An Andersonian Deontic Logic with Contextualized Sanctions M. - PowerPoint PPT Presentation

DSL : grammar/ interpretation (2) S A : “ A causes a sanction” “ A causes liability to a sanction” “ A causes a violation (of a norm)” “(The validity of) A represents a reason for a sanction” “A sanction is in place due to (the validity of) A ” F A = df A ⇒ S A M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

DSL : grammar/ interpretation (2) S A : “ A causes a sanction” “ A causes liability to a sanction” “ A causes a violation (of a norm)” “(The validity of) A represents a reason for a sanction” “A sanction is in place due to (the validity of) A ” F A = df A ⇒ S A O A = df F ¬ A M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

DSL : grammar/ interpretation (2) S A : “ A causes a sanction” “ A causes liability to a sanction” “ A causes a violation (of a norm)” “(The validity of) A represents a reason for a sanction” “A sanction is in place due to (the validity of) A ” F A = df A ⇒ S A O A = df F ¬ A S A : “ A holds and causes a sanction” M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

DSL : grammar/ interpretation (2) S A : “ A causes a sanction” “ A causes liability to a sanction” “ A causes a violation (of a norm)” “(The validity of) A represents a reason for a sanction” “A sanction is in place due to (the validity of) A ” F A = df A ⇒ S A O A = df F ¬ A S A : “ A holds and causes a sanction” A ⇒ S A : “ If A were to hold, it would cause a sanction” M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

DSL : axiomatization (1) DSL is obtained by adding to S5 the following axiom schemata and rules: S A ⊃ A (SR) S A ⊃ ( A ⇒ S A ) (S ⇒ ) ( A ⇒ S ( A ∧ B )) ⊃ ( A ⇒ S A ) (SW) ( S A ∧ S B ) ⊃ S ( A ∧ B ) (S ∧ ) (( A ∨ B ) ⇒ S ( A ∨ B )) ≡ (( A ⇒ S A ) ∧ ( B ⇒ S B )) (S ∨ ) If ⊢ EL A ⊃ B then ⊢ S A ⊃ S B (S ⊃ ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 7 / 25

DSL : axiomatization (2) S A ⊃ A (SR) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

DSL : axiomatization (2) S A ⊃ A (SR) If A causes a sanction, then A . M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

DSL : axiomatization (2) S A ⊃ A (SR) If A causes a sanction, then A . S A ⊃ ( A ⇒ S A ) (S ⇒ ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

DSL : axiomatization (2) S A ⊃ A (SR) If A causes a sanction, then A . S A ⊃ ( A ⇒ S A ) (S ⇒ ) If A causes a sanction in some world, then A causes a sanction in every accessible world in which A holds. M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

DSL : axiomatization (2) S A ⊃ A (SR) If A causes a sanction, then A . S A ⊃ ( A ⇒ S A ) (S ⇒ ) If A causes a sanction in some world, then A causes a sanction in every accessible world in which A holds. S A ≡ ( F A ∧ A ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

DSL : axiomatization (2) S A ⊃ A (SR) If A causes a sanction, then A . S A ⊃ ( A ⇒ S A ) (S ⇒ ) If A causes a sanction in some world, then A causes a sanction in every accessible world in which A holds. S A ≡ ( F A ∧ A ) (( A ⇒ B ) ∧ (( A ∧ B ) ⇒ S ( A ∧ B ))) ⊃ ( A ⇒ S A ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

DSL : axiomatization (2) S A ⊃ A (SR) If A causes a sanction, then A . S A ⊃ ( A ⇒ S A ) (S ⇒ ) If A causes a sanction in some world, then A causes a sanction in every accessible world in which A holds. S A ≡ ( F A ∧ A ) (( A ⇒ B ) ∧ (( A ∧ B ) ⇒ S ( A ∧ B ))) ⊃ ( A ⇒ S A ) If: A necessitates B , and A ∧ B , if it holds, would cause a sanction, then A is itself sufficient to cause the sanction. M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

DSL : axiomatization (2) S A ⊃ A (SR) If A causes a sanction, then A . S A ⊃ ( A ⇒ S A ) (S ⇒ ) If A causes a sanction in some world, then A causes a sanction in every accessible world in which A holds. S A ≡ ( F A ∧ A ) (( A ⇒ B ) ∧ (( A ∧ B ) ⇒ S ( A ∧ B ))) ⊃ ( A ⇒ S A ) If: A necessitates B , and A ∧ B , if it holds, would cause a sanction, then A is itself sufficient to cause the sanction. ( A ⇒ S ( A ∧ B )) ⊃ ( A ⇒ S A ) (SW) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

