Dynamic Proof Theories For Normative Reasoning on the Basis of - - PowerPoint PPT Presentation

Dynamic Proof Theories For Normative Reasoning on the Basis of Consistency Considerations

Christian Straßer and Joke Meheus and Mathieu Beirlaen and Frederik Van De Putte

Centre for Logic and Philosophy of Science Ghent University, Belgium {Christian.Strasser, Joke.Meheus, Mathieu.Beirlaen, frvdeput.Vandeputte}@UGent.be

September 14, 2012

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Central Problem

Answer the question: What are “actual”/“all-things- considered”/“proper”/ etc. obligations given:

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Central Problem

Answer the question: What are “actual”/“all-things- considered”/“proper”/ etc. obligations given:

◮ a set of norms

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Central Problem

Answer the question: What are “actual”/“all-things- considered”/“proper”/ etc. obligations given:

◮ a set of norms

Oa, O¬a′, Ob, O(a|c), . . .

2/29

Central Problem

Answer the question: What are “actual”/“all-things- considered”/“proper”/ etc. obligations given:

◮ a set of norms

Oa, O¬a′, Ob, O(a|c), . . .

◮ (optionally) a set of facts

2/29

Central Problem

Answer the question: What are “actual”/“all-things- considered”/“proper”/ etc. obligations given:

◮ a set of norms

Oa, O¬a′, Ob, O(a|c), . . .

◮ (optionally) a set of facts

a, d, . . .

2/29

Central Problem

Answer the question: What are “actual”/“all-things- considered”/“proper”/ etc. obligations given:

◮ a set of norms

Oa, O¬a′, Ob, O(a|c), . . .

◮ (optionally) a set of facts

a, d, . . .

◮ (optionally) a set of constraints

2/29

Central Problem

Answer the question: What are “actual”/“all-things- considered”/“proper”/ etc. obligations given:

◮ a set of norms

Oa, O¬a′, Ob, O(a|c), . . .

◮ (optionally) a set of facts

a, d, . . .

◮ (optionally) a set of constraints

¬(a ∧ a′), . . .

2/29

Central Problem

Answer the question: What are “actual”/“all-things- considered”/“proper”/ etc. obligations given:

◮ a set of norms

Oa, O¬a′, Ob, O(a|c), . . .

◮ (optionally) a set of facts

a, d, . . .

◮ (optionally) a set of constraints

¬(a ∧ a′), . . . Problem: There may be conflicts/inconsistencies.

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Input: Obligations Constraints a ¬(a ∧ a′) a′ b

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Input: Obligations Constraints a ¬(a ∧ a′) a′ b Obligations Constraints a ¬(a ∧ a′) a′ b Consistent chunks: a, b a′, b

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Input: Obligations Constraints a ¬(a ∧ a′) a′ b Obligations Constraints a ¬(a ∧ a′) a′ b Consistent chunks: a, b a′, b Further reasoning: a ∧ b, a ∨ b, a ∨ a′ a′ ∧ b, a ∨ b, a ∨ a′

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Input: Obligations Constraints a ¬(a ∧ a′) a′ b Obligations Constraints a ¬(a ∧ a′) a′ b Consistent chunks: a, b a′, b Further reasoning: a ∧ b, a ∨ b, a ∨ a′ a′ ∧ b, a ∨ b, a ∨ a′ What to do with the chunks?

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Input: Obligations Constraints a ¬(a ∧ a′) a′ b Obligations Constraints a ¬(a ∧ a′) a′ b Consistent chunks: a, b a′, b Further reasoning: a ∧ b, a ∨ b, a ∨ a′ a′ ∧ b, a ∨ b, a ∨ a′ What to do with the chunks?

◮ intersection

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Input: Obligations Constraints a ¬(a ∧ a′) a′ b Obligations Constraints a ¬(a ∧ a′) a′ b Consistent chunks: a, b a′, b Further reasoning: a ∧ b, a ∨ b, a ∨ a′ a′ ∧ b, a ∨ b, a ∨ a′ What to do with the chunks?

◮ intersection ◮ union

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back to IO

Input: Obligations Facts (a, b) a (a, c) (b, ¬c)

4/29

back to IO

Input: Obligations Facts (a, b) a (a, c) (b, ¬c) Obligations Facts (a, b) a (a, c) (b, ¬c) Consistent chunks: (a, b), (a, c) (a, b), (b, ¬c) (a, c), (b, ¬c)

4/29

back to IO

Input: Obligations Facts (a, b) a (a, c) (b, ¬c) Obligations Facts (a, b) a (a, c) (b, ¬c) Consistent chunks: (a, b), (a, c) (a, b), (b, ¬c) (a, c), (b, ¬c) Further reasoning: (a, b∧c), b, c, b ∨ c b, ¬c, b ∨ c c, b ∨ c

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back to IO

Input: Obligations Facts (a, b) a (a, c) (b, ¬c) Obligations Facts (a, b) a (a, c) (b, ¬c) Consistent chunks: (a, b), (a, c) (a, b), (b, ¬c) (a, c), (b, ¬c) Further reasoning: (a, b∧c), b, c, b ∨ c b, ¬c, b ∨ c c, b ∨ c

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◮ flat normative/knowledge/etc. base

◮ Rescher-Manor, Horty 5/29

◮ flat normative/knowledge/etc. base

◮ Rescher-Manor, Horty

◮ conditional normative/knowledge/etc. base

◮ Input/Output logic (Makinson, Van Der Torre) 5/29

◮ flat normative/knowledge/etc. base

◮ Rescher-Manor, Horty

◮ conditional normative/knowledge/etc. base

◮ Input/Output logic (Makinson, Van Der Torre)

lack of proof theory adaptive logics

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Idea: apply If •O then O. “as much as possible”.

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Idea: apply If •O explicit obligation then O. “as much as possible”.

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Idea: apply If •O explicit obligation then O “actual” obligation . “as much as possible”.

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Idea: apply If •O explicit obligation then O “actual” obligation . “as much as possible”. Monadic case: If •OA then OA.

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Idea: apply If •O explicit obligation then O “actual” obligation . “as much as possible”. Monadic case: If •OA then OA. Dyadic case: If •O(A, B) then O(A, B).

