\documentclass{slides}
\usepackage{color}

\begin{document}
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\begin{slide}
  \begin{center}
    \textcolor{red}{HUMAN-TYPE COMMON SENSE NEEDS EXTENSIONS TO
      LOGIC} \\
    John McCarthy, Stanford University  \end{center}
$\bullet$ Logical AI \textcolor{blue}{(artificial intelligence)} is
based on programs that represent facts about the world in languages of
mathematical logic and decide what actions will achieve goals by
logical reasoning.  A lot has been accomplished with logic as is.

$\bullet$ This was Leibniz's goal, and I think we'll eventually
achieve it.  When he wrote \textcolor{blue}{Let us calculate}, maybe
he imagined that the AI problem would be solved and not just that of a
logical language for expressing common sense facts.  We can have a
language adequate for expressing common sense facts and reasoning
before we have the ideas needed for human-level AI.

$\bullet$ It's a \textcolor{blue}{disgrace} that logicians have
forgotten Leibniz's goal, but there's an excuse.  Nonmonotonic
reasoning is needed for common sense, but it can yield conclusions
that aren't true in all models of the premises---just the preferred
models.

$\bullet$ Almost 50 years work has gone into logical AI and its rival,
AI based on imitating neurophysiology.  Both have achieved some
success, but neither is close to human-level intelligence.

$\bullet$ The \textcolor{red}{common sense informatic situation}, in
contrast to \textcolor{red}{bounded informatic situations}, is key
to human-level AI.

%$\bullet$ I'll discuss extensions to present mathematical logic needed for
%\textcolor{blue}{human-type} common sense.

$\bullet$ First order languages will do, especially if a heavy duty
axiomatic set theory is included, e.g. $A \times B$, $A^B$, list
operations, and recursive definition are directly included.  To make
reasoning as concise as human informal set-theoretic reasoning, many
theorems of set theory need to be taken as axioms.

\vfill\end{slide}


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\begin{slide}
  \begin{center}
    \textcolor{red}{THREE KINDS OF EXTENSION: more may be needed.} \\
  \end{center}
$\bullet$ \textcolor{blue}{Non-monotonic reasoning}.  G\"odel's
completeness theorem tells us that logical deduction cannot be
extended if we demand truth in all interpretations of the premises.
Non-monotonic reasoning is relative to a variety of notions of
\emph{preferred interpretation.}

$\bullet$ \textcolor{blue}{Approximate objects}.  Many entities with
which commonsense reasoning deals do not admit if-and-only-if
definitions.  Attempts to give them if-and-only-if definitions lead to
confusion.

$\bullet$ \textcolor{blue}{Extensive reification}.  Contrary to some
philosophical opinion, common sense requires lots of reification, e.g.
of actions, attitudes, beliefs, concepts, contexts, intentions, hopes,
and even whole theories.  Modal logic is insufficient.

\vfill\end{slide}

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\begin{slide}
  \begin{center}
    \textcolor{red}{THE COMMON SENSE INFORMATIC SITUATION} \\
  \end{center}
  $\bullet$ By the \emph{informatic situation} of an animal, person or
  computer program, I mean the kinds of information and reasoning
  methods available to it.
  
  $\bullet$ The \emph{common sense informatic situation } is that of a
  human with ordinary abilities to observe, ordinary innate knowledge,
  and ordinary ability to reason, especially about the consequences of
  events that might occur including the consequences of actions it
  might take.

$\bullet$ Specialized information, like science and about human
institutions such as law, can be learned and embedded in a person's
common sense information. 

$\bullet$ Scientific theories and almost all AI common sense theories
are based on \textcolor{blue}{bounded information situations} in which
the entities and the information about them are limited by their human
designers.

% c:\Documents and Settings\jmc\ftp\aiphil2.tex=50801 approximate
$\bullet$ When such a scientific theory or an AI common sense theory
of the kinds that have been developed proves inadequate,
\textcolor{blue}{its designers examine it from the outside} and make a
better theory.  For a human's common sense as a whole
\textcolor{blue}{there is no outside}.  AI common sense also has to be
\textcolor{blue}{extendable from within}.

