Developing Default Logic as a Theory of Reasonsusers.umiacs.umd.edu/~horty/courses/readings/ddl-talk... · 2012-09-26 · vidual reasons. The function may not be simply addi-tive,

Developing Default Logic as a

Theory of Reasons

John Horty

University of Marylandwww.umiacs.umd.edu/users/horty

1

Introduction

1. Tools for understanding deliberation/justification:

Standard deductive logic

Decision theory (plus probability, inductivelogic)

2. But ordinairly, we seem to focus on reasons—inboth deliberation and justification

3. Examples:

We should eat at Obelisk tonight

Racoons have been in the back yard again

4. This could be an allusion, or an abbreviation, orheuristic

But it could also be right . . .

. . . an idea common in epistemology, and especiallyin ethics

2

5. Common questions about reasons:

Relation between reasons and motivation?

Relation between reasons and desires?

Relation between reasons and values?

Objectivity of reason?

6. A different question:

How do reasons support actions or conclu-sions?

What is the mechanism of support?

3

7. One answer (numerical weighing):

. . . my way is to divide half a sheet ofpaper by a line into two columns; writingover the one Pro, and over the other Con

Then . . . I put down under the differentheads short hints of the different motives,that at different times occur to me, for oragainst the measure.

When I have thus got them all togetherin one view, I endeavor to estimate theirrespective weights; and where I find two,one on each side, that seem equal, I strikethem both out.

If I find a reason pro equal to some tworeasons con, I strike out the three. If I judgesome two reasons con, equal to three rea-sons pro, I strike out the five; and thus pro-ceeding I find at length where the balancelies . . .

(Benjamin Franklin,letter to Joseph Priestley)

Objection: too simple, optimistic

4

8. Another answer (qualitative weighing):

Each reason is associated with a metaphor-ical weight. This weight need not be any-thing so precise as a number; it may be anentity of some vaguer sort.

The reasons for you to φ and those foryou not to φ are aggregated or weighedtogether in some way. The aggregate issome function of the weights of the indi-vidual reasons.

The function may not be simply addi-tive, as it is in the mechanical case. It maybe a complicated function, and the specificnature of the reasons may influence it.

Finally, the aggregate comes out in favorof your φing, and that is why you oughtto φ.

(John Broome, Reasons)

Objection: What are these vaguer entities? Whatis the aggregating function?

5

9. An analogy:

Compare

(1) How do reasons support conclusions?

to

(2) How do premises support conclusions?

Answers to (2) provided by Frege and Tarski

We want an answer to (1) at the same level ofrigor, evaluated by the same standards

10. My answer:

Reasons are (provided by) defaults

The logic of defaults tells us how reasonssupport conclusions

6

11. Talk outline:

Prioritized default logic

Extensions, scenarios

Triggering, conflict, defeat

Binding defaults, proper scenarios

Elaborating the theory

Variable priorities

Undercutting (exclusionary) defeat

Exclusion and priorities

Applications and open questions

Moral particularism

Floating conclusions

7

Fixed priority default theories

1. Notation:

Propositions: A, B, C, . . . ,>

Background language: ∧, ∨, ¬, ⇒

Consequence: `

Logical closure: Th(E) = {A : E ` A}

2. Example:

Tweety is a bird

Therefore, Tweety is able to fly

Why? There is a default that birds fly

Tweety is a bird

Tweety is a penguin

Therefore, Tweety is not able to fly

Because there is a (stronger) default that penguinsdon’t fly

3. Default rules: X → Y

Example: B(t) → F(t)

Instance of: B(x) → F(x) (“Birds fly”)

8

4. Premise and conclusion:

If δ = X → Y , then

Prem(δ) = X

Conc(δ) = Y

If D set of defaults, then

Conc(D) = {Conc(δ) : δ ∈ D}

5. Priority ordering on defaults (strict, partial)

δ < δ′ means: δ′ stronger than δ

6. Priorities have different sources:

Specificity

Reliability

Authority

Our own reasoning

For now, take priorities as fixed, leading to . . .

