Beliefs and Defeasible Reasoning

In an environment of real-world complexity, intelligent agents must be able to form beliefs and make decisions against a background of pervasive ignorance. Given this situation, reasoning cannot be confined to logical deductions. In an environment of real-world complexity, it is necessary for intelligent agents to reason defeasibly. They need to draw conclusions that are made reasonable by the current evidence and be prepared to withdraw those conclusions and to draw new conclusions when they acquire new information about the world.

Information that can mandate the retraction of the conclusion of a defeasible argument constitutes a defeater for the argument. There are two kinds of defeaters. One kind of defeater is a rebutting defeater. It attacks the argument by attacking its conclusion. It provides a reason to think the conclusion of the argument is false. The other kind of defeater is an undercutting defeater. This defeater does not attack the argument by attacking the conclusion. Instead, it provides a reason to think that the premise is not a good reason for the conclusion.


An Example of Defeasible Reasoning

An example helps to illustrate how defeasible reasoning works.

John Pollock (who did some the fundamental work on the subject) gives the following example consisting of three arguments:






In the first argument, the agent observes a number of white swans. This provides a defeasible reason to believe that all swans are white.

In the second argument, an ornithologist (Herbert) informs the agent that not all swans are white. Now there are two arguments in play.

People do not always speak truly, so the fact that the ornithologist says that not all swans are white does not entail that not all swans are white. Nevertheless, because the information comes from an expert (an ornithologist), the agent now has a defeasible reason to believe that not all swans are white. This reason is a rebutting defeater.

Because rebutting defeaters are symmetrical (hence the double arrow in the diagram), the conclusions of both of the first two arguments are under attack.

In the third argument, Simon (whom the agent regards as very reliable) says that Herbert (the ornithologist) is incompetent.

Now the agent a reason that defeats the argument for the conclusion that all swans are not white. Note, however, that even if Herbert (the ornithologist) is incompetent, he might be still right that not all swans are white. The new information gives the agent a defeater, but not a rebutting defeater. It gives him an undercutting defeater.

Given the inference graph, what is it rational for the agent to believe? Beliefs and conclusions are not the same thing. When, in the course of reasoning, an agent constructs an argument, he has the conclusion in mind but he need not believe that it is true. Constructing arguments is one thing. Deciding which conclusions to accept is another.


Argumentation Semantics

A criterion which determines defeat-statuses of the conclusions is a semantics for defeasible reasoning.

To begin to understand what a semantics for defeasible reasoning is, it is helpful to abstract from the internal structure of arguments.

What remains in the argument when the internal structure is abstracted away is an argumentation framework. An argumentation framework is a set of (abstract) arguments and a binary defeat relation between these arguments. More formally, it is a pair <Arg, Def>, where Arg is a set of arguments and Def is a subset of Arg x Arg.

Argument A defeats argument B just in case <A, B> is in the set Def.


An example makes it a little clearer how an argumentation semantics works.

Suppose there are three arguments, A, B, and C. Suppose that B defeats A and that C defeats B. These relationships among the arguments may be pictured as follows:



     A <----  B, B <------ C


Given these relations between the arguments, what should the agent believe?

In light of the defeat relationships among the arguments, it seems reasonable to think that the agent can accept arguments A and C but not argument B. Here is the explanation. Argument B defeats argument A, but argument C defeats B. In this way, C reinstates A. Further, no argument defeats C. So arguments A and C are undefeated.

A procedure that takes an argumentation framework and determines which of the arguments can and cannot be accepted is an argumentation semantics.


Labeling Arguments In, Out, or Undecided

One way to provide a semantics is in terms of labeling arguments in, out, or undecided. Here are the conditions:

An argument is in iff all its defeaters are out.

An argument is out iff all it has at least one defeater that is in.

An argument is undecided iff it is neither in nor out.


