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Normative Reasoning and Deontic Logics

[@RUB – SS2015]

Christian Straßer

Institute for Philosophy II, Ruhr-University BochumCentre for Logic and Philosophy of Science

Ghent University, BelgiumChristian.Strasser@UGent.be

April 11, 2015

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Modal Logics – Kripkean Frames, Informally

◮ How to express in a formally precise way modifiers?

◮ We have a proposition like “She bakes a cake” and put itwithin the scope of a modifier, e.g.:

◮ Possibly/Necessarily, she bakes a cake.◮ It should be that she bakes a cake.◮ I know/believe that she bakes a cake.

◮ idea: accessible worlds

◮ accessibility relates to the modifier in question◮ accessible worlds are possible worlds◮ accessible worlds are ethically/legally/etc. ideal worlds◮ accessible worlds are states/worlds compatible with my

doxastic/epistemic state

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Modal Logics – Kripkean Frames, More Formally

◮ a model is a tuple M = 〈W ,R , v〉 where

◮ W is a set of points often called “worlds”

◮ R ⊆ W ×W is a relation often called “accessibility relation”

◮ v : W ×A → {0, 1} is an assignment function◮ often also: v : A → ℘(W ) or v : W → ℘(A)

◮ atoms get their truth value at each world w ∈ W in a modelM via the assignment just as expected:

M,w |= A (where A ∈ A) iff v(w ,A) = 1 (resp. iff w ∈ v(A),resp. iff A ∈ v(w))

◮ the classical connectives are interpreted at each world in amodel just as expected:

M,w |= ¬A iff M,w 6|= AM,w |= A ∧ B iff M,w |= A and M,w |= BM,w |= A ∨ B iff M,w |= A or M,w |= Betc.

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Modal Logics – Kripkean Frames, More Formally◮ we now also have a unary modal operator �:

M,w |= �A iff for all w ′ for which (w ,w ′) ∈ R (read: “for allworlds w ′ accessible from w”): M,w ′ |= Ae.g., M,w |= �A where � represents necessity means that Aholds in all possible worldse.g., M,w |= �A where � represents normativity means thatA holds in all ideal worlds

◮ etc.◮ duality principle: ♦ = ¬�¬

◮ ♦ represents possibility◮ ♦ represents permission◮ etc.

◮ truth for ♦:M,w |= ♦A iff there is a w ′ such that (w ,w ′) ∈ R andM,w ′ |= A

◮ What do you think: M,w |= �A implies M,w |= ♦A?

◮ Show: M,w |= �(A ∧ B) implies M,w |= �A ∧�B

◮ Show: M,w |= �A,�B implies M,w |= �(A ∧ B)4/43

How to define an entailment relation? The modal logic K

◮ Answer: as usual!

◮ only: where M = 〈W ,R , v〉: M |= A iff for all w ∈ W ,M,w |= A.

◮ We say M is a model of Γ iff M |= A for all A ∈ Γ

◮ Γ A iff all models of Γ are models of A◮ often the semantics is phrased with a so-called

actual/designated world◮ a model is a quadruple M = 〈W ,R , v , a〉◮ difference to before: M |= A iff M, a |= A.◮ rest as before: Γ A iff all models of Γ are models of A

◮ these two represenational formats aresemi-expressive/equivalent: try to show this!

◮ What do you think: �A K ♦A?

◮ Show: �(A ∧ B) K �A ∧�B

◮ Show: �A,�B K �(A ∧ B)

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Axiomatizing K

N ⊢ A implies ⊢ �A

K ⊢ �(A ⊃ B) ⊃ (�A ⊃ �B)

◮ Show: �(A ∧ B) ⊢K �A ∧�B

◮ Show: �A,�B ⊢K �(A ∧ B)

◮ Show: �A,♦B ⊢K ♦(A ∧ B)

◮ Show: �A ∨�B ,♦¬B K �A

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Strengthening Kripkean Semantics: Frame Conditions

◮ requiring R to be reflexive (i.e., (w ,w) ∈ R for all w): T “=K + ⊢ �A ⊃ A”