$\bullet$ This problem is unsolved in general, and one purpose of this
lecture is to propose some ideas for extending common sense knowledge
from within.  The key point is that in the common sense informatic
situation, any set of facts is subject to elaboration.

\vfill\end{slide}

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\begin{slide}
  \begin{center}
    \textcolor{red}{THE COMMON SENSE INFORMATIC SITUATION (2)} \\
  \end{center}
The common sense informatic situation has at least the following
features.

$\bullet$  In contrast to bounded informatic situations, it is open to new
information.  Thus a person in a supermarket for steaks for dinner may 
phone an airline to find whether a guest will arrive in time for
dinner and will need a steak.

$\bullet$ Common sense knowledge and reasoning often involves
ill-defined entities.  Thus the concepts of my obligations or my
beliefs, though important, are ill-defined. Leibniz might have needed
to express logically, ``If Marlborough wins at Blenheim, Louis XIV
won't be able to make his grandson king of Spain.''  The concepts used
and their relations to previously known entities can take arbitrary
forms.

$\bullet$ Much common sense knowledge has been learned by evolution, e.g. the
semi-permanence of three dimensional objects and is available to young 
babies \cite{Spelke}.

$\bullet$ Our knowledge of the effects of actions and other events that permits
planning has an incomplete form.

$\bullet$ We do much of our common sense thinking in bounded contexts
in which ill-defined concepts become more precise.  A story about a
physics exam problem provides a nice example.

 
\vfill\end{slide}

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\begin{slide}
  \begin{center}
    \textcolor{red}{COMMON SENSE INFORMATIC SITUATION---PHYSICS EXAMPLE} \\
  \end{center}
% c:\Documents and Settings\jmc\ftp\human.tex=10493
  
  A nice example of what happens when a student doesn't do the
  \textcolor{blue}{nonmonotonic reasoning} that puts a problem in its
  intended bounded context was discussed in the \emph{American Journal
    of Physics}.  \textcolor{red}{Problem}: find the height of a
  building using a barometer.
  
  $\bullet$ \textcolor{red}{Intended answer}: Multiply the difference
  in pressures by the ratio of densities of mercury and air.
  
  $\bullet$ In the \textcolor{blue}{bounded context} intended by the
  examiner, the above is the only correct answer, but in the
  \textcolor{blue}{common sense informatic situation}, there are
  others.  The article worried about this but involved no explicit
  notion of nonmonotonic reasoning or of context.  Computers solving
  the problem will need explicit nonmonotonic reasoning to identify
  the intended context.
  
\vfill\end{slide}
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\begin{slide}
  \begin{center}
    \textcolor{red}{UNINTENDED COMMON SENSE ANSWERS}
  \end{center}

(1) Drop the barometer from the top of the building and measure the
time before it hits the ground.

(2) Measure the height and length of the shadow of the barometer and
the shadow of the building.
  
(3) Rappel down the building with the barometer as a yardstick.

(4) Lower the barometer on a string till it reaches the ground and
  measure the string.
  
(5) Sit on the barometer and multiply the stories by ten feet.

(6) Tell the janitor, ``I'll give you this fine barometer if you'll
tell me the height of the building.''

(7) Sell the barometer and buy a GPS.

$\bullet$ The limited theory intended by the examiners requires
elaboration to admit the new solutions, and these elaborations are not
just adding sentences.

$\bullet$ We consider two common sense theories that have been
developed (the first now and the second if there's time).  Imbedding
them properly in the common sense informatic situation will require
some extensions to logic---at least nonmonotonic reasoning.

