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

Adaptive Proofs A line: formula condition l A ∧ B l’,. . . ,l”; R ∆ line- justification number Conditional rule: ∆ 1 A 1 . . . . . . If A 1 , . . . , A n ⊢ LLL B ∨ Dab (∆): A n ∆ n B ∆ 1 ∪ . . . ∪ ∆ n ∪ ∆ 9/29

Adaptive Proofs A line: formula condition l A ∧ B l’,. . . ,l”; R ∆ line- justification number Conditional rule: ∆ 1 A 1 . . . . . . If A 1 , . . . , A n ⊢ LLL B ∨ Dab (∆): A n ∆ n B ∆ 1 ∪ . . . ∪ ∆ n ∪ ∆ collect abnormalities 9/29

Adaptive Proofs A line: formula condition A ∧ B l l’,. . . ,l”; R ∆ line- justification number Conditional rule: A 1 ∆ 1 . . . . . . If A 1 , . . . , A n ⊢ LLL B ∨ Dab (∆) : A n ∆ n ∆ 1 ∪ . . . ∪ ∆ n ∪ ∆ B add new condition collect abnormalities 9/29

1 • O a PREM ∅ • O a ′ 2 PREM ∅ 3 • O c PREM ∅ � ¬ ( a ∧ a ′ ) 4 PREM ∅ premise introduction 10/29

1 • O a PREM ∅ • O a ′ 2 PREM ∅ 3 • O c PREM ∅ � ¬ ( a ∧ a ′ ) 4 PREM ∅ {• O a ∧ ¬ O a } 5 O a 1; RC Note: • O a ⊢ LLL O a ∨ ( • O a ∧ ¬ O a ) 10/29

1 • O a PREM ∅ • O a ′ 2 PREM ∅ 3 • O c PREM ∅ � ¬ ( a ∧ a ′ ) 4 PREM ∅ {• O a ∧ ¬ O a } 5 O a 1; RC O( a ∨ a ′ ) 6 5; RU {• O a ∧ ¬ O a } unconditional rule Recall: O is KD -modality A 1 ∆ 1 . . . . . . If A 1 , . . . , A n ⊢ LLL B then ∆ n A n ∆ 1 ∪ . . . ∪ ∆ n B 10/29

1 • O a PREM ∅ • O a ′ 2 PREM ∅ 3 • O c PREM ∅ � ¬ ( a ∧ a ′ ) 4 PREM ∅ {• O a ∧ ¬ O a } 5 O a 1; RC O( a ∨ a ′ ) 6 5; RU {• O a ∧ ¬ O a } {• O a ′ ∧ ¬ O a ′ } O a ′ 7 2; RC {• O a ′ ∧ ¬ O a ′ } O( a ∨ a ′ ) 8 7; RU analogous to lines 5 and 6 10/29

1 • O a PREM ∅ • O a ′ 2 PREM ∅ 3 • O c PREM ∅ � ¬ ( a ∧ a ′ ) 4 PREM ∅ {• O a ∧ ¬ O a } 5 O a 1; RC O( a ∨ a ′ ) 6 5; RU {• O a ∧ ¬ O a } {• O a ′ ∧ ¬ O a ′ } O a ′ 7 2; RC {• O a ′ ∧ ¬ O a ′ } O( a ∨ a ′ ) 8 7; RU ! a ∨ ! a ′ 9 1,2,4; RU ∅ We shortcut: this follows by O-aggregation and OIC ! A = df • O A ∧ ¬ O A 10/29

1 • O a PREM ∅ • O a ′ 2 PREM ∅ 3 • O c PREM ∅ � ¬ ( a ∧ a ′ ) 4 PREM ∅ {• O a ∧ ¬ O a } 5 O a 1; RC O( a ∨ a ′ ) 6 5; RU {• O a ∧ ¬ O a } {• O a ′ ∧ ¬ O a ′ } O a ′ 7 2; RC {• O a ′ ∧ ¬ O a ′ } O( a ∨ a ′ ) 8 7; RU ! a ∨ ! a ′ 9 1,2,4; RU ∅ ◮ One of our assumptions is false! We need a retraction mechanism. 10/29

