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

SLIDE 1

An Andersonian Deontic Logic with Contextualized Sanctions

M. Beirlaen and C. Straßer

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

June 5 2012, Trends in Logic XI

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 1 / 25

SLIDE 2

Anderson’s reduction

Idea: define deontic in terms of alethic operators

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 2 / 25

SLIDE 3

Anderson’s reduction

Idea: define deontic in terms of alethic operators Method: sanction constant s

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 2 / 25

SLIDE 4

Anderson’s reduction

Idea: define deontic in terms of alethic operators Method: sanction constant s KDA: K plus the schema: ¬s (s)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 2 / 25

SLIDE 5

Anderson’s reduction

Idea: define deontic in terms of alethic operators Method: sanction constant s KDA: K plus the schema: ¬s (s) FA =df (A ⊃ s)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 2 / 25

SLIDE 6

Anderson’s reduction

Idea: define deontic in terms of alethic operators Method: sanction constant s KDA: K plus the schema: ¬s (s) FA =df (A ⊃ s) OA =df F¬A = (¬A ⊃ s)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 2 / 25

SLIDE 7

Anderson’s reduction

Idea: define deontic in terms of alethic operators Method: sanction constant s KDA: K plus the schema: ¬s (s) FA =df (A ⊃ s) OA =df F¬A = (¬A ⊃ s) SDL is the deontic fragment of KDA (Åqvist ’02)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 2 / 25

SLIDE 8

Outlook

Contextualizing sanctions: from s to S

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 3 / 25

SLIDE 9

Outlook

Contextualizing sanctions: from s to S Causal view: SA means ‘A causes (liability to) a sanction’

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 3 / 25

SLIDE 10

Outlook

Contextualizing sanctions: from s to S Causal view: SA means ‘A causes (liability to) a sanction’ The logic DSL:

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 3 / 25

SLIDE 11

Outlook

Contextualizing sanctions: from s to S Causal view: SA means ‘A causes (liability to) a sanction’ The logic DSL:

Study the logical properties of sanctions

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 3 / 25

SLIDE 12

Outlook

Contextualizing sanctions: from s to S Causal view: SA means ‘A causes (liability to) a sanction’ The logic DSL:

Study the logical properties of sanctions Conflict-tolerant

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 3 / 25

SLIDE 13

Outlook

Contextualizing sanctions: from s to S Causal view: SA means ‘A causes (liability to) a sanction’ The logic DSL:

Study the logical properties of sanctions Conflict-tolerant New insights into the ‘paradoxes’ of deontic logic

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 3 / 25

SLIDE 14

Outlook

Contextualizing sanctions: from s to S Causal view: SA means ‘A causes (liability to) a sanction’ The logic DSL:

Study the logical properties of sanctions Conflict-tolerant New insights into the ‘paradoxes’ of deontic logic

Structure:

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 3 / 25

SLIDE 15

Outlook

Contextualizing sanctions: from s to S Causal view: SA means ‘A causes (liability to) a sanction’ The logic DSL:

Study the logical properties of sanctions Conflict-tolerant New insights into the ‘paradoxes’ of deontic logic

Structure:

(I) DSL: definition

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 3 / 25

SLIDE 16

Outlook

Contextualizing sanctions: from s to S Causal view: SA means ‘A causes (liability to) a sanction’ The logic DSL:

Study the logical properties of sanctions Conflict-tolerant New insights into the ‘paradoxes’ of deontic logic

Structure:

(I) DSL: definition (II) DSL: further properties

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 3 / 25

SLIDE 17

Outlook

Contextualizing sanctions: from s to S Causal view: SA means ‘A causes (liability to) a sanction’ The logic DSL:

Study the logical properties of sanctions Conflict-tolerant New insights into the ‘paradoxes’ of deontic logic

Structure:

(I) DSL: definition (II) DSL: further properties (III) DSL and the ‘paradoxes’

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 3 / 25

SLIDE 18

Outlook

Contextualizing sanctions: from s to S Causal view: SA means ‘A causes (liability to) a sanction’ The logic DSL:

Study the logical properties of sanctions Conflict-tolerant New insights into the ‘paradoxes’ of deontic logic

Structure:

(I) DSL: definition (II) DSL: further properties (III) DSL and the ‘paradoxes’ (IV) Work in progress