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Adaptive logic in the standard format

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Adaptive logic in the standard format

1. Lower Limit Logic supraclassical core logic (reflexive, monotonic, transitive)

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Adaptive logic in the standard format

1. Lower Limit Logic Lower Limit Logic supraclassical core logic (reflexive, monotonic, transitive) 2. Abnormalities characterized by a logical form, in our case Ω = {•O ∧ ¬O | O is an O-formula}

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Adaptive logic in the standard format

1. Lower Limit Logic Lower Limit Logic supraclassical core logic (reflexive, monotonic, transitive) 2. Abnormalities Abnormalities characterized by a logical form, in our case Ω = {•O ∧ ¬O | O is an O-formula} 3. Strategy e.g., minimal abnormality and reliability

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The Lower Limit Logic for the Monadic Case

◮ classical connectives ∧, ∨, ⊃, ≡, ¬

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The Lower Limit Logic for the Monadic Case

◮ classical connectives ∧, ∨, ⊃, ≡, ¬ ◮ deontic operator O: e.g., KD-operator

8/29

The Lower Limit Logic for the Monadic Case

◮ classical connectives ∧, ∨, ⊃, ≡, ¬ ◮ deontic operator O: e.g., KD-operator

◮ no nestings 8/29

The Lower Limit Logic for the Monadic Case

◮ classical connectives ∧, ∨, ⊃, ≡, ¬ ◮ deontic operator O: e.g., KD-operator

◮ no nestings

◮ a modal operator for constraints: e.g., K-operator

8/29

The Lower Limit Logic for the Monadic Case

◮ classical connectives ∧, ∨, ⊃, ≡, ¬ ◮ deontic operator O: e.g., KD-operator

◮ no nestings

◮ a modal operator for constraints: e.g., K-operator ◮ “explicitness” operator •

8/29

The Lower Limit Logic for the Monadic Case

◮ classical connectives ∧, ∨, ⊃, ≡, ¬ ◮ deontic operator O: e.g., KD-operator

◮ no nestings

◮ a modal operator for constraints: e.g., K-operator ◮ “explicitness” operator •

◮ where OA is well-formed, so is •OA 8/29

The Lower Limit Logic for the Monadic Case

◮ classical connectives ∧, ∨, ⊃, ≡, ¬ ◮ deontic operator O: e.g., KD-operator

◮ no nestings

◮ a modal operator for constraints: e.g., K-operator ◮ “explicitness” operator •

◮ where OA is well-formed, so is •OA

◮ “Ought-implies-can”: ⊢ A ⊃ ¬O¬A

8/29

The Lower Limit Logic for the Monadic Case

◮ classical connectives ∧, ∨, ⊃, ≡, ¬ ◮ deontic operator O: e.g., KD-operator

◮ no nestings

◮ a modal operator for constraints: e.g., K-operator ◮ “explicitness” operator •

◮ where OA is well-formed, so is •OA

◮ “Ought-implies-can”: ⊢ A ⊃ ¬O¬A ◮ classical propositional logic

8/29

The Lower Limit Logic for the Monadic Case

◮ classical connectives ∧, ∨, ⊃, ≡, ¬ ◮ deontic operator O: e.g., KD-operator

◮ no nestings

◮ a modal operator for constraints: e.g., K-operator ◮ “explicitness” operator •

◮ where OA is well-formed, so is •OA

◮ “Ought-implies-can”: ⊢ A ⊃ ¬O¬A ◮ classical propositional logic ◮ • is a “dummy”

“adaptive meaning”

8/29

Adaptive Proofs

A line: l line- number A ∧ B l’,. . . ,l”; R ∆

9/29

Adaptive Proofs

A line: l line- number A ∧ B formula l’,. . . ,l”; R ∆

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Adaptive Proofs

A line: l line- number A ∧ B formula l’,. . . ,l”; R justification ∆

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Adaptive Proofs

A line: l line- number A ∧ B formula l’,. . . ,l”; R justification ∆ condition

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Adaptive Proofs

A line: l line- number A ∧ B formula l’,. . . ,l”; R justification ∆ condition Conditional rule: If A1, . . . , An ⊢LLL B ∨ Dab (∆): A1 ∆1 . . . . . . An ∆n B ∆1 ∪ . . . ∪ ∆n ∪ ∆

9/29

Adaptive Proofs

A line: l line- number A ∧ B formula l’,. . . ,l”; R justification ∆ condition Conditional rule: If A1, . . . , An ⊢LLL B ∨ Dab (∆): A1 ∆1 . . . . . . An ∆n B ∆1 ∪ . . . ∪ collect abnormalities ∆n ∪ ∆

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Adaptive Proofs

A line: l line- number A ∧ B formula l’,. . . ,l”; R justification ∆ condition Conditional rule: If A1, . . . , An ⊢LLL B ∨ Dab (∆) add new condition : A1 ∆1 . . . . . . An ∆n B ∆1 ∪ . . . ∪ collect abnormalities ∆n ∪ ∆

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1

• Oa

PREM ∅ 2

• Oa′

PREM ∅ 3

• Oc

PREM ∅ 4 ¬(a ∧ a′) PREM premise introduction ∅

10/29

1

• Oa

PREM ∅ 2

• Oa′

PREM ∅ 3

• Oc

PREM ∅ 4 ¬(a ∧ a′) PREM ∅ 5 Oa 1; RC Note: •Oa ⊢LLL Oa ∨ (•Oa ∧ ¬Oa) {•Oa ∧ ¬Oa}

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1

• Oa

PREM ∅ 2

• Oa′

PREM ∅ 3

• Oc

PREM ∅ 4 ¬(a ∧ a′) PREM ∅ 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU unconditional rule Recall: O is KD-modality If A1, . . . , An ⊢LLL B then A1 ∆1 . . . . . . An ∆n B ∆1 ∪ . . . ∪ ∆n {•Oa ∧ ¬Oa}

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1

• Oa

PREM ∅ 2

• Oa′

PREM ∅ 3

• Oc

PREM ∅ 4 ¬(a ∧ a′) PREM ∅ 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU {•Oa ∧ ¬Oa} 7 Oa′ 2; RC {•Oa′ ∧ ¬Oa′} 8 O(a ∨ a′) 7; RU analogous to lines 5 and 6 {•Oa′ ∧ ¬Oa′}

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1

• Oa

PREM ∅ 2

• Oa′

PREM ∅ 3

• Oc

PREM ∅ 4 ¬(a ∧ a′) PREM ∅ 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU {•Oa ∧ ¬Oa} 7 Oa′ 2; RC {•Oa′ ∧ ¬Oa′} 8 O(a ∨ a′) 7; RU {•Oa′ ∧ ¬Oa′} 9 !a ∨ !a′ We shortcut: !A =df •OA ∧ ¬OA 1,2,4; RU this follows by O-aggregation and OIC ∅

10/29

1

• Oa

PREM ∅ 2

• Oa′

PREM ∅ 3

• Oc

PREM ∅ 4 ¬(a ∧ a′) PREM ∅ 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU {•Oa ∧ ¬Oa} 7 Oa′ 2; RC {•Oa′ ∧ ¬Oa′} 8 O(a ∨ a′) 7; RU {•Oa′ ∧ ¬Oa′} 9 !a ∨ !a′ 1,2,4; RU ∅

◮ One of our assumptions is false! We need a retraction

mechanism.

