Prima Facie Norms, Normative Conflicts, and Dilemmasxp.3pts.net/pdf/teach/conflicts.pdf · Prima Facie Norms, Normative Conßicts, and Dilemmas 13 As Horty presents the case, he imagines

IntroductionAccommodating conflicts

Resolving conflicts

Prima Facie Norms, Normative Conflicts, andDilemmas

by L. Goble

Lecturer: X. Parent

by L. Goble Prima Facie Norms, Normative Conflicts, and Dilemmas


Resolving conflicts

Chapter summary

Normative conflicts

OA, OB but ¬♦(A ∧ B) (¬: not / ♦:possible)

Problem:

Horn1: they are common-placeHorn 2: ... and ruled out by core principles of deontic logic

Aim of chapter

Roadmap of all the solutions available from literatureAssess them



Resolving conflicts

Lecture layout

Lecture layout (with required reading)

1© How come conflicts are ruled out by deontic logic?

Section 1 “The dilemma of normative conflicts”

2© Acommodating conflicts

Sections 5.1 to 5.3the problem will be introduced, but not resolved

3© Conflicts resolution mechanismsIntroduction to Section 4.3 on prima facie oughts

last paragraph may be skipped

Section 4.3.2 “Defeasible inference”



Resolving conflicts

Examples from text

Ada’s predicament (opening paragraph)

2 Lou Goble

4.3.3 Overriding prima facie oughts . . . . . . . . 434.4 Basic prima facie oughts revisited . . . . . . . . . . . 46

4.4.1 The criteria . . . . . . . . . . . . . . . . . . . 514.4.2 Derived prima facie oughts . . . . . . . . . . 53

4.5 A hybrid account . . . . . . . . . . . . . . . . . . . . . 544.6 Reprise . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Conflict 1: Revisionist strategies . . . . . . . . . . . . . . . . 575.1 Non-Kantian systems . . . . . . . . . . . . . . . . . . 585.2 Non-aggregative systems . . . . . . . . . . . . . . . . . 59

5.2.1 Consistent aggregation . . . . . . . . . . . . 635.3 Non-distributive systems . . . . . . . . . . . . . . . . . 65

5.3.1 Consistent distribution . . . . . . . . . . . . 675.3.2 Unconflicted distribution . . . . . . . . . . . 69

5.4 Systems with limited replacement . . . . . . . . . . . 755.5 Reprise . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Conflict 2: More radical strategies . . . . . . . . . . . . . . . 816.1 Paraconsistent deontic logics . . . . . . . . . . . . . . 816.2 Two-phase deontic logic . . . . . . . . . . . . . . . . . 866.3 An imperatival approach . . . . . . . . . . . . . . . . 906.4 Adaptive deontic logics . . . . . . . . . . . . . . . . . 986.5 Reprise . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Ada promised to take her son to the circus Friday afternoon, and sopresumably she ought to spend that afternoon with him at the circus. Italso happens, however, that there is an important meeting of her committeethat same afternoon, and she ought to be present for that. She cannotdo both. Ada seems stuck; whatever she does, it seems she will not dosomething she ought to do.

1 The dilemma of normative conflicts

Ada faces a normative conflict. Her situation is typical of what we willbe discussing in this chapter. Generally speaking, there is a normativeconflict when an agent ought to do a number of things, each of which ispossible for the agent, but it is impossible for the agent to do them all. Theprospect of such conflicts poses a dilemma for a logic of normative concepts.On one hand, they appear to be commonplace. We have all, no doubt,found ourselves in positions like Ada’s from time to time. The literatureon normative conflicts and moral dilemmas is replete with examples, fromthe pedestrian to the poignant; we will see some classic ones below. On

Logical representation: p, m, Oc , Oa, ¬♦(c ∧ a)

♦: possible



Resolving conflicts

Examples from text

Ross’s example (p. 13)