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 3 / 25

SLIDE 19

Part I The logic DSL: definition

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 4 / 25

SLIDE 20

DSL: grammar/ interpretation(1)

W := P | ¬W | W ∨ W

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 5 / 25

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DSL: grammar/ interpretation(1)

W := P | ¬W | W ∨ W WS := W | SW | ¬WS | WS ∨ WS

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 5 / 25

SLIDE 22

DSL: grammar/ interpretation(1)

W := P | ¬W | W ∨ W WS := W | SW | ¬WS | WS ∨ WS WS

:= WS | WS | ¬WS | WS ∨ WS

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 5 / 25

SLIDE 23

DSL: grammar/ interpretation(1)

W := P | ¬W | W ∨ W WS := W | SW | ¬WS | WS ∨ WS WS

:= WS | WS | ¬WS | WS ∨ WS

∧, ⊃, ≡, defined as usual
M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 5 / 25

SLIDE 24

DSL: grammar/ interpretation(1)

W := P | ¬W | W ∨ W WS := W | SW | ¬WS | WS ∨ WS WS

:= WS | WS | ¬WS | WS ∨ WS

∧, ⊃, ≡, defined as usual

A ⇒ B =df (A ⊃ B)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 5 / 25

SLIDE 25

DSL: grammar/ interpretation(1)

W := P | ¬W | W ∨ W WS := W | SW | ¬WS | WS ∨ WS WS

:= WS | WS | ¬WS | WS ∨ WS

∧, ⊃, ≡, defined as usual

A ⇒ B =df (A ⊃ B) A: “A holds in every world in which our norms hold”(Mares ’92)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 5 / 25

SLIDE 26

DSL: grammar/ interpretation(1)

W := P | ¬W | W ∨ W WS := W | SW | ¬WS | WS ∨ WS WS

:= WS | WS | ¬WS | WS ∨ WS

∧, ⊃, ≡, defined as usual

A ⇒ B =df (A ⊃ B) A: “A holds in every world in which our norms hold”(Mares ’92) : S5-modality (equivalence class of accessible worlds)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 5 / 25

SLIDE 27

DSL: grammar/ interpretation (2)

SA:

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

SLIDE 28

DSL: grammar/ interpretation (2)

SA:

“A causes a sanction”

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

SLIDE 29

DSL: grammar/ interpretation (2)

SA:

“A causes a sanction” “A causes liability to a sanction”

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

SLIDE 30

DSL: grammar/ interpretation (2)

SA:

“A causes a sanction” “A causes liability to a sanction” “A causes a violation (of a norm)”

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

SLIDE 31

DSL: grammar/ interpretation (2)

SA:

“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”

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

SLIDE 32

DSL: grammar/ interpretation (2)

SA:

“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”

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

SLIDE 33

DSL: grammar/ interpretation (2)

SA:

“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”

FA =df A ⇒ SA

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

SLIDE 34

DSL: grammar/ interpretation (2)

SA:

“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”

FA =df A ⇒ SA OA =df F¬A

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

SLIDE 35

DSL: grammar/ interpretation (2)

SA:

“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”

FA =df A ⇒ SA OA =df F¬A SA: “A holds and causes a sanction”

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 6 / 25

SLIDE 36

DSL: grammar/ interpretation (2)

SA:

“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”

FA =df A ⇒ SA OA =df F¬A SA: “A holds and causes a sanction” A ⇒ SA: “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

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DSL: axiomatization (1)

DSL is obtained by adding to S5 the following axiom schemata and rules: SA ⊃ A (SR) SA ⊃ (A ⇒ SA) (S⇒) (A ⇒ S(A ∧ B)) ⊃ (A ⇒ SA) (SW) (SA ∧ SB) ⊃ S(A ∧ B) (S∧) ((A ∨ B) ⇒ S(A ∨ B)) ≡ ((A ⇒ SA) ∧ (B ⇒ SB)) (S∨) If ⊢EL A ⊃ B then ⊢ SA ⊃ SB (S⊃)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 7 / 25

SLIDE 38

DSL: axiomatization (2)

SA ⊃ A (SR)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

SLIDE 39

DSL: axiomatization (2)