The Three Arguments (A,B,C) Example Revisited

Consider the example pictured above in involving the three arguments A, B, and C. In this example,


     Arg = {A,B,C}, Def = {<B, A>, <C,B>}

     A <----  B,  B <------ C

C is in because all its defeaters are out. (It is trivially true that all of C's defeaters are out because C does not have any defeaters.)

C defeats B, and C is in. Therefore, B is out.

Finally, A is in because all of its defeaters are out.



Another Example: the Nixon Diamond

Here is another example, commonly referred to as the "Nixon Diamond." There are two arguments:

Argument A: Nixon is a Quaker. Therefore, he is against war.

Argument B: Nixon is a Republican. Therefore, he is for war.

(Richard Nixon was the 37th President. As a boy, he went to Quaker meetings and played the piano at services. In politics, he was a hawk on the war in Vietnam. He won (in 1972) in a landslide against Democratic Senator George McGovern of South Dakota, who was calling for an immediate end to the war.)

This example yields an argumentation framework <Arg, Def> where Arg = {A, B} and Def = {<A, B>, <B, A>}.

This framework gets its name because it was originally depicted as follows:


Nixon Diamond

The defeat relations between the arguments A and B is



      A <---- B, B <----- A


Given the argumentation framework, there are three possible complete labelings:

1. A = in, B = out

2. A = out, B = in

3. A = undecided, B = undecided


Which labeling is correct?

The answer, it seems, is that the third is correct. Is not rational to accept either argument.


Another Example

Here is another example: Arg = {A, B, C}, Def = {<B,A>, <C, B>, <A, C>}. It may be depicted this way



     A <----- B, B <------ C, C <----- A


In this framework, A = undecided, B = undecided, and C = undecided.



Complete Extensions

The another (slightly more formal) approach to argumentation semantics is in terms of what are called complete extensions.

In this approach,

A set of arguments is conflict-free iff it does not contain any arguments A and B such that A defeats B.

A set of arguments defends an argument C iff each defeater of C is defeated by an argument in the set.

Furthermore,

F: 2Arg → 2Arg such that F(Args) = {A | Args defends A}, where ArgsArg.

Given these definitions,

Args is a complete extension iff Args = F(Args).


To understand what a complete extension is, it helps to consider the examples. First, consider the framework


     Arg = {A,B,C}, Def = {<B, A>, <C, B>}

     A <----  B. B <------ C

In this framework, there is just one complete extension: {A,C}. It is a complete extension since it is conflict-free and defends exactly itself. Next, consider the Nixon Diamond:


      Arg = {A,B}, Def = {<A,B>, <B,A>} 

      A <---- B,  B <----- A

In this framework, there is just one complete extension: { }. Finally, consider the remaining example framework:

 
     Arg = {A, B, C}, Def = {<B,A>, <C,B>, <A,C>} 
   
     A <----- B, B <------ C, C <----- A

In this framework, there are three complete extensions: {A}, {B}, { }.


When there is just one complete extension, it is the set of in arguments. When there is more than one complete extension, the minimal complete extension is the set of in arguments. The minimal complete extension is called the grounded extension. There is always exactly one grounded semantics in an argumentation framework.


Some of John Pollock's Ideas

Suppose that a person reads something in the newspaper. What makes it reasonable for him or her to belief that the report is true?

"[O]n what basis do I believe what I read in the newspaper? Certainly not that everything printed in the newspaper is true. No one believes that. But I do believe that it is probable that what is printed in the newspaper is true, and that justifies me in believing individual newspaper reports" (John Pollock, Thinking about Acting: Logical Foundations for Rational Decision Making, 109).

The inference here is defeasible. It is what Pollock calls Statistical Syllogism. Initially, it may be stated as follows

     
     the probability that a thing is a B given that it is an A  >  1/2       c is A 
    -----------------------------------------------------------------------------------
                                        c is a B
     

It seems natural to think that in this rule the higher the probability, the stronger the reason. Further, it seems natural to think that the circumstances matter. That is to say, it seems natural to think that in some circumstances the probability could be greater than 1/2 but still not high enough for the agent to believe the conclusion on the basis of the premises.