DSL : axiomatization (3) ? S ( A ∧ B ) ⊃ S A M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

DSL : axiomatization (3) ? S ( A ∧ B ) ⊃ S A Example: Little Peter is told by his mom to either wash the dishes or bring out the garbage, otherwise he is not allowed to watch his favorite TV show this evening. M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

DSL : axiomatization (3) ? S ( A ∧ B ) ⊃ S A Example: Little Peter is told by his mom to either wash the dishes or bring out the garbage, otherwise he is not allowed to watch his favorite TV show this evening. S ( ¬ W ∧ ¬ G ) �⊃ S ¬ W M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

DSL : axiomatization (3) S ( A ∧ B ) �⊃ S A Example: Little Peter is told by his mom to either wash the dishes or bring out the garbage, otherwise he is not allowed to watch his favorite TV show this evening. S ( ¬ W ∧ ¬ G ) �⊃ S ¬ W M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

DSL : axiomatization (3) S ( A ∧ B ) �⊃ S A Example: Little Peter is told by his mom to either wash the dishes or bring out the garbage, otherwise he is not allowed to watch his favorite TV show this evening. S ( ¬ W ∧ ¬ G ) �⊃ S ¬ W ? S A ⊃ S ( A ∧ B ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

DSL : axiomatization (3) S ( A ∧ B ) �⊃ S A Example: Little Peter is told by his mom to either wash the dishes or bring out the garbage, otherwise he is not allowed to watch his favorite TV show this evening. S ( ¬ W ∧ ¬ G ) �⊃ S ¬ W ? S A ⊃ S ( A ∧ B ) If S ( A ∧ B ) , then the validity of A ∧ B is a necessary condition for it being the cause of a sanction. M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

DSL : axiomatization (3) S ( A ∧ B ) �⊃ S A Example: Little Peter is told by his mom to either wash the dishes or bring out the garbage, otherwise he is not allowed to watch his favorite TV show this evening. S ( ¬ W ∧ ¬ G ) �⊃ S ¬ W S A �⊃ S ( A ∧ B ) If S ( A ∧ B ) , then the validity of A ∧ B is a necessary condition for it being the cause of a sanction. M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

DSL : axiomatization (3) S ( A ∧ B ) �⊃ S A Example: Little Peter is told by his mom to either wash the dishes or bring out the garbage, otherwise he is not allowed to watch his favorite TV show this evening. S ( ¬ W ∧ ¬ G ) �⊃ S ¬ W S A �⊃ S ( A ∧ B ) If S ( A ∧ B ) , then the validity of A ∧ B is a necessary condition for it being the cause of a sanction. ( S A ∧ S B ) ⊃ S ( A ∧ B ) (S ∧ ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

DSL : axiomatization (3) S ( A ∧ B ) �⊃ S A Example: Little Peter is told by his mom to either wash the dishes or bring out the garbage, otherwise he is not allowed to watch his favorite TV show this evening. S ( ¬ W ∧ ¬ G ) �⊃ S ¬ W S A �⊃ S ( A ∧ B ) If S ( A ∧ B ) , then the validity of A ∧ B is a necessary condition for it being the cause of a sanction. ( S A ∧ S B ) ⊃ S ( A ∧ B ) (S ∧ ) If A causes a sanction and B causes a sanction, then A ∧ B causes a sanction. M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

DSL : axiomatization (4) (( A ∨ B ) ⇒ S ( A ∨ B )) ≡ (( A ⇒ S A ) ∧ ( B ⇒ S B )) (S ∨ ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

DSL : axiomatization (4) (( A ∨ B ) ⇒ S ( A ∨ B )) ≡ (( A ⇒ S A ) ∧ ( B ⇒ S B )) (S ∨ ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

DSL : axiomatization (4) (( A ∨ B ) ⇒ S ( A ∨ B )) ≡ (( A ⇒ S A ) ∧ ( B ⇒ S B )) (S ∨ ) If he were to press button a or button b , Homer would cause a meltdown. M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

DSL : axiomatization (4) (( A ∨ B ) ⇒ S ( A ∨ B )) ≡ (( A ⇒ S A ) ∧ ( B ⇒ S B )) (S ∨ ) If he were to press button a or button b , Homer would cause a meltdown. ( a ∨ b ) ⇒ S ( a ∨ b ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