\vfill\end{slide}

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\begin{slide}
  \begin{center}
    \textcolor{red}{A WELL-KNOWN COMMON SENSE THEORY} \\
  \end{center}
%  Here are examples of present common sense AI.
%  
  Here's the main axiom of the \emph{blocks world}, a favorite domain
  for logical AI research.
\begin{equation}
  \label{eq:blocks1}
  \begin{array}[l]{l}
\quad\quad\quad\quad Clear(x,s) \land Clear(y,s) \rightarrow On(x,y,Result(Move(x,y)),s),\\
\mbox{with the definition}\\
\quad\quad\quad\quad Clear(x,s) \equiv (\forall z)\lnot On(z,x) \lor x = Table.
  \end{array}
\end{equation}
Only one block can be on another.
A version that reifies relevant fluents and in which the variable $l$
ranges over locations is
\begin{equation}
  \label{eq:blocks2}
  \begin{array}[l]{l}
Holds(Clear(Top(x)),s) \land Holds(Clear(l),s)\\
\quad\quad\quad \rightarrow
Holds(At(x,l),Result(Move(x,l)),s).
  \end{array}
\end{equation}
This reified version permits quantification over the first argument of
$Holds$.
More axioms than there is time to present are needed in order to
permit inferring in a particular initial situation that a certain plan
will achieve a goal, e.g. to infer
\begin{displaymath}
  \label{eq:blocks3}
  \begin{array}[l]{l}
\quad\quad On(Block1,Block2,\\
Result(Move(Block2,Top(Block2,Result(Move(Block3,Table,S0))))),
  \end{array}
\end{displaymath}
where we have $On(Block3,Block1,S0)$ and therefore $Block3$ has to be
moved before $Block1$ can be moved.

More elaborate versions of the blocks world have been studied, and
there are applications (Reiter and Levesque) to the control of robots.
However, each version is designed by a human and can be extended only
by a human.

We'll discuss the well known example of the stuffy room if there's time.
\vfill\end{slide}


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%\begin{slide}
%  \begin{center}
%    \textcolor{red}{MORE COMMON SENSE IN LOGIC} \\
%  \end{center}

%$\bullet$ 

%%open mind, open heart
  
%\vfill\end{slide}


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\begin{slide}
  \begin{center}
    \textcolor{red}{NEED FOR NON-MONOTONICITY} \\
  \end{center}
%
$\bullet$ Human-level common sense theories and the programs that use
them must elaborate themselves.  For this extensions to logic are
needed, but G\"odel showed that first order logic is complete.  New
conclusions given by extended inference rules would be false in some
interpretations---\textcolor{blue}{but not in preferred interpretations.}

$\bullet$ We humans do nonmonotonic reasoning in many circumstances.
\textcolor{red}{1} The only blocks on the table are those mentioned.
\textcolor{red}{2} A bird may be assumed to fly.  \textcolor{red}{3
  }The meeting may be assumed to be on Wednesday.  \textcolor{red}{4
  }The only \textcolor{blue}{things wrong with the boat} are those
that may be inferred from the facts you know.  \textcolor{red}{5 }In
planning one's day, one \textcolor{blue}{doesn't even think about}
getting hit by a meteorite.

$\bullet$ Deduction is monotonic in the following sense. Let $A$ be a
set of sentences, $p$ a sentence such that \textcolor{red}{$A \vdash
  p$}, and $B$ a set of sentences such that \textcolor{red}{$A \subset
  B$}, then we will also have \textcolor{red}{$B \vdash p$}.
Increasing the set of premises can never reduce the set of deductive
conclusions.

If we nonmonotonically conclude that $B1$ and $B2$ are the only blocks
on the table and now want to mention another block $B3$, we must do
the nonmonotonic reasoning all over again.  Thus nonmonotonic
reasoning is applied to the \textcolor{blue}{whole} set of facts---not
to a subset.

$\bullet$ The word \textcolor{blue}{but} in English blocks certain
nonmonotonic reasoning.  \textcolor{blue}{``The meeting is on
  Wednesday but not at the usual time.''}

$\bullet$ Nonmonotonic reasoning is not subsumed under probabilistic
reasoning---neither in theory nor in practice.  Often it's the reverse.

$\bullet$ Many formalizations of nonmonotonic reasoning have been
studied, including circumscription, default logic, negation as failure
in logic programming.  We'll discuss circumscription, which involves
minimization in logical AI and so is analogous to minimization in
other sciences.