10/29

1

• Oa

PREM ∅ 2

• Oa′

PREM ∅ 3

• Oc

PREM ∅ 4 ¬(a ∧ a′) PREM ∅ 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU {•Oa ∧ ¬Oa} 7 Oa′ 2; RC {•Oa′ ∧ ¬Oa′} 8 O(a ∨ a′) 7; RU {•Oa′ ∧ ¬Oa′} 9 !a ∨ !a′ 1,2,4; RU ∅

◮ One of our assumptions is false! We need a retraction

mechanism.

◮ marking of lines which are retracted

10/29

1

• Oa

PREM ∅ 2

• Oa′

PREM ∅ 3

• Oc

PREM ∅ 4 ¬(a ∧ a′) PREM ∅ 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU {•Oa ∧ ¬Oa} 7 Oa′ 2; RC {•Oa′ ∧ ¬Oa′} 8 O(a ∨ a′) 7; RU {•Oa′ ∧ ¬Oa′} 9 !a ∨ !a′ 1,2,4; RU ∅

◮ One of our assumptions is false! We need a retraction

mechanism.

◮ marking of lines which are retracted ◮ determined by the minimal disjunctions of abnormalities which

are derived on the empty condition

10/29

1

• Oa

PREM ∅ 2

• Oa′

PREM ∅ 3

• Oc

PREM ∅ 4 ¬(a ∧ a′) PREM ∅ 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU {•Oa ∧ ¬Oa} 7 Oa′ 2; RC {•Oa′ ∧ ¬Oa′} 8 O(a ∨ a′) 7; RU {•Oa′ ∧ ¬Oa′} 9 !a ∨ !a′ 1,2,4; RU ∅

◮ One of our assumptions is false! We need a retraction

mechanism.

◮ marking of lines which are retracted ◮ determined by the minimal disjunctions of abnormalities which

are derived on the empty condition

◮ exact definition depends on the strategy

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Reliability and Minimal Abnormality

. . . . . . . . . . . . 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU {•Oa ∧ ¬Oa} 7 Oa′ 2; RC {•Oa′ ∧ ¬Oa′} 8 O(a ∨ a′) 7; RU {•Oa′ ∧ ¬Oa′} 9 !a ∨ !a′ 1,2,4; RU ∅

◮ Reliability assumption contains a member of a minimal

disjunction of abnormalities ⇒ retract

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Reliability and Minimal Abnormality

. . . . . . . . . . . . 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU {•Oa ∧ ¬Oa} 7 Oa′ 2; RC {•Oa′ ∧ ¬Oa′} 8 O(a ∨ a′) 7; RU {•Oa′ ∧ ¬Oa′} 9 !a ∨ !a′ 1,2,4; RU ∅

◮ Reliability assumption contains a member of a minimal

disjunction of abnormalities ⇒ retract

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Reliability and Minimal Abnormality

. . . . . . . . . . . . 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU {•Oa ∧ ¬Oa} 7 Oa′ 2; RC {•Oa′ ∧ ¬Oa′} 8 O(a ∨ a′) 7; RU {•Oa′ ∧ ¬Oa′} 9 !a ∨ !a′ 1,2,4; RU ∅

◮ Reliability assumption contains a member of a minimal

disjunction of abnormalities ⇒ retract

◮ application context: where conflict is likely to be a sign of

erroneous issuing of norms by the authority

◮ e.g., authority may have made a mistake in issuing Oa′ (that

explains the conflict)

◮ there may additionally be a high cost for realizing an erroneous

norm (e.g., a big financial investment)

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Reliability and Minimal Abnormality

. . . . . . . . . . . . 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU {•Oa ∧ ¬Oa} 7 Oa′ 2; RC {•Oa′ ∧ ¬Oa′} 8 O(a ∨ a′) 7; RU {•Oa′ ∧ ¬Oa′} 9 !a ∨ !a′

• {!a}, {!a′}
• 1,2,4; RU

∅

◮ Reliability assumption contains a member of a minimal

disjunction of abnormalities ⇒ retract

◮ application context: where conflict is likely to be a sign of

erroneous issuing of norms by the authority

◮ e.g., authority may have made a mistake in issuing Oa′ (that

explains the conflict)

◮ there may additionally be a high cost for realizing an erroneous

norm (e.g., a big financial investment)

◮ Minimal Abnormality

◮ minimal choice sets 11/29

Reliability and Minimal Abnormality

. . . . . . . . . . . . 5 Oa 1; RC {•Oa ∧ ¬Oa} 6 O(a ∨ a′) 5; RU {•Oa ∧ ¬Oa} 7 Oa′ 2; RC {•Oa′ ∧ ¬Oa′} 8 O(a ∨ a′) 7; RU {•Oa′ ∧ ¬Oa′} 9 !a ∨ !a′ 1,2,4; RU ∅

◮ Reliability assumption contains a member of a minimal

disjunction of abnormalities ⇒ retract

◮ application context: where conflict is likely to be a sign of

erroneous issuing of norms by the authority

◮ e.g., authority may have made a mistake in issuing Oa′ (that

explains the conflict)

◮ there may additionally be a high cost for realizing an erroneous

norm (e.g., a big financial investment)

◮ Minimal Abnormality

◮ minimal choice sets ◮ A is derived “safely” if for each minimal choice ϕ, A is derived

• n a condition ∆ such that ϕ ∩ ∆ = ∅

11/29

Counting Strategy

1

• Oa

PREM ∅ 2

• Ob

PREM ∅ 3

• Oc

PREM ∅ 4

• Od

PREM ∅

12/29

Counting Strategy

1

• Oa

PREM ∅ 2

• Ob

PREM ∅ 3

• Oc

PREM ∅ 4

• Od

PREM ∅ 5 (a ⊃ (¬b ∧ ¬c ∧ ¬d)) PREM ∅

12/29

Counting Strategy

1

• Oa

PREM ∅ 2

• Ob

PREM ∅ 3

• Oc

PREM ∅ 4

• Od

PREM ∅ 5 (a ⊃ (¬b ∧ ¬c ∧ ¬d)) PREM ∅ 6 !a ∨ !b 1,2,5; RU ∅ 7 !a ∨ !c 1,3,5; RU ∅ 8 !a ∨ !d 1,4,5; RU ∅

12/29

Counting Strategy

1

• Oa

PREM ∅ 2

• Ob

PREM ∅ 3

• Oc

PREM ∅ 4

• Od

PREM ∅ 5 (a ⊃ (¬b ∧ ¬c ∧ ¬d)) PREM ∅ 6 !a ∨ !b 1,2,5; RU ∅ 7 !a ∨ !c 1,3,5; RU ∅ 8 !a ∨ !d 1,4,5; RU ∅ 9 Oa 1; RC {!a} 10 Ob 2; RC {!b} 11 Oc 3; RC {!c} 12 Od 4; RC {!d}

