Philosophy, Computing, and Artificial Intelligence

PHI 319. Reasoning and Knowledge.

"A common misconception about reasoning is that reasoning is deducing, and in good reasoning the conclusions follow logically from the premises. It is now generally recognized both in philosophy and in AI that nondeductive reasoning is at least as common as deductive reasoning, and a reasonable epistemology must accommodate both. For instance, inductive reasoning is not deductive, and in perception, when one judges the color of something on the basis of how it looks to him, he is not reasoning deductively. Such reasoning is defeasible, in the sense that the premises taken by themselves may justify us in accepting the conclusion, but when additional information is added, that conclusion may no longer be justified. For example, some thing’s looking red to me may justify me in believing that it is red, but if I subsequently learn that the object is illuminated by red lights and I know that that can make things look red when they are not, then I cease to be justified in believing that the object is red" (John L. Pollock, "Defeasible Reasoning." Cognitive Science, 11, 1987, 481-518).

"We can combine all of a cognizer’s reasoning into a single inference graph and regard that as a representation of those aspects of his cognitive state that pertain to reasoning. The hardest problem in a theory of defeasible reasoning is to give a precise account of how the structure of the cognizer’s inference graph determines what he should believe. Such an account is called a “semantics” for defeasible reasoning, although it is not a semantics in the same sense as, for example, a semantics for first-order logic. If a cognizer reasoned only deductively, it would be easy to pro- vide an account of what he should believe. In that case, a cognizer should believe all and only the conclusions of his arguments (assuming that the premises are somehow initially justified). However, if an agent reasons defeasibly, then the conclusions of some of his arguments may be defeaters for other arguments, and so he should not believe the conclusions of all of them. ... We want a general account of how it is determined which conclusions should be believed, or to use philosophical parlance, which conclusions are “justified” and which are not. This distinction enforces a further distinction between beliefs and conclusions. When a cognizer constructs an argument, he entertains the conclusion and he entertains the propositions comprising the intervening steps, but he need not believe them. Constructing arguments is one thing. Deciding which conclusions to accept is another" (John L. Pollock, "A Recursive Semantics for Defeasible Reasoning." Argumentation in Artificial Intelligence, edited by Guillermo Simari and Iyad Rahwan (Springer, 2009), 173-197).
In an environment of real-world complexity, agents must be able to form beliefs and make decisions against a background of pervasive ignorance. Reasoning cannot be confined to logical deductions. 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.

Beliefs and Defeasible Reasoning

Arguments can be defeated in two ways.

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. It attacks the argument by providing a reason to think that the premise is not a good reason for the conclusion.

John Pollock (who did some the fundamental work on defeasible reasoning) gives this example:

In argument (1), the agent observes n white swans. These observations constitute premises in an argument that provides a defeasible reason to believe that all swans are white.

In argument (2), an ornithologist (Herbert) informs the agent that not all swans are white. Since ornithologists are reliable sources of information about birds, this second argument provides the agent with a defeasible reason to believe that no all swans are white.

At this point, the conclusions of the first two arguments rebut each other.

In argument (3), Simon (whom the agent regards as reliable) says that Herbert is incompetent. This argument provides the agent with the agent a reason that undercuts argument (2).

"What will prove to be a pivotal observation is that, at least in humans, the computation of defeat statuses, and more generally the computation of degrees of justification, is a subdoxastic process. That is, we do not reason explicitly about how to assign defeat statuses or degree of justification to conclusions. Rather, there is a computational process going on in the background that simply assigns degrees of justification as our reasoning proceeds. If we knew how to perform this computation by explicitly reasoning about inference-graphs and degrees of justification, that would make it easier to find a theory that correctly describes how the computation works. But instead, the computation of degrees of justification is done without our having an explicit awareness of anything but its result. Similar subdoxastic processes occur in many parts of human cognition. For example, the output of the human visual system is a percept, already parsed into lines, edges, corners, objects, etc. The input is a time course of retinal stimulation. But we cannot introspect how the computation of the percept works. To us, it is a black box. Similarly, the process of computing degrees of justification is a black box that operates in the background as we construct arguments in the foreground. Two important characteristics of such background computations are, first, that the output has to be computable on the basis of the input, and second, that the computation has to be fast. In the case of vision, the computation of the percept takes roughly 500 milliseconds. If it took longer, in many cases vision would not apprise us of our surroundings quickly enough for us to react to them. Similarly, degrees of justification must be computable on the basis of readily observable features of conclusions and their place in the agent's inference-graph. And the computational complexity of the computation must be such that the computation can keep up with the process of constructing arguments. As we will see, these simple observations impose serious restrictions on what theories of defeasible reasoning might be correct as descriptions of human reasoning..." (John L. Pollock, "Defeasible Reasoning and Degrees of Justification." Argument & Computation, 1:1, 2010, 7-22). Given these three arguments, what is it rational for the agent to believe?

