CONCEPTS OF LOGICAL AI

John McCarthy

Computer Science Department

Stanford University

Stanford, CA 94305

jmc@cs.stanford.edu

http://www-formal.stanford.edu/jmc/

2000 Apr 17, 2:07 p.m.

Abstract

Logical AI involves representing knowledge of an agent’s world,

its goals and the current situation by sentences in logic. The agent

decides what to do by inferring that a certain action or course of

action is appropriate to achieve the goals. We characterize briefly a

large number of concepts that have arisen in research in logical AI.

Reaching human-level AI requires programs that deal with the

common sense informatic situation. Human-level logical AI requires

extensions to the way logic is used in formalizing branches of math-

ematics and physical science. It also seems to require extensions to

the logics themselves, both in the formalism for expressing knowledge

and the reasoning used to reach conclusions.

A large number of concepts need to be studied to achieve logical

AI of human level. This article presents candidates. The references,

though numerous, to articles concerning these concepts are still insuf-

ficient, and I’ll be grateful for more, especially for papers available on

the web.

This article is available in several forms via http://www-formal.stanford.edu/jmc/concepts-ai.html.

Introduction

Logical AI involves representing knowledge of an agent’s world, its goals

and the current situation by sentences in logic. The agent decides what to

do by inferring that a certain action or course of action was appropriate to

achieve the goals. The inference may be monotonic, but the nature of the

world and what can be known about it often requires that the reasoning be

nonmonotonic.

Logical AI has both epistemological problems and heuristic problems.

The former concern the knowledge needed by an intelligent agent and how

it is represented. The latter concerns how the knowledge is to be used to

decide questions, to solve problems and to achieve goals. These are discussed

in [MH69]. Neither the epistemological problems nor the heuristic problems

of logical AI have been solved. The epistemological problems are more fun-

damental, because the form of their solution determines what the heuristic

problems will eventually be like.1

This article has links to other articles of mine. I’d like to supplement the

normal references by direct links to such articles as are available.

2 A LOT OF CONCEPTS

The uses of logic in AI and other parts of computer science that have been

undertaken so far do not involve such an extensive collection of concepts.

However, it seems to me that reaching human level AI will involve all of the

following—and probably more.

Logical AI Logical AI in the sense of the present article was proposed in

[McC59] and also in [McC89]. The idea is that an agent can represent

knowledge of its world, its goals and the current situation by sentences

in logic and decide what to do by inferring that a certain action or

course of action is appropriate to achieve its goals.

Logic is also used in weaker ways in AI, databases, logic programming,

hardware design and other parts of computer science. Many AI systems

1Thus the heuristics of a chess program that represents “My opponent has an open file

for his rooks.” by a sentence will be different from those of a present program which at

most represents the phenomenon by the value of a numerical co-efficient in an evaluation

function.

represent facts by a limited subset of logic and use non-logical programs

as well as logical inference to make inferences. Databases often use

only ground formulas. Logic programming restricts its representation

to Horn clauses. Hardware design usually involves only propositional

logic. These restrictions are almost always justified by considerations

of computational efficiency.

Epistemology and Heuristics In philosophy, epistemology is the study of

knowledge, its form and limitations. This will do pretty well for AI also,

provided we include in the study common sense knowledge of the world

and scientific knowledge. Both of these offer difficulties philosophers

haven’t studied, e.g.

they haven’t studied in detail what people or

machines can know about the shape of an object the field of view,

remembered from previously being in the field of view, remembered

from a description or remembered from having been felt with the hands.

This is discussed a little in [MH69].

Most AI work has concerned heuristics, i.e. the algorithms that solve

problems, usually taking for granted a particular epistemology of a

particular domain, e.g. the representation of chess positions.

Bounded Informatic Situation Formal theories in the physical sciences

deal with a bounded informatic situation. Scientists decide informally

in advance what phenomena to take into account. For example, much

celestial mechanics is done within the Newtonian gravitational theory

and does not take into account possible additional effects such as out-

gassing from a comet or electromagnetic forces exerted by the solar

wind. If more phenomena are to be considered, scientists must make a

new theories—and of course they do.

Most AI formalisms also work only in a bounded informatic situation.

What phenomena to take into account is decided by a person before

the formal theory is constructed. With such restrictions, much of the

reasoning can be monotonic, but such systems cannot reach human

level ability. For that, the machine will have to decide for itself what

information is relevant, and that reasoning will inevitably be partly

nonmonotonic.