9

P

F

B

d 1 < d

2

d 1

d 2

7. A fixed priority default theory is a tuple

〈W ,D, <〉

where W contains ordinary statements, D containsdefaults, and < is an ordering

8. Example (Tweety Triangle):

W = {P, P ⇒ B}D = {δ1, δ2}δ1 = B → F

δ2 = P → ¬Fδ1 < δ2

(P = Penguin, B = Bird, F = Flies)

9. Main question: what can we conclude from sucha theory?

10

10. An extension E of 〈W ,D, <〉 is a belief set an idealreasoner might settle on, based this information

Usually defined directly, but we take roundaboutroute . . .

11. A scenario based on 〈W ,D, <〉 is some subset S ofthe defaults D

12. A proper scenario is the “right” subset of defaults

13. An extension E based on 〈W ,D, <〉 is a set

E = Th(W ∪ Conc(S))

where S is a proper scenario

11

P

F

B

d 1 < d

2

d 1

d 2

14. Returning to example: 〈W ,D, <〉 where

W = {P, P ⇒ B}D = {δ1, δ2}δ1 = B → Fδ2 = P → ¬Fδ1 < δ2

Four possible scenarios:

S1 = ∅S2 = {δ1}S3 = {δ2}S4 = {δ1, δ2}

But only S3 proper (“right”), so extension is

E3 = Th(W ∪ Conc(S3))= Th({P, P ⊃ B} ∪ {¬F})= Th({P, P ⊃ B,¬F}),

15. Immediate goal: specify proper scenarios

12

Binding defaults

1. If defaults provide reasons, binding defaults pro-vide good reasons—forceful, or persuasive, in acontext of a scenario

Defined through preliminary concepts:

Triggering

Conflict

Defeat

2. Triggered defaults:

TriggeredW ,D,<(S) = {δ ∈ D : W ∪ Conc(S) ` Prem(δ)}

3. Example: 〈W ,D, <〉 with

W = {B}D = {δ1, δ2}δ1 = B → Fδ2 = P → ¬F

δ1 < δ2

Then

TriggeredW ,D,<(∅) = {δ1}

4. Terminology question: What are reasons?

Answer: Reasons are antecedents oftriggered defaults

13

R Q

P

d 1 d

2

5. Conflicted defaults:

ConflictedW ,D,<(S) = {δ ∈ D : W ∪ Conc(S) ` ¬Conc(δ)}

6. Example (Nixon Diamond):

Take 〈W ,D, <〉 with

W = {Q, R}D = {δ1, δ2}δ1 = Q → Pδ2 = R → ¬P

< = ∅.

(Q = Quaker, R = Republican, P = Pacifist)

Then

TriggeredW ,D,<(∅) = {δ1, δ2}ConflictedW ,D,<(∅) = ∅

But

ConflictedW ,D,<({δ1}) = {δ2}ConflictedW ,D,<({δ2}) = {δ1}

14

P

F

B

d 1 < d

2

d 1

d 2

7. Basic idea: A default is defeated if there is astronger reason supporting a contrary conclusion

DefeatedW ,D,<(S) = {δ ∈ D : ∃ δ′ ∈ TriggeredW ,D,<(S).

(1) δ < δ′

(2) Conc(δ′) ` ¬Conc(δ)}.

8. Example of defeat (Tweety, again):

〈W ,D, <〉 where

W = {P, P ⇒ B}D = {δ1, δ2}δ1 = B → Fδ2 = P → ¬F

δ1 < δ2

Here, δ1 is defeated:

DefeatedW ,D,<(∅) = {δ1}

15

P

F

B

d 1 < d

2

d 1

d 2

9. Finally, binding defaults:

BindingW ,D,<(S) = {δ ∈ D : δ ∈ TriggeredW ,D,<(S)

δ 6∈ ConflictedW ,D,<(S)

δ 6∈ DefeatedW ,D,<(S)}

10. Stable scenarios: S is stable just in case

S = BindingW ,D,<(S)

11. Example (Tweety, yet again): four scenarios

S1 = ∅S2 = {δ1}S3 = {δ2}S4 = {δ1, δ2}

Only S3 = {δ2} is stable, because

S3 = BindingW ,D,<(S3)

16

Three complications

1. Complication #1: Can we just identify the properscenarios with the stable scenarios?

Almost . . . but not quite

2. Problem is “groundedness”


W = ∅D = {δ1}δ1 = A → A< = ∅.