Prima Facie Norms, Normative Conflicts, and Dilemmas 13

As Horty presents the case, he imagines that the two premises stem fromdi↵erent sources, (i) from the laws of Smith’s country and (ii) from hisreligious convictions or conscience. While that is accidental to the fullsignificance of the example, here it is useful. If obligations stemming fromseparate sources should be represented by separate operators, we shouldrepresent the premises of the argument by, for example,

i)0 O`(f _ s)ii)0 Oc¬f

But now there is no way to represent the conclusion. That Smith performalternative service is not required by the laws of his country, nor by hisreligious convictions. Even if what one ought to do is often determined bydi↵erent sources or authorities, insofar as propositions of what one ought todo serve as guides to action or as standards of evaluation of an agent’s overallactions, there must be a common ought derived from those separate sources.We want to be able to represent the inference of the Smith Argument by

i)⇤ O(f _ s)ii)⇤ O¬f

) iii)⇤ Os

where the O represents that common ought. If there are cases like OmA andO`B, when ¬3(A^B), then there still seems to be a normative conflict, OAand OB, for that common sense of ought, for which the appeal to separatedomains or separate authorities or, more generally, separate contexts has nopurchase. Hence the challenge remains to explain the examples of conflictwhen they apply either to the ought of a single domain or to the commonought derived from the oughts of di↵erent domains.

In all that follows we shall assume that the operator O represents aunivocal sense of ‘ought’ in its setting. When distinctions are appropriate,they will be indicated by subscripts, superscripts or other devices.

There is another way the oughts in the examples might be consideredambiguous. At the end of Section 2.1, I mentioned the distinction, familiarfrom ethical theory, between prima facie and all-things-considered obliga-tions. When Ross [1930] introduced this distinction, or this terminology forit, he illustrated it (p. 18) with the well-known example of a person whohas promised to meet a friend for some trivial purpose, but who could alsohelp the victims of a serious accident, though only by breaking the promise.Because of the promise, presumably the person ought to meet the friend.Because of the need of the accident’s victims, presumably the person oughtto help them. Each is possible, but they are not jointly possible. Becausethis example is so common in the literature, and because we will refer to itfrequently, we henceforth call it simply the Ross Example.Logical representation: p, a, Om, Oh, ¬♦(m ∧ h)



Resolving conflicts

Why are conflicts ruled out by deontic logic?Argument 1

Pausible laws

(P) OA→ ♦A (♦: possible) (ought-can)(And) (OA ∧ OB)→ O(A ∧ B) (aggregation/agglomeration)

NB: Goble uses the label C for the second.

Problem

Assume OA and OB hold, but ¬♦(A ∧ B)

OA ∧ OBAnd

O(A ∧ B)P♦(A ∧ B)



Resolving conflicts


Pausible laws



Problem


OA ∧ OBAnd

O(A ∧ B)P♦(A ∧ B)



Resolving conflicts


Pausible laws



Problem


OA ∧ OBAnd

O(A ∧ B)P♦(A ∧ B)



Resolving conflicts


Pausible laws



Problem


OA ∧ OBAnd

O(A ∧ B)P♦(A ∧ B)



Resolving conflicts


Pausible laws

D) OA→ ¬O¬A (principle of seriality)W) �(A→ B)→ (OA→ OB) (�: necessary)



Resolving conflicts


Pausible laws


About W:

W is a shorthand for Weakening (my own terminology)

Goble uses the label NM

antecedent is very strong: B necessary for A

�(dentist→ bus) - I have no car. etc



Resolving conflicts


Pausible laws


About W:

W is a shorthand for Weakening (my own terminology)

Goble uses the label NM

Compare with O(A→ B)→ (OA→ OB)

A=pay; B= receipt



Resolving conflicts


Pausible laws


Problem




Resolving conflicts


Pausible laws


Problem

¬♦(A ∧ B)

�(A→ ¬B) OAW

O¬BD ¬O¬¬B

PL ¬OB



Resolving conflicts

Why are conflicts ruled out by deontic logic?Summary

In a nutshell

The logic makes the following set inconsistent:

¬♦(A ∧ B)OAOB



Resolving conflicts


In a nutshell


¬♦(A ∧ B)OAOB

Methodological remark

you can make the same points using ` (‘prove’) and ⊥(contradiction) instead � and ♦:�-idiom `-idiom

�(A → B) A ` B¬♦(A ∧ B) A,B ` ⊥♦(A ∧ B) A,B 6` ⊥

During the lecture, I will occasionnally switch idiom



Resolving conflicts


In a nutshell


¬♦(A ∧ B)OAOB

Historical remark

Problem initially raised for DSL - it has all the above laws

The author states the problem in its full generality



Resolving conflicts

Part IHow to accommodate the existence of conflicts



Resolving conflicts

Conflict: Revisionist strategies

Revisionist strategies (sections 6 and 7)

At least one of [(P), (And)] and [(W), (D)] must go

Systems Law rejected

Non-kantian PNon-aggregative AndNon-distributive W? D

Given And/W, D is equivalent to P



Resolving conflicts

Desiderata

Desideratum 1 (conflicts are logically consistent)

¬♦(A ∧ B),OA,OB 0 ⊥ (⊥ : contradiction)Arguments 1 and 2 must be blocked.

Desideratum 2 (Avoid deontic explosion)

The logic should not contain (DEX), or anything like it.DEX) ¬♦(A ∧ B),OA,OB ` OC

Desideratum 3

The logic should explain in a plausible way the apparent validity ofseveral paradigm arguments, including the Smith Argument.



Resolving conflicts

Desiderata


¬♦(A ∧ B),OA,OB 0 ⊥ (⊥ : contradiction)Arguments 1 and 2 must be blocked.

Desideratum 2 (Avoid deontic explosion)

The logic should not contain (DEX), or anything like it.DEX) ¬♦(A ∧ B),OA,OB ` OC

Smith argument (Horty)i) Smith ought to fight in the army or perform alternative national

service. — O(f ∨ s)ii) Smith ought not to fight in the army. — O¬f

∴ iii) Smith ought to perform alternative national service. — Os



Resolving conflicts

Non-kantian systems

Lemmon (1962)

(P) OA→ ♦A rejectedArgument 1 blocked, because P directly involved in the derivation.Argument 2 blocked, because P ⇔ D

Deontic explosion

DEX) ¬♦(A ∧ B),OA,OB ` OC holds



Resolving conflicts

Non-kantian systems

Lemmon (1962)


Deontic explosion


Exercise: show DEX using And and WHint: ¬♦(A ∧ B)↔ �((A ∧ B)→ C )



Resolving conflicts

Non-kantian systems

Lemmon (1962)


Deontic explosion


¬♦(A ∧ B)

�((A ∧ B)→ C )OA OB (And)O(A ∧ B)

(W)OC



Resolving conflicts

Non-aggregative systems


(And) (OA ∧ OB)→ O(A ∧ B) rejected



Resolving conflicts




Implementation - multi-relational semantics

Intuition: conflicts result from clashes between normative systems

R: a set of serial binary deontic alternativeness relations R1,R2, ...

w |= OA iff there is an Ri ∈ R such that:w ′ |= A for all w ′ with wRiw

′



Resolving conflicts




SDL is mono-relational

w

w'w''

R R

OA

A A¬

¬

A

A

we write w → w ′ to express that wRw ′

intuitively: w ′ is a ‘good’ successor of w



Resolving conflicts




SDL is mono-relational

w

w'w''

R R

OA

A A¬

¬

A

A

good bad

we write w → w ′ to express that wRw ′

intuitively: w ′ is a ‘good’ successor of w



Resolving conflicts




Going multi-relational:

famility of accessibility relations R1,R2, ... (one per normativesystem)

w

w'w''

R R

OA

A A¬A

11R2

R2



Resolving conflicts




Going multi-relational:

w |= OA iff there is an Ri ∈ R such that:

w ′ |= A for all w ′ with wRi−→ w ′

w

w'w''

