John Pollock’s Work on Defeasible

Reasoning

John Horty

University of Maryland(and currently, Stanford CASBS)

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A bit of history

1. In Philosophy:

Hart 1948

“It is not usually possible to define a legalconcept such as . . . “contract” by specify-ing the necessary and sufficient conditionsfor its application . . . such concepts canonly be explained with the aid of a list ofexceptions or negative examples . . .”

Chisholm 1957, 1964

“These questions concern the defeasibilityof moral requirements.”

(Loui, interview: Chisholm says that hiscolleague, John Ladd, “who was quite takenby Hart,” had used the term)

Toulmin 1958

Gauthier 1961

“practical principles are defeasible. (I takethe term from Professor H. L. A Hart.)”

Gettier, Goldman, Klein, Lehrer, Paxton, Swain . . .

Pollock, 1967, 1970, 1971, 1974

Rescher 1977

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2. In Artificial Intelligence:

Doyle 1979

McCarthy 1980

Reiter 1980

McDermott and Doyle 1980

Touretzky 1986

Horty, Thomason, Touretzky 1987

Loui 1987

Pollock 1987

Nute 1988

Then, an explosion of research . . .

3. A quote:

“I believe that I developed the first formalsemantics for defeasible reasoning in 1979,but I did not initially publish it because,being ignorant of AI, I did not think anyonewould be interested. That semantics wasfinally published in 1986” (Pollock, 2007)

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4. Pollock’s work:

• Defeasible reasoning, Cognitive Science, 1987

• A theory of defeasible reasoning, International

Journal of Intelligent Sytems, 1991

• Self-defeating arguments, Minds and Machines,1991

• How to reason defeasibly, Artificial Intelligence,1992

• Justification and defeat, Artificial Intelligence,1994

• Cognitive Carpentry, MIT Press, 1995

• Defeasible reasoning with variable degrees ofjustification, Artificial Intelligence, 2002

• Defeasible reasoning, in Reasoning: Studies of

Human Inference and its Foundations, Adlerand Rips (eds), 2007

• A recursive semantics for defeasible reason-ing, in Argumentation in Artificial Intelligence,Rahwan and Simari (eds) 2009

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Pollock’s theory: constant features

1. There are ordinary statements:

Penguin(Tweety)

Quaker(Nixon)

Republican(Nixon)

along with strict rules of inference:

A ∧ B ⇒ A

Triangular(x) ⇒ Trilateral(x)

Penguin(x) ⇒ Bird(x)

But there are also defeasible (or prima facie) rules:

Looks.red(x) → Is.red(x)

Bird(x) → Flys(x)

Penguin(x) → ¬Fly(x)

Quaker(x) → Pacifist(x)

Republican(x) → ¬Pacifist(x)

2. Where W is a set of ordinary facts and rules, Da set of defeasible rules, and < an ordering onthe defeasible rules, we can refer to the structure〈W ,D, <〉 as a default theory

The semantic question is then: What should weconclude from a given default theory?

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3. Default theories can be represented through infer-ence graphs

Example (Nixon Diamond):

W = {Q, R}D = {r1, r2}r1 = Q → P

r2 = R → ¬P< = ∅.

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

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Another example (Tweety Triangle):

W = {P, P ⇒ B}D = {r1, r2}r1 = B → Fr2 = P → ¬Fr1 < r2

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

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4. Two different structures:

Inference graphs

Defeat graphs

We’ve just seen some inference graphs

Defeat graphs depict defeat relations among argu-ments, so we begin with . . .

5. Arguments:

An argument is a sequence of tuples 〈Pi, Ji, Li, Si〉such that for each i either

(a) Pi is an axiom or a member of W, in whichcase

i. Ji is either axiom or W

ii. Li is ∅

iii. Si is ∞

(b) Pi follows from previous Pj1and Pj2

by MP, inwhich case

i. Ji is MP

ii. Li is {j1, j2}

iii. Si is the weaker of {Sj1, Sj2

}

(c) Pi follows from previous Pj1. . . Pjn

by defeasiblerule r, in which case

i. Ji is r

ii. Li is {j1 . . . jn}

iii. Si is the weakest of {Sj1. . . Sjn

} ∪ {r}

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6. Examples:

From the Nixon Diamond

Argument α

1. 〈Q,W ,∅,∞〉

2. 〈P, r1, {1}, r1〉

Argument β

1. 〈R,W ,∅,∞〉

2. 〈¬P, r2, {1}, r2〉

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From the Tweety Triangle

Argument α

1. 〈P,W , ∅,∞〉

2. 〈P ⇒ B,W , ∅,∞〉

3. 〈B, MP, {1,2},∞〉

4. 〈F, r1, {3}, r1〉

Argument β

1. 〈P,W , ∅,∞〉

2. 〈¬F, r2, {3}, r2〉

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7. Rebutting defeat:

An argument line 〈P, J, L, S〉 is rebutted by an ar-gument line 〈P ′, J ′, L′, S ′〉 iff

• J is a defeasible rule, and

• P ′ is ¬P , and

• ¬(S ′ < S)

An argument α is rebutted by an argument β iffsome line of β rebuts some line of α.

8. Undercutting defeat:

An argument line 〈P, J, L, S〉 is undercut by an ar-gument line 〈P ′, J ′, L′, S ′〉 iff

• J is a defeasible rule r, and

• P ′ is Out(r), and

• ¬(S ′ < S)

An argument α is undercut by an argument β iffsome line of β undercut some line of α.

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9. Example (Drug 1):

W = {LR, D1}D = {r1, r2}r1 = LR → R

r2 = D1 → Out(r1)r1 < r2

(LR = Looks.red, R = Is.red, D1 = Drug 1)

Here, α is undercut by β

Argument α

1. 〈LR,W , ∅,∞〉

2. 〈R, r1, {1}, r1〉

Argument β

1. 〈D1,W , ∅,∞〉

2. 〈Out(r1), r2, {1}, r2〉

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10. Defeat:

An argument α is defeated by an argumentβ iff α is rebutted or undercut by β

11. Defeat graph: a graph with

• Nodes α, β, γ, . . . representing arguments

• Edges ; representing defeat relations

α ; β means: α defeats β

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Admissibility semantics

1. Due to Phan Minh Dung, 1995

2. Some simple definitions:

• We’ve seen α ; β

• Γ ; β means: ∃α(α ∈ Γ ∧ α ; β)

• Γ ; ∆ means: ∃α∃β(α ∈ Γ ∧ β ∈ ∆ ∧ α ; β)

• Γ is consistent means: ¬(Γ ; Γ)

3. Defense:

• Γ defends α means: ∀β(β ; α ⇒ Γ ; β)

• F(Γ) = {α : Γ defends α}

Note: F is monotonic

Γ ⊆ Γ′ ⇒ F(Γ) ⊆ F(Γ′)

4. Admissible sets:

• Γ admissible means: consistent and F(Γ) ⊆ Γ

(ie, Γ defends all of its own members)

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5. Example (Drugs 1 and 2):

W = {D1, D2, LR}D = {r1, r2, r3}r1 = LR → Rr2 = D1 → Out(r1)r3 = D2 → Out(r2)r1 < r2 < r3

(LR = Looks.red, R = Is.red, D1 = Drug1, D2 =Drug2 )

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6. Various extension concepts:

An admissible set Γ is:

• Preferred iff: Γ is a maximal admissible set

• Complete iff: F(Γ) = Γ

• Grounded iff: the minimal complete set

• Stable iff: α 6∈ Γ ⇒ Γ ; α

Note 1: Only the grounded extension is unique

Note 2: If everything is finite, let

Γ0 = ∅Γi+1 = F(Γi)

Then grounded extension can be calculated as

Γ =⋃

n Γn

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7. Status assignment: assigns to zero or more argu-ments either in or out, subject to conditions:

• An argument is in iff all defeaters are out,

• An argument is out iff some defeater is in

8. Correlation with extension concepts:

• Complete: any status assignment

• Grounded: maximal undecided

• Preferred: maximal in (maximal out)

• Stable: every argument in or out

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9. Exercise: calculate all complete extensions for eachof these graphs. What complete extensions aregrounded, stable, preferred?

