfrom the current Γ)

e.g. diagnoses, engineering,conversations, . . .

modelled by LArs[Meheus(2010)]

Theoretical Abduction

Γ Pa

Γ Qa

Γ Pa ∨ Qa

reasoning aimed at furtherresearch to find the actualcause (extend Γ)

e.g. science, criminalinvestigations, . . .

modelled by MLAs

[Gauderis(2011)]


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1 Modeling Abduction by means of Adaptive Logics

AbductionThe Idea behind Adaptive Logics for AbductionTwo Types of Factual Abduction

2 The Logic MLAs

3 Three Complications for Modelling Abduction in ScienceInsufficient ArgumentsAnomaliesHierarchical Background Knowledge


Theoretical Abduction Reconsidered

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Allowing the consequences Pa and Qawhile preventing the conjunction Pa ∧ Qa

is not a convincing model for theoretical abduction.



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complex proof dynamics

⇒ In large premise sets it is hard to keep track of which formulas can beconjoined together.



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countra-intuitive⇒ CL-rules should be applicable to all elements of the consequence set.



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Allowing the consequencesPa

andQa

while preventing the conjunction Pa ∧ Qa




countra-intuitive⇒ CL-rules should be applicable to all elements of the consequence set.

not mirroring natural human reasoning⇒ asserting that “Pa is the case and Qa is also the case, but not both” is

not what people do when they consider two different explanations.


by going modal

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... by going modal


by going modal

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... by going modal

Idea: If we represent abduced hypotheses by possibilities, then


by going modal

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... by going modal


Γ ♦Pa

Γ ♦Qa

Γ ♦Pa ∧ ♦Qa


... by going modal

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... by going modal


Γ ♦Pa

Γ ♦Qa

Γ ♦Pa ∧ ♦Qa

Γ ♦(Pa ∧ Qa) in any standard modal logic.


... by going modal

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... by going modal


Γ ♦Pa

Γ ♦Qa

Γ ♦Pa ∧ ♦Qa

Γ ♦(Pa ∧ Qa) in any standard modal logic.

But, therefore we will need to reformulate our premise set to

Γ =

(∀x )(Px ⊃ Rx ),(∀x )(Qx ⊃ Rx ),Ra

Because deductive consequences of the premise set need to be able torefute untenable hypotheses.


Language Schema

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g g S

LM , the language of our logic, is L (of CL) extended with the modal

operator

(♦

is defined as ¬

¬).

1

W is the set of closed formulas of CL


Language Schema

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g g


operator

(♦

is defined as ¬

¬).

W M , the set of closed formulas of LM , is the smallest set that satisfies:1

1 if A ∈ W , then A, A ∈ W M

2 if A ∈ W M , then ¬A ∈ W M

3 if A, B ∈ W M , then A ∧ B , A ∨ B , A ⊃ B , A ≡ B ∈ W M

1



Language Schema

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g g


operator

(♦

is defined as ¬

¬).

W M , the set of closed formulas of LM , is the smallest set that satisfies:1

1 if A ∈ W , then A, A ∈ W M

2 if A ∈ W M , then ¬A ∈ W M

3 if A, B ∈ W M , then A ∧ B , A ∨ B , A ⊃ B , A ≡ B ∈ W M

W Γ, the subset of W M , the elements of which can act as premises in ourlogic:

W Γ = A | A ∈ W

1



Proof Theory

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y


Proof Theory

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undefeasible reasoning steps: characterized by a monotonic logic, thelower limit logic; in our case this is the modal logic D.

(RU) A1, ...,An MLAs B if A1, ...,An D B


Proof Theory

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defeasible reasoning steps: characterized by a set of abnormalities Ω(the elements of which (we call them Θi ) are defined by a logical form):

(RC) A1, ..., An MLAs B if A1, ..., An D B ∨ Θ1 ∨ . . . ∨ Θn


Proof Theory

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Every formula has a recursively defined set of conditions associatedthat are assumed to be false:

premises : ∅ RU-results : union of conditions of A1, ..., An

RC-results : union of Θ1, . . . , Θn and conditions of A1, ...,An


Proof Theory

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Every formula has a recursively defined set of conditions associatedthat are assumed to be false:

premises : ∅ RU-results : union of conditions of A1, ..., An

RC-results : union of Θ1, . . . , Θn and conditions of A1, ...,An

defeated reasoning steps: if the strategy shows them to be unreliable.In our case we have the simple strategy:

a formula is unreliable if an element of the set of conditions isunconditionally derived


Set of Abnormalities

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What are the conditions to derive ♦A(β) from a set of premises Γ?