R R

OA

A A¬A

11R2

R2



Resolving conflicts



¬♦(A ∧ B),OA,OB 0 ⊥



Resolving conflicts



¬♦(A ∧ B),OA,OB 0 ⊥

A, ¬ B A, ¬ B

OA OBA, ¬ B¬

R 2R1

∧¬ ♢（）A B



Resolving conflicts


Desideratum 2 (no deontic explosion)

¬♦(A ∧ B),OA,OB 6` OC



Resolving conflicts


Desideratum 2 (no deontic explosion)

¬♦(A ∧ B),OA,OB 6` OC

A ¬ B A ¬ B

OA OBA, ¬ B¬

R 2R1

A ¬ B C C¬C

C¬A ¬

B

w

∧¬ ♢（）A BOC false



Resolving conflicts


Desideratum 3 (Smith argument)

We want O(A ∨ B),O¬A ` OB



Resolving conflicts


Desideratum 3 (Smith argument)

We want O(A ∨ B),O¬A ` OB

A ¬ B A ¬ B

O¬A

R 2R1

A B

A ¬B

w

¬

O(A∨B)OB false at w



Resolving conflicts

Non-distributive systems


(W) �(A→ B)→ (OA→ OB) rejected - along with (P).



Resolving conflicts




Why call them “distributive”?

(W) implies O distributes over ∧:

O(A ∧ B)→ OA



Resolving conflicts




Desideratum 3 - Smith Argument

Below: a proof of O(A ∨ B),OA ` OB

O(A ∨ B) O¬A(And)

O((A ∨ B) ∧ ¬A) �(((A ∨ B) ∧ ¬A)→ B)(W)

OB



Resolving conflicts




Desideratum 3 - Smith Argument

O(A ∨ B) O¬A(And)

O((A ∨ B) ∧ ¬A) �(((A ∨ B) ∧ ¬A)→ B)(W)

OB

Enough to disqualify the approach.



Resolving conflicts

Let’s take stock!

Summary

By rejecting one of the rules, the logic is either too strong(desideratum 2 not met) or too weak (desideratum 3 not met)

What do we do?



Resolving conflicts

Let’s take stock!

Summary


What do we do?

Mainstream solution

Keep all the rules, but restrict their application

Explains why more and more have been moving away frompossible-worlds semantics (SDL, DSDL).



Resolving conflicts

Let’s take stock!

Summary


What do we do?

Main research challenge

Find the best way to do it

consistency proviso “A is consistent with B” or ♦(A ∧ B)

permissions

Explains why more and more have been moving away frompossible-worlds semantics (SDL, DSDL).



Resolving conflicts

Restricted And

Restricted aggregation:

©A ©B A consistent with B

©(A ∧ B)



Resolving conflicts

Restricted And



©(A ∧ B)

Deontic explosion

OA O¬A(And)O(A ∧ ¬A) A ∧ ¬A ` B

(W)OB



Resolving conflicts

Restricted And



©(A ∧ B)

Deontic explosion not avoided

OA(W)O(A ∨ B) O¬A

(And)O((A ∨ B) ∧ ¬A)

(W)OB



Resolving conflicts

Part IIConflict resolution mecanisms for obligations



Resolving conflicts

Overall picture2 types of obligations (Ross, 1930)

Ceteris paribus principle: a hedged generalization

Other things being equal, lying is wrongConditional

what to do in the absence of furthercomplications

May conflict with each others

Prima facie obligation: as if I have that duty

May conflictStill binding

All-things-considered obligation: what I must do after balancing allthe conflicting prima facie obligations

Duty proper



Resolving conflicts

Overall picture2 types of obligations (Ross, 1930)

Hybrid approach

Prima facie obligations can conflict; all-things-consideredobligations cannot

Core principles of deontic logic

revised for prima facie obligationsmaintained for all-things-considered obligations



Resolving conflicts

Prima facie obligations

Preliminary step

In a given scenario, what are the prima facie obligations?