SA ⊃ 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

SLIDE 40

DSL: axiomatization (2)

SA ⊃ A (SR) If A causes a sanction, then A. SA ⊃ (A ⇒ SA) (S⇒)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

SLIDE 41

DSL: axiomatization (2)

SA ⊃ A (SR) If A causes a sanction, then A. SA ⊃ (A ⇒ SA) (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

SLIDE 42

DSL: axiomatization (2)

SA ⊃ A (SR) If A causes a sanction, then A. SA ⊃ (A ⇒ SA) (S⇒) If A causes a sanction in some world, then A causes a sanction in every accessible world in which A holds. SA ≡ (FA ∧ A)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

SLIDE 43

DSL: axiomatization (2)

SA ⊃ A (SR) If A causes a sanction, then A. SA ⊃ (A ⇒ SA) (S⇒) If A causes a sanction in some world, then A causes a sanction in every accessible world in which A holds. SA ≡ (FA ∧ A) ((A ⇒ B) ∧ ((A ∧ B) ⇒ S(A ∧ B))) ⊃ (A ⇒ SA)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

SLIDE 44

DSL: axiomatization (2)

SA ⊃ A (SR) If A causes a sanction, then A. SA ⊃ (A ⇒ SA) (S⇒) If A causes a sanction in some world, then A causes a sanction in every accessible world in which A holds. SA ≡ (FA ∧ A) ((A ⇒ B) ∧ ((A ∧ B) ⇒ S(A ∧ B))) ⊃ (A ⇒ SA) 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

SLIDE 45

DSL: axiomatization (2)

SA ⊃ A (SR) If A causes a sanction, then A. SA ⊃ (A ⇒ SA) (S⇒) If A causes a sanction in some world, then A causes a sanction in every accessible world in which A holds. SA ≡ (FA ∧ A) ((A ⇒ B) ∧ ((A ∧ B) ⇒ S(A ∧ B))) ⊃ (A ⇒ SA) 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 ⇒ SA) (SW)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 8 / 25

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DSL: axiomatization (3)

S(A ∧ B)

?

⊃ SA

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

SLIDE 47

DSL: axiomatization (3)

S(A ∧ B)

?

⊃ SA 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

SLIDE 48

DSL: axiomatization (3)

S(A ∧ B)

?

⊃ SA 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

SLIDE 49

DSL: axiomatization (3)

S(A ∧ B) ⊃ SA 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

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DSL: axiomatization (3)

S(A ∧ B) ⊃ SA 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 SA

?

⊃ S(A ∧ B)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

SLIDE 51

DSL: axiomatization (3)

S(A ∧ B) ⊃ SA 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 SA

?

⊃ 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

SLIDE 52

DSL: axiomatization (3)

S(A ∧ B) ⊃ SA 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 SA ⊃ 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

SLIDE 53

DSL: axiomatization (3)

S(A ∧ B) ⊃ SA 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 SA ⊃ 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. (SA ∧ SB) ⊃ S(A ∧ B) (S∧)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 9 / 25

SLIDE 54

DSL: axiomatization (3)

S(A ∧ B) ⊃ SA 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 SA ⊃ 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. (SA ∧ SB) ⊃ 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

SLIDE 55

DSL: axiomatization (4)

((A ∨ B) ⇒ S(A ∨ B)) ≡ ((A ⇒ SA) ∧ (B ⇒ SB)) (S∨)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

SLIDE 56

DSL: axiomatization (4)

((A ∨ B) ⇒ S(A ∨ B)) ≡ ((A ⇒ SA) ∧ (B ⇒ SB)) (S∨)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

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DSL: axiomatization (4)

((A ∨ B) ⇒ S(A ∨ B)) ≡ ((A ⇒ SA) ∧ (B ⇒ SB)) (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

SLIDE 58

DSL: axiomatization (4)

((A ∨ B) ⇒ S(A ∨ B)) ≡ ((A ⇒ SA) ∧ (B ⇒ SB)) (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

SLIDE 59

DSL: axiomatization (4)

((A ∨ B) ⇒ S(A ∨ B)) ≡ ((A ⇒ SA) ∧ (B ⇒ SB)) (S∨) F(A ∨ B) ≡ (FA ∧ FB)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

SLIDE 60

DSL: axiomatization (4)

((A ∨ B) ⇒ S(A ∨ B)) ≡ ((A ⇒ SA) ∧ (B ⇒ SB)) (S∨) F(A ∨ B) ≡ (FA ∧ FB) Sa

?