1 • O a PREM ∅ • O a ′ 2 PREM ∅ 3 • O c PREM ∅ � ¬ ( a ∧ a ′ ) 4 PREM ∅ {• O a ∧ ¬ O a } 5 O a 1; RC O( a ∨ a ′ ) 6 5; RU {• O a ∧ ¬ O a } {• O a ′ ∧ ¬ O a ′ } O a ′ 7 2; RC {• O a ′ ∧ ¬ O a ′ } O( a ∨ a ′ ) 8 7; RU ! a ∨ ! a ′ 9 1,2,4; RU ∅ ◮ One of our assumptions is false! We need a retraction mechanism. ◮ marking of lines which are retracted 10/29

1 • O a PREM ∅ • O a ′ 2 PREM ∅ 3 • O c PREM ∅ � ¬ ( a ∧ a ′ ) 4 PREM ∅ {• O a ∧ ¬ O a } 5 O a 1; RC O( a ∨ a ′ ) 6 5; RU {• O a ∧ ¬ O a } {• O a ′ ∧ ¬ O a ′ } O a ′ 7 2; RC {• O a ′ ∧ ¬ O a ′ } O( a ∨ a ′ ) 8 7; RU ! a ∨ ! a ′ 9 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 • O a PREM ∅ • O a ′ 2 PREM ∅ 3 • O c PREM ∅ � ¬ ( a ∧ a ′ ) 4 PREM ∅ {• O a ∧ ¬ O a } 5 O a 1; RC O( a ∨ a ′ ) 6 5; RU {• O a ∧ ¬ O a } {• O a ′ ∧ ¬ O a ′ } O a ′ 7 2; RC {• O a ′ ∧ ¬ O a ′ } O( a ∨ a ′ ) 8 7; RU ! a ∨ ! a ′ 9 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 10/29

Reliability and Minimal Abnormality . . . . . . . . . . . . {• O a ∧ ¬ O a } 5 O a 1; RC O( a ∨ a ′ ) 6 5; RU {• O a ∧ ¬ O a } {• O a ′ ∧ ¬ O a ′ } O a ′ 7 2; RC {• O a ′ ∧ ¬ O a ′ } O( a ∨ a ′ ) 8 7; RU ! a ∨ ! a ′ ∅ 9 1,2,4; RU ◮ Reliability assumption contains a member of a minimal disjunction of abnormalities ⇒ retract 11/29

Reliability and Minimal Abnormality . . . . . . . . . . . . {• O a ∧ ¬ O a } � 5 O a 1; RC O( a ∨ a ′ ) � 6 5; RU {• O a ∧ ¬ O a } {• O a ′ ∧ ¬ O a ′ } O a ′ � 7 2; RC {• O a ′ ∧ ¬ O a ′ } O( a ∨ a ′ ) � 8 7; RU ! a ∨ ! a ′ ∅ 9 1,2,4; RU ◮ Reliability assumption contains a member of a minimal disjunction of abnormalities ⇒ retract 11/29

Reliability and Minimal Abnormality . . . . . . . . . . . . {• O a ∧ ¬ O a } 5 O a 1; RC O( a ∨ a ′ ) 6 5; RU {• O a ∧ ¬ O a } {• O a ′ ∧ ¬ O a ′ } O a ′ 7 2; RC {• O a ′ ∧ ¬ O a ′ } O( a ∨ a ′ ) 8 7; RU ! a ∨ ! a ′ ∅ 9 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 O a ′ (that explains the conflict) ◮ there may additionally be a high cost for realizing an erroneous norm (e.g., a big financial investment) 11/29