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Pollock’s semantics

1. The 1987 theory, preliminary version:

• All arguments are in at level 0

• An argument is in at level n + 1 iff it is notdefeated by any argument in at level n

• An argument is justified (or: ultimately unde-feated) iff there is some m such that, for everyn ≥ m, the argument is in at level n

2. A slight reformulation:

Define

G(Γ) = {α : ¬(Γ ; α)}

Where ∆ is entire set of arguments, let

∆0 = ∆∆i+1 = G(∆i)

Then define

α justified iff ∃m∀n ≥ m(α ∈ ∆n)

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3. Example (Drugs 1 and 2, again):

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4. Fact (Dung, 1995):

An argument is justified iff it belongs to thegrounded extension

Recall:

The grounded extension is the minimal fixed pointof F , where

F(Γ) = {α : Γ defends α}

= {α : ∀β(β ; α ⇒ Γ ; β)}

Compare

G(Γ) = {α : ¬(Γ ; α)}

= {α : ∀β(β ∈ Γ ⇒ ¬(β ; α))}

and note that

(1) F(Γ) = G(G(Γ))

(2) G(∆) = F(∅),

where ∆ is the entire set of arguments

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5. Example (zombie arguments):

W = {Q, R}D = {r1, r2, r3, r4}r1 = Q → P

r2 = R → ¬Pr3 = P → Sr4 = > → ¬S< = ∅.

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6. Refining the preliminary version: consider

W = {Q, R, B}D = {r1, r2}r1 = Q → P

r2 = R → ¬Pr3 = B → F< = ∅.

Now have:

1. 〈Q, fact, ∅, S〉

2. 〈P, r1, {1}, S〉

3. 〈R, fact, ∅, S〉

4. 〈¬P, r2, {3}, S〉

5. 〈¬F, logic, {2,4}, S〉

Argument self-defeating iff one step defeats an-other. Pollock’s initial solution: simply rule out allself-defeating arguments. This is 1987

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7. Now, what’s wrong with 1987?

Example (Unreliable John):

W = {JA(U(j))}D = {r1, r2}r1 = JA(U(j)) → U(j)

r2 = U(j) → Out(r1)

< = ∅.

(JA(X) = John asserts X, U(y) = y is unreliable,j = John)

Now have:

1. 〈JA(U(j)), fact, ∅, S〉

2. 〈U(j), r1, {1}, S〉

3. 〈Out(r1), r2, {2}, S〉

So this is a kind of self-defeat we can’t rule out

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8. The 1994/1995 theory:

A mapping σ of nodes from an inference graph toin and out is a partial status assignment iff

• σ assigns in to each initial node;

• σ assigns in to a non-initial node iff it assignsin to all immediate ancestors and out to alldefeaters;

• σ assigns out to a non-initial node iff it as-signs out to an immediate ancestor or in to adefeater.

A status assignment is a maximal partial statusassignment

A node is: justified iff in in all status assignments;overruled iff out or undecided in all status assign-ments; defensible iff in in some but not all statusassignments

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9. Pollock writes:

“In earlier publications, I proposed that de-feat could be analyzed as defeat among ar-guments, rather than inference nodes . . . Inow feel that obscured the proper treat-ment of self-defeat. I see no way to recastthe present analysis in terms of a defeatrelation between arguments, as opposed tonodes, which are argument steps rather thancomplete arguments” (Pollock 1994, p 393)

10. But there is a way (Jakobovits, 1999):

Pollock’s 1994/1995 analysis for inferencegraphs is equivalent to Dung’s preferred se-mantics for defeat graphs

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11. Now what motivates the further change?

We approach in stages . . .

Stage A (John asserts P):

W = {JA(U(j)), JA(P)}D = {r1, r2, r3, r4}r1 = JA(U(j)) → U(j)

r2 = U(j) → Out(r1)

r3 = JA(P) → P

r4 = U(j) → Out(r3)

< = ∅.

(JA(X) = John asserts X, U(y) = y is unreliable,j = John)

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Stage B (Adding Susan):

W = {JA(U(s)), SA(U(j)), JA(P)}D = {r1, r2, r3, r4, r5, r6}r1 = JA(U(j)) → U(s)

r2 = U(j) → Out(r1)

r3 = JA(P) → P

r4 = U(j) → Out(r3)

r5 = SA(U(j)) → U(j)

r6 = U(s) → Out(r5)

< = ∅.

(JA(X) = John asserts X, SA(X) = Susan assertsX, U(y) = y is unreliable, j = John, s = Susan)

A quote:

We get the right answer, but we get it ina different way than we do for even-lengthdefeat cycles . . . This difference has alwaysbothered me (Pollock, 2002)

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Stage A1 (John and Donald):

W = {JA(U(j)), JA(P)}D = {r1, r2, r3, r4, r7}r1 = JA(U(j)) → U(j)

r2 = U(j) → Out(r1)

r3 = JA(P) → P

r4 = U(j) → Out(r3)

r7 = DA(¬P) → ¬P

< = ∅.