Then S1 = {δ1} is a stable scenario, but shouldn’tbe proper

The belief set generated by S1 is

Th(W ∪ Conc(S)) = Th({A})

but that’s not right!

17

3. Solution:

Let

ThS(W) =Formulas provable from W when ordi-nary inference rules supplemented withdefaults from S

Then given theory 〈W ,D, <〉, define scenario S asgrounded in W iff

Th(W ∪ Conc(S)) = ThS(W)

Finally, given 〈W ,D, <〉, define S as proper scenariobased on this theory iff

S is (i) stable and (ii) grounded in W

18

d 1 < d

2

d 1

d 2

A

4. Complication #2: Some theories have no properscenarios, and so no extensions

Example: 〈W ,D, <〉 with

W = ∅D = {δ1, δ2}δ1 = > → Aδ2 = A → ¬A

δ1 < δ2

5. Options:

Syntactic restrictions to rule out “viciouscycles”

Generalize definition of proper scenario, us-ing tools from truth theory

Live with it (benign choice if we like “skep-tical” theory)

19

R Q

P

d 1 d

2

6. Complication #3: Some theories have multiple

proper scenarios, and so multiple extensions

Example: Nixon Diamond, again


W = {Q, R}D = {δ1, δ2}δ1 = Q → Pδ2 = R → ¬P< = ∅.

Then two proper scenarios

S1 = {δ1}S2 = {δ2}

and so two extensions:

E1 = Th({Q, R, P})E2 = Th({Q, R,¬P})

So . . . what should we conclude?

20

7. Consider three options:

#1. Choice: pick an arbitrary proper scenario

Sensible, actually

But hard to codify as a consequence rela-tion

#2. Brave/credulous: give some weight to any con-clusion A contained in some extension

• Epistemic version (crazy): Endorse A when-ever A is contained in some extension

Example: P and ¬P in Nixon case

• Epistemic version (not crazy): Endorse B(A)—A is “believable”—whenever A is containedin some extension

Example: B(P) and B(¬P) in Nixon case

• Practical version: Endorse ©(A)—A is an“ought”—whenever A is contained in someextension

Example: ©(P) and ©(¬P) in Nixon case

#3. Cautious/“Skeptical”: endorse A as conclusionwhenever A contained in every extension

Defines consequence relation, and not weird:supports neither P nor ¬P in Nixon case

Note: most popular option, but some prob-lems . . .

21

Elaborating default logic

1. Discuss here only two things:

Ability to reason about priorities

Treatment of “undercutting” or “exclusion-ary” defeat

2. Begin with first problem

So far, fixed priorities on default rules

But we can reason about default priorities . . . andthen use the priorities we arrive at to control ourreasoning

22

3. Five steps:

#1. Add priority statements (δ7 < δ9) toobject language

#2. Introduce new variable priority default theories

〈W ,D〉

with priority statements now belonging to Wand D

#3. Add strict priority axioms to W:

δ < δ′ ⇒ ¬(δ′ < δ)

(δ < δ′ ∧ δ′ < δ′′) ⇒ δ < δ′′

#4. Lift priorities from object to meta language

δ <S δ′ iff W ∪ Conc(S) ` δ < δ′.