Reliability and Minimal Abnormality . . . . . . . . . . . . {• O a ∧ ¬ O a } � 5 O a 1; RC O( a ∨ a ′ ) 6 5; RU {• O a ∧ ¬ O a } {• O a ′ ∧ ¬ O a ′ } O a ′ � 7 2; RC { ! a } , { ! a ′ } � � {• O a ′ ∧ ¬ O a ′ } O( a ∨ a ′ ) 8 7; RU ! a ∨ ! a ′ ∅ 9 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 O a ′ (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 . . . . . . . . . . . . {• O a ∧ ¬ O a } � 5 O a 1; RC O( a ∨ a ′ ) 6 5; RU {• O a ∧ ¬ O a } {• O a ′ ∧ ¬ O a ′ } O a ′ � 7 2; RC {• O a ′ ∧ ¬ O a ′ } O( a ∨ a ′ ) 8 7; RU ! a ∨ ! a ′ ∅ 9 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 O a ′ (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 on a condition ∆ such that ϕ ∩ ∆ = ∅ 11/29

Counting Strategy 1 • O a PREM ∅ 2 • O b PREM ∅ 3 • O c PREM ∅ 4 • O d PREM ∅ 12/29

Counting Strategy 1 • O a PREM ∅ 2 • O b PREM ∅ 3 • O c PREM ∅ 4 • O d PREM ∅ 5 � ( a ⊃ ( ¬ b ∧ ¬ c ∧ ¬ d )) PREM ∅ 12/29

Counting Strategy 1 • O a PREM ∅ 2 • O b PREM ∅ 3 • O c PREM ∅ 4 • O d PREM ∅ 5 � ( a ⊃ ( ¬ b ∧ ¬ c ∧ ¬ d )) PREM ∅ ! a ∨ ! b ∅ 6 1,2,5; RU 7 ! a ∨ ! c 1,3,5; RU ∅ ! a ∨ ! d ∅ 8 1,4,5; RU 12/29

Counting Strategy 1 • O a PREM ∅ 2 • O b PREM ∅ 3 • O c PREM ∅ 4 • O d PREM ∅ 5 � ( a ⊃ ( ¬ b ∧ ¬ c ∧ ¬ d )) PREM ∅ ! a ∨ ! b ∅ 6 1,2,5; RU 7 ! a ∨ ! c 1,3,5; RU ∅ ! a ∨ ! d ∅ 8 1,4,5; RU � 9 O a 1; RC { ! a } 10 O b 2; RC { ! b } 11 O c 3; RC { ! c } 12 O d 4; RC { ! d } 12/29

Counting Strategy 1 • O a PREM ∅ 2 • O b PREM ∅ 3 • O c PREM ∅ 4 • O d PREM ∅ 5 � ( a ⊃ ( ¬ b ∧ ¬ c ∧ ¬ d )) PREM ∅ ! a ∨ ! b ∅ 6 1,2,5; RU 7 ! a ∨ ! c 1,3,5; RU ∅ ! a ∨ ! d ∅ 8 1,4,5; RU � 9 O a 1; RC { ! a } 10 O b 2; RC { ! b } 11 O c 3; RC { ! c } 12 O d 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 • O a , • O b , • O c , • O d , � ( a ⊃ ( ¬ b ∧ ¬ c ∧ ¬ d )) 13/29

Duality between Maximal Consistent Subsets and the Minimal Choice Sets • O a , • O b , • O c , • O d , � ( 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 • O a , • O b , • O c , • O d , � ( a ⊃ ( ¬ b ∧ ¬ c ∧ ¬ d )) Maximal consistent subsets: Maximal choice sets: 1. { a } 1. { ! b , ! c , ! d } 2. { b , c , d } 2. { ! a } 13/29

Duality between Maximal Consistent Subsets and the Minimal Choice Sets • O a , • O b , • O c , • O d , � ( a ⊃ ( ¬ b ∧ ¬ c ∧ ¬ d )) Maximal consistent subsets: Maximal choice sets: 1. { a } 1. { ! b , ! c , ! d } 2. { b , c , d } 2. { ! a } Where O and C are sets of propositional formulas: ◮ Let Γ O , C = {• O A | 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 • O a , • O b , • O c , • O d , � ( a ⊃ ( ¬ b ∧ ¬ c ∧ ¬ d )) Maximal consistent subsets: Maximal choice sets: 1. { a } 1. { ! b , ! c , ! d } 2. { 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 Γ O , C such that O ′ = O \ { A | ! A ∈ ϕ } 13/29