12/29

Counting Strategy

1

• Oa

PREM ∅ 2

• Ob

PREM ∅ 3

• Oc

PREM ∅ 4

• Od

PREM ∅ 5 (a ⊃ (¬b ∧ ¬c ∧ ¬d)) PREM ∅ 6 !a ∨ !b 1,2,5; RU ∅ 7 !a ∨ !c 1,3,5; RU ∅ 8 !a ∨ !d 1,4,5; RU ∅ 9 Oa 1; RC {!a} 10 Ob 2; RC {!b} 11 Oc 3; RC {!c} 12 Od 4; RC {!d}

◮ marking like for Minimal Abnormality: just now consider the

quantitatively minimal choice sets

◮ minimal choice sets (w.r.t. ⊂): {!a} and {!b, !c, !d} ◮ minimal choice sets (w.r.t. cardinality): {!a} 12/29

Duality between Maximal Consistent Subsets and the Minimal Choice Sets

• Oa, •Ob, •Oc, •Od, (a ⊃ (¬b ∧ ¬c ∧ ¬d))

13/29

Duality between Maximal Consistent Subsets and the Minimal Choice Sets

• Oa, •Ob, •Oc, •Od, (a ⊃ (¬b ∧ ¬c ∧ ¬d))

Maximal consistent subsets:

• 1. {a}
• 2. {b, c, d}

13/29

Duality between Maximal Consistent Subsets and the Minimal Choice Sets

• Oa, •Ob, •Oc, •Od, (a ⊃ (¬b ∧ ¬c ∧ ¬d))

Maximal consistent subsets:

• 1. {a}
• 2. {b, c, d}

Maximal choice sets:

• 1. {!b, !c, !d}
• 2. {!a}

13/29

Duality between Maximal Consistent Subsets and the Minimal Choice Sets

• Oa, •Ob, •Oc, •Od, (a ⊃ (¬b ∧ ¬c ∧ ¬d))

Maximal consistent subsets:

• 1. {a}
• 2. {b, c, d}

Maximal choice sets:

• 1. {!b, !c, !d}
• 2. {!a}

Where O and C are sets of propositional formulas:

◮ Let ΓO,C = {•OA | A ∈ O} ∪ {A | A ∈ C}. ◮ We say that O′ ⊆ O is consistent w.r.t. C iff O′ ∪ C is

consistent.

◮ O′ is ≺-maximally consistent w.r.t. C iff it is consistent w.r.t.

C and there is no O′′ ≺ O′ that is consistent w.r.t. C.

13/29

Duality between Maximal Consistent Subsets and the Minimal Choice Sets

• Oa, •Ob, •Oc, •Od, (a ⊃ (¬b ∧ ¬c ∧ ¬d))

Maximal consistent subsets:

• 1. {a}
• 2. {b, c, d}

Maximal choice sets:

• 1. {!b, !c, !d}
• 2. {!a}

Note the following duality:

◮ for each maximal consistent subset O′ w.r.t. C there is a

maximal choice set ϕ of ΓO,C such that O′ = O \ {A |!A ∈ ϕ}

13/29

Duality between Maximal Consistent Subsets and the Minimal Choice Sets

• Oa, •Ob, •Oc, •Od, (a ⊃ (¬b ∧ ¬c ∧ ¬d))

Maximal consistent subsets:

• 1. {a}
• 2. {b, c, d}

Maximal choice sets:

• 1. {!b, !c, !d}
• 2. {!a}

Note the following duality:

◮ for each maximal consistent subset O′ w.r.t. C there is a

maximal choice set ϕ of ΓO,C such that O′ = O \ {A |!A ∈ ϕ}

◮ and vice versa

13/29

Theorem

ΓO,C ⊢ALm OA iff A is implied by all ⊂-maximally consistent subsets of O.

14/29

Theorem

ΓO,C ⊢ALm OA iff A is implied by all ⊂-maximally consistent subsets of O.

Theorem

ΓO,C ⊢ALc OA iff A is implied by all ≺card-maximally consistent subsets of O w.r.t. C.

14/29

Theorem

ΓO,C ⊢ALm OA iff A is implied by all ⊂-maximally consistent subsets of O.

Theorem

ΓO,C ⊢ALc OA iff A is implied by all ≺card-maximally consistent subsets of O w.r.t. C.

Definition

A ∈ O is free in O w.r.t. C iff A ∈ O′ for all ⊂-maximally consistent subsets O′ of O w.r.t. C.

14/29

Theorem

ΓO,C ⊢ALm OA iff A is implied by all ⊂-maximally consistent subsets of O.

Theorem

ΓO,C ⊢ALc OA iff A is implied by all ≺card-maximally consistent subsets of O w.r.t. C.

Definition

A ∈ O is free in O w.r.t. C iff A ∈ O′ for all ⊂-maximally consistent subsets O′ of O w.r.t. C E.g., b is free in O = {a, ¬a, b} w.r.t. C = ∅. .

14/29

Theorem

ΓO,C ⊢ALm OA iff A is implied by all ⊂-maximally consistent subsets of O.

Theorem

ΓO,C ⊢ALc OA iff A is implied by all ≺card-maximally consistent subsets of O w.r.t. C.

Definition

A ∈ O is free in O w.r.t. C iff A ∈ O′ for all ⊂-maximally consistent subsets O′ of O w.r.t. C.

Theorem

ΓO,C ⊢ALr OA iff A is implied by the set of free members of O w.r.t. C.

14/29

Taking into account implicit obligations

15/29

Taking into account implicit obligations

E.g., •O(a ∧ b), •O¬a AL Ob.

◮ new operator: ◦ characterized by

◮ If A ⊢CL B then ⊢ •OA ⊃ ◦OB. 15/29

Taking into account implicit obligations

E.g., •O(a ∧ b), •O¬a AL Ob.

◮ new operator: ◦ characterized by

◮ If A ⊢CL B then ⊢ •OA ⊃ ◦OB.

1

• O(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅

15/29

Taking into account implicit obligations

E.g., •O(a ∧ b), •O¬a AL Ob.

◮ new operator: ◦ characterized by

◮ If A ⊢CL B then ⊢ •OA ⊃ ◦OB.

1

• O(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅ 3 O(a ∧ b) 1; RC {!a}

15/29

Taking into account implicit obligations

E.g., •O(a ∧ b), •O¬a AL Ob.

◮ new operator: ◦ characterized by

◮ If A ⊢CL B then ⊢ •OA ⊃ ◦OB.

1

• O(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅ 3 O(a ∧ b) 1; RC {!a} 4 Ob 3; RU {!a}

15/29

Taking into account implicit obligations

E.g., •O(a ∧ b), •O¬a AL Ob.

◮ new operator: ◦ characterized by

◮ If A ⊢CL B then ⊢ •OA ⊃ ◦OB.

1

• O(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅ 3 O(a ∧ b) 1; RC {!a} 4 Ob 3; RU {!a} 5 O¬a 2; RC {!(¬a)}

15/29

Taking into account implicit obligations

E.g., •O(a ∧ b), •O¬a AL Ob.