$\bullet$ There are also general theories of nonmonotonic reasoning
\cite{KLM02}.  Unfortunately, the ones I have seen are not oriented
towards common sense.
% Lehmann and Magidor

\vfill\end{slide}


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\begin{slide}
  \begin{center}
    \textcolor{red}{CIRCUMSCRIPTION} \\
  \end{center}
  Circumscription is a form of minimization in logic, perhaps a
  logical analog of calculus of variations.  We minimize a predicate
  $P$ with one or more arguments.  We are allowed to vary $Z$, a
  vector of predicates or domain elements with respect to an ordering
  $P < P'$.  We use a notation proposed by Vladimir Lifschitz where
  $CIRC[A;P;Z]$ is defined by
\begin{displaymath}
  \label{eq:a}
  \begin{array}[l]{l}
A(P,Z) \land \lnot(\exists p\ z)[A(p,z)
\land p < P]
  \end{array}
\end{displaymath}
Any unmentioned symbols are thus assumed constant for the purposes of
circumscription. Often $p \leq p' \equiv (\forall x)(p(x) \rightarrow p'(x))$.

I don't know whether circumscription admits anything analogous to
Lagrange multipliers.

\vfill\end{slide}

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\begin{slide}
  \begin{center}
    \textcolor{red}{AN EXAMPLE OF CIRCUMSCRIPTION} \\
  \end{center}
Let $A$ be the following axiom concerning objects that fly.
\begin{displaymath}
  \label{eq:a}
  \begin{array}[l]{l}
\lnot Ab1(x) \rightarrow \lnot Flies(x)\\
Bird(x) \rightarrow Ab1(x) \\
Bird(x) \land \lnot Ab2(x) \rightarrow Flies(x)\\
Penguin(x) \rightarrow Bird(x) \\
Penguin(x) \rightarrow Ab2(x) \\
Penguin(x) \land \lnot Ab3(x) \rightarrow \lnot Flies(x).
  \end{array}
\end{displaymath}

$Circ[A;(Ab1,Ab2,Ab3);(Flies)]$ lets us infer that the flying objects
are the birds that aren't penguins.

Now add to $A$ the assertions $Bat(x) \rightarrow Ab1(x)$ and
$Bat(x) \land \lnot Ab2(x) \rightarrow Flies(x)$ and do the
circumscription again.  The flying objects are now bats and the
birds that are not penguins.
\vfill\end{slide}



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\begin{slide}
  \begin{center}
    \textcolor{red}{REIFYING PROPOSITIONS AND INDIVIDUAL CONCEPTS} \\
  \end{center}
%
   \textcolor{blue}{...it seems that hardly anybody
    proposes to use different variables for propositions and for
    truth-values, or different variables for individuals and
    individual concepts.}---(Carnap 1956, p. 113) [Church 1951, perhaps?]
  
  $\bullet$ It is customary to assert the necessity, truth or
  knowledge of propositions in some form of modal logic, but modal
  logic is weaker than ordinary language which can treat concepts as
  objects.
  
  $\bullet$ We propose \textcolor{red}{abstract spaces of concepts} to
  provide flexibility.  Thus we can have $pp\ AAnd\ qq = qq\ AAnd\ pp$
  when convenient.  Expressions denoting concepts have doubled initial
  letters.

$\bullet$ Also human-level common sense needs functions from things to
concepts of them.  Here's an example.
\begin{displaymath}
  \begin{array}[l]{l}
  Denot(NNumber(PPlanets)) = Number(Planets),\\
  \lnot Knew(Kepler,CComposite(NNumber(PPlanets))),\\
  Knew(Kepler,CComposite(CConcept1(Denot(NNumber(PPlanets))))).
  \end{array}
\end{displaymath}
$\bullet$ Here $Denot(xx)$ is the thing $xx$ denotes.
  
$\bullet$ We can also define $Exists(xx) \equiv (\exists
x)Denotes(xx,x)$ so that $\lnot Exists(PPegasus)$ asserts that Pegasus
doesn't exist.

$\bullet$ $NNumber(PPlanets)$ is the concept of the number of planets,
and $CConcept1(number)$ is a standard concept of that number.
%
\vfill\end{slide}

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\begin{slide}
  \begin{center}
    \textcolor{red}{APPROXIMATE OR PARTLY DEFINED CONCEPTS} \\
  \end{center}

$\bullet$ Humans language expresses and humans often think in terms of
concepts that are only partly defined.  Examples: the snow and rocks
that constitute Mount Everest, the wants of the United States.
$Wants(U.S.,DDemocratic(IIraq))$.
Mathematical concepts are an exception.