⊃ S(a ∨ b)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

SLIDE 61

DSL: axiomatization (4)

((A ∨ B) ⇒ S(A ∨ B)) ≡ ((A ⇒ SA) ∧ (B ⇒ SB)) (S∨) F(A ∨ B) ≡ (FA ∧ FB) Pressing button a causes a

meltdown. Pressing button b does

not. Sa

?

⊃ S(a ∨ b)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

SLIDE 62

DSL: axiomatization (4)

((A ∨ B) ⇒ S(A ∨ B)) ≡ ((A ⇒ SA) ∧ (B ⇒ SB)) (S∨) F(A ∨ B) ≡ (FA ∧ FB) Pressing button a causes a

meltdown. Pressing button b does

not. Sa ⊃ S(a ∨ b) (Sa ∨ Sb)

?

⊃ S(a ∨ b)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

SLIDE 63

DSL: axiomatization (4)

((A ∨ B) ⇒ S(A ∨ B)) ≡ ((A ⇒ SA) ∧ (B ⇒ SB)) (S∨) F(A ∨ B) ≡ (FA ∧ FB) Pressing button a causes a

meltdown. Pressing button b does

not. Sa ⊃ S(a ∨ b) (Sa ∨ Sb) ⊃ S(a ∨ b)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 10 / 25

SLIDE 64

DSL: axiomatization (5)

Suppose: If ⊢CL A ⊃ B then ⊢ SA ⊃ SB

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 11 / 25

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DSL: axiomatization (5)

Suppose: If ⊢CL A ⊃ B then ⊢ SA ⊃ SB Then, since S(A ∨ B) ⊃ (SA ∨ SB): SA ⊃ (S(A ∧ B) ∨ S(A ∧ ¬B))

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 11 / 25

SLIDE 66

DSL: axiomatization (5)

Suppose: If ⊢CL A ⊃ B then ⊢ SA ⊃ SB Then, since S(A ∨ B) ⊃ (SA ∨ SB): SA ⊃ (S(A ∧ B) ∨ S(A ∧ ¬B)) Solution: If ⊢EL A ⊃ B then ⊢ SA ⊃ SB (S⊃)

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 11 / 25

SLIDE 67

DSL: axiomatization (5)

Suppose: If ⊢CL A ⊃ B then ⊢ SA ⊃ SB Then, since S(A ∨ B) ⊃ (SA ∨ SB): SA ⊃ (S(A ∧ B) ∨ S(A ∧ ¬B)) Solution: If ⊢EL A ⊃ B then ⊢ SA ⊃ SB (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

SLIDE 68

DSL: axiomatization (5)

Suppose: If ⊢CL A ⊃ B then ⊢ SA ⊃ SB Then, since S(A ∨ B) ⊃ (SA ∨ SB): SA ⊃ (S(A ∧ B) ∨ S(A ∧ ¬B)) Solution: If ⊢EL A ⊃ B then ⊢ SA ⊃ SB (S⊃) Let (τ, σ) ∈ {(∧, ∨), (∨, ∧)}. EL is defined by: ¬(A τ B) ⊣⊢(¬A σ ¬B) (1) ¬¬A ⊣⊢ A (2) ((A τ B) τ C) ⊣⊢(A τ(B τ C)) (3) (A τ B) ⊣⊢(B τ A) (4) (A τ(B σ C)) ⊣⊢((A τ B) σ(A τ C)) (5) (A τ A) ⊣⊢ A (6) ((A τ ¬A) σ B) ⊢ B (7) ((A τ B) σ A) ⊢ A (8) ((A τ ¬A) τ B) ⊢ (A τ ¬A) (9) If A ⊢ B then C ⊢ CA/B (10) Where CA/B is the product of substituting any amount of subformulas A in C by B.