Resolving conflicts


Preliminary step


KNOWLEDGE BASE

input output

facts

ceteris paribus principles

prima facie obligations



Resolving conflicts


Preliminary step


We write Γ |∼ A to say that Γ outputs A

|∼: a defeasible consequence relation (called “snake”)

On top of a deontic logic with all the core principles`d : consequence relation of the base logic

Output A is a sentence of the form Opf B

To avoid clusters of symbols, we say

Opf B is ‘in’, when Γ |∼ Opf B

Opf B is ‘out’, when Γ 6|∼ Opf B

Question

How to define in/out (aka |∼) using `d?



Resolving conflicts


Preliminary step


Faithfull maximal consistent subset


other information in �d, this set �d+c may well be inconsistent. As a result,we cannot say simply that A is supported by � just in case it is a logicalconsequence of �d+c. Instead we look at maximal consistent subsets of �d+c,and the information on which they all agree, and we expect they all agreeabout the information in �d, that they are faithful to the basic informationgiven.

• A set �⇤ is a faithful maximal consistent subset of �d+c if and only if(i) �d ✓ �⇤ and (ii) �⇤ ✓ �d+c, and (iii) �⇤ is consistent (as definedby `d) and (iv) there is no B 2 �d+c such that B /2 �⇤ and �⇤ [ {B}is consistent.

(i) provides faithfulness, (ii) inclusion in �d+c, (iii) consistency, and (iv)maximality within �d+c. This gives our first definition for � |⇠ A thatrepresents the defeasible inference of A from �.

Def 6) � |⇠ A if and only if �⇤ `d A, for each �⇤ that is afaithful maximal consistent subset of �d+c.

We can apply this definition to the Ross Example to see its virtues,and its limitations. Given just the fact of the promise to meet and theprinciple that if one has promised to meet then ceteris paribus one oughtto meet, one concludes one ought to meet. � = {p, p � Om}. Hence�d = {p}, �c = {p � Om}, C(�c) = {Om}, �d+c = {p, Om}, and that is theonly faithful maximal consistent extension of itself. Since {p, Om} `d Om,� |⇠ Om. The same applies with respect to the accident and helping thevictims. If � = {a, a � Oh} then � |⇠ Oh.

On the other hand, given both facts of the promise and the accidentand the two principles, one expects not to infer Om. Here � = {p, a, p �Om, a � Oh, ¬3(m ^ h)}, and �d = {p, a,¬3(m ^ h)} and �c = {p �Om, a � Oh}. C(�c) = {Om, Oh}. Since O follows all the core principles,Om ^ Oh `d 3(m ^ h). Hence �d `d ¬(Om ^ Oh). As a result, Omand Oh cannot both be in any faithful maximal consistent subset of �d+c.Instead there are two such subsets, �1 = {p, a,¬3(m ^ h), Om} and �2 ={p, a,¬3(m ^ h), Oh}. Since Om is not a consequence of both of those,� |⇠6 Om, as it should be.

Less satisfying, (Def 6) also yields � |⇠6 Oh. The problem is that it doesnot take relations of priority of prima facie oughts into consideration. Allthe consequent oughts are treated equally when they should not be. Thatleads to our second proposal that will take such relations into account.

So far, however, we have no account of when one certeris paribus principletakes priority over another. Following common practice, we now take thatto be determined by specificity, that more specific principles take precedence

Figure : Page 41

Γd+c = Γd ∪ C(Γc)C(∆) = {C : ∃B(B � C ∈ ∆)}



Resolving conflicts


Preliminary step


Credulous approach


other information in �d, this set �d+c may well be inconsistent. As a result,we cannot say simply that A is supported by � just in case it is a logicalconsequence of �d+c. Instead we look at maximal consistent subsets of �d+c,and the information on which they all agree, and we expect they all agreeabout the information in �d, that they are faithful to the basic informationgiven.