DSL : axiomatization (4) (( A ∨ B ) ⇒ S ( A ∨ B )) ≡ (( A ⇒ S A ) ∧ ( B ⇒ S B )) (S ∨ ) F ( A ∨ B ) ≡ ( F A ∧ F B ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

DSL : axiomatization (4) (( A ∨ B ) ⇒ S ( A ∨ B )) ≡ (( A ⇒ S A ) ∧ ( B ⇒ S B )) (S ∨ ) F ( A ∨ B ) ≡ ( F A ∧ F B ) ? S a ⊃ S ( a ∨ b ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

DSL : axiomatization (4) (( A ∨ B ) ⇒ S ( A ∨ B )) ≡ (( A ⇒ S A ) ∧ ( B ⇒ S B )) (S ∨ ) F ( A ∨ B ) ≡ ( F A ∧ F B ) Pressing button a causes a meltdown. Pressing button b does not. ? S a ⊃ S ( a ∨ b ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

DSL : axiomatization (4) (( A ∨ B ) ⇒ S ( A ∨ B )) ≡ (( A ⇒ S A ) ∧ ( B ⇒ S B )) (S ∨ ) F ( A ∨ B ) ≡ ( F A ∧ F B ) Pressing button a causes a meltdown. Pressing button b does not. S a �⊃ S ( a ∨ b ) ? ( S a ∨ S b ) ⊃ S ( a ∨ b ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

DSL : axiomatization (4) (( A ∨ B ) ⇒ S ( A ∨ B )) ≡ (( A ⇒ S A ) ∧ ( B ⇒ S B )) (S ∨ ) F ( A ∨ B ) ≡ ( F A ∧ F B ) Pressing button a causes a meltdown. Pressing button b does not. S a �⊃ S ( a ∨ b ) ( S a ∨ S b ) �⊃ S ( a ∨ b ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

DSL : axiomatization (5) Suppose: If ⊢ CL A ⊃ B then ⊢ S A ⊃ S B M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 11 / 25

DSL : axiomatization (5) Suppose: If ⊢ CL A ⊃ B then ⊢ S A ⊃ S B Then, since S ( A ∨ B ) ⊃ ( S A ∨ S B ) : S A ⊃ ( S ( A ∧ B ) ∨ S ( A ∧ ¬ B )) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 11 / 25

DSL : axiomatization (5) Suppose: If ⊢ CL A ⊃ B then ⊢ S A ⊃ S B Then, since S ( A ∨ B ) ⊃ ( S A ∨ S B ) : S A ⊃ ( S ( A ∧ B ) ∨ S ( A ∧ ¬ B )) Solution: If ⊢ EL A ⊃ B then ⊢ S A ⊃ S B (S ⊃ ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 11 / 25

DSL : axiomatization (5) Suppose: If ⊢ CL A ⊃ B then ⊢ S A ⊃ S B Then, since S ( A ∨ B ) ⊃ ( S A ∨ S B ) : S A ⊃ ( S ( A ∧ B ) ∨ S ( A ∧ ¬ B )) Solution: If ⊢ EL A ⊃ B then ⊢ S A ⊃ S B (S ⊃ ) Let ( τ, σ ) ∈ { ( ∧ , ∨ ) , ( ∨ , ∧ ) } . EL is defined by: M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 11 / 25

DSL : axiomatization (5) Suppose: If ⊢ CL A ⊃ B then ⊢ S A ⊃ S B Then, since S ( A ∨ B ) ⊃ ( S A ∨ S B ) : S A ⊃ ( S ( A ∧ B ) ∨ S ( A ∧ ¬ B )) Solution: If ⊢ EL A ⊃ B then ⊢ S A ⊃ S B (S ⊃ ) Let ( τ, σ ) ∈ { ( ∧ , ∨ ) , ( ∨ , ∧ ) } . EL is defined by: (6) (1) ( A τ A ) ⊣⊢ A ¬ ( A τ B ) ⊣⊢ ( ¬ A σ ¬ B ) (7) (2) (( A τ ¬ A ) σ B ) ⊢ B ¬¬ A ⊣⊢ A (8) (3) (( A τ B ) σ A ) ⊢ A (( A τ B ) τ C ) ⊣⊢ ( A τ ( B τ C )) (9) (4) (( A τ ¬ A ) τ B ) ⊢ ( A τ ¬ A ) ( A τ B ) ⊣⊢ ( B τ A ) (5) If A ⊢ B then C ⊢ C A / B (10) ( A τ ( B σ C )) ⊣⊢ (( A τ B ) σ ( A τ C )) Where C A / B is the product of substituting any amount of subformulas A in C by B . M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 11 / 25