(JA(X) = John asserts X, DA(X) = Donald as-serts X, U(y) = y is unreliable j = John)

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Stage B1 (John, Susan, and Donald):

W = {JA(U(s)), SA(U(j)), JA(P)}D = {r1, r2, r3, r4, r5, r6, r7}r1 = JA(U(j)) → U(s)

r2 = U(j) → Out(r1)

r3 = JA(P) → P

r4 = U(j) → Out(r3)

r5 = SA(U(j)) → U(j)

r6 = U(s) → Out(r5)

r7 = DA(¬P) → ¬P

< = ∅.

(JA(X) = John asserts X, SA(X) = Susan assertsX, DA(X) = Donald asserts X, U(y) = y is unre-liable j = John, s = Susan)

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12. So, the 2009 semantics is motivated by conflictingbehaviors of odd and even defeat loops

Another quote:

“This, I take it, is a problem. Although itmight not be clear which inference-graphis producing the right answer, the right an-swer ought to be the same in both inferencegraphs. Thus the semantics is getting oneof them wrong.” (Pollock 2009)

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13. I feel that this is a misplaced concern. Odd andeven cycles are just different: odd cycles are para-doxes, even cycles are dilemmas.

This holds in many areas.

Odd cycle:

John: What I am now saying is false

Even cycle:

John: What Susan is now saying is falseSusan: What John is now saying is false

Odd cycle:

John: What Susan is now saying is falseSusan: What Jason is now saying is falseJason: What John is now saying is false

Odd cycle:

John: What Susan is now saying is falseSusan: What Jason is now saying is falseJason: What Sara is now saying is falseSara: What John is now saying is false

Etc

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Some open issues

1. Although defeasible reasoning spans normative andnon-normative domains (ethics, law, epistemology),Pollock concentrates only on non-normative rea-soning.

As a result: he misses some important generaliza-tions, and ducks some difficult issues

(a) Pollock uses weakest-link ordering to calculatestrengths, but in many normative domains thelast-link ordering is more natural. Consider:

W = ∅D = {r1, r2, r3}Captain (r1) = > → HMajor (r2) = > → ¬W

Colonel (r3) = H → Wr1 < r2 < r3

(H = Heater on, W = Window open)

Does this work in some epistemic domains aswell?

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(b) More natural in normative domains to think ofstrengths as not totally ordered, which leadsto many problems

Consider:

W = ∅D = {r1, r2, r3, r4}Captain (r1) = > → A

Major (r2) = > → ¬APriest (r3) = > → ABishop (r4) = > → ¬Ar1 < r2

r3 < r4

Are there epistemic problems like this?

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2. How do we reason about priorities?

W = {Q, R}D = {r1, r2, r3, r4}r1 = Q → Pr2 = R → ¬Pr3 = > → r2 < r1

r4 = > → r1 < r2

< = ∅ (initially)

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3. Is the reason relation necessarily “formal,” or canit be “material”?

A quote:

. . . prima facie reasons are supposed to belogical relationships between concepts. It isa necessary feature of the concept red thatsomething’s looking red to me gives me aprima facie reason for thinking it is red.

By contrast, Reiters’s defaults often repre-sent contingent generalizations. If we knowthat most birds can fly, then the inferencefrom being a bird to flying may be adoptedas a default. [In my theory] the latter in-ference is instead handled in terms of thefollowing reason schema:

“Most A’s are B’s and this is an A” isa prima facie reason for “This is a B”

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4. For Pollock: arguments all “present” at once, andthen evaluated together

I prefer: interleaving argument construction andevaluation

W = {¬(A ∧ B)}D = {r1, r2, r3}Priest (r1) = > → ABishop (r2) = > → B

Cardinal (r3) = A → ¬Br1 < r2 < r3

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5. Issues surrounding reason accrual

Consider: I am invited to a wedding and . . .

The wedding falls at an inconvenient timeAunt Olive will be thereAunt Petunia will be thereAunt Olive and Aunt Petunia will be there

Is the fourth reason really independent of the sec-ond and third?

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6. Undercutting by weaker defaults

Take 〈W ,D〉 with

W = {A, B}D = {r1, r2}r1 = A → Out(r2)r2 = B → Pr1 < r2

Can r2 be undercut by r1?

Not for Pollock, because . . .

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. (Pollock 1995, p 104)

But I disagree

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Conclusions

1. It helps to look at Pollock’s work on defeasibilitythrough the lens of admissibility semantics

2. The 1987 theory is sensible

3. The 1994/1995 theory is also sensible

4. Many open issues in this area, and a lot of inter-esting work to do

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