$\bullet$ Syntactically, approximate concepts are handled by weak
axioms, e.g.
\begin{displaymath}
  \begin{array}[l]{l}
\ldots \rightarrow Wants(U.S.,pp) \\
\quad\quad\quad\quad \mbox{and}\\
\ldots \rightarrow \lnot Wants(U.S.,pp).
  \end{array}
\end{displaymath}

$\bullet$ In general, there is no fact of the matter, even
undiscovered, exactly characterizing $Wants(U.S.,pp)$.

$\bullet$ The semantic situation seems similar.  In some
interpretations $Wants(U.S.,pp)$ is true, and in others it is false,
but these needn't match up, although they shouldn't be contradictory.
Defining the semantics of approximate concepts seems puzzling.

$\bullet$ A concept that is approximate in general, can be precise in a
limited context.  The barometer problem shows that.

\vfill\end{slide}

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\begin{slide}
  \begin{center}
    \textcolor{red}{CONTEXTS AS OBJECTS---1} \\
  \end{center}
% ~/ftp/approximate.tex
% ~/ftp/context.tex
  $\bullet$ Everything a person says, writes, or thinks is in a
  context, and the meanings of what one says is relative to the
  context.  Attempts to define terms free of context are usually
  incompletely successful outside mathematics.
  
  $\bullet$ People switch from one context to another rather
  automatically.  I propose contexts as objects---members of suitable
  abstract spaces.

$\bullet$ The are two main formulas.  $Ist(c,pp)$ asserts that the
proposition $pp$ is true in the context $c$.  $Value(c,tterm)$ gives
the value of the individual concept $tterm$ in the context $c$.  Using
$Value$ requires that there be a domain associated with $c$.


\vfill\end{slide}

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\begin{slide}
  \begin{center}
    \textcolor{red}{CONTEXTS AS OBJECTS---2} \\
  \end{center}
$\bullet$ $Ist$ can be compounded.  We can have $Ist(c1,IIst(c2,pp))$
and $Ist(c1,VValue(c2,term)\ Equals\ a)$.

$\bullet$ An alternative notation to $Ist(c,p)$ is
\begin{displaymath}
c: pp,
\end{displaymath}
and likewise
\begin{displaymath}
c1:cc2: pp.
\end{displaymath}

\vfill\end{slide}


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\begin{slide}
  \begin{center}
    \textcolor{red}{EXAMPLES} \\
  \end{center}
$\bullet$ Here are some semi-formal examples.
\begin{displaymath}
  \label{eq:context1}
  \begin{array}[l]{l}
Ist(Conan\ Doyle,DDetective(HHolmes))\\
Ist(U.S. medical history, DDoctor(HHolmes))\\
Ist(U.S. literature, PPoet(HHolmes)) \\
Value(U.S. literature,HHolmes) = Value(U.S. medical history,HHolmes)\\
Ist(U.S. legal history,JJudge(HHolmes)) \\
Value(U.S. literature,HHolmes) = Father(Value(U.S. legal,HHolmes)).
  \end{array}
\end{displaymath}

$\bullet$ Here's an example of \emph{lifting} a theory in which the
predicates $On$ and $Above$ have two arguments to a situation calculus
theory in which they have three arguments.  [An application of abstract
group theory would provide bigger examples.]



\vfill\end{slide}
% c:\Documents and Settings\jmc\ftp\aiphil2.tex=50801
% c:\Documents and Settings\jmc\slides\context\contapp-sli.tex=4839
% c:\Documents and Settings\jmc\ftp\context2.tex=3574  AF-GE
% c:\Documents and Settings\jmc\ftp\human.tex=9399 height of building
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\begin{slide}
  \begin{center}
    \textcolor{red}{\emph{On} AND \emph{Above} WITH TWO AND THREE ARGUMENTS} \\
  \end{center}

  To describe the two argument $Above\hbox{-}theory$, we write 

\medskip

\noindent $Above\hbox{-}theory:$
%
\begin{displaymath}
(\forall x y)(On(x,y)\rightarrow Above(x,y)),
\end{displaymath}
%
\begin{displaymath}
(\forall x y z)(Above(x,y)\land Above(y,z)\rightarrow Above(x,z)),
\end{displaymath}
%
$$\rm{etc.}$$
%
which stands for
%
\begin{displaymath}
C0: Ist(Above\hbox{-}theory,(\forall x y)(On(x,y)\rightarrow Above(x,y)))
\label{onabove1}
\end{displaymath}
%
$$\rm{etc.}$$
\vfill\end{slide}