DSL : axiomatization (6) Some examples: M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 12 / 25

DSL : axiomatization (6) Some examples: A ∧ ¬ ( B ∨ C ) EL ⊣⊢ EL A ∧ ( ¬ B ∧ ¬ C ) EL ⊣⊢ EL ( A ∧ ¬ C ) ∧ ¬ B M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 12 / 25

DSL : axiomatization (6) Some examples: A ∧ ¬ ( B ∨ C ) EL ⊣⊢ EL A ∧ ( ¬ B ∧ ¬ C ) EL ⊣⊢ EL ( A ∧ ¬ C ) ∧ ¬ B A ∨ ¬ ( B ∧ C ) EL ⊣⊢ EL A ∨ ( ¬ B ∨ ¬ C ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 12 / 25

DSL : axiomatization (6) Some examples: A ∧ ¬ ( B ∨ C ) EL ⊣⊢ EL A ∧ ( ¬ B ∧ ¬ C ) EL ⊣⊢ EL ( A ∧ ¬ C ) ∧ ¬ B A ∨ ¬ ( B ∧ C ) EL ⊣⊢ EL A ∨ ( ¬ B ∨ ¬ C ) A ∧ A EL ⊣⊢ EL A EL ⊣⊢ EL A ∨ A EL ⊣⊢ EL A ∨ ( A ∧ A ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 12 / 25

DSL : axiomatization (6) Some examples: A ∧ ¬ ( B ∨ C ) EL ⊣⊢ EL A ∧ ( ¬ B ∧ ¬ C ) EL ⊣⊢ EL ( A ∧ ¬ C ) ∧ ¬ B A ∨ ¬ ( B ∧ C ) EL ⊣⊢ EL A ∨ ( ¬ B ∨ ¬ C ) A ∧ A EL ⊣⊢ EL A EL ⊣⊢ EL A ∨ A EL ⊣⊢ EL A ∨ ( A ∧ A ) ¬ ( A ∧ B ) EL ⊣⊢ EL ¬ A ∨ ¬ B M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 12 / 25

DSL : axiomatization (6) Some examples: A ∧ ¬ ( B ∨ C ) EL ⊣⊢ EL A ∧ ( ¬ B ∧ ¬ C ) EL ⊣⊢ EL ( A ∧ ¬ C ) ∧ ¬ B A ∨ ¬ ( B ∧ C ) EL ⊣⊢ EL A ∨ ( ¬ B ∨ ¬ C ) A ∧ A EL ⊣⊢ EL A EL ⊣⊢ EL A ∨ A EL ⊣⊢ EL A ∨ ( A ∧ A ) ¬ ( A ∧ B ) EL ⊣⊢ EL ¬ A ∨ ¬ B ( A ∨ ¬ A ) ∧ B ⊢ EL B M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 12 / 25

DSL : axiomatization (6) Some examples: A ∧ ¬ ( B ∨ C ) EL ⊣⊢ EL A ∧ ( ¬ B ∧ ¬ C ) EL ⊣⊢ EL ( A ∧ ¬ C ) ∧ ¬ B A ∨ ¬ ( B ∧ C ) EL ⊣⊢ EL A ∨ ( ¬ B ∨ ¬ C ) A ∧ A EL ⊣⊢ EL A EL ⊣⊢ EL A ∨ A EL ⊣⊢ EL A ∨ ( A ∧ A ) ¬ ( A ∧ B ) EL ⊣⊢ EL ¬ A ∨ ¬ B ( A ∨ ¬ A ) ∧ B ⊢ EL B B � EL ( A ∨ ¬ A ) ∧ B M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 12 / 25

DSL : axiomatization (6) Some examples: A ∧ ¬ ( B ∨ C ) EL ⊣⊢ EL A ∧ ( ¬ B ∧ ¬ C ) EL ⊣⊢ EL ( A ∧ ¬ C ) ∧ ¬ B A ∨ ¬ ( B ∧ C ) EL ⊣⊢ EL A ∨ ( ¬ B ∨ ¬ C ) A ∧ A EL ⊣⊢ EL A EL ⊣⊢ EL A ∨ A EL ⊣⊢ EL A ∨ ( A ∧ A ) ¬ ( A ∧ B ) EL ⊣⊢ EL ¬ A ∨ ¬ B ( A ∨ ¬ A ) ∧ B ⊢ EL B B � EL ( A ∨ ¬ A ) ∧ B EL is the fragment of CL that does not allow for the introduction of new propositional variables M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 12 / 25

