\documentclass[12pt]{slides}\usepackage{color}%\usepackage{graphics}% usage:  \textcolor{cyan}{ Reasoning about action}% for 3400 display, compile on Powerbook 3400% configs on Oztex, select A4 landscape\setlength{\textwidth}{8in}\setlength{\textheight}{6in}\setlength{\topmargin}{0in}% colors include cyan, red, green, yellow (pale) magenta\begin{document}%\figure{}\begin{slide}\begin{center}	    \textcolor{red}{CONCEPTS AS OBJECTS} \\        John McCarthy\\	Computer Science Department\\%	Stanford University\\	jmc@cs.stanford.edu \\	\textcolor{red}{http://www-formal.stanford.edu/jmc/} \\%	\today\\        \vspace{0.3in}\end{center}\begin{enumerate}        \item Concepts (including propositions) as objects%    in a first order theory.%\item Individual concepts are the main case.\item Functions from objects to concepts of them.\item Concepts and propositions are not a natural kind.\begin{itemize}    \item There are a variety of useful spaces of concepts.\item Concepts are (usually) approximate entities.\end{itemize}\end{enumerate}\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{slide}\begin{center}	    \textcolor{red}{Concepts and propositions---1} \\        \vspace{0.3in}\end{center}{\it ``...it seems that hardly anybody proposes to use different variablesfor propositions and for truth-values, or different variables forindividuals and individual concepts.''}