◮ requiring R to be serial (i.e., for all w there is a w ′ such that(w ,w ′) ∈ R): KD “= K + ⊢ �A ⊃ ♦A”

◮ requiring that the accessibility relation ≤ is a partial order(reflexivity, transitivity, antisymmetry): intuitionistic logic

◮ additional requirement: w ≤ w ′ implies v(w) ⊆ v(w ′)◮ negation is a modal operator: M,w |= ¬A iff there is no w ′

such that (w ,w ′) ∈ R and w ′ |= A◮ implication: M,w |= A → B iff for all w ′ for which

(w ,w ′) ∈ R , M,w ′ |= A implies M,w |= B .

Standard Deontic Logic is KD.

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Deontic Logic Paradox

◮ Natural language representation: NR

◮ formal representation: FR

◮ Paradox 1:◮ NR implies A◮ FR does not imply A

◮ Paradox 2:◮ FR implies A.◮ NR does not imply A.

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Some Problems

◮ ⊢ A ⊃ B implies ⊢ OA ⊃ OB◮ Ross’ Paradox: Ol (you’re supposed to post the letter) implies

O(l ∨ b) (you’re supposed to post the letter or burn the city)◮ The birthday cake: O(i1 ∧ . . . ∧ in) implies Oij (1 ≤ j ≤ n)?

◮ problems with conditional obligations◮ specificity: m ⊃ O¬f but (m ∧ a) ⊃ Of . Suppose m. Then we

get triviality in SDL◮ similar: contrary-to-duty (Forrester paradox, Chisholm

paradox, see later)

◮ deontic conflicts: e.g., OA ∧ O¬A

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Deontic Dilemmas

Example: The dilemma of Sartre’s pupil

◮ Obligation 1: stay with the ill mother

◮ Obligation 2: join the forces to fight the Nazis

Formal definition

◮ Two obligations: OA, OB

◮ both are possible: ♦A, ♦B

◮ they cannot jointly be realized: ¬♦(A ∧ B)

They are often characterized by

◮ obligations with equal force

◮ incommensurable obligations

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Deontic Explosions

Conflict(A,B) ⊢ ⊥

Example 1: Principle D

D OA → ¬O¬AECQ A ∧ ¬A → ⊥

1 OA2 O¬A3 OA → ¬O¬A D

4 ¬O¬A 1, 3;MP

5 O¬A ∧ ¬O¬A 2, 4;∧ − intro6 ⊥ 5;ECQ

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Deontic Explosions

Starting point:1 OA2 OB3 ¬♦(A ∧ B)

Example 2: Aggregation and Kant’s “ought implies can”

AND OA ∧ OB → O(A ∧ B)Kant’s principle OA → ♦A

ECQ A ∧ ¬A → ⊥

4 O(A ∧ B) 1, 2;AND5 ♦(A ∧ B) 3, 4; Kant’s principle6 ⊥ 5, 6;ECQ

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Deontic Explosions

Starting point:1 OA2 OB3 ¬♦(A ∧ B)

Example 3: Distribution

RM �(A → B) → (OA → OB)D OA → ¬O¬A

ECQ A ∧ ¬A → ⊥

4 �(A → ¬B) 35 O¬B 1, 4;RM6 ¬O¬B 2;D7 ⊥ 5, 6;ECQ

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DEX0 Conflict implies triviality (anything follows)

DEX1 Conflict implies that anything is obligatory

◮ Show that AND and ⊢ ¬O⊥ implies DEX0

◮ Show that AND and RM’ (see below) implies DEX1

RM’ If ⊢ A ⊃ B then ⊢ OA ⊃ OB .