Part II The logic DSL : further properties M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 13 / 25

Deontic properties of DSL The following properties fail in DSL : O A , O ¬ A ⊢ B O A , O ¬ A ⊢ O B If ⊢ CL A ⊃ B , then ⊢ DSL O A ⊃ O B If ⊢ CL A ≡ B , then ⊢ DSL O A ≡ O B O A ⊢ O ( A ∨ B ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 14 / 25

Deontic properties of DSL The following properties fail in DSL : O A , O ¬ A ⊢ B O A , O ¬ A ⊢ O B If ⊢ CL A ⊃ B , then ⊢ DSL O A ⊃ O B If ⊢ CL A ≡ B , then ⊢ DSL O A ≡ O B O A ⊢ O ( A ∨ B ) The following properties hold in DSL : O ( A ∧ B ) ⊢ O A ∧ O B O A , O B ⊢ O ( A ∨ B ) O A , O B ⊢ O ( A ∧ B ) O ( A ∨ B ) , O ¬ A ⊢ O B O ( A ∨ B ) , ¬ � A ⊢ O B If ⊢ CL A , then ⊢ O A If ⊢ EL A ≡ B , then ⊢ O A ≡ O B M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 14 / 25

Alternative axiomatization of DSL W O � := W | O W | � W O � | ¬W O � | W O � ∨ W O � M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 15 / 25

Alternative axiomatization of DSL W O � := W | O W | � W O � | ¬W O � | W O � ∨ W O � DSLO is axiomatized by strengthening S5 by: M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 15 / 25

Alternative axiomatization of DSL W O � := W | O W | � W O � | ¬W O � | W O � ∨ W O � DSLO is axiomatized by strengthening S5 by: ( O A ∧ O B ) ⊃ O ( A ∧ B ) (AND) O ( A ∧ B ) ⊃ O A (ADE) ( O A ∧ O B ) ⊃ O ( A ∨ B ) (OR) (( B ⇒ A ) ∧ O ( A ∨ B )) ⊃ O A (DINH) O A ⊃ � O A (ON) ( ¬ A ⇒ O A ) ⊃ O A (OW) If A ⊢ EL B then ⊢ O A ⊃ O B (EINH) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 15 / 25

Alternative axiomatization of DSL W O � := W | O W | � W O � | ¬W O � | W O � ∨ W O � DSLO is axiomatized by strengthening S5 by: ( O A ∧ O B ) ⊃ O ( A ∧ B ) (AND) O ( A ∧ B ) ⊃ O A (ADE) ( O A ∧ O B ) ⊃ O ( A ∨ B ) (OR) (( B ⇒ A ) ∧ O ( A ∨ B )) ⊃ O A (DINH) O A ⊃ � O A (ON) ( ¬ A ⇒ O A ) ⊃ O A (OW) If A ⊢ EL B then ⊢ O A ⊃ O B (EINH) S A = df O ¬ A ∧ A M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 15 / 25

Alternative axiomatization of DSL W O � := W | O W | � W O � | ¬W O � | W O � ∨ W O � DSLO is axiomatized by strengthening S5 by: ( O A ∧ O B ) ⊃ O ( A ∧ B ) (AND) O ( A ∧ B ) ⊃ O A (ADE) ( O A ∧ O B ) ⊃ O ( A ∨ B ) (OR) (( B ⇒ A ) ∧ O ( A ∨ B )) ⊃ O A (DINH) O A ⊃ � O A (ON) ( ¬ A ⇒ O A ) ⊃ O A (OW) If A ⊢ EL B then ⊢ O A ⊃ O B (EINH) S A = df O ¬ A ∧ A F A = df O ¬ A M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 15 / 25