#5. Proper scenarios for new default theories:

S is a proper scenario based on 〈W ,D〉

iff

S is a proper scenario based on 〈W ,D, <S〉

23

R Q

P

d 2

d 1

d 2 < d

1

d 3

d 4

d 1 < d

2

4. Example (Extended Nixon Diamond):

Consider 〈W ,D〉 where

W contains Q, P

D contains

δ1 = Q → P

δ2 = R → ¬P

δ3 = > → δ2 < δ1

δ4 = > → δ1 < δ2

δ5 = > → δ4 < δ3

Then unique proper scenario is

S = {δ1, δ3, δ5}

24

5. Example (Perfected Security Interest):

Consider 〈W ,D〉 where

W contains

Possession

¬Documents

Later(δSMA, δUCC )

Federal(δSMA)

State(δUCC )

D contains

δUCC = Possession → Perfected

δSMA = ¬Documents → ¬Perfected

δLP = Later(δ, δ′) → δ < δ′

δLS = Federal(δ) ∧ State(δ′) → δ′ < δ

δLSLP = > → δLS < δLP

Unique proper scenario is

S = {δLSLP, δLP , δUCC}

25

6. Undercutting defeat (epistemology), compared torebutting defeat

Example:

The object looks red

My reliable friend says it is not red

Drug 1 makes everything look red

7. Exclusionary reasons (practical reasoning)

Example (Colin’s dilemma, from Raz):

Should son go to private school??

The school provides good education

He’ll meet fancy friends

The school is expensive

Decision would undermine public education

Promise: only consider son’s interests . . .

8. How can this be represented?

One view (Pollock): undercutting a separate formof defeat

My suggestion:

Only ordinary (rebutting) defeat

Enhance the language slightly

Tweak the notion of triggering

26

9. Four steps:

#1. New predicate Out, so that Out(δ) means thatδ is undercut, or excluded

#2. Introduce new exclusionary default theories astheories in a language containing Out.

#3. Lift notion of exclusion from object to metalanguage: where S is scenario based on theorywith W as hard information

δ ∈ ExcludedS iff W ∪ Conc(S) ` Out(δ).

#4. Only defaults that are not excluded can be trig-gered:

TriggeredW ,D,<(S) = {δ 6∈ ExcludedS and

W ∪ Conc(S) ` Prem(δ)}

27

S L

R

d 2

d 1

D1

d 1 < d

2

d 1 < d

3

d 3

10. Example: For ordinary rebutting defeat, take 〈W ,D〉where

W contains L, S, and δ1 < δ2, δ1 < δ3

D contains

δ1 = L → R

δ2 = S → ¬R

δ3 = D1 → Out(δ1)

(L = Looks red, R = Red, S = Statement byfriend, D1 = Drug 1)

So proper scenario is

S = {δ2}

generating the extension

E = Th(W ∪ {¬R})

28

S L

R

d 2

d 1

D1

d 1 < d

2

d 1 < d

3

d 3

11. Example: For undercutting, or exclusionary, de-feat, take 〈W ,D〉 where

D contains

δ1 = L → R

δ2 = S → ¬R

δ3 = D1 → Out(δ1)

W contains L, D1, and δ1 < δ2, δ1 < δ3


S = {δ3}


E = Th(W ∪ {Out(δ1)})

29

d 1 < d

2

d 1 < d

3 < d

4

S L

R

d 2

d 1

D1

d 3

D2

d 4

12. Example: Drug 2 is an antidote to Drug 1, so foran excluder excluder, take 〈W ,D〉 where

W contains L, D1, D2, δ1 < δ2, and δ1 < δ3 < δ4

D contains

δ1 = L → R

δ2 = S → ¬R

δ3 = D1 → Out(δ1)

δ4 = D2 → Out(δ3)


S = {δ1, δ4}


E = Th(W ∪ {R, Out(δ3)})

30

14. Example (Colin’s dilemma, simplified):

Let D contain

δ1 = E → Sδ2 = U → ¬Sδ3 = ¬Welfare(δ2) → Out(δ2)

(E = Provides good education, S = Send son toprivate school, U = Undermine support for publiceducation)

The default δ3 is itself an instance of

¬Welfare(δ) → Out(δ),

Let W contain E, U , and ¬Welfare(δ2)

Then proper scenario is

S = {δ1, δ3}


E = Th(W ∪ {S,Out(δ2)})

31

Exclusion and priorities

1. Can exclusion be defined in terms of priority ad-justment?

Many people think so...