• A set �⇤ is a faithful maximal consistent subset of �d+c if and only if(i) �d ✓ �⇤ and (ii) �⇤ ✓ �d+c, and (iii) �⇤ is consistent (as definedby `d) and (iv) there is no B 2 �d+c such that B /2 �⇤ and �⇤ [ {B}is consistent.

(i) provides faithfulness, (ii) inclusion in �d+c, (iii) consistency, and (iv)maximality within �d+c. This gives our first definition for � |⇠ A thatrepresents the defeasible inference of A from �.

Def 6) � |⇠ A if and only if �⇤ `d A, for each �⇤ that is afaithful maximal consistent subset of �d+c.

We can apply this definition to the Ross Example to see its virtues,and its limitations. Given just the fact of the promise to meet and theprinciple that if one has promised to meet then ceteris paribus one oughtto meet, one concludes one ought to meet. � = {p, p � Om}. Hence�d = {p}, �c = {p � Om}, C(�c) = {Om}, �d+c = {p, Om}, and that is theonly faithful maximal consistent extension of itself. Since {p, Om} `d Om,� |⇠ Om. The same applies with respect to the accident and helping thevictims. If � = {a, a � Oh} then � |⇠ Oh.

On the other hand, given both facts of the promise and the accidentand the two principles, one expects not to infer Om. Here � = {p, a, p �Om, a � Oh, ¬3(m ^ h)}, and �d = {p, a,¬3(m ^ h)} and �c = {p �Om, a � Oh}. C(�c) = {Om, Oh}. Since O follows all the core principles,Om ^ Oh `d 3(m ^ h). Hence �d `d ¬(Om ^ Oh). As a result, Omand Oh cannot both be in any faithful maximal consistent subset of �d+c.Instead there are two such subsets, �1 = {p, a,¬3(m ^ h), Om} and �2 ={p, a,¬3(m ^ h), Oh}. Since Om is not a consequence of both of those,� |⇠6 Om, as it should be.

Less satisfying, (Def 6) also yields � |⇠6 Oh. The problem is that it doesnot take relations of priority of prima facie oughts into consideration. Allthe consequent oughts are treated equally when they should not be. Thatleads to our second proposal that will take such relations into account.

So far, however, we have no account of when one certeris paribus principletakes priority over another. Following common practice, we now take thatto be determined by specificity, that more specific principles take precedence

some

Figure : Page 41



Resolving conflicts


Preliminary step


Credulous approach

Opf A is

{in if ∃Γ∗s.t.Γ∗ `d A

out otherwise.

N.B.: Γ∗ must be a faithfull maximal consistent subset of Γd+c



Resolving conflicts


Preliminary step


Promise exampleΓd = {p, a,¬♦(m ∧ h)}Γc = {p � Om, a� Oh}

Γd+c = {p, a,¬♦(m ∧ h),Om,Oh} `d ⊥

Γ?1 = {p, a,¬♦(m ∧ h),Om} 6`d ⊥, and maximally so

Γ?2 = {p, a,¬♦(m ∧ h),Oh} 6`d ⊥, and maximally so



Resolving conflicts


Preliminary step


Promise exampleΓd = {p, a,¬♦(m ∧ h)}Γc = {p � Om, a� Oh}

Γd+c = {p, a,¬♦(m ∧ h),Om,Oh} `d ⊥

Γ?1 = {p, a,¬♦(m ∧ h),Om} 6`d ⊥, and maximally so

Opfm is ‘in’ , since Γ?1 `d Om

Γ?2 = {p, a,¬♦(m ∧ h),Oh} 6`d ⊥, and maximally so

Opf h is ‘in’ , since Γ?2 `d Oh



Resolving conflicts

Conflicts resolution mechanismsIncremental procedure - A standard in AI

Incremental procedure, p. 45

42 Lou Goble

over less specific ones. A proposition A is more specific than another, B, ifA necessitates B but B does not necessitate A. In the context of a given �we take that A necessitates B with respect to � to mean that the necessitieswithin � together with A entail B, i.e., if �n = {⇤C : ⇤C 2 �d}, then Anecessitates B for � just in case �n, A `d B.