Alternative axiomatization of DSL W O � := W | O W | � W O � | ¬W O � | W O � ∨ W O � DSLO is axiomatized by strengthening S5 by: ( O A ∧ O B ) ⊃ O ( A ∧ B ) (AND) O ( A ∧ B ) ⊃ O A (ADE) ( O A ∧ O B ) ⊃ O ( A ∨ B ) (OR) (( B ⇒ A ) ∧ O ( A ∨ B )) ⊃ O A (DINH) O A ⊃ � O A (ON) ( ¬ A ⇒ O A ) ⊃ O A (OW) If A ⊢ EL B then ⊢ O A ⊃ O B (EINH) S A = df O ¬ A ∧ A F A = df O ¬ A Theorem Γ ⊢ DSL A iff Γ ⊢ DSLO A. M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 15 / 25

Part III DSL and the ‘paradoxes’ M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 16 / 25

Ross’ paradox P : “posting the letter” B : “burning the letter” M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 17 / 25

Ross’ paradox P : “posting the letter” B : “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii): M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 17 / 25

Ross’ paradox P : “posting the letter” B : “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii): (i) O P M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 17 / 25

Ross’ paradox P : “posting the letter” B : “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii): (i) O P (ii) O ( P ∨ B ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 17 / 25

Ross’ paradox P : “posting the letter” B : “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii): (i) ¬ P ⇒ S ¬ P (ii) ¬ ( P ∨ B ) ⇒ S ¬ ( P ∨ B ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 17 / 25

Ross’ paradox P : “posting the letter” B : “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii): (i) ¬ P ⇒ S ¬ P (ii) ¬ ( P ∨ B ) ⇒ S ¬ ( P ∨ B ) O P � DSL O ( P ∨ B ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 17 / 25

Ross’ paradox P : “posting the letter” B : “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii): (i) ¬ P ⇒ S ¬ P (ii) ¬ ( P ∨ B ) ⇒ S ¬ ( P ∨ B ) O P � DSL O ( P ∨ B ) ¬ P ⇒ S ¬ P � DSL ¬ ( P ∨ B ) ⇒ S ¬ ( P ∨ B ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 17 / 25

Ross’ paradox P : “posting the letter” B : “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii): (i) ¬ P ⇒ S ¬ P (ii) ¬ ( P ∨ B ) ⇒ S ¬ ( P ∨ B ) O P � DSL O ( P ∨ B ) ¬ P ⇒ S ¬ P � DSL ¬ ( P ∨ B ) ⇒ S ¬ ( P ∨ B ) S ¬ P � DSL S ¬ ( P ∨ B ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 17 / 25

Ross’ paradox P : “posting the letter” B : “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii): (i) ¬ P ⇒ S ¬ P (ii) ¬ ( P ∨ B ) ⇒ S ¬ ( P ∨ B ) O P � DSL O ( P ∨ B ) ¬ P ⇒ S ¬ P � DSL ¬ ( P ∨ B ) ⇒ S ¬ ( P ∨ B ) S ¬ P � DSL S ¬ ( P ∨ B ) S ¬ P � DSL S ( ¬ P ∧ ¬ B ) M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 17 / 25

Ross’ paradox P : “posting the letter” B : “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii): (i) ¬ P ⇒ S ¬ P (ii) ¬ ( P ∨ B ) ⇒ S ¬ ( P ∨ B ) O P � DSL O ( P ∨ B ) ¬ P ⇒ S ¬ P � DSL ¬ ( P ∨ B ) ⇒ S ¬ ( P ∨ B ) S ¬ P � DSL S ¬ ( P ∨ B ) S ¬ P � DSL S ( ¬ P ∧ ¬ B ) ¬ P ⇒ S ¬ P ⊢ DSL ¬ ( P ∨ B ) ⇒ S ¬ P M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 17 / 25

The good Samaritan H : “ x helps y who has been robbed” R : “ y has been robbed” M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 18 / 25

The good Samaritan H : “ x helps y who has been robbed” R : “ y has been robbed” The good Samaritan paradox concerns the inference from (i) and (ii) to (iii) (in KDA ): M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 18 / 25

The good Samaritan H : “ x helps y who has been robbed” R : “ y has been robbed” The good Samaritan paradox concerns the inference from (i) and (ii) to (iii) (in KDA ): (i) H ⇒ R M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 18 / 25

The good Samaritan H : “ x helps y who has been robbed” R : “ y has been robbed” The good Samaritan paradox concerns the inference from (i) and (ii) to (iii) (in KDA ): (i) H ⇒ R (ii) R ⇒ s M. Beirlaen and C. Straßer (Ghent) Deontic Logic with Contextualized Sanctions Trends in Logic XI 18 / 25