Perry: “an exclusionary reason is simply thespecial case where one or more first-orderreasons are treated as having zero weight”

Dancy: “If we are happy with the idea thata reason can be attenuated . . . , why shouldwe fight shy of supposing that it can bereduced to nothing”

Schroeder: “undercutting” is best analyzedas an extreme case of attenuation in thestrength of reasons; he refers to this thesisas the “undercutting hypothesis”

Horty: developed a formal theory of ex-clusion as the assignment to a default ofa priority that falls below some particularthreshold

2. But this idea entials

Downward closure for exclusion: if δ is ex-cluded and a δ′ < δ, then δ′ is excluded.

32

B A

P

d 2

d 1

C

d 1 < d

2

d 2 < d

3

d 3

3. Example:

Take 〈W ,D〉 with

W = {A, B, C, δ1 < δ2, δ2 < δ3}

D = {δ1, δ2, δ3}

δ1 = A → P

δ2 = B → ¬P

δ3 = C → Out(δ2)

Then the proper scenario is

S = {δ1, δ3}


E = Th(W ∪ {P,Out(δ2)})

So downward closure fails, but is that right?

33

B A

P

d 2

d 1

C

d 1 < d

2

d 2 < d

3

d 3

4. First interpretation (mathematicians):

Priority ordering represents reliability of themathematicians

P = Some conjecture

A = First mathematician’s assertion thathe has proved P

B = Second mathematician’s assertion thatshe has proved ¬P

C = Third mathematician’s assertion thatsecond mathematician is too unreliableto be trusted

Here downward closure seems to hold

34

B A

P

d 2

d 1

C

d 1 < d

2

d 2 < d

3

d 3

5. Second interpretation (officers):

Captain < Major < Colonel

P = Some action to perform (or not)

A = Captain’s command to perform P

B = Major’s command not to perform P

C = Colonel’s command to ignore Major’scommand

Here downward closure seems to fail

35

B A

P

d 2

d 1

C

d 1 < d

2

d 2 < d

3

d 3

6. So if downward closure fails, what do we do whenwe want downward closure?

Answer: Supplement hard information with

(Out(δ) ∧ δ′ < δ) ⊃ Out(δ′),

This give us the proper scenario

S = {δ3}


E = Th(W ∪ {Out(δ2)})

36

B A

P

d 1 < d

2

d 1 d

2

7. Next question: a default cannot be defeated by aweaker default, but can it be excluded by a weakerdefault?

Yes, on current account.

Take 〈W ,D〉 with

W = {A, B, δ1 < δ2}D = {δ1, δ2}δ1 = A → Out(δ2)δ2 = B → P


S = {δ1}


E = Th(W ∪ {Out(δ2)})

37

B A

P

d 1 < d

2

d 1 d

2

8. Pollock’s answer: No

It seems apparent that any adequate ac-count of justification must have the conse-quence that if a belief is unjustified relativeto a particular degree of justification, thenit is unjustified relative to any highter de-gree of justification. (Cognitive Carpentry,p 104)

9. I disagree: different standards of legal evidence,jailhouse snitch

10. But what do we do in cases where we do want thisconstraint? (Military officer interpretation)

38

B A

P

d 1 < d

2

d 1 d

2

11. My (tentative) suggeston: suppose defaults areprotected from exclusion

Begin with 〈W ,D〉, where

W = {A, B, δ1 < δ2}D = {δ1, δ2}δ1 = A → Out(δ2)δ2 = B → P ∧ ¬Out(δ2)


S = {δ2}


E = Th(W ∪ {P ∧ ¬Out(δ2)})

39

An application: moral particularism

1. Dancy’s argument for moral particularism

Four steps:

1. Principles state the force of reasons

2. Reasons vary in force (valence, polarity) fromcontext to context

3. There are no reasons with constant force

4. Moral principles must be wrong, and so cannotguide our action

2. OK, why believe Step 2 ??

Examples, such as . . .