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DSL: axiomatization (6)

Some examples:

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DSL: axiomatization (6)

Some examples: A ∧ ¬(B ∨ C) EL⊣⊢EL A ∧ (¬B ∧ ¬C) EL⊣⊢EL(A ∧ ¬C) ∧ ¬B

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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)

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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)

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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

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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

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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

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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

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Part II The logic DSL: further properties

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Deontic properties of DSL

The following properties fail in DSL: OA, O¬A ⊢ B OA, O¬A ⊢ OB If ⊢CL A ⊃ B, then ⊢DSL OA ⊃ OB If ⊢CL A ≡ B, then ⊢DSL OA ≡ OB OA ⊢ O(A ∨ B)

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Deontic properties of DSL

The following properties fail in DSL: OA, O¬A ⊢ B OA, O¬A ⊢ OB If ⊢CL A ⊃ B, then ⊢DSL OA ⊃ OB If ⊢CL A ≡ B, then ⊢DSL OA ≡ OB OA ⊢ O(A ∨ B) The following properties hold in DSL: O(A ∧ B) ⊢ OA ∧ OB OA, OB ⊢ O(A ∨ B) OA, OB ⊢ O(A ∧ B) O(A ∨ B), O¬A ⊢ OB O(A ∨ B), ¬A ⊢ OB If ⊢CL A, then ⊢ OA If ⊢EL A ≡ B, then ⊢ OA ≡ OB

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Alternative axiomatization of DSL

WO

:= W | OW | WO | ¬WO | WO ∨ WO

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Alternative axiomatization of DSL

WO

:= W | OW | WO | ¬WO | WO ∨ WO

DSLO is axiomatized by strengthening S5 by:
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Alternative axiomatization of DSL

WO

:= W | OW | WO | ¬WO | WO ∨ WO

DSLO is axiomatized by strengthening S5 by:

(OA ∧ OB) ⊃ O(A ∧ B) (AND) O(A ∧ B) ⊃ OA (ADE) (OA ∧ OB) ⊃ O(A ∨ B) (OR) ((B ⇒ A) ∧ O(A ∨ B)) ⊃ OA (DINH) OA ⊃ OA (ON) (¬A ⇒ OA) ⊃ OA (OW) If A ⊢EL B then ⊢ OA ⊃ OB (EINH)

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Alternative axiomatization of DSL

WO

:= W | OW | WO | ¬WO | WO ∨ WO

DSLO is axiomatized by strengthening S5 by:

(OA ∧ OB) ⊃ O(A ∧ B) (AND) O(A ∧ B) ⊃ OA (ADE) (OA ∧ OB) ⊃ O(A ∨ B) (OR) ((B ⇒ A) ∧ O(A ∨ B)) ⊃ OA (DINH) OA ⊃ OA (ON) (¬A ⇒ OA) ⊃ OA (OW) If A ⊢EL B then ⊢ OA ⊃ OB (EINH) SA =df O¬A ∧ A

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Alternative axiomatization of DSL

WO

:= W | OW | WO | ¬WO | WO ∨ WO

DSLO is axiomatized by strengthening S5 by:

(OA ∧ OB) ⊃ O(A ∧ B) (AND) O(A ∧ B) ⊃ OA (ADE) (OA ∧ OB) ⊃ O(A ∨ B) (OR) ((B ⇒ A) ∧ O(A ∨ B)) ⊃ OA (DINH) OA ⊃ OA (ON) (¬A ⇒ OA) ⊃ OA (OW) If A ⊢EL B then ⊢ OA ⊃ OB (EINH) SA =df O¬A ∧ A FA =df O¬A

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Alternative axiomatization of DSL

WO

:= W | OW | WO | ¬WO | WO ∨ WO

DSLO is axiomatized by strengthening S5 by:

(OA ∧ OB) ⊃ O(A ∧ B) (AND) O(A ∧ B) ⊃ OA (ADE) (OA ∧ OB) ⊃ O(A ∨ B) (OR) ((B ⇒ A) ∧ O(A ∨ B)) ⊃ OA (DINH) OA ⊃ OA (ON) (¬A ⇒ OA) ⊃ OA (OW) If A ⊢EL B then ⊢ OA ⊃ OB (EINH) SA =df O¬A ∧ A FA =df O¬A

Theorem

Γ ⊢DSL A iff Γ ⊢DSLO A.