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Many “Bad” CombinationsAND

OA ∧ OB → O(A ∧ B)NM

�(A → B) → (OA → OB)

KP

OA → ♦AECQ

A ∧ ¬A → ⊥D

OA → ¬O¬A

Approaches for logics dealing with deontic explosions:

◮ Restricting/Rejecting ECQ – going paraconsistent◮ Da Costa&Carnielli (1986) [4], Beirlaen et al [3, 2, 1]

◮ Restricting AND: Goble’s logic P drawback

◮ Goble (2000,2013) [7, 10], Meheus et al (2010) [13], Van DePutte&Straßer (2012) [20]

◮ Restricting RM: Goble’s logics DPM

◮ Goble (2009,2013) [9, 10], Straßer (2010) [15, 16],Straßer&Beirlaen&Meheus (2012) [19] 15/43

Some other approaches

◮ “Contextualizing”: Van Fraassen (1973) [5], Horty (2003)[11], Makinson&Van der Torre (2001) [12]

◮ argumentation theory: Oren et al. (2008) [14], Gabbay (2012)[6], Straßer&Arieli (2014) [18]

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Restricting Aggregation [7, 8]

◮ preferential semantics:◮ M = 〈W , 〈≤a〉a∈W , v〉 where ≤a are preorders on their fields

(reflexive and transitive)◮ the field of a relation: F ≤a= {b ∈ W | there is a c ∈ W such

that either b ≤a c or c ≤a b}◮ define: M,w |= OA iff there is a w ′ ∈ F ≤w such that for all

w ′′ for which w ′ ≤w w ′′, M,w ′′ |= A◮ where ≤a are also connected (for all w 6= w ′ in F ≤a, w ≤ w ′

or w ′ ≤ w): this semantics characterizes SDL (and as we willsee below also a dyadic version of SDL)

◮ multi-relational semantics (generalization of Kripkeansemantics)

◮ M = 〈W ,R, v〉 where R is a non-empty family of serialaccessibility relations R

◮ each R represents a normative standard/value system/etc.:◮ M,w |= OA iff there is a R ∈ R such that M,w |= A for all

w ′ ∈ W for which (w ,w ′) ∈ R

◮ both systems characterize the same consequence relation

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System P

PC If A is a classical tautology then A is an axiom of P

RM If ⊢ A ⊃ B then ⊢ OA ⊃ OB

N ⊢ O⊤

P ⊢ ¬O¬⊤

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Restricting Inheritance

Replace the inheritance principle

RM if ⊢ A → B then ⊢ OA → OB

by a restricted version:

RPM if ⊢ A → B then ⊢ PA → (OA → OB)

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Goble’s Logic DPM.1

DPM.1 Axiomsall axioms of classical propositional calculus andRPM if ⊢ A → B then ⊢ PA → (OA → OB)RE if ⊢ A ↔ B then ⊢ OA ↔ OBN ⊢ O⊤

AND ⊢ (OA ∧ OB) → O(A ∧ B)

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Semantics

Neighborhood Frame

A neighbourhood frame F is a pair 〈W ,O〉 in which W is anon-empty set of points, e.g., possible worlds, and O is a functionassigning every a ∈ W a set, Oa, of subsets of W ; i.e., Oa ⊆ ℘W .

ModelsA model, M, is a pair 〈F , v〉 where F is a neighborhood frame〈W ,O〉, and v is a function assigning every atomic formula p of La subset of W ,i.e., v(p) ⊆ W . A satisfaction relation |= is definedas follows.

Tp) M, a |= p iff a ∈ v(p)T¬) M, a |= ¬A iff M, a 2 AT∧) M, a |= A ∧ B iff M, a |= A and M, a |= BT∨) M, a |= A ∨ B iff M, a |= A or M, a |= BTO) M, a |= OA iff |A|M ∈ Oa

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Semantics

For DPM.1: For all X ,Y ⊆ W and all a ∈ W :

a) W ∈ Oa

b) If X ∈ Oa and Y ∈ Oa then X ∩ Y ∈ Oa

c) If X ⊆ Y and X ∈ Oa and −X /∈ Oa then Y ∈ Oa

Condition a) validates N, condition b) validates AND andcondition c) validates RPM.

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Problem with the Weakenings of SDL

◮ they are weak!

◮ solution: adaptive strengthening

◮ e.g., in the context of P: apply aggregation conditionally asmuch as possible

◮ e.g., in the context of DPM: apply inheritance conditionallyand as much as possible

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Problems with Conditional Obligations

Asparagus – Specificity

◮ Being served a meal, you ought not to eat with fingers.