◮ new operator: ◦ characterized by

◮ If A ⊢CL B then ⊢ •OA ⊃ ◦OB.

1

• O(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅ 3 O(a ∧ b) 1; RC {!a} 4 Ob 3; RU {!a} 5 O¬a 2; RC {!(¬a)} 6

• Ob

1; RU ∅

15/29

Taking into account implicit obligations

E.g., •O(a ∧ b), •O¬a AL Ob.

◮ new operator: ◦ characterized by

◮ If A ⊢CL B then ⊢ •OA ⊃ ◦OB.

1

• O(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅ 3 O(a ∧ b) 1; RC {!a} 4 Ob 3; RU {!a} 5 O¬a 2; RC {!(¬a)} 6

• Ob

1; RU ∅ 7 Ob 6; RC {†b} †b = ◦Ob ∧ ¬Ob?

15/29

Taking into account implicit obligations

E.g., •O(a ∧ b), •O¬a AL Ob.

◮ new operator: ◦ characterized by

◮ If A ⊢CL B then ⊢ •OA ⊃ ◦OB.

1

• O(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅ 3 O(a ∧ b) 1; RC {!a} 4 Ob 3; RU {!a} 5 O¬a 2; RC {!(¬a)} 6

• Ob

1; RU ∅ 7 Ob 6; RC {†b} 8 O(a ∨ ¬b) 1; RC {†(a ∨ ¬b)} 9 O(¬a ∨ ¬b) 2; RC {†(¬a ∨ ¬b)} 10 O¬b 8,9; RU {†(a ∨ ¬b), †(¬a ∨ ¬b)

15/29

Taking into account implicit obligations

1

• O(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅ 3 O(a ∧ b) 1; RC {!a} 4 Ob 3; RU {!a} 5 O¬a 2; RC {!(¬a)} 6

• Ob

1; RU ∅ 7 Ob 6; RC {†b} 8 O(a ∨ ¬b) 1; RC {†(a ∨ ¬b)} 9 O(¬a ∨ ¬b) 2; RC {†(¬a ∨ ¬b)} 10 O¬b 8,9; RU {†(a ∨ ¬b), †(¬a ∨ ¬b) 11 †b ∨ †(a ∨ ¬b) ∨ †(¬a ∨ ¬b) 1,2; RU ∅ 12 !(a ∧ b) ∨ !¬a 1,2; RU ∅ 13 †a ∨ †¬a 1,2; RU ∅ 14 †(a ∨ ¬b) ∨ †(¬a ∨ ¬b) 13; RU ∅

◮ †(

I Ai) =

• O

I Ai ∧ ¬O I Ai

• ∨

∅=J⊂I

• O

J Aj ∧ ¬O J Aj

• ◮ e.g.,

†(a∨¬b) = (◦O(a∨¬b)∧¬O(a∨¬b))∨(◦Oa∧¬Oa)∨(◦O¬b∧¬O¬b)

15/29

Permissions

◮ add new connective P

16/29

Permissions

◮ add new connective P ◮ besides the former abnormalities add {•PA ∧ ¬PA}

16/29

Permissions

◮ add new connective P ◮ besides the former abnormalities add {•PA ∧ ¬PA} ◮ or the more complicated

• P

I Ai ∧ ¬P I Ai

• ∨

∅=J⊂I

• P

J Aj ∧ ¬P J Aj

• in

combination with If A ⊢CL B, then • PA ⊢ ◦PA.

16/29

Permissions

◮ add new connective P ◮ besides the former abnormalities add {•PA ∧ ¬PA} ◮ or the more complicated

• P

I Ai ∧ ¬P I Ai

• ∨

∅=J⊂I

• P

J Aj ∧ ¬P J Aj

• in

combination with If A ⊢CL B, then • PA ⊢ ◦PA.

16/29

Permissions

◮ add new connective P ◮ besides the former abnormalities add {•PA ∧ ¬PA} ◮ or the more complicated

• P

I Ai ∧ ¬P I Ai

• ∨

∅=J⊂I

• P

J Aj ∧ ¬P J Aj

• in

combination with If A ⊢CL B, then • PA ⊢ ◦PA. 1

• P(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅

16/29

Permissions

◮ add new connective P ◮ besides the former abnormalities add {•PA ∧ ¬PA} ◮ or the more complicated

• P

I Ai ∧ ¬P I Ai

• ∨

∅=J⊂I

• P

J Aj ∧ ¬P J Aj

• in

combination with If A ⊢CL B, then • PA ⊢ ◦PA. 1

• P(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅ 3

• Pa

1; RU ∅ 4

• Pb

1; RU ∅ 5

• O¬a

2; RU ∅

16/29

Permissions

◮ add new connective P ◮ besides the former abnormalities add {•PA ∧ ¬PA} ◮ or the more complicated

• P

I Ai ∧ ¬P I Ai

• ∨

∅=J⊂I

• P

J Aj ∧ ¬P J Aj

• in

combination with If A ⊢CL B, then • PA ⊢ ◦PA. 1

• P(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅ 3

• Pa

1; RU ∅ 4

• Pb

1; RU ∅ 5

• O¬a

2; RU ∅ 6 (◦O¬a ∧ ¬O¬a) ∨ (◦Pa ∧ ¬Pa) 3,5; RU ∅

16/29

Permissions

◮ add new connective P ◮ besides the former abnormalities add {•PA ∧ ¬PA} ◮ or the more complicated

• P

I Ai ∧ ¬P I Ai

• ∨

∅=J⊂I

• P

J Aj ∧ ¬P J Aj

• in

combination with If A ⊢CL B, then • PA ⊢ ◦PA. 1

• P(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅ 3

• Pa

1; RU ∅ 4

• Pb

1; RU ∅ 5

• O¬a

2; RU ∅ 6 (◦O¬a ∧ ¬O¬a) ∨ (◦Pa ∧ ¬Pa) 3,5; RU ∅ 7 !¬a∨(•P(a∧b)∧¬P(a∧b)) 1,2; RU ∅

16/29

Permissions

◮ add new connective P ◮ besides the former abnormalities add {•PA ∧ ¬PA} ◮ or the more complicated

• P

I Ai ∧ ¬P I Ai

• ∨

∅=J⊂I

• P

J Aj ∧ ¬P J Aj

• in

combination with If A ⊢CL B, then • PA ⊢ ◦PA. 1

• P(a ∧ b)

PREM ∅ 2

• O¬a

PREM ∅ 3

• Pa

1; RU ∅ 4

• Pb

1; RU ∅ 5

• O¬a

2; RU ∅ 6 (◦O¬a ∧ ¬O¬a) ∨ (◦Pa ∧ ¬Pa) 3,5; RU ∅ 7 !¬a∨(•P(a∧b)∧¬P(a∧b)) 1,2; RU ∅ 8 Pb 4; RC {◦Pb ∧ ¬Pb}

16/29

Excursus: Discussive Context and Rescher-Manor Consequence Relations

◮ Discussant 1 states: a

17/29

Excursus: Discussive Context and Rescher-Manor Consequence Relations

◮ Discussant 1 states: a ◮ Discussant 2 states: ¬a ∨ b

17/29

Excursus: Discussive Context and Rescher-Manor Consequence Relations

◮ Discussant 1 states: a ◮ Discussant 2 states: ¬a ∨ b ◮ Question: should we derive b?