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\begin{slide}
  \begin{center}
    \textcolor{red}{LIFTING $Above\mbox{-}theory$} \\
  \end{center}
  
        We want to apply $Above\hbox{-}theory$ in a context $C$ in
which $On$ and $Above$ have a third argument denoting a situation. 
We have
%
\begin{displaymath}
C:\quad\quad\quad\quad (\forall x\ y\ s)(On(x,y,s) \equiv Ist(C1(s),On(x,y))),\
\end{displaymath}
%
%$$\rm{etc.},$$
%
thus associating a context $C1(s)$ with each situation $s$.
%
We also need
%
\begin{displaymath}
  \begin{array}[l]{l}
C0:\quad\quad\quad\quad Ist(C,(\forall p\ s)(Ist(Above\hbox{-}theory,p)
\rightarrow Ist(C1(s),p))),  
\end{array}
\end{displaymath}
%
which abbreviates to
%
\begin{displaymath}
C:\quad\quad\quad\quad (\forall p\ s)(Ist(Above\hbox{-}theory,p)\rightarrow Ist(C1(s),p)),\label{c2}
\end{displaymath}
\noindent giving finally

C:\quad\quad\quad\quad $On(x,y,s) \rightarrow Above(x,y,s)$
\vfill\end{slide}


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\begin{slide}
  \begin{center}
    \textcolor{red}{APPROXIMATE ENTITIES CAN BE PRECISE IN LIMITED CONTEXTS} \\
  \end{center}
% c:\Documents and Settings\jmc\ftp\appcon.tex
% ~\ftp\appcon.tex


$\bullet$ Owning, buying and selling, e.g. of a house or a business,
are such complicated concepts in general that a complete axiomatic theory
is out of reach.  However, reasonably complete theories are possible
and used in limited contexts, e.g. while shopping in a supermarket.
\begin{displaymath}
  \label{eq:a}
  \begin{array}[l]{l}
InMarket(s) \land Ist(C(Market),Owns(x,Result(Buys(x),s)))\\
\rightarrow Owns(x,Result(Buys(x),s))
  \end{array}
\end{displaymath}
%
$\bullet$ The AI \emph{drosophila} theories sampled above are also valid in
limited contexts.

$\bullet$ The \textcolor{blue}{lifting relations} between the
sentences true in limited contexts and those valid in more general
contexts need to be explored.

\vfill\end{slide}
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\begin{slide}
  \begin{center}
    \textcolor{red}{CONSCIOUSNESS AND SELF AWARENESS} \\
  \end{center}
$\bullet$ Much self awareness is simple enough not to require any
extensions to logic, e.g. sensations of hunger. or of the positions of
ones limbs.

$\bullet$ However, knowledge and belief, especially assertions of
non-knowledge involve formulas analogous to reflexion principles.
asserting truth.

$\bullet$ In discussing what self awareness a robot requires, I found
it helpful to reify hopes, fears, promises, beliefs, what one thinks a
concept denotes, intentions, prohibitions, likes and dislikes, its own
abilities and those of others, and many more.  The doctrine, common
among philosophers and mathematicians, advocating minimizing the set
of concepts, seems to me to be mistaken.

$\bullet$ When a human or robot needs to refer to the whole of its
knowledge, the situation becomes more complicated, and there are
possibilities for paradox, e.g. with

$\bullet$ $\lnot Know(I,SSitting(PPresidentBush,NNow))$.

$\bullet$ $\lnot Know(Putin,SSitting(PPresidentBush,NNow))$.

$\bullet$ Kraus, Perlis, and Horty treated formulas like the above
expressing non-knowledge.

$\bullet$ One way of avoiding paradox may be to allow reference to ones
knowledge up to the present time.  This is analogous to the restricted
comprehension principle.