◮ Being served asparagus, you ought to eat with fingers.

◮ You’re being served asparagus.

What if we model this via classical implication?

◮ m ⊃ O¬f

◮ (m ∧ a) ⊃ Of

◮ m ∧ a

Problem: This is classically inconsistent (if SDL models O).

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Problems with Conditional Obligations 2

Chisholm’s Paradox: Contrary to Duty Obligations

◮ John ought not to impregnate Suzy Mae.

◮ If John impregnates Suzy Mae, he ought to marry her.

◮ If John doesn’t impregnate Suzy Mae, he ought not to marryher.

◮ John impregnates Suzy Mae.

Desiderata of a Formal Modeling

◮ logical independence

◮ non-triviality

◮ symmetry/non-ad-hoc modeling

◮ detachment possible

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Chisholm continued

Option 1

◮ ⊤ ⊃ O¬i

◮ i ⊃ Om

◮ ¬i ⊃ O¬m

◮ i

What’s the problem?

Option 2

◮ O(⊤ ⊃ ¬i)

◮ O(i ⊃ m)

◮ O(¬i ⊃ ¬m)

◮ i

What’s the problem?

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Chisholm continued

Option 3

◮ O(⊤ ⊃ ¬i)

◮ O(i ⊃ m)

◮ ¬i ⊃ O¬m

◮ i

What’s the problem?

Option 4

◮ O(⊤ ⊃ ¬i)

◮ i ⊃ Om

◮ O(¬i ⊃ ¬m)

◮ i

What’s the problem?

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Some Approaches to Conditional Obligations

◮ use of binary modal operators

◮ default logic approach

◮ Input/Output logic

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Using preferential semantics for a dyadic version of SDL

◮ M = 〈W ,≤, v〉 where ≤ is a connected preorder (reflexiveand transitive)

◮ M,w |= O(B/A) iff there is a w ′ ∈ F ≤w such thatM,w ′ |= A ∧ B and for any w ′′ for which w ′ ≤w w ′′:M,w ′′ |= A implies M,w ′′ |= B .

◮ axiomatized by:

RCE If ⊢ A ≡ A′ then ⊢ O(B/A) ≡ O(B/A′)RCM If ⊢ B ⊃ C then ⊢ O(B/A) ⊃ O(C/A)CK O(B ⊃ C/A) ⊃ (O(B/A) ⊃ O(C/A))CD O(B/A) ⊃ ¬O(¬B/A)CN O(⊤/⊤)

CO∧ O(B/A) ⊃ O(A ∧ B/A)trans ((A ≥ B) ∧ (B ≥ C )) ⊃ (A ≥ C ) where

A ≥ B =df ¬O(¬A/A ∨ B) (read: “A is at least as good asB”)

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SDDL and a Problem with Specificity

Problem: we get Rational Monotonicity:

RM (O(A/B) ∧ P(C/B)) ⊃ O(A/B ∧ C )

e.g., (O(¬f /m) ∧ P(a/m)) ⊃ O(¬f /m ∧ a)

◮ Various solutions have been proposed to weaken themonotonicity principle further and using different semantics(such as neighborhood semantics).

◮ All that I know generate other problematic examples (see [17,Part IV])

◮ One option: instead of “hard-coding” a weakenedmonotonicity principle, “go adaptive”. I.e., apply monotonicity(O(A/B) ⊃ O(A/B ∧ C )) defeasibly “as much as possible”.

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The Detachment Problem

◮ Van Eck: “How can we take seriously a conditional obligationif it cannot, by way of detachment, lead to an unconditionalobligation?” [21]

◮ SDDL has not means for detachment

◮ general problem: the possibility of specificity cases andCTD-cases means that we cannot apply MP naively toconditional obligations

◮ how to deal with this problem then? E.g.,◮ Input/Output logic (Makinson / Van Der Torre)◮ Adaptive logics (the Gent crew)◮ Default Logic (Horty)

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