Degree of Justification and Practical Importance

To demonstrate that the circumstances matter, Pollock uses the story of the vacationing captain to show that "[t]he practical importance of a question (i.e., our degree of interest in it) determines how justified we must be in an answer before we can rest content with that answer" (Cognitive Carpentry. A Blueprint for How to Build a Person, 48).

In the first part of the story, the captain is on a cruise vacation. He is a passenger and has no other role. He wonders how many lifeboats are on the ship. Pollock says that "[t]o answer this question, [the captain] might simply consult the descriptive brochure passed out to all the passengers." In the second part of the story, there is an accident. The ship is in danger of sinking. The officers of the ship, in including its captain, are incapacitated. The vacationing captain becomes aware of the situation and assumes command. At this point, he cannot simply consult the brochure to learn the number of lifeboats on board. Pollock says that "it becomes very important for him to know whether there are enough lifeboats" onboard and that the captain "must either count them himself or have them counted by someone he regards as reliable" because now "[t]he importance of the question makes it incumbent on him to have a very good reason for his believed answer."

Pollock does not supply much detail, but the idea seems to be that because the number of lifeboats onboard the ship is not relevant to the decisions the captain will make as a passenger on a cruise vacation, its degree of interest is low for him before the accident. The sorts of things he will decide are where to eat, which shows to attend, and so on. In making these decisions, the number of lifeboats on board the ship does not matter one way or another. Its degree of interest is low enough that the degree of justification required to "rest content" with an answer is minimal. Given that he is content with the answer, he can frame his decision problems in a certain way. He can treat the number of lifeboats, whatever the brochure says it is, as part of the way the world is. After the accident, this is no longer true. Now the number of lifeboats on board matters much more to him. According to Pollock, unless the captain can increase his degree of justification by counting the lifeboats or by having someone reliable count them for him, he will have to treat the proposition as merely probable. In this case, when he is deciding whether to give the order to abandon ship to await rescue in the lifeboats, he will not be able treat the number of lifeboats on the ship as part of the way the world is. To make his decision, the captain will have to take into account his uncertainty about the number of lifeboats.


The Use of the word 'Know' in Pollock's Example

When the captain first boards the ship, he consults the safety brochure and thereby comes to believe that there are n lifeboats on the ship. After the accident, when he has assumed command of the ship, it would be natural for him to give the following order to one of the crew.

"I think there are n lifeboats onboard, but I need to be sure. Count them and report back to me."

This suggests that after the accident the captain still believes what he previously believed: that the number is what the brochure says it is. He retains this belief when he takes command, but because he no longer "knows," he thinks that he needs more justification if he is to use this belief to frame his decision problem.

What does 'know' mean here? (What does it mean in Levesque's claim that "[t]hinking is bringing to bear what you know on what you are doing" (Thinking as Computation, 3).)

"We do not ordinarily require of someone who claims to know that he should have the kind of reason and justification for his belief which allows him to rule out all incompatible beliefs, that knowledge has to be firm or certain exactly in the sense that somebody who really knows cannot be argued our of his belief on the basis of assumptions incompatible with it. It seems ordinarily we only expect satisfaction of these standards to the extent and degree which is proportional to the importance we attribute to the matter in question. And thus, following common usage, a skeptic might well be moved to say, in perfect consistency with his skepticism, that he knows this or that. There is no reason that the skeptic should not follow the common custom to mark the fact that he is saying what he is saying having given the matter appropriate consideration in the way one ordinarily goes about doing this, by using the verb 'to know'" (Michael Frede, "The Skeptic's Two Kinds of Assent," 211).


What we have accomplished in this lecture

We have looked at the connection between defeasible reasoning and the conditions under which it is rational for an agent to hold a belief.









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