Duality between Maximal Consistent Subsets and the Minimal Choice Sets • O a , • O b , • O c , • O d , � ( a ⊃ ( ¬ b ∧ ¬ c ∧ ¬ d )) Maximal consistent subsets: Maximal choice sets: 1. { a } 1. { ! b , ! c , ! d } 2. { 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 Γ O , C such that O ′ = O \ { A | ! A ∈ ϕ } ◮ and vice versa 13/29

Theorem Γ O , C ⊢ AL m O A iff A is implied by all ⊂ -maximally consistent subsets of O . 14/29

Theorem Γ O , C ⊢ AL m O A iff A is implied by all ⊂ -maximally consistent subsets of O . Theorem Γ O , C ⊢ AL c O A iff A is implied by all ≺ card -maximally consistent subsets of O w.r.t. C . 14/29

Theorem Γ O , C ⊢ AL m O A iff A is implied by all ⊂ -maximally consistent subsets of O . Theorem Γ O , C ⊢ AL c O A 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 Γ O , C ⊢ AL m O A iff A is implied by all ⊂ -maximally consistent subsets of O . Theorem Γ O , C ⊢ AL c O A 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 Γ O , C ⊢ AL m O A iff A is implied by all ⊂ -maximally consistent subsets of O . Theorem Γ O , C ⊢ AL c O A 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 Γ O , C ⊢ AL r O A 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 O b . ◮ new operator: ◦ characterized by ◮ If A ⊢ CL B then ⊢ • O A ⊃ ◦ O B . 15/29

Taking into account implicit obligations E.g., • O( a ∧ b ) , • O ¬ a � AL O b . ◮ new operator: ◦ characterized by ◮ If A ⊢ CL B then ⊢ • O A ⊃ ◦ O B . • O( a ∧ b ) ∅ 1 PREM 2 • O ¬ a PREM ∅ 15/29

Taking into account implicit obligations E.g., • O( a ∧ b ) , • O ¬ a � AL O b . ◮ new operator: ◦ characterized by ◮ If A ⊢ CL B then ⊢ • O A ⊃ ◦ O B . • O( a ∧ b ) ∅ 1 PREM 2 • O ¬ a PREM ∅ O( a ∧ b ) { ! a } 3 1; RC 15/29

Taking into account implicit obligations E.g., • O( a ∧ b ) , • O ¬ a � AL O b . ◮ new operator: ◦ characterized by ◮ If A ⊢ CL B then ⊢ • O A ⊃ ◦ O B . • O( a ∧ b ) ∅ 1 PREM 2 • O ¬ a PREM ∅ O( a ∧ b ) { ! a } 3 1; RC 4 O b 3; RU { ! a } 15/29

Taking into account implicit obligations E.g., • O( a ∧ b ) , • O ¬ a � AL O b . ◮ new operator: ◦ characterized by ◮ If A ⊢ CL B then ⊢ • O A ⊃ ◦ O B . • O( a ∧ b ) ∅ 1 PREM 2 • O ¬ a PREM ∅ O( a ∧ b ) { ! a } 3 1; RC 4 O b 3; RU { ! a } 5 O ¬ a 2; RC { !( ¬ a ) } 15/29

Taking into account implicit obligations E.g., • O( a ∧ b ) , • O ¬ a � AL O b . ◮ new operator: ◦ characterized by ◮ If A ⊢ CL B then ⊢ • O A ⊃ ◦ O B . • O( a ∧ b ) ∅ 1 PREM 2 • O ¬ a PREM ∅ O( a ∧ b ) { ! a } 3 1; RC 4 O b 3; RU { ! a } 5 O ¬ a 2; RC { !( ¬ a ) } 6 ◦ O b 1; RU ∅ 15/29