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♦A(β) is a possible explanation for some Γ D B (β)



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B (β) is not a tautology (otherwise: anything is a hypothesis)


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B (β) is not a tautology (otherwise: anything is a hypothesis)

♦A(β) has no redundant part (otherwise: anything is a hypothesis)


Ω =

∀α

A(α) ⊃ B (α)

∧

B (β) ∧ ¬A(β)

∨ (∀α)B (α)

∨

n

i =1(∀α)(A−1i

(α) ⊃ B (α)) |No predicate that occurs in B occurs in A,

α ∈ V , β ∈ C, A, B ∈ F


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Example



Γ =

(∀x )(Px ⊃ Rx ),(∀x )(Qx ⊃ Rx ),(∀x )(Px ⊃ Sx ),Ra,¬Sa


Example

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Γ =


1 (∀x )(Px ⊃ Rx ) PREM ∅2 Ra PREM ∅


Example

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Γ =


1 (∀x )(Px ⊃ Rx ) PREM ∅2 Ra PREM ∅3 ♦Pa 1,2;RC !Pa Ra


Example

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Γ =


1 (∀x )(Px ⊃ Rx ) PREM ∅2 Ra PREM ∅3 ♦Pa 1,2;RC !Pa Ra4 (∀x )(Px ⊃ Sx ) PREM ∅


Example

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Γ =


1 (∀x )(Px ⊃ Rx ) PREM ∅2 Ra PREM ∅3 ♦Pa 1,2;RC !Pa Ra4 (∀x )(Px ⊃ Sx ) PREM ∅5 ♦Sa 3,4;RU !Pa Ra


Example ( )( ) ( )(Q ) ( )( S )

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Γ =



6 ¬Sa PREM ∅


Example (∀ )(P R ) (∀ )(Q R ) (∀ )(P S )

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Γ =



6 ¬Sa PREM ∅7 ¬Pa 4,6;RU ∅


Example (∀ )(P R ) (∀ )(Q R ) (∀ )(P S )

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Γ =



6 ¬Sa PREM ∅7 ¬Pa 4,6;RU ∅8

(∀x )(Px ⊃ Rx ) ∧ (Ra ∧ ¬Pa)

1,2,7;RU ∅


Example (∀ )(P ⊃ R ) (∀ )(Q ⊃ R ) (∀ )(P ⊃ S )

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Γ =



6 ¬Sa PREM ∅7 ¬Pa 4,6;RU ∅8

(∀x )(Px ⊃ Rx ) ∧ (Ra ∧ ¬Pa)

1,2,7;RU ∅

9 !Pa Ra 8;RU ∅


Example (∀x)(Px ⊃ Rx) (∀x)(Qx ⊃ Rx) (∀x)(Px ⊃ Sx)

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Γ =


1 (∀x )(Px ⊃ Rx ) PREM ∅2 Ra PREM ∅3 ♦Pa 1,2;RC !Pa Ra 9

4 (∀x )(Px ⊃ Sx ) PREM ∅5 ♦Sa 3,4;RU !Pa Ra 9

6 ¬Sa PREM ∅7 ¬Pa 4,6;RU ∅8

(∀x )(Px ⊃ Rx ) ∧ (Ra ∧ ¬Pa)

1,2,7;RU ∅

9 !Pa Ra 8;RU ∅



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Γ =




6 ¬Sa PREM ∅7 ¬Pa 4,6;RU ∅8

(∀x )(Px ⊃ Rx ) ∧ (Ra ∧ ¬Pa)

1,2,7;RU ∅

9 !Pa Ra 8;RU ∅

10 (∀x )(Qx ⊃ Rx ) PREM ∅



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Γ =




6 ¬Sa PREM ∅7 ¬Pa 4,6;RU ∅8

(∀x )(Px ⊃ Rx ) ∧ (Ra ∧ ¬Pa)

1,2,7;RU ∅

9 !Pa Ra 8;RU ∅

10 (∀x )(Qx ⊃ Rx ) PREM ∅

11 ♦Qa 2,10;RC !Qa Ra



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Γ =




6 ¬Sa PREM ∅7 ¬Pa 4,6;RU ∅8

(∀x )(Px ⊃ Rx ) ∧ (Ra ∧ ¬Pa)

1,2,7;RU ∅

9 !Pa Ra 8;RU ∅

10 (∀x )(Qx ⊃ Rx ) PREM ∅

11 ♦Qa 2,10;RC !Qa Ra12 (∀x )((Qx ∧ Sx ) ⊃ Rx ) 10;RU ∅



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Γ =




6 ¬Sa PREM ∅7 ¬Pa 4,6;RU ∅8

(∀x )(Px ⊃ Rx ) ∧ (Ra ∧ ¬Pa)

1,2,7;RU ∅

9 !Pa Ra 8;RU ∅

10 (∀x )(Qx ⊃ Rx ) PREM ∅

11 ♦Qa 2,10;RC !Qa Ra12 (∀x )((Qx ∧ Sx ) ⊃ Rx ) 10;RU ∅13 ♦(Qa ∧ Sa) 2,10;RC !(Qa ∧ Sa) Ra


Example (∀x)(Px ⊃ Rx),(∀x)(Qx ⊃ Rx),(∀x)(Px ⊃ Sx),

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Γ =




6 ¬Sa PREM ∅7 ¬Pa 4,6;RU ∅8

(∀x )(Px ⊃ Rx ) ∧ (Ra ∧ ¬Pa)