17/29

Excursus: Discussive Context and Rescher-Manor Consequence Relations

◮ Discussant 1 states: a ◮ Discussant 2 states: ¬a ∨ b ◮ Question: should we derive b? ◮ Problem: Discussant 2 may not agree with/support a but

nevertheless not state ¬a (e.g., due to lack of knowledge)

17/29

Excursus: Discussive Context and Rescher-Manor Consequence Relations

◮ Discussant 1 states: a ◮ Discussant 2 states: ¬a ∨ b ◮ Question: should we derive b? ◮ Problem: Discussant 2 may not agree with/support a but

nevertheless not state ¬a (e.g., due to lack of knowledge)

◮ in our framework this can be expressed (under different

readings of O and P)

17/29

Excursus: Discussive Context and Rescher-Manor Consequence Relations

◮ Discussant 1 states: a ◮ Discussant 2 states: ¬a ∨ b ◮ Question: should we derive b? ◮ Problem: Discussant 2 may not agree with/support a but

nevertheless not state ¬a (e.g., due to lack of knowledge)

◮ in our framework this can be expressed (under different

readings of O and P)

◮ Discussant 1: •Oa (she explicitly supports a) 17/29

Excursus: Discussive Context and Rescher-Manor Consequence Relations

◮ Discussant 1 states: a ◮ Discussant 2 states: ¬a ∨ b ◮ Question: should we derive b? ◮ Problem: Discussant 2 may not agree with/support a but

nevertheless not state ¬a (e.g., due to lack of knowledge)

◮ in our framework this can be expressed (under different

readings of O and P)

◮ Discussant 1: •Oa (she explicitly supports a) ◮ Discussant 2: •O(¬a ∨ b) (he explicitly supports ¬a ∨ b) 17/29

Excursus: Discussive Context and Rescher-Manor Consequence Relations

◮ Discussant 1 states: a ◮ Discussant 2 states: ¬a ∨ b ◮ Question: should we derive b? ◮ Problem: Discussant 2 may not agree with/support a but

nevertheless not state ¬a (e.g., due to lack of knowledge)

◮ in our framework this can be expressed (under different

readings of O and P)

◮ Discussant 1: •Oa (she explicitly supports a) ◮ Discussant 2: •O(¬a ∨ b) (he explicitly supports ¬a ∨ b) ◮ Discussant 2:

• P¬a

note: this is different from •O¬a! (he explicitly states lack of support for ¬a)

17/29

Excursus: Discussive Context and Rescher-Manor Consequence Relations

◮ Discussant 1 states: a ◮ Discussant 2 states: ¬a ∨ b ◮ Question: should we derive b? ◮ Problem: Discussant 2 may not agree with/support a but

nevertheless not state ¬a (e.g., due to lack of knowledge)

◮ in our framework this can be expressed (under different

readings of O and P)

◮ Discussant 1: •Oa (she explicitly supports a) ◮ Discussant 2: •O(¬a ∨ b) (he explicitly supports ¬a ∨ b) ◮ Discussant 2:

• P¬a (he explicitly states lack of support for

¬a)

◮ we get: O(¬a ∨ b) (e.g., “¬a ∨ b is supportable for all

discussants”)

17/29

Excursus: Discussive Context and Rescher-Manor Consequence Relations

◮ Discussant 1 states: a ◮ Discussant 2 states: ¬a ∨ b ◮ Question: should we derive b? ◮ Problem: Discussant 2 may not agree with/support a but

nevertheless not state ¬a (e.g., due to lack of knowledge)

◮ in our framework this can be expressed (under different

readings of O and P)

◮ Discussant 1: •Oa (she explicitly supports a) ◮ Discussant 2: •O(¬a ∨ b) (he explicitly supports ¬a ∨ b) ◮ Discussant 2:

• P¬a (he explicitly states lack of support for

¬a)

◮ we get: O(¬a ∨ b) (e.g., “¬a ∨ b is supportable for all

discussants”)

◮ but not Ob. 17/29

Introducing Indexes

◮ instead of • [◦] we use •i [◦i] where i ∈ I for some index set I

18/29

Introducing Indexes

◮ instead of • [◦] we use •i [◦i] where i ∈ I for some index set I ◮ Various interpretations possible

18/29

Introducing Indexes

◮ instead of • [◦] we use •i [◦i] where i ∈ I for some index set I ◮ Various interpretations possible

◮ •iOa

“Authority i issues obligation a”

18/29

Introducing Indexes

◮ instead of • [◦] we use •i [◦i] where i ∈ I for some index set I ◮ Various interpretations possible

◮ •iOa

“Authority i issues obligation a”

◮ •iOa

“An authority of importance i issues an obligation”

18/29

Introducing Indexes

◮ instead of • [◦] we use •i [◦i] where i ∈ I for some index set I ◮ Various interpretations possible

◮ •iOa

“Authority i issues obligation a”

◮ •iOa

“An authority of importance i issues an obligation”

◮ •iOa

“The obligation a was issued at time point i.”

18/29

i indicates a specific authority

◮ abnormalities: Ω = I Ωi where Ωi = {•iOA ∧ ¬A}

19/29

i indicates a specific authority

◮ abnormalities: Ω = I Ωi where Ωi = {•iOA ∧ ¬A}

similar for the strengthenings/variants

19/29

i indicates a specific authority

◮ abnormalities: Ω = I Ωi where Ωi = {•iOA ∧ ¬A} ◮ e.g.,

• 1Oa,
• 2Oa,
• 3Oa,
• 4O¬a

⊢ALc Oa

19/29

i indicating time

◮ use lexicographic/reverse-lexicographic ALs

1

• 1O(a ∧ c)

PREM ∅ 2

• 2O¬a

PREM ∅ 3

• 3Ob

PREM ∅

20/29

i indicating time

◮ use lexicographic/reverse-lexicographic ALs

1

• 1O(a ∧ c)

PREM ∅ 2

• 2O¬a

PREM ∅ 3

• 3Ob

PREM ∅ 4 !1(a ∧ c) ∨ !2¬a where !iA =df •iOA ∧ ¬OA 1,2; RU ∅

20/29

i indicating time

◮ use lexicographic/reverse-lexicographic ALs

1

• 1O(a ∧ c)