Taking into account implicit obligations E.g., • O( a ∧ b ) , • O ¬ a � AL O b . ◮ new operator: ◦ characterized by ◮ If A ⊢ CL B then ⊢ • O A ⊃ ◦ O B . • O( a ∧ b ) ∅ 1 PREM 2 • O ¬ a PREM ∅ O( a ∧ b ) { ! a } 3 1; RC 4 O b 3; RU { ! a } 5 O ¬ a 2; RC { !( ¬ a ) } 6 ◦ O b 1; RU ∅ 7 O b 6; RC {† b } † b = ◦ O b ∧ ¬ O b ? 15/29

Taking into account implicit obligations E.g., • O( a ∧ b ) , • O ¬ a � AL O b . ◮ new operator: ◦ characterized by ◮ If A ⊢ CL B then ⊢ • O A ⊃ ◦ O B . • O( a ∧ b ) ∅ 1 PREM 2 • O ¬ a PREM ∅ O( a ∧ b ) { ! a } 3 1; RC 4 O b 3; RU { ! a } 5 O ¬ a 2; RC { !( ¬ a ) } 6 ◦ O b 1; RU ∅ 7 O b 6; RC {† b } 8 O( a ∨ ¬ b ) 1; RC {† ( a ∨ ¬ b ) } 9 O( ¬ a ∨ ¬ b ) 2; RC {† ( ¬ a ∨ ¬ b ) } O ¬ b {† ( a ∨ ¬ b ) , † ( ¬ a ∨ ¬ b ) 10 8,9; RU 15/29

Taking into account implicit obligations • O( a ∧ b ) ∅ 1 PREM 2 • O ¬ a PREM ∅ O( a ∧ b ) { ! a } 3 1; RC 4 O b 3; RU { ! a } 5 O ¬ a 2; RC { !( ¬ a ) } 6 ◦ O b 1; RU ∅ 7 O b 6; RC {† b } � 8 O( a ∨ ¬ b ) 1; RC {† ( a ∨ ¬ b ) } � 9 O( ¬ a ∨ ¬ b ) 2; RC {† ( ¬ a ∨ ¬ b ) } O ¬ b {† ( a ∨ ¬ b ) , † ( ¬ a ∨ ¬ b ) � 10 8,9; RU 11 † b ∨ † ( a ∨ ¬ b ) ∨ † ( ¬ a ∨ ¬ b ) 1,2; RU ∅ !( a ∧ b ) ∨ ! ¬ a ∅ 12 1,2; RU 13 † a ∨ †¬ a 1,2; RU ∅ 14 † ( a ∨ ¬ b ) ∨ † ( ¬ a ∨ ¬ b ) 13; RU ∅ ◮ † ( � � ◦ O � I A i ∧ ¬ O � � ∨ � � ◦ O � J A j ∧ ¬ O � � I A i ) = I A i J A j ∅� = J ⊂ I ◮ e.g., † ( a ∨¬ b ) = ( ◦ O( a ∨¬ b ) ∧¬ O( a ∨¬ b )) ∨ ( ◦ O a ∧¬ O a ) ∨ ( ◦ O ¬ b ∧¬ O ¬ b ) 15/29

Permissions ◮ add new connective P 16/29

Permissions ◮ add new connective P ◮ besides the former abnormalities add {• P A ∧ ¬ P A } 16/29

Permissions ◮ add new connective P ◮ besides the former abnormalities add {• P A ∧ ¬ P A } ◮ or the more complicated �� ◦ P � I A i ∧ ¬ P � � ∨ � � ◦ P � J A j ∧ ¬ P � �� I A i J A j in ∅� = J ⊂ I combination with If A ⊢ CL B , then • P A ⊢ ◦ P A . 16/29

Permissions ◮ add new connective P ◮ besides the former abnormalities add {• P A ∧ ¬ P A } ◮ or the more complicated �� ◦ P � I A i ∧ ¬ P � � ∨ � � ◦ P � J A j ∧ ¬ P � �� I A i J A j in ∅� = J ⊂ I combination with If A ⊢ CL B , then • P A ⊢ ◦ P A . 16/29