\vfill\end{slide}


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\begin{slide}
  \begin{center}
    \textcolor{red}{COMMON SENSE IN MATHEMATICS} \\
  \end{center}
%
  $\bullet$ In mathematical writing, the text between the formulas is
  essential to understanding the formulas.  Introductions often contain
  no formulas.  \textcolor{red}{Understanding this text is
    mathematical common sense.}  Its formal expression should be more
  straightforward than the common sense of (say) history.
  
  $\bullet$---from G\"odel, \emph{Collected Works}, p. 147, we have

\newpage Similarly, proofs, \textcolor{blue}{from a
    formal point of view}, are nothing but finite sequences of
  formulas (with certain specifiable properties).  Of course, for
  metamathematical considerations, \textcolor{blue}{it does not matter
    what objects are chosen for primitive signs}, and we shall assign
  natural numbers to this use.  Consequently, a formula will be a
  finite sequence of natural numbers, and a proof array a finite
  sequence of finite sequences of natural numbers.  The
  metamathematical notions (propositions) thus become notions
  (propositions) about natural numbers of sequences of them; therefore
  they can (at least in part) be expressed by the symbols of PM
  itself.  In particular, it can be shown that the notions
  ``formula'', ``proof'', and ``provable formula'' can be defined in
  the system PM; that is, we can find a formula $F(v)$ with one free
  variable $v$ (of the type of a number sequence) such that $F(v)$,
  \textcolor{blue}{interpreted according to the meaning of the terms
    of PM}, says: $v$ is a provable formula.  We \textcolor{blue}{now
    construct} an undecidable proposition of the system PM, that is, a
  proposition $A$ for which neither $A$ nor \emph{not}-$A$ is
  provable, \textcolor{blue}{in the following manner.}

  $\bullet$ False mathematical counterfactual: If $2^{2^5}+1$ were prime,
  twice it would be prime.

  $\bullet$ ``the notion that the continuum hypothesis is analogous to
  the parallel axiom'', ``G\"odel's incompleteness theorems demolished
  Hilbert's program.'', ``Russell's first reaction to the paradox,
  which he discovered on reading Frege's work, was the `vicious circle
  principle' which declared \ldots meaningless'',
  
\vfill\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{slide}
%  \begin{center}
%    \textcolor{red}{TRANSCENDING THE OUTERMOST CONTEXT} \\
%  \end{center}
%This is the opposite of the previous use of contexts.

%$\bullet$ This often involves introducing and reifying new
%entities---as in introducing the concept of an electron or of a metric
%space.

%$\bullet$ This is analogous to Tarski's definition of truth or
%Turing's and Feferman's extensions of Peano arithmetic.

%$\bullet$ ``I'll never know whether the continuum hypothesis is true.''
%\vfill\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
  \begin{center}
    \textcolor{red}{AI RESEARCH ON COMMON SENSE IN LOGIC} \\
  \end{center}
  
%  $\bullet$ Leibniz, Boole, and Frege wanted to apply mathematical
%  logic to human affairs.  Few logicians today work towards this.
%  
  $\bullet$ Much has been done to express common sense knowledge and
  reasoning in logic.  However, present axiomatic AI theories require
  human modification whenever they are to be elaborated.
  \textcolor{blue}{Human-level AI systems must modify their own
    theories.}
  
  $\bullet$ There are biennial conferences on knowledge representation
  and also triennial workshops on common sense.  CYC is a mostly
  proprietary database of more than a million common sense facts.
  expressed a syntactically sugared mathematical logic.  Its reasoning
  facilities have proved difficult to use.
  
  $\bullet$ Automatic theorem proving and interactive theorem proving
  have had considerable success in \emph{bounded} mathematical and AI
  domains.

\vfill\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
  \begin{center}
    \textcolor{red}{DIFFICULTIES AND CONCLUSIONS} \\
  \end{center}

$\bullet$ \textcolor{red}{We will eventually have human-level logical
  AI.}

$\bullet$ Sooner if we have help from logicians in devising ways of
representing common sense theories and extending them.

$\bullet$ The above are almost surely not the only kinds of extensions
to logic needed for dealing with common sense knowledge and reasoning.

$\bullet$ We discussed the following kinds of extensions.  
(\textcolor{red}{1}) Formal non-monotonic reasoning,
(\textcolor{red}{2}) Reification, especially of concepts and
contexts---and even theories, (\textcolor{red}{3}) Approximate
entities without if-and-only-if definitions.