Context A: Things are normal

Object looks redThis is a reason for taking it to be red

Context B: I have taken Drug #3, whichflips red and blue

Object looks redThis is a reason for taking it to be blue

3. Go back to theory of reasons:

• Reasons are antecedents of triggered defaults

40

4. First representation: 〈W ,D〉 where

D contains

δ1 = L → R

δ2 = L ∧ D3 → B

W contains L, D3, ¬(R ∧ B), and δ1 < δ2

Proper scenario is

S1 = {δ2}


E = Th(W ∪ {B})

So on this representation

L is, in fact, a reason for R

It is just defeated by L ∧ D3

But, Dancy:

It is not as if it is some reason for me tobelieve that there is something red beforeme, but that as such . . . it is overwhelmedby contrary reasons. It is no longer any rea-

son at all to believe that there is somethingred before me . . .

41

5. Second representation: 〈W ,D〉 where

D contains

δ1 = L → R

δ2 = L ∧ D3 → B

δ3 = D3 → Out(δ1)

W contains L, D3, ¬(B ∧ R), and δ1 < δ2

Proper scenario is

S2 = {δ2, δ3}


E = Th(W ∪ {B,Out(δ1)})

So on this representation

L is not a reason for R

9. Upshot: reason holism does not imply absence ofa principled framework.

42

6. Dancy’s practical example

Context A: Things are normal

I borrowed a book from youThis is a reason for returning it to you

Context B: Book is stolen from library

I borrowed a book from youThis is not a reason for returning it to you

Dancy, again:

It is not that I have some reason to returnit to you and more reason to put it back inthe library. I have no reason at all to returnit to you.

7. My intuitions are different: five possibilities

8. But all viewpoints can be represented, and what isa reason and what isn’t in each case can be decidedin a systematic way, by appeal to principles

43

An open question: floating conclusions

1. Getting from scenario S to extension E

Direct route:

E = Th(W ∪ Conc(S))

Indirect route:

Defaults are rules of inferenceConstruct arguments to support conclusionsExamples:

> ⇒ A → B ⇒ ¬C

> ⇒ Q → P

> ⇒ R → ¬P

support conclusion ¬C, P , ¬P .

So, first form argument extension ΦThen take conclusions supported by Φ

2. Function ∗ maps arguments into conclusions:

∗α = conclusion supported by α

∗Φ = {∗α : α ∈ Φ}

44

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3. Consider multiple argument extensions

Φ1 = {> ⇒ Q,> ⇒ R,> ⇒ Q → P}

Φ2 = {> ⇒ Q,> ⇒ R,> ⇒ R → ¬P}

4. Skeptical—or “intersect extensions”—option nowbifurcates

Alternative #1:

∗(⋂

{Φ : Φ is an argument extension of Γ})

Alternative #2:⋂

{∗Φ : Φ is an argument extension of Γ}

5. In this case, same result:

{Q,R}

But not always, due to the phenomena of floating

conclusions

45

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Argument Extensions:

Φ1 = { > ⇒ Q, > ⇒ R,

> ⇒ Q → D,

> ⇒ Q → D 6⇒ H,

> ⇒ Q → D ⇒ E }

Φ2 = { > ⇒ Q, > ⇒ R,

> ⇒ R → H,

> ⇒ R → H 6⇒ D,

> ⇒ R → H ⇒ E }

Alternative #1 yields:

{Q, R}


{Q, R, E}

46

6. Conventional view is that floating conclusions should

be accepted (so Alternative #2 is correct).

Ginsberg:

Given that both hawks and doves are politi-cally [extreme], Nixon certainly should be aswell.” (Essentials of Artificial Intelligence,

1993)

Makinson and Schlechta:

It is an oversimplification to take a proposi-tion A as acceptable . . . iff it is supportedby some [argument] path α in the intersec-tion of all extensions. Instead A must betaken as acceptable iff it is in the inter-section of all outputs of extensions, wherethe output of an extension is the set of allpropositions supported by some path withinit. (Artificial Intelligence, 1991)

Stein:

The difficulty lies in the fact that some con-clusions may be true in every credulous ex-tension, but supported by different [argu-ment] paths in each. Any path-based the-ory must either accept one of these paths,and be unsound, or reject all such paths,and with them the ideally skeptical conclu-sion (Resolving Ambiguity . . . , 1991)