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Part III DSL and the ‘paradoxes’

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Ross’ paradox

P: “posting the letter” B: “burning the letter”

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Ross’ paradox

P: “posting the letter” B: “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii):

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Ross’ paradox

P: “posting the letter” B: “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii): (i) OP

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Ross’ paradox

P: “posting the letter” B: “burning the letter” Ross’ paradox concerns the validity of the inference from (i) to (ii): (i) OP (ii) O(P ∨ B)

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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)

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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) OP DSL O(P ∨ B)

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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) OP DSL O(P ∨ B) ¬P ⇒ S¬P DSL ¬(P ∨ B) ⇒ S¬(P ∨ B)

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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) OP DSL O(P ∨ B) ¬P ⇒ S¬P DSL ¬(P ∨ B) ⇒ S¬(P ∨ B) S¬P DSL S¬(P ∨ B)

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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) OP 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)

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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) OP 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

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The good Samaritan

H: “x helps y who has been robbed” R: “y has been robbed”

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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):

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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

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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

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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 H ⇒ R, R ⇒ s ⊢KDA H ⇒ s

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SLIDE 102

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 (iii) H ⇒ s H ⇒ R, R ⇒ s ⊢KDA H ⇒ s

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SLIDE 103

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 ⇒ SR (iii) H ⇒ SH H ⇒ R, R ⇒ s ⊢KDA H ⇒ s

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SLIDE 104

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 ⇒ SR (iii) H ⇒ SH H ⇒ R, R ⇒ s ⊢KDA H ⇒ s H ⇒ R, R ⇒ SR ⊢DSL H ⇒ SR

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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 ⇒ SR (iii) H ⇒ SH H ⇒ R, R ⇒ s ⊢KDA H ⇒ s H ⇒ R, R ⇒ SR ⊢DSL H ⇒ SR H ⇒ R, R ⇒ SR DSL H ⇒ SH

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Part IV Work in progress

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Permissions

PA =df ¬O¬A

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Permissions

PA =df ¬O¬A PA ≡ (A ∧ ¬s)

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Permissions

PA =df ¬O¬A PA ≡ (A ∧ ¬s) PA ≡ (A ∧ ¬SA)

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Permissions

PA =df ¬O¬A PA ≡ (A ∧ ¬s) PA ≡ (A ∧ ¬SA) Suppose OA ∧ ¬O(A ∨ B)

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Permissions

PA =df ¬O¬A PA ≡ (A ∧ ¬s) PA ≡ (A ∧ ¬SA) Suppose OA ∧ ¬O(A ∨ B)

¬O(A ∨ B) ≡ ¬O¬(¬A ∧ ¬B) ≡ P(¬A ∧ ¬B)

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Permissions

PA =df ¬O¬A PA ≡ (A ∧ ¬s) PA ≡ (A ∧ ¬SA) Suppose OA ∧ ¬O(A ∨ B)

¬O(A ∨ B) ≡ ¬O¬(¬A ∧ ¬B) ≡ P(¬A ∧ ¬B)

Hence OA ∧ ¬O(A ∨ B) ⊢ P(¬A ∧ ¬B)

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Permissions

PA =df ¬O¬A PA ≡ (A ∧ ¬s) PA ≡ (A ∧ ¬SA) Suppose OA ∧ ¬O(A ∨ B)

¬O(A ∨ B) ≡ ¬O¬(¬A ∧ ¬B) ≡ P(¬A ∧ ¬B)

Hence OA ∧ ¬O(A ∨ B) ⊢ P(¬A ∧ ¬B) Alternative: P

I Ai
=
I Ai ∧ ¬

∅=J⊆I S

J Aj
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Permissions

PA =df ¬O¬A PA ≡ (A ∧ ¬s) PA ≡ (A ∧ ¬SA) Suppose OA ∧ ¬O(A ∨ B)

¬O(A ∨ B) ≡ ¬O¬(¬A ∧ ¬B) ≡ P(¬A ∧ ¬B)

Hence OA ∧ ¬O(A ∨ B) ⊢ P(¬A ∧ ¬B) Alternative: P

I Ai
=
I Ai ∧ ¬

∅=J⊆I S

J Aj
Strong/positive permission, free choice permission?
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Future work

Devise a semantics for DSL

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Future work

Devise a semantics for DSL Contextualize Kanger’s constant q, abbreviating that ‘all normative demands are met’

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Future work

Devise a semantics for DSL Contextualize Kanger’s constant q, abbreviating that ‘all normative demands are met’

‘Dual’ to Anderson’s reduction, yet different properties in our setting

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Thank you!