Argumentation Semantics

An argumentation framework is a set of (abstract) arguments and a binary defeat relation. More formally, it is a pair <Arg, Def>, where Arg is a set of arguments and Def is a subset of Arg x Arg that is the extension of the defeat relation between the arguments in Arg.

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

An example makes it a little clearer how this 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?

The answer, it seems, is that the agent can accept arguments A and C but not argument B. Argument B defeats argument A, but argument C defeats B. In this way, C reinstates argument A. Further, no argument defeats argument C. So arguments A and C are undefeated.

A procedure to determine which arguments can be accepted is an argumentation semantics.

The semantics that gives the answer for arguments A, B, C works in terms of three conditions:

• An argument is in iff all its defeaters are out.
•An argument is out iff it has at least one defeater that is in.
•An argument is undecided iff it is neither in nor out.

In this example, Arg = {A,B,C} and Def = {<B, A>, <C,B>}.

C is in because all its defeaters are out. (All of C's defeaters are out because C has no defeaters.)

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

A is in because all of its defeaters are out.

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 McGovern was calling for an immediate end to the war. 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.

The argumentation framework is <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 are

A <---- B      B <---- A

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

1. A = in, B = out
2. A = out, B = in
3. A = undecided, B = undecided

Which labelling is correct?

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

Degree of Justification and Degree of Importance

"[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 L. Pollock, Thinking about Acting: Logical Foundations for Rational Decision Making, 109).

It seems natural to think that the higher the probability that what is printed in newspapers is true, 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 much greater than 1/2 but still not high enough for the agent to believe the newspaper report.

The Story of the Vacationing Ship Captain

"The 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. For example, consider a ship's captain on a busman's holiday" (John L. Pollock, Cognitive Carpentry. A Blueprint for How to Build a Person, 48. MIT Press, 1995).

"[T]here is a difference between a conclusion being 'justified simpliciter' and having a degree of justification greater than 0. Justification simpliciter requires the degree of justification to pass a threshold, but the threshold is contextually determined and not fixed by logic alone" (John L. Pollock, "Defeasible Reasoning and Degrees of Justification." Argument & Computation, 1:1, 2010, 7-22).
To show that the circumstances matter, Pollock tells a story about a vacationing ship captain.

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."

Because the number of lifeboats is not relevant to the decisions the captain will make as a passenger on a cruise vacation, its degree of interest for him is low. 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 the ship matters much more to the vacationing captain. Unless he can increase his degree of justification by counting the lifeboats or by having someone reliable count them for him, when he is thinking about what to do, he will have to treat the proposition as merely probable. 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.

What the Captain Knows and When he Knows it

"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). 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. Does he know there are n lifeboats on the ship?

If 'know' means that one has given the matter the appropriate consideration given its importance, then it seems that the captain knows that there are n lifeboats on the ship.

After the accident, when the captain has assumed command, Pollock says that "it becomes very important for him to know whether there are enough lifeboats." The importance he attributes to the matter has increased. Consulting the brochure is not appropriate consideration.

It may be that this is how Levesque uses 'know' when he says that "[t]hinking is bringing to bear what you know on what you are doing" (Hector Levesque, Thinking as Computation, 3).

What we have Accomplished in this Lecture

We considered the connection between defeasible reasoning and the conditions under which it is rational for an agent to hold a belief. We looked at an argumentation semantics for answering this question. We considered degrees of justification and importance in connection with knowledge.

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