One example is the “blocks world” where the position of a block x is

entirely characterized by a sentence At(x, l) or On(x, y), where l is a

location or y is another block.

Another example is the Mycin [DS77] expert system in which the ontol-

ogy (objects considered) includes diseases, symptoms, and drugs, but

not patients (there is only one), doctors or events occurring in time.

See [McC83] for more comment.

Common Sense Knowledge of the World As first discussed in [McC59],

humans have a lot of knowledge of the world which cannot be put in

the form of precise theories. Though the information is imprecise, we

believe it can still be put in logical form. The Cyc project [LG90] aims

at making a large base of common sense knowledge. Cyc is useful,

but further progress in logical AI is needed for Cyc to reach its full

potential.

Common Sense Informatic Situation In general a thinking human is in

what we call the common sense informatic situation, as distinct from

the bounded informatic situation. The known facts are necessarily in-

complete. We live in a world of middle-sized object which can only

be partly observed. We only partly know how the objects that can

be observed are built from elementary particles in general, and our

information is even more incomplete about the structure of particu-

lar objects. These limitations apply to any buildable machines, so the

problem is not just one of human limitations.2

In many actual situations, there is no a priori limitation on what facts

are relevant.

It may not even be clear in advance what phenomena

should be taken into account. The consequences of actions cannot be

fully determined. The common sense informatic situation necessitates

the use of approximate concepts that cannot be fully defined and the

use of approximate theories involving them. It also requires nonmono-

tonic reasoning in reaching conclusions. Many AI texts and articles

effectively assume that the information situation is bounded—without

even mentioning the assumption explicitly.

There is an infamous physics exam problem of using a barometer to

determine the height of a building. The student is expected to propose

measuring the air pressure at the top and bottom, but what about the

2Science fiction and scientific and philosophical speculation have often indulged in the

Laplacian fantasy of super beings able to predict the future by knowing the positions and

velocities of all the particles. That isn’t the direction to go. Rather they would be better

at using the information that is available to the senses.

following other answers. (1) Drop the barometer from the top of the

building and measure the time before it hits. (2) Lower the barome-

ter on a string till it reaches the ground and measure the string. (3)

Compare the shadow of the barometer with the shadow of the build-

ing. (4) Offer the barometer to the janitor in exchange for information

about the height. In this common sense informatic situation, there may

be still other solutions to the problem taking into account additional

phenomena.

The common sense informatic situation often includes some knowledge

about the system’s mental state as discussed in [McC96a].

One key problem in formalizing the common sense informatic situation

is to make the axiom sets elaboration tolerant2.

Epistemologically Adequate Languages A logical language for use in

the common sense informatic situation must be capable of expressing

directly the information actually available to agents. For example,

giving the density and temperature of air and its velocity field and

the Navier-Stokes equations does not practically allow expressing what

a person or robot actually can know about the wind that is blowing.

We and robots can talk about its direction, strength and gustiness

approximately, and can give a few of these quantitities numerical values

with the aid of instruments if instruments are available, but we have

to deal with the phenomena even when no numbers can be obtained.

The idea of epistemological adequacy was introduced in [MH69].

Robot We can generalize the notion of a robot as a system with a variant

of the physical capabilities of a person, including the ability to move

around, manipulate objects and perceive scenes, all controlled by a

computer program. More generally, a robot is a computer-controlled

system that can explore and manipulate an environment that is not

part of the robot itself and is, in some important sense, larger than the

robot. A robot should maintain a continued existence and not reset

itself to a standard state after each task. From this point of view, we

can have a robot that explores and manipulates the Internet without

it needing legs, hands and eyes. The considerations of this article that

mention robots are intended to apply to this more general notion. The

internet robots discussed so far are very limited in their mentalities.

Qualitative Reasoning This concerns reasoning about physical processes

when the numerical relations required for applying the formulas of

physics are not known. Most of the work in the area assumes that

information about what processes to take into account are provided by

the user. Systems that must be given this information often won’t do

human level qualitative reasoning. See [De90] and [Kui94].

Common Sense Physics Corresponds to people’s ability to make deci-

sions involving physical phenomena in daily life, e.g. deciding that

the spill of a cup of hot coffee is likely to burn Mr. A, but Mr. B is

far enough to be safe.