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References

1. Alan Ross Anderson (1958). The reduction of deontic logic to alethic modal logic

(Mind, vol. 67, pp. 100-103).

2. Lennart Åqvist (2002). Deontic logic (D. Gabbay & F. Guenthner (Eds.): Handbook
f Philosophical Logic (2nd Edition), Vol. 8, Kluwer Academic Publishers).
3. Stig Kanger (1971, first published 1957). New foundations for ethical theory (R.

Hilpinen (Ed.): Deontic Logic: Introductory and Systematic Readings, D. Reidel Publishing Company, Dordrecht, pp. 36-58).

4. Ed Mares (1992). Andersonian Deontic Logic (Theoria, vol. 58, pp. 3-20).
5. Christian Straßer and Mathieu Beirlaen (2012). An Andersonian Deontic Logic with

Contextualized Sanctions (T. Agotnes, J. Broersen & D. Elgesem (Eds.): DEON 2012, LNAI 7393, Springer-Verlag, Berlin-Heidelberg).

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Explicit Disjunctive Obligations

(a) You should post the letter. (b) Thus, implicitly: You should post the letter or burn it. (a’) You should post the letter or email it. (c) You cannot post the letter. (b’) You cannot post the letter. (e.g. the post is closed al- ready) (d) You should burn the letter. (c’) You should email it. (d) is counter-intuitive Note that we get in DSL: O(P ∨ E) ¬P OE

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Explicit Disjunctive Obligations

(a) You should post the letter. (b) Thus, implicitly: You should post the letter or burn it. (a’) You should post the letter or email it. (c) You cannot post the letter. (b’) You cannot post the letter. (e.g. the post is closed al- ready) (d) You should burn the letter. (c’) You should email it. (d) is counter-intuitive (c’) is intuitive Note that we get in DSL: O(P ∨ E) ¬P OE

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Explicit Disjunctive Obligations

(a) You should post the letter. (b) Thus, implicitly: You should post the letter or burn it. (a’) You should post the letter or email it. (c) You cannot post the letter. (b’) You cannot post the letter. (e.g. the post is closed al- ready) (d) You should burn the letter. (c’) You should email it. (d) is counter-intuitive (c’) is intuitive explicit disjunctions have a different logic than derived ones Note that we get in DSL: O(P ∨ E) ¬P OE

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Kanger: A contextualized fulfillment-logic

in Kanger’s framework there is an atomic fulfillment proposition q.

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Kanger: A contextualized fulfillment-logic

in Kanger’s framework there is an atomic fulfillment proposition q. we contextualize this ala DSL: OA =df A ⇒ QA

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SLIDE 125

Kanger: A contextualized fulfillment-logic

in Kanger’s framework there is an atomic fulfillment proposition q. we contextualize this ala DSL: OA =df A ⇒ QA interesting for inseparable norms: e.g., Anne ought to sing and dance (Lou Goble), shopping list for a cake, etc.

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 25 / 25

SLIDE 126

Kanger: A contextualized fulfillment-logic

in Kanger’s framework there is an atomic fulfillment proposition q. we contextualize this ala DSL: OA =df A ⇒ QA interesting for inseparable norms: e.g., Anne ought to sing and dance (Lou Goble), shopping list for a cake, etc. from (S ∧ D) ⇒ Q(S ∧ D) we cannot derive S ⇒ QS and D ⇒ QD

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 25 / 25

SLIDE 127

Kanger: A contextualized fulfillment-logic

in Kanger’s framework there is an atomic fulfillment proposition q. we contextualize this ala DSL: OA =df A ⇒ QA interesting for inseparable norms: e.g., Anne ought to sing and dance (Lou Goble), shopping list for a cake, etc. from (S ∧ D) ⇒ Q(S ∧ D) we cannot derive S ⇒ QS and D ⇒ QD in contrast: O(S ∧ D) ⊢DSL OS

M. Beirlaen and C. Straßer (Ghent)

Deontic Logic with Contextualized Sanctions Trends in Logic XI 25 / 25

SLIDE 128