PREM ∅ 2

• 2O¬a

PREM ∅ 3

• 3Ob

PREM ∅ 4 !1(a ∧ c) ∨ !2¬a 1,2; RU ∅ 5

• 1Oa

1; RU ∅ 6

• 1Oc

1; RU ∅ 7

• 2O¬a

2; RU ∅

20/29

i indicating time

◮ use lexicographic/reverse-lexicographic ALs

1

• 1O(a ∧ c)

PREM ∅ 2

• 2O¬a

PREM ∅ 3

• 3Ob

PREM ∅ 4 !1(a ∧ c) ∨ !2¬a 1,2; RU ∅ 5

• 1Oa

1; RU ∅ 6

• 1Oc

1; RU ∅ 7

• 2O¬a

2; RU ∅ 8 †1a ∨ †2¬a where †iA =df ◦iOA ∧ ¬OA 5,7; RU ∅

20/29

i indicating time

◮ use lexicographic/reverse-lexicographic ALs

1

• 1O(a ∧ c)

PREM ∅ 2

• 2O¬a

PREM ∅ 3

• 3Ob

PREM ∅ 4 !1(a ∧ c) ∨ !2¬a 1,2; RU ∅ 5

• 1Oa

1; RU ∅ 6

• 1Oc

1; RU ∅ 7

• 2O¬a

2; RU ∅ 8 †1a ∨ †2¬a 5,7; RU ∅ 9 O¬a 2; RC {!2¬a} 10 O(a ∧ c) 1; RC {!1(a ∧ c)} 11 Oa 5; RC {†1a} 12 Oc 6; RC {†1c} 13 Ob 3; RC {!3b} Concerning the reverse-lexicographic order on Ω2, {!1(a ∧ c), †1a} is the minimal choice set at this stage.

20/29

i indicating the degree of authority

i5 i4 i2 i3 i1

21/29

i indicating the degree of authority

i5 i4 i2 i3 i1

◮ Suppose we have: •i1Oa, •i4O¬a, •i2Ob, •i3Ob′,

• i5Oc, ¬(b ∧ b′).

21/29

i indicating the degree of authority

i5 i4 i2 i3 i1 Oc O¬a Ob Ob′ Oa

◮ Suppose we have: •i1Oa, •i4O¬a, •i2Ob, •i3Ob′,

• i5Oc, ¬(b ∧ b′).

21/29

i indicating the degree of authority

i5 i4 i2 i3 i1 Oc O¬a Ob Ob′ Oa

◮ Suppose we have: •i1Oa, •i4O¬a, •i2Ob, •i3Ob′,

• i5Oc, ¬(b ∧ b′).

◮ minimal disjunctions of abnormalities:

◮ !i1a ∨ !i4¬a ◮ !i2b ∨ !i3b′ 21/29

i indicating the degree of authority

i5 i4 i2 i3 i1 Oc O¬a Ob Ob′ Oa

◮ Suppose we have: •i1Oa, •i4O¬a, •i2Ob, •i3Ob′,

• i5Oc, ¬(b ∧ b′).

◮ minimal disjunctions of abnormalities:

◮ !i1a ∨ !i4¬a ◮ !i2b ∨ !i3b′

◮ choice sets: partial order on I imposes partial order on Ω2

{!i1a, !i2b} {!i1a, !i3b′} {!i4¬a, !i2b} {!i4¬a, !i3b′}

21/29

i indicating the degree of authority

i5 i4 i2 i3 i1 Oc O¬a Ob Ob′ Oa

◮ Suppose we have: •i1Oa, •i4O¬a, •i2Ob, •i3Ob′,

• i5Oc, ¬(b ∧ b′).

◮ minimal disjunctions of abnormalities:

◮ !i1a ∨ !i4¬a ◮ !i2b ∨ !i3b′

◮ choice sets: partial order on I imposes partial order on Ω2

{!i1a, !i2b} {!i1a, !i3b′} {!i4¬a, !i2b} {!i4¬a, !i3b′}

◮ we get: Oa, O(b ∨ b′), Oc

21/29

More complex combinations

◮ i•jOA ◮ i indicates time ◮ j indicates the degree of authority

22/29

The dyadic case: Illustration of the main idea

Factual Input A1 A2 A3 . . . Conditional Input

• O(A1, D1)
• O(A2, D2)
• O(A3, D3)

. . . Constraints C1 C2 C3 . . .

23/29

The dyadic case: Illustration of the main idea

Factual Input A1 A2 A3 . . . Conditional Input

• O(A1, D1)
• O(A2, D2)
• O(A3, D3)

. . . Constraints C1 C2 C3 . . .

23/29

The dyadic case: Illustration of the main idea

Factual Input A1 A2 A3 . . . Conditional Input

• O(A1, D1)
• O(A2, D2)
• O(A3, D3)

. . . Constraints C1 C2 C3 . . .

23/29

The dyadic case: Illustration of the main idea

Factual Input A1 A2 A3 . . . Conditional Input

• O(A1, D1)
• O(A2, D2)
• O(A3, D3)

. . . Constraints C1 C2 C3 . . . Actualized Conditionals O(A2, D2) . . . actualize “as much as possible”

23/29

The dyadic case: Illustration of the main idea

Factual Input A1 A2 A3 . . . Conditional Input

• O(A1, D1)
• O(A2, D2)
• O(A3, D3)

. . . Constraints C1 C2 C3 . . . Actualized Conditionals O(A2, D2) . . . Output OD2 . . . detach actualize “as much as possible”

• n the

23/29

The dyadic case: Illustration of the main idea

Factual Input A1 A2 A3 . . . Conditional Input

• O(A1, D1)
• O(A2, D2)
• O(A3, D3)

. . . Constraints C1 C2 C3 . . . Actualized Conditionals O(A2, D2) . . . Output OD2 . . . detach actualize “as much as possible”

• n the

23/29

Lower Limit Logic

◮ mostly as for the monadic case

24/29

Lower Limit Logic

◮ mostly as for the monadic case ◮ detachment principle:

⊢ (A ∧ O(A, B)) ⊃ OB

24/29

Lower Limit Logic

◮ mostly as for the monadic case ◮ detachment principle:

⊢ (A ∧ O(A, B)) ⊃ OB

◮ some axioms characterizing O(A, B) such as:

⊢ O(A, B) ∧ O(A′, B) ⊃ O(A ∨ A′, B)

24/29

Simple example for reliability

Γ = {i1 ∧ i2, i3, •O(i1, a ∧ b), •O(i2, ¬a ∧ b), •O(i3, c), •O(i1, ¬d), d}. 1 i1 ∧ i2 PREM ∅ 2 i1 1; RU ∅ 3 i3 PREM ∅ 4

• O(i1, a ∧ b)

PREM ∅ 5

• O(i2, ¬a ∧ b)