Permissions ◮ add new connective P ◮ besides the former abnormalities add {• P A ∧ ¬ P A } ◮ or the more complicated �� ◦ P � I A i ∧ ¬ P � � ∨ � � ◦ P � J A j ∧ ¬ P � �� I A i J A j in ∅� = J ⊂ I combination with If A ⊢ CL B , then • P A ⊢ ◦ P A . 1 • P( a ∧ b ) PREM ∅ 2 • O ¬ a PREM ∅ 16/29

Permissions ◮ add new connective P ◮ besides the former abnormalities add {• P A ∧ ¬ P A } ◮ or the more complicated �� ◦ P � I A i ∧ ¬ P � � ∨ � � ◦ P � J A j ∧ ¬ P � �� I A i J A j in ∅� = J ⊂ I combination with If A ⊢ CL B , then • P A ⊢ ◦ P A . 1 • P( a ∧ b ) PREM ∅ 2 • O ¬ a PREM ∅ 3 ◦ P a 1; RU ∅ 4 ◦ P b 1; RU ∅ ◦ O ¬ a ∅ 5 2; RU 16/29

Permissions ◮ add new connective P ◮ besides the former abnormalities add {• P A ∧ ¬ P A } ◮ or the more complicated �� ◦ P � I A i ∧ ¬ P � � ∨ � � ◦ P � J A j ∧ ¬ P � �� I A i J A j in ∅� = J ⊂ I combination with If A ⊢ CL B , then • P A ⊢ ◦ P A . 1 • P( a ∧ b ) PREM ∅ 2 • O ¬ a PREM ∅ 3 ◦ P a 1; RU ∅ 4 ◦ P b 1; RU ∅ ◦ O ¬ a ∅ 5 2; RU 6 ( ◦ O ¬ a ∧ ¬ O ¬ a ) ∨ ( ◦ P a ∧ 3,5; RU ∅ ¬ P a ) 16/29

Permissions ◮ add new connective P ◮ besides the former abnormalities add {• P A ∧ ¬ P A } ◮ or the more complicated �� ◦ P � I A i ∧ ¬ P � � ∨ � � ◦ P � J A j ∧ ¬ P � �� I A i J A j in ∅� = J ⊂ I combination with If A ⊢ CL B , then • P A ⊢ ◦ P A . 1 • P( a ∧ b ) PREM ∅ 2 • O ¬ a PREM ∅ 3 ◦ P a 1; RU ∅ 4 ◦ P b 1; RU ∅ ◦ O ¬ a ∅ 5 2; RU 6 ( ◦ O ¬ a ∧ ¬ O ¬ a ) ∨ ( ◦ P a ∧ 3,5; RU ∅ ¬ P a ) 7 ! ¬ a ∨ ( • P( a ∧ b ) ∧¬ P( a ∧ b )) 1,2; RU ∅ 16/29

Permissions ◮ add new connective P ◮ besides the former abnormalities add {• P A ∧ ¬ P A } ◮ or the more complicated �� ◦ P � I A i ∧ ¬ P � � ∨ � � ◦ P � J A j ∧ ¬ P � �� I A i J A j in ∅� = J ⊂ I combination with If A ⊢ CL B , then • P A ⊢ ◦ P A . 1 • P( a ∧ b ) PREM ∅ 2 • O ¬ a PREM ∅ 3 ◦ P a 1; RU ∅ 4 ◦ P b 1; RU ∅ ◦ O ¬ a ∅ 5 2; RU 6 ( ◦ O ¬ a ∧ ¬ O ¬ a ) ∨ ( ◦ P a ∧ 3,5; RU ∅ ¬ P a ) 7 ! ¬ a ∨ ( • P( a ∧ b ) ∧¬ P( a ∧ b )) 1,2; RU ∅ 8 P b 4; RC {◦ P b ∧ ¬ P b } 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