1,2,7;RU ∅

9 !Pa Ra 8;RU ∅

10 (∀x )(Qx ⊃ Rx ) PREM ∅

11 ♦Qa 2,10;RC !Qa Ra12 (∀x )((Qx ∧ Sx ) ⊃ Rx ) 10;RU ∅13 ♦(Qa ∧ Sa) 2,10;RC !(Qa ∧ Sa) Ra14 !(Qa ∧ Sa) Ra 10;RU ∅


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Outline





2 The Logic MLAs




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Extending the Language Schema

L f fi L (

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LM 1, the language of the first extension, is LS 5 (with the modal operatorsa and ♦a) extended with the extra modal operator (♦ is defined as

¬¬).


Extending the Language Schema

L h l f h fi i i L ( i h h d l

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LM 1, the language of the first extension, is LS 5 (with the modal operatorsa and ♦a) extended with the extra modal operator (♦ is defined as

¬¬).

W M 1, the set of closed formulas of LM 1, is the smallest set that satisfies:

1 if A ∈ W S 5, then A, A ∈ W M 1

2 if A ∈ W M 1, then ¬A ∈ W M 1

3 if A, B ∈ W M 1, then A ∧ B , A ∨ B , A ⊃ B , A ≡ B ∈ W M 1


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We add an extra possibility operator to our language ♦a which behaves

according to S5 and change the set of abnormalities in the following way:

ΩM1 =

∀α

A(α) ⊃ ♦aB (α)

∧

B (β) ∧ ¬A(β)

∨ (∀α)B (α)

∨n

i =1(∀α)(A−1i (α) ⊃ ♦aB (α)) |No predicate that occurs in B occurs in A,

α ∈ V , β ∈ C, A, B ∈ F


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Example

Γ =(∀x )(Px ⊃ ♦aRx ),(∀x )(Qx ⊃ Rx ),Ra


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Example

Γ =(∀x )(Px ⊃ ♦aRx ),(∀x )(Qx ⊃ Rx ),Ra

1 (∀x )(Px ⊃ ♦aRx ) PREM ∅2 Ra PREM ∅3 ♦Pa 1,2;RC !Pa Ra


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Example

Γ =(∀x )(Px ⊃ ♦aRx ),(∀x )(Qx ⊃ Rx ),Ra

1 (∀x )(Px ⊃ ♦aRx ) PREM ∅2 Ra PREM ∅3 ♦Pa 1,2;RC !Pa Ra4 (∀x )(Qx ⊃ Rx ) PREM ∅


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Example

Γ =(∀x )(Px ⊃ ♦aRx ),(∀x )(Qx ⊃ Rx ),Ra

1 (∀x )(Px ⊃ ♦aRx ) PREM ∅2 Ra PREM ∅3 ♦Pa 1,2;RC !Pa Ra4 (∀x )(Qx ⊃ Rx ) PREM ∅5 (∀x )(Qx ⊃ ♦aRx ) 4;RU ∅6 ♦Qa 2,10;RC !Qa Ra


Outline

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2 The Logic MLA

s



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Outline





2 The Logic MLA

s



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Example



Γ = ♦b (∀x )(Px ⊃ Rx ),♦2

b (∀x )(Qx ⊃ ¬Rx ),

Ra,Qa


Example

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Γ = ♦b (∀x )(Px ⊃ Rx ),♦2

b (∀x )(Qx ⊃ ¬Rx ),

Ra,Qa

1 ♦b (∀x )(Px ⊃ Rx ) PREM ∅2 (∀x )(Px ⊃ Rx ) 1;RCK1

!¬Px , !Rx 3 Ra PREM ∅

4 ♦Pa 2,3;RCMLA

s

!¬Px , !Rx , !Pa Ra


Example

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Γ = ♦b (∀x )(Px ⊃ Rx ),♦2

b (∀x )(Qx ⊃ ¬Rx ),

Ra,Qa

1 ♦b (∀x )(Px ⊃ Rx ) PREM ∅2 (∀x )(Px ⊃ Rx ) 1;RCK1

!¬Px , !Rx 3 Ra PREM ∅

4 ♦Pa 2,3;RCMLA

s

!¬Px , !Rx , !Pa Ra

5 ♦2b (∀x )(Qx ⊃ ¬Rx ) PREM ∅

6 (∀x )(Qx ⊃ ¬Rx ) 5;RCK2!¬Qx , !¬Rx 9

7 Qa PREM ∅8 ♦¬Ra 5,7;RU ∅

9 !¬Ra 3,8;RU ∅


Did ik B t

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Diderik Batens.Adaptive Logics and Dynamic Proofs. A Study in the Dynamics of

Reasoning.

Forthcoming, 2011.

Tjerk Gauderis.Modelling abduction in science by means of a modal adaptive logic.

Forthcoming , 2011.

Joke Meheus.A formal logic for the abductions of singular hypotheses.Forthcoming , 2010.


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presentation clmps july 2011

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