It differs from qualitative physics, as studied

by most researchers in qualitative reasoning, in that the system doing

the reasoning must itself use common sense knowledge to decide what

phenomena are relevant in the particular case. See [Hay85] for one view

of this.

Expert Systems These are designed by people, i.e. not by computer pro-

grams, to take a limited set of phenomena into account. Many of them

do their reasoning using logic, and others use formalisms amounting

to subsets of first order logic. Many require very little common sense

knowledge and reasoning ability. Restricting expressiveness of the rep-

resentation of facts is often done to increase computational efficiency.

Knowledge Level Allen Newell ([New82] and [New93]) did not advocate

(as we do here) using logic as the way a system should represent its

knowledge internally. He did say that a system can often be appropri-

ately described as knowing certain facts even when the facts are not

represented by sentences in memory. This view corresponds to Daniel

Dennett’s intentional stance [Den71], reprinted in [Den78], and was

also proposed and elaborated in [McC79].

Elaboration Tolerance A set of facts described as a logical theory needs

to be modifiable by adding sentences rather than only by going back

to natural language and starting over. For example, we can modify the

missionaries and cannibals problem by saying that there is an oar on

each bank of the river and that the boat can be propelled with one oar

carrying one person but needs two oars to carry two people. Some for-

malizations require complete rewriting to accomodate this elaboration.

Others share with natural language the ability to allow the elaboration

by an addition to what was previously said.

There are degrees of elaboration tolerance. A state space formalization

of the missionaries and cannibals problem in which a state is repre-

sented by a triplet (m c b) of the numbers of missionaries, cannibals

and boats on the initial bank is less elaboration tolerant than a situation

calculus formalism in which the set of objects present in a situation is

not specified in advance. In particular, the former representation needs

surgery to add the oars, whereas the latter can handle it by adjoining

more sentences—as can a person. The realization of elaboration toler-

ance requires nonmonotonic reasoning. See [McC97].

Robotic Free Will Robots need to consider their choices and decide which

of them leads to the most favorable situation. In doing this, the robot

considers a system in which its own outputs are regarded as free vari-

ables, i.e. it doesn’t consider the process by which it is deciding what

to do. The perception of having choices is also what humans con-

sider as free will. The matter is discussed in [MH69] and is roughly

in accordance with the philosophical attitude towards free will called

compatibilism, i.e. the view that determinism and free will are compat-

ible.

Reification To refify an entity is to “make a thing” out of it (from Latin

re for thing). From a logical point of view, things are what variables

can range over. Logical AI needs to reify hopes, intentions and “things

wrong with the boat”. Some philosophers deplore reification, referring

to a “bloated ontology”, but AI needs more things than are dreamed of

in the philosophers’ philosophy. In general, reification gives a language

more expressive power, because it permits referring to entities directly

that were previously mentionable only in a metalanguage.

Ontology In philosophy, ontology is the branch that studies what things

exist. W.V.O. Quine’s view is that the ontology is what the variables

range over. Ontology has been used variously in AI, but I think Quine’s

usage is best for AI. “Reification” and “ontology” treat the same phe-

nomena. Regrettably, the word “ontology” has become popular in AI

in much vaguer senses. Ontology and reification are basically the same

concept.

Approximate Concepts Common sense thinking cannot avoid concepts

without clear definitions. Consider the welfare of an animal. Over a

period of minutes, the welfare is fairly well defined, but asking what

will benefit a newly hatched chick over the next year is ill defined. The

exact snow, ice and rock that constitutes Mount Everest is ill defined.

The key fact about approximate concepts is that while they are not

well defined, sentences involving them may be quite well defined. For

example, the proposition that Mount Everest was first climbed in 1953

is definite, and its definiteness is not compromised by the ill-definedness

of the exact boundaries of the mountain. See [McC99b].

There are two ways of regarding approximate concepts. The first is to

suppose that there is a precise concept, but it is incompletely known.

Thus we may suppose that there is a truth of the matter as to which

rocks and ice constitute Mount Everest. If this approach is taken, we

simply need weak axioms telling what we do know but not defining the

concept completely.

The second approach is to regard the concept as intrinsically approxi-

mate. There is no truth of the matter. One practical difference is that

we would not expect two geographers independently researching Mount

Everest to define the same boundary. They would have to interact, be-

cause the boundaries of Mount Everest are yet to be defined.3

Approximate Theories Any theory involving approximate concepts is an

approximate theory. We can have a theory of the welfare of chickens.