Pollock:

(Defeasible reasoning, unpublished) makesit clear that desire for floating conclusionsmotivated 1995 semantics

47

7. Yacht example:

• Both (elderly) parents have $500K

• I want a yacht, requires large deposit, balancedue later—otherwise, lose deposit

• Utilities determine conditional preferences:

– If I will inherit at least half a million dollars,I should place a deposit on the yacht

– Otherwise, I should not place a deposit

So decision hinges on truth of

F ∨ M

• Brother says: ”Father will leave his money tome, but Mother is leaving her money to you”

BA(¬F ∧ M )

• Sister says: ”Mother will leave her money tome, but Father is leaving his money to you”

SA(F ∧ ¬M )

• Both brother and sister reliable, so have de-faults:

BA(¬F ∧ M ) → (¬F ∧ M )

SA(F ∧ ¬M ) → (F ∧ ¬M )

48

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SA(F ∧ ¬M)

F ∨ M

BA(¬F ∧ M)

>

¬F ∧ M F ∧ ¬M

Argument Extensions:

Φ1 = { > ⇒ BA(¬F ∧ M ),> ⇒ SA(F ∧ ¬M ),> ⇒ BA(¬F ∧ M ) → ¬F ∧ M ,> ⇒ BA(¬F ∧ M ) → ¬F ∧ M 6⇒ F ∧ ¬M ,

> ⇒ BA(¬F ∧ M ) → ¬F ∧ M ⇒ F ∨ M }

Φ2 = { > ⇒ BA(¬F ∧ M ),> ⇒ SA(F ∧ ¬M ),> ⇒ SA(F ∧ ¬M ) ⇒ F ∧ ¬M ,> ⇒ SA(F ∧ ¬M ) ⇒ F ∧ ¬M 6⇒ ¬F ∧ M ,

> ⇒ SA(F ∧ ¬M ) ⇒ F ∧ ¬M ⇒ F ∨ M }


{BA(¬F ∧ M ), SA(F ∧ ¬M )}


{BA(¬F ∧ M ), SA(F ∧ ¬M ), F ∨ M }

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8. Other examples:

• Military example:

– You want to press ahead if enemy has re-treated from defensive position

– Spy 1 says: enemy retreating over moun-tains, diversionary force feigns retreat alongriver

– Spy 2 says: enemy retreating along river,diversionary force feigns retreat over moun-tains

• Economics example:

– Economic health, low inflation, strong growth

– Prediction 1: strong growth will trigger highinflation, leading to recession

– Prediction 2: inflation will continue to de-cline, resulting in deflation and so recession

• Ginsberg’s original example:

– Why not suppose that the extreme tenden-cies serve to moderate each other?

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9. Why accept floating conclusions?

Maybe an analogy between

B ⊃ A

C ⊃ A

B ∨ C

A

and (supposing two extensions, E1 and E2)

A ∈ E1 (so : E1 ⊃ A)

A ∈ E2 (so : E2 ⊃ A)

E1 ∨ E2 ??????

A

First argument relies on premise that A∨ B; mustskeptical reasoner suppose “E1 ∨ E2”?

Sometimes appropriate to think

• One or another extension must be (entirely)right—we just don’t know which

But other times

• Real possibility that they might all be wrong

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Norwegian name

Iceskating

Born Holland

>

Dutch Norwegian

10. Prakken’s example:

Brygt Rykkje was born in Holland

Brygt Rykkje has a Norwegian name

Brygt Rykkje is Dutch

Brygt Rykkje is Norwegian

Here we do like the floating conclusion

11. Open question: how do we distinguish cases inwhich we do like floating conclusion from cases inwhich we don’t?

52

Conclusions

1. What’s good:

It is an actual theory of reasons and theirinteraction

Plausible results in many cases

Plausible resolution to some existing prob-lems

Helps to frame some new problems

2. What’s bad, or still open:

The real definition of defeat is ugly

Lumps together practical and epistemicreasons—an insight or a mistake?

Other linguistic issues, such as generics

.

.

.

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