PREM ∅

25/29

Simple example for reliability

Γ = {i1 ∧ i2, i3, •O(i1, a ∧ b), •O(i2, ¬a ∧ b), •O(i3, c), •O(i1, ¬d), d}. 1 i1 ∧ i2 PREM ∅ 2 i1 1; RU ∅ 3 i3 PREM ∅ 4

• O(i1, a ∧ b)

PREM ∅ 5

• O(i2, ¬a ∧ b)

PREM ∅ 6 !(i1, a ∧ b) ∨ !(i2, ¬a ∧ b) !(A, B) =df •O(A, B) ∧ ¬O(A, B) 1,4,5; RU ∅

25/29

Simple example for reliability

Γ = {i1 ∧ i2, i3, •O(i1, a ∧ b), •O(i2, ¬a ∧ b), •O(i3, c), •O(i1, ¬d), d}. 1 i1 ∧ i2 PREM ∅ 2 i1 1; RU ∅ 3 i3 PREM ∅ 4

• O(i1, a ∧ b)

PREM ∅ 5

• O(i2, ¬a ∧ b)

PREM ∅ 6 !(i1, a ∧ b) ∨ !(i2, ¬a ∧ b) 1,4,5; RU ∅

67

O(i1, a ∧ b) 4;RC {!(i1, a ∧ b)}

68

O(a ∧ b) 2,4; RC {!(i1, a ∧ b)}

25/29

Simple example for reliability

Γ = {i1 ∧ i2, i3, •O(i1, a ∧ b), •O(i2, ¬a ∧ b), •O(i3, c), •O(i1, ¬d), d}. 1 i1 ∧ i2 PREM ∅ 2 i1 1; RU ∅ 3 i3 PREM ∅ 4

• O(i1, a ∧ b)

PREM ∅ 5

• O(i2, ¬a ∧ b)

PREM ∅ 6 !(i1, a ∧ b) ∨ !(i2, ¬a ∧ b) 1,4,5; RU ∅

67

O(i1, a ∧ b) 4;RC {!(i1, a ∧ b)}

68

O(a ∧ b) 2,4; RC {!(i1, a ∧ b)} 9

• O(i3, c)

PREM ∅ 10 Oc 3,9; RC {!(i3, c)}

25/29

Simple example for reliability

Γ = {i1 ∧ i2, i3, •O(i1, a ∧ b), •O(i2, ¬a ∧ b), •O(i3, c), •O(i1, ¬d), d}. 1 i1 ∧ i2 PREM ∅ 2 i1 1; RU ∅ 3 i3 PREM ∅ 4

• O(i1, a ∧ b)

PREM ∅ 5

• O(i2, ¬a ∧ b)

PREM ∅ 6 !(i1, a ∧ b) ∨ !(i2, ¬a ∧ b) 1,4,5; RU ∅

67

O(i1, a ∧ b) 4;RC {!(i1, a ∧ b)}

68

O(a ∧ b) 2,4; RC {!(i1, a ∧ b)} 9

• O(i3, c)

PREM ∅ 10 Oc 3,9; RC {!(i3, c)} 11

• O(i1, ¬d)

PREM ∅ 12 O¬d 2,11; RU {!(i1, ¬d)}

25/29

Simple example for reliability

Γ = {i1 ∧ i2, i3, •O(i1, a ∧ b), •O(i2, ¬a ∧ b), •O(i3, c), •O(i1, ¬d), d}. 1 i1 ∧ i2 PREM ∅ 2 i1 1; RU ∅ 3 i3 PREM ∅ 4

• O(i1, a ∧ b)

PREM ∅ 5

• O(i2, ¬a ∧ b)

PREM ∅ 6 !(i1, a ∧ b) ∨ !(i2, ¬a ∧ b) 1,4,5; RU ∅

67

O(i1, a ∧ b) 4;RC {!(i1, a ∧ b)}

68

O(a ∧ b) 2,4; RC {!(i1, a ∧ b)} 9

• O(i3, c)

PREM ∅ 10 Oc 3,9; RC {!(i3, c)}

1511

• O(i1, ¬d)

PREM ∅ 12 O¬d 2,11; RU {!(i1, ¬d)} 13 d PREM ∅ 14 ¬O¬d 13; RU ∅ 15 !(i1, ¬d) 1,11,14; RU ∅

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Excursus: Input/Output logics

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Naive approach to produce output

• ut(G, A) = Cn{B | for some A ∈ CnCL(A), (A, B) ∈ G}

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Naive approach to produce output

• ut(G, A) = Cn{B | for some A ∈ CnCL(A), (A, B) ∈ G}

         a, c ∧ e, f ⊃ q . . .          CnCL a → b c → d e → f . . . b d f . . . CnCL facts classical closure trigger + detach

• utput

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Naive approach to produce output

• ut(G, A) = Cn{B | for some A ∈ CnCL(A), (A, B) ∈ G}

         a, c ∧ e, f ⊃ q . . .          CnCL a → b c → d e → f . . . b d f . . . CnCL facts classical closure trigger + detach

• utput

Problem: conflicting output (i.e., conflicting obligations or conflicting with constraints)

Solution: work with consistent chunks 27/29

◮ we have representational theorems for all the 8 standard

I/O-systems with constraints

◮ hence, we provide a proof theory for I/O-logics

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Summary

Obligations Facts Constraints   ⇒ What are the “actual” obligations?

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Summary

Obligations Facts Constraints   ⇒ What are the “actual” obligations?

• OA

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Summary

Obligations Facts Constraints   ⇒ What are the “actual” obligations?

• OA

adaptively actualize “as much as possible” OA

translate

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Summary

Obligations Facts Constraints   ⇒ What are the “actual” obligations?

• OA

adaptively actualize “as much as possible” OA

translate

◮ adaptive strategies disambiguate this

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Summary

Obligations Facts Constraints   ⇒ What are the “actual” obligations?

• OA

adaptively actualize “as much as possible” OA

translate

◮ adaptive strategies disambiguate this ◮ more or less cautious variants

29/29

Summary

Obligations Facts Constraints   ⇒ What are the “actual” obligations?

• OA

adaptively actualize “as much as possible” OA

translate

◮ adaptive strategies disambiguate this ◮ more or less cautious variants ◮ monadic or dyadic variant

29/29

Summary

Obligations Facts Constraints   ⇒ What are the “actual” obligations?

• OA

adaptively actualize “as much as possible” OA

translate

◮ adaptive strategies disambiguate this ◮ more or less cautious variants ◮ monadic or dyadic variant ◮ various strengthenings and enhancements (time, degree of

authority, etc.)

29/29

Summary

Obligations Facts Constraints   ⇒ What are the “actual” obligations?

• OA

adaptively actualize “as much as possible” OA

translate

◮ adaptive strategies disambiguate this ◮ more or less cautious variants ◮ monadic or dyadic variant ◮ various strengthenings and enhancements (time, degree of

authority, etc.)

◮ representational theorems for approaches ala Rescher/Manor,

Horty, and I/O-logic with constraints

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