$\bullet$ There is a particular difficulty in extending a theory
defined in a limited context to a more general context if the theory
requires nonmonotonic reasoning, e.g. if the set of blocks is to be
minimized.

$\bullet$ Human-level logical AI will also require language for
expressing facts about methods effective in reasoning about particular
subjects.

$\bullet$ Articles discussing these questions are available in
http://www-formal.stanford.edu/jmc/.
  
\vfill\end{slide}

 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
 \begin{slide}
        \begin{center}
                        \textcolor{red}{STUFFY ROOM AXIOMS} \\
%                       \vspace{0.3in}
                \end{center}
Effect axioms:
\begin{displaymath}
  \label{eq:ste1}
  \begin{array}[l]{l}
Blocked1(Result(Block1,s)) \\
Blocked2(Result(Block2,s)) \\
\lnot Blocked1(Result(Unblock1,s)) \\
\lnot Blocked2(Result(Unblock2,s)) \\
Stuffy(Result(Getstuffy,s)) \\
\lnot Stuffy(Result(Ungetstuffy,s)) \\
  \end{array}
\end{displaymath}
Occurrence axioms:
\begin{displaymath}
  \label{eq:sto1}
  \begin{array}[l]{l}
Blocked1(s) \land Blocked2(s) \land \lnot Stuffy(s)\\ \rightarrow
Occurs(Getstuffy,s) \\
(\lnot Blocked1(s) \lor \lnot Blocked2(s)) \land Stuffy(s)\\ \rightarrow
Occurs(Ungetstuffy,s)
\end{array}
\end{displaymath}
 \end{slide}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
 \begin{slide}
        \begin{center}
            \textcolor{red}{AN ELABORATION GIVING OSCILLATING STUFFINESS} \\
                \end{center}
Suppose Bob is unhappy when the room is stuffy, but Alice is unhappy
when the room is cold.  The stuffy room axioms tolerate adding the
following axioms which makes Vent1 oscillate between open and closed.
%
\begin{displaymath}
    \begin{array}{c}
        Stuffy(s) \rightarrow Occurs(Does(Bob,Unblock1),s)  \\
        Unblocked1(s) \rightarrow Occurs(Getcold(Alice),s), \\
        Cold(Alice,(Result(Getcold,s)), \\
        Cold(Alice,s) \rightarrow Occurs(Does(Alice,Block1),s).
    \end{array}
    \label{eq:osc}
\end{displaymath}
 
Alas, these axioms hold in a bounded domain.  Common sense requires
logic in which they inhabit an extendable context.
\vfill \end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\end{document}
csis
CYC

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
  \begin{center}
    \textcolor{red}{} \\
  \end{center}
  
\vfill\end{slide}



$\bullet$ Express facts about domain dependent reasoning, including. those
like Polya proposed.

Notices AMS April 2006
The Iincompleteness theorem - Martin Davis

How Godel transformed set theory Juliet Floyd and Akihiro Kanamori
Cantor 1873 Every infinite set of real numbers either is countable or has the
power of the continuum.
In 1873 Cantor established that the continuum is not countable.
There is no bijection between the natural numbers
$N = \{0,1,2,3,\ldots\}$ and the real numbers $R$, since for any
countable sequence of real one can specify nested intervals so that
any real in the intersection will not be in the sequence.

Almost two decades after his initial 1873 proof, Cantor in 1891 came
to his celebrated \emph{diagonal argument}.
$P(x) = \{y|y \subset x\}$ has a higher power than $x$.

What are some states of mind in which one (e.g. Cantor) accepts the
first result but doesn't have the second.  Common sense in mathematics
must be able to represent such states of mind.

Fri Oct  6 17:37:01 2006
Fef's explicit math
Successive Church articles on concepts
Fef grumbled that Godel extract was purely motivational
admitted that some of the text had a larger role.
We discussed "We will construct ...
Grisha noted that contexts are modality
Nobody reacted to dog's concept of her puppies
Need to express some math texts formally
/u/ftp/jmc/asl-sli.tex  What AI needs from mathematical logic