However, its notions don’t make sense if pushed too far. For example,

animal rights people assign some rights to chickens but cannot define

them precisely.

It is not presently apparent whether the expression

of approximate theories in mathematical logical languages will require

any innovations in mathematical logic. See [McC99b].

Ambiguity Tolerance Assertions often turn out to be ambiguous with the

ambiguity only being discovered many years after the assertion was

enunciated. For example, it is a priori ambiguous whether the phrase

“conspiring to assault a Federal official” covers the case when the crim-

inals mistakenly believe their intended victim is a Federal official. An

3Regarding a concept as intrinsically approximate is distinct from either regarding it

as fully defined by nature or fully defined by human convention.

ambiguity in a law does not invalidate it in the cases where it can be

considered unambiguous. Even where it is formally ambiguous, it is

subject to judicial interpretation. AI systems will also require means

of isolating ambiguities and also contradictions. The default rule is that

the concept is not ambiguous in the particular case. The ambiguous

theories are a kind of approximate theory.

Causal Reasoning A major concern of logical AI has been treating the

consequences of actions and other events. The epistemological problem

concerns what can be known about the laws that determine the results

of events. A theory of causality is pretty sure to be approximate.

Situation Calculus Situation calculus is the most studied formalism for

doing causal reasoning. A situation is in principle a snapshot of the

world at an instant. One never knows a situation—one only knows

facts about a situation. Events occur in situations and give rise to new

situations. There are many variants of situation calculus, and none

of them has come to dominate. [MH69] introduces situation calculus.

[GLR91] is a 1991 discussion.

Fluents Functions of situations in situation calculus. The simplest fluents

are propositional and have truth values. There are also fluents with

values in numerical or symbolic domains. Situational fluents take on

situations as values.

Frame Problem This is the problem of how to express the facts about the

effects of actions and other events in such a way that it is not necessary

to explicitly state for every event, the fluents it does not affect. Murray

Shanahan [Sha97] has an extensive discussion.

Qualification Problem This concerns how to express the preconditions for

actions and other events. That it is necessary to have a ticket to fly

on a commercial airplane is rather unproblematical to express. That it

is necessary to be wearing clothes needs to be kept inexplicit unless it

somehow comes up.

Ramification Problem Events often have other effects than those we are

immediately inclined to put in the axioms concerned with the particular

kind of event.

Projection Given information about a situation, and axioms about the ef-

fects of actions and other events, the projection problem is to determine

facts about future situations.

It is assumed that no facts are avail-

able about future situations other than what can be inferred from the

“known laws of motion” and what is known about the initial situation.

Query: how does one tell a reasoning system that the facts are such

that it should rely on projection for information about the future.

Planning The largest single domain for logical AI has been planning, usu-

ally the restricted problem of finding a finite sequence of actions that

will achieve a goal. [Gre69a] is the first paper to use a theorem prover to

do planning. Planning is somewhat the inverse problem to projection.

Narrative A narrative tells what happened, but any narrative can only

tell a certain amount. What narratives can tell, how to express that

logically, and how to elaborate narratives is given a preliminary logical

treatment in [McC95b] and more fully in [CM98a]. [PR93] and [RM94]

are relevant here. A narrative will usually give facts about the future of

a situation that are not just consequences of projection from an initial

situation. [While we may suppose that the future is entirely determined

by the initial situation, our knowledge doesn’t permit inferring all the

facts about it by projection. Therefore, narratives give facts about the

future beyond what follows by projection.]

Understanding A rather demanding notion is most useful. In particular,

fish do not understand swimming, because they can’t use knowledge to

improve their swimming, to wish for better fins, or to teach other fish.

See the section on understanding in [McC96a]. Maybe fish do learn

to improve their swimming, but this presumably consists primarily of

the adjustment of parameters and isn’t usefully called understanding.

I would apply understanding only to some systems that can do hypo-

thetical reasoning—if p were true, then q would be true. Thus Fortran

compilers don’t understand Fortran.

Consciousness, awareness and introspection Human level AI systems

will require these qualities in order to do tasks we assign them. In order

to decide how well it is doing, a robot will need to be able to examine

its goal structure and the structure of its beliefs from the outside. See

[McC96a].

Intention to do something Intentions as objects are discussed briefly in

[McC89] and [McC96a].

Mental situation calculus The idea is that there are mental situations,

mental fluents and mental events that give rise to new mental sit-

uations. The mental events include observations and inferences but

also the results of observing the mental situation up to the current

time. This allows drawing the conclusion that there isn’t yet informa-

tion needed to solve a certain problem, and therefore more information

must be sought outside the robot or organism. [SL93] treats this and

so does [McC96a].

Discrete processes Causal reasoning is simplest when applied to processes

in which discrete events occur and have definite results. In situation

calculus, the formulas s(cid:48) = result(e, s) gives the new situation s(cid:48) that

results when the event e occurs in situation s. Many continuous pro-

cesses that occur in human or robot activity can have approximate

theories that are discrete.

Continuous Processes Humans approximate continuous processes with rep-resentations that are as discrete as possible. For example, “Junior read

a book while on the airplane from Glasgow to London.” Continuous

processes can be treated in the situation calculus, but the theory is so

far less successful than in discrete cases. We also sometimes approxi-

mate discrete processes by continuous ones.

[Mil96] and [Rei96] treat

this problem.

Non-deterministic events Situation calculus and other causal formalisms

are harder to use when the effects of an action are indefinite. Often

result(e, s) is not usefully axiomatizable and something like occurs(e, s)

must be used.

Concurrrent Events Formalisms treating actions and other events must

allow for any level of dependence between events. Complete indepen-

dence is a limiting case and is treated in [McC95b].

Conjunctivity It often happens that two phenomena are independent. In

that case, we may form a description of their combination by taking

the conjunction of the descriptions of the separate phenomena. The

description language satisfies conjunctivity if the conclusions we can

draw about one of the phenomena from the combined description are

the same as the conjunctions we could draw from the single description.

For example, we may have separate descriptions of the assassination

of Abraham Lincoln and of Mendel’s contemporaneous experiments

with peas. What we can infer about Mendel’s experiments from the

conjunction should ordinarily be the same as what we can infer from

just the description of Mendel’s experiments. Many formalisms for

concurrent events don’t have this property, but conjunctivity itself is

applicable to more than concurrent events.

To use logician’s language, the conjunction of the two theories should be

a conservative extension of each of the theories. Actually, we may settle

for less. We only require that the inferrable sentences about Mendel (or

about Lincoln) in the conjunction are the same. The combined theory

may admit inferring other sentences in the language of the separate

theory that weren’t inferrable in the separate theories.

Learning Making computers learn presents two problems—epistemological

and heuristic. The epistemological problem is to define the space of

concepts that the program can learn. The heuristic problem is the

actual learning algorithm. The heuristic problem of algorithms for

learning has been much studied and the epistemological mostly ignored.

The designer of the learning system makes the program operate with a

fixed and limited set of concepts. Learning programs will never reach

human level of generality as long as this approach is followed. [McC59]

says, “A computer can’t learn what it can’t be told.” We might

correct this, as suggested by Murray Shanahan, to say that it can

only learn what can be expressed in the language we equip it with.

To learn many important concepts, it must have more than a set of

weights. [MR94] and [BM95] present some progress on learning within

a logical language. The many kinds of learning discussed in [Mit97]

are all, with the possible exception of inductive logic programming,

very limited in what they can represent—and hence can conceivably

learn. [McC99a] presents a challenge to machine learning problems and

discovery programs to learn or discovery the reality behind appearance.

Representation of Physical Objects We aren’t close to having an epis-

temologically adequate language for this. What do I know about my

pocket knife that permits me to recognize it in my pocket or by sight

or to open its blades by feel or by feel and sight? What can I tell oth-

ers about that knife that will let them recognize it by feel, and what

information must a robot have in order to pick my pocket of it?

Representation of Space and Shape We again have the problem of an

epistemologically adequate representation. Trying to match what a hu-

man can remember and reason about when out of sight of the scene

is more what we need than some pixel by pixel representation. Some

problems of this are discussed in [McC95a] which concerns the Lem-

mings computer games. One can think about a particular game and

decide how to solve it away from the display of the position, and this ob-

viously requires a compact representation of partial information about

a scene.

Discrimination, Recognition and Description Discrimination is the de-

ciding which category a stimulus belongs to among a fixed set of cate-

gories, e.g. decide which letter of the alphabet is depicted in an image.

Recognition involves deciding whether a stimulus belongs to the same

set, i.e. represents the same object, e.g. a person, as a previously seen

stimulus. Description involves describing an object in detail appropri-

ate to performing some action with it, e.g. picking it up by the handle

or some other designated part. Description is the most ambitious of

these operations and has been the forte of logic-based approaches.

Logical Robot [McC59] proposed that a robot be controlled by a program

that infers logically that a certain action will advance its goals and then

does that action. This approach was implemented in [Gre69b], but the

program was very slow. Shortly greater speed was obtained in systems

like STRIPS at the cost of limiting the generality of facts the robot

takes into account. See [Nil84], [LRL+97], and [Sha96].

Declarative Expression of Heuristics [McC59] proposes reasoning be con-trolled by domain-dependent and problem-dependent heuristics expresseddeclaratively. Expressing heuristics declaratively means that a sentence

about a heuristic can be the result of reasoning and not merely some-

thing put in from the outside by a person. Josefina Sierra [Sie98b],

[Sie98a], [Sie98c], [Sie99] has made some recent progress.

Logic programming Logic programming isolates a subdomain of first or-

der logic that has nice computational properties. When the facts are

described as a logic program, problems can often be solved by a stan-

dard program, e.g. a Prolog interpreter, using these facts as a program.

Unfortunately, in general the facts about a domain and the problems

we would like computers to solve have that form only in special cases.

Useful Counterfactuals “If another car had come over the hill when you

passed that Mercedes, there would have been a head-on collision.”

One’s reaction to believing that counterfactual conditional sentence is

quite different from one’s reaction to the corresponding material con-

ditional. Machines need to represent such sentences in order to learn

from not-quite-experiences. See [CM98b].

Formalized Contexts Any particular bit of thinking occurs in some con-

text. Humans often specialize the context to particular situations or

theories, and this makes the reasoning more definite, sometimes com-

pletely definite. Going the other way, we sometimes have to generalize

the context of our thoughts to take some phenomena into account.

It has been worthwhile to admit contexts as objects into the ontology of

logical AI. The prototype formula ist(c, p) asserts that the proposition

p is true in the context c. The formal theory is discussed in [McC93],

[MB98] and in papers by Saˇsa Buvaˇc, available in [Buv95].

Rich and Poor Entities A rich entity is one about which a person or ma-

chine can never learn all the facts. The state of the reader’s body is

a rich entity. The actual history of my going home this evening is a

rich entity, e.g. it includes the exact position of my body on foot and

in the car at each moment. While a system can never fully describe

a rich entity, it can learn facts about it and represent them by logical

sentences.

Poor entities occur in plans and formal theories and in accounts of

situations and events and can be fully prescribed. For example, my

plan for going home this evening is a poor entity, since it does not

contain more than a small, fixed amount of detail. Rich entities are

often approximated by poor entities. Indeed some rich entities may be

regarded as inverse limits of trees of poor entities. (The mathematical

notion of inverse limit may or may not turn out to be useful, although

I wouldn’t advise anyone to study the subject quite yet just for its

possible AI applications.)

Nonmonotonic Reasoning Both humans and machines must draw con-

clusions that are true in the “best” models of the facts being taken

into account. Several concepts of best are used in different systems.

Many are based on minimizing something. When new facts are added,

some of the previous conclusions may no longer hold. This is why the

reasoning that reached these conclusions is called nonmonotonic.

Probabilistic Reasoning Probabilistic reasoning is a kind of nonmono-

tonic reasoning.

If the probability of one sentence is changed, say

given the value 1, other sentences that previously had high probability

may now have low or even 0 probability. Setting up the probabilistic

models, i.e defining the sample space of “events” to which probabilities

are to be given often involves more general nonmonotonic reasoning,

but this is conventionally done by a person informally rather than by

a computer.

In the open common sense informatic situation, there isn’t any apparent

overall sample space. Probabilistic theories may formed by limiting the

space of events considered and then establishing a distribution. Lim-

iting the events considered should be done by whatever nonmonotonic

reasoning techniques are developed techniques for limiting the phenom-

ena taken into account. (You may take this as a confession that I don’t

know these techniques.) In forming distributions, there would seem to

be a default rule that two events e1 and e2 are to be taken as indepen-

dent unless there is a reason to do otherwise. e1 and e2 can’t be just

any events but have to be in some sense basic events.

Circumscription A method of nonmonotonic reasoning involving minimiz-

ing predicates (and sometimes domains). It was introduced in [McC77],

[McC80] and [McC86]. An up-to-date discussion, including numerous

variants, is [Lif94].

Default Logic A method of nonmonotonic reasoning introduced in [Rei80]

that is the main survivor along with circumscription.

Yale Shooting Problem This problem, introduced in [HM86], is a simple

Drosophila for nonmonotonic reasoning. The simplest formalizations

of causal reasoning using circumscription or default logic for doing the

nonmonotonic reasoning do not give the result that intuition demands.

Various more recent formalizations of events handle the problem ok.

The Yale shooting problem is likely to remain a benchmark problem

for formalizations of causality.

Design Stance Daniel Dennett’s idea [Den78] is to regard an entity in terms

of its function rather than in terms of its physical structure. For exam-

ple, a traveller using a hotel alarm clock need not notice whether the

clock is controlled by a mechanical escapement, the 60 cycle power line

or by an internal crystal. We formalize it in terms of (a) the fact that

it can be used to wake the traveller, and (b) setting it and the noise it

makes at the time for which it is set.

Physical Stance We consider an object in terms of its physical structure.

This is needed for actually building it or repairing it but is often un-

necessary in making decisions about how to use it.

Intentional Stance Dennett proposes that sometimes we consider the be-

havior of a person, animal or machine by ascribing to it belief, desires

and intentions. This is discussed in [Den71] and [Den78] and also in

[McC79].

Relation between logic and calculation and various data structures

Murray Shanahan recommends putting in something about this.

Creativity Humans are sometimes creative—perhaps rarely in the life of

an individual and among people. What is creativity? We consider

creativity as an aspect of the solution to a problem rather than as

attribute of a person (or computer program).

A creative solution to a problem contains a concept not present in

the functions and predicates in terms of which the problem is posed.

[McC64] and [McC]discuss the mutilated checkerboard problem.

The problem is to determine whether a checkerboard with two diago-

nally opposite squares can be removed can be covered with dominoes,

each of which covers two rectilinearly adjacent squares. The standard

proof that this can’t be done is creative relative to the statement of the

problem. It notes that a domino covers two squares of opposite color,

but there are 32 squares of one color and 30 of the other color to be

colored.

Colors are not mentioned in the statement of the problem, and their

introduction is a creative step relative to this statement. For a mathe-

matician of moderate experience (and for many other people), this bit

of creativity is not difficult. We must, therefore, separate the concept

of creativity from the concept of difficulty.

Before we can have creativity we must have some elaboration tolerance2.

Namely, in the simple languagge of A tough nut . . ., the colors of the

squares cannot even be expressed. A program confined to this language

could not even be told the solution. As discussed in [McC96b], Zermelo-

Frankel set theory is an adequate language. In general, set theory, in

a form allowing definitions may have enough elaboration tolerance in

general. Regard this as a conjecture that requires more study.

How it happened Consider an action like buying a pack of cigarettes on a

particular occasion and the subactions thereof. It would be a mistake

to regard the relation between the action and its subactions as like

that between a program and its subroutines. On one occasion I might

have bought the cigarettes from a machine. on a second occasion at a

supermarket, and on a third occasion from a cigarettelegger, cigarettes

having become illegal.

3 Alphabetical Index

Ambiguity Tolerance

Approximate Concepts

Approximate Theories

Bounded Informatic Situation

Causal Reasoning

Circumscription

Common Sense Informatic Situation

Common Sense Knowledge of the World

Common Sense Physics

Concurrrent Events

Conjunctivity

Consciousness, awareness and introspection

Continuous Processes

Creativity

Declarative Expression of Heuristics

Default Logic

Design Stance

Discrete processes

Discrimination, Recognition and Description

Elaboration Tolerance

Epistemologically Adequate Languages

Epistemology and Heuristics

Expert Systems

Fluents

Formalized Contexts

Frame Problem

How it happened

Intention to do something

Intentional Stance

Knowledge Level

Learning

Logic programming

Logical Robot

Mental situation calculus

Narrative

Non-deterministic events

Nonmonotonic Reasoning

Ontology

Physical Stance

Planning

Probabilistic Reasoning

Projection

Qualification Problem

Qualitative Reasoning

Ramification Problem

Reification

Relation between logic and calculation and various data structures

Representation of Physical Objects

Representation of Space and Shape

Rich and Poor Entities

Robotic Free Will

Situation Calculus

Understanding

Useful Counterfactuals

Yale Shooting Problem

4 REMARKS

0775.

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I am grateful to Murray Shanahan for many useful suggestions, encouraging

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