The relative strength of conditionals is determined by the specificity oftheir antecedents.

• A � B is stronger than C � D for � — (A � B) =� (C � D) —just in case �n, A `d C and �n, C 0d A.

This relation, =�, is a strict partial order, irreflexive and transitive.This relation orders the conditionals in �c somewhat, but it could easily

be that neither (A � B) =� (C � D) nor (C � D) =� (A � B). Leta complete strict order, �, over �c be said to ‘respect’ =� just in case=� ✓ �. Such relations merely fill in the gaps left by =�. There canbe many such respectful total orders for a given �. These determine the‘preferred’ maximal consistent subsets of �d+c that define � |⇠ A, preferredbecause they do take priorities into account.

For a complete strict order, �, over �c that respects =�, let the orderingof �c by � be hc1, . . . , ci, . . .i, so that ci < cj in the sequence just in caseci � cj .

47 Then define �� from a sequence of subsets, �i�, of �d+c, thus:

• �0� = �d;

• for 0 i < j = i+1, �j� = �i

�[{C(cj)} if that is consistent, otherwise

�j� = �i

�;

• �� =S

�i�.

Each such set, ��, is a faithful maximal consistent subset of �d+c, andso is appropriate to apply to define � |⇠ A.

Def 7) � |⇠ A i↵ �� `d A, for each complete strict order, �, thatrespects =�.

(Def 7) di↵ers from (Def 6) only when �c contains more than one member.Hence the little examples from the start of the Ross Example are treated thesame as before. For the full Ross Example, however, in which we supposealso (p ^ a) � Oh 2 � in order to catch the priority of the obligationto help over the obligation to meet the friend, then we see, �d = {p, a,

47We ignore for convenience the possibility that �c might be empty; that case is easilyincorporated into the account to follow.



Resolving conflicts

Conflicts resolution mechanismsIncremental procedure - A standard in AI

Question

In a given scenario, what are the all-things-considered (ATC)obligations?

Basic idea

Priority relation on the obligationsPick up a “preferred” faithfull maximal consistent subsetNB: priority relation is given



Resolving conflicts

Preferred faithfull maximal consistent subset

Let Γc = {c1, c2, ..., } be totally ordered by Aci A cj : ci is stronger than (overrides, etc) cj

Rearrange the ci ’s in Γc by decreasing strenght



Resolving conflicts


Φ: ‘preferred’ faithfull maximal consistent subsetΦ: defined from a sequence Φ0, Φ1,..., of subsets of Γd+c

Φ0 = Γd

For 1 ≤ i ,Φi =

{Φi−1 ∪ {C(ci )} if that is consistent,

Φi−1 otherwise.

Put Φ =⋃

Φi

OatcA is

{in if Φ =

⋃Φi `d OA

out otherwise.



Resolving conflicts


Promise example

Γd = {p, a,¬♦(m ∧ h)}Γc = {p � Om︸︷︷︸

c1

, a� Oh︸︷︷︸c2

with c2 A c1

Here Γd+c = {p, a,¬♦(m ∧ h),Om,Oh} `d ⊥



Resolving conflicts


Promise example

Γd = {p, a,¬♦(m ∧ h)}Γc = {a� Oh, p � Om} 1© Commute

Here Γd+c = {p, a,¬♦(m ∧ h),Om,Oh}

2© Calculate Φ (preferred faithfull maximal consistent subset)

Φ0 = Γd = {p, a,¬♦(m ∧ h)}Φ1 = {p, a,¬♦(m ∧ h)︸︷︷︸

Φ0

,Oh}

Φ2 = Φ1...

Φ = Φ1

3©ATC obligationsOatch in, since Φ `d OhOatcm out, since Φ 6`d Om



Resolving conflicts

Completion of an order

Back to the def from text

42 Lou Goble

over less specific ones. A proposition A is more specific than another, B, ifA necessitates B but B does not necessitate A. In the context of a given �we take that A necessitates B with respect to � to mean that the necessitieswithin � together with A entail B, i.e., if �n = {⇤C : ⇤C 2 �d}, then Anecessitates B for � just in case �n, A `d B.

The relative strength of conditionals is determined by the specificity oftheir antecedents.

• A � B is stronger than C � D for � — (A � B) =� (C � D) —just in case �n, A `d C and �n, C 0d A.

This relation, =�, is a strict partial order, irreflexive and transitive.This relation orders the conditionals in �c somewhat, but it could easily

be that neither (A � B) =� (C � D) nor (C � D) =� (A � B). Leta complete strict order, �, over �c be said to ‘respect’ =� just in case=� ✓ �. Such relations merely fill in the gaps left by =�. There canbe many such respectful total orders for a given �. These determine the‘preferred’ maximal consistent subsets of �d+c that define � |⇠ A, preferredbecause they do take priorities into account.

For a complete strict order, �, over �c that respects =�, let the orderingof �c by � be hc1, . . . , ci, . . .i, so that ci < cj in the sequence just in caseci � cj .

47 Then define �� from a sequence of subsets, �i�, of �d+c, thus:

• �0� = �d;

• for 0 i < j = i+1, �j� = �i

�[{C(cj)} if that is consistent, otherwise

�j� = �i

�;

• �� =S

�i�.

Each such set, ��, is a faithful maximal consistent subset of �d+c, andso is appropriate to apply to define � |⇠ A.

Def 7) � |⇠ A i↵ �� `d A, for each complete strict order, �, thatrespects =�.

(Def 7) di↵ers from (Def 6) only when �c contains more than one member.Hence the little examples from the start of the Ross Example are treated thesame as before. For the full Ross Example, however, in which we supposealso (p ^ a) � Oh 2 � in order to catch the priority of the obligationto help over the obligation to meet the friend, then we see, �d = {p, a,

47We ignore for convenience the possibility that �c might be empty; that case is easilyincorporated into the account to follow.



Resolving conflicts


What if gaps in the ranking?

ConsiderΓd = ∅Γc = {c1, c2, c3, c4}

Captain(c1) > � OAMajor(c2) > � O¬APriest(c3) > � OABishop(c4) > � O¬A

c2 A c1 c4 A c3



Resolving conflicts


Completion

Let A be a partial order. � is said a completion of A if

� extends (“respects”) A, viz A⊆�� is a total order

� just fills in the gaps left in A

There might be many such completions



Resolving conflicts


Completion





c2 A c1 c4 A c3



Resolving conflicts


Completion





c2 A c1 c4 A c3



Resolving conflicts


Completion




There might be many such completions �.

Each � gives its own faithfull maximal consistent subset.

Subscript notation: Φ�

OatcA is

{in if Φ� `d OA for all completion �out otherwise.



Resolving conflicts


Completion




There might be many such completions �.

Captain/Priest example: the procedure yields correct result

Oatc¬A in



Resolving conflicts

Non-triggerred obligations

ConsiderΓd = {¬(A ∧ B)}Γc = {c1, c2, c3}

Priest(c1) > � OABishop(c2) > � OBCardinal(c3) A� O¬B

c3 A c2 A c1

A: heating onB: window open

OatcA and Oatc¬B both inOatcB out

Is it as it should be?



Resolving conflicts

Non-triggerred obligations

ConsiderΓd = {¬(A ∧ B)}Γc = {c1, c2, c3}

Priest(c1) > � OABishop(c2) > � OBCardinal(c3) A� O¬B

c3 A c2 A c1

A: heating onB: window open

OatcA and Oatc¬B both inOatcB out

Is it as it should be?


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Prima Facie Norms, Normative Conflicts, and Dilemmasxp.3pts.net/pdf/teach/conflicts.pdf · Prima Facie Norms, Normative Conßicts, and Dilemmas 13 As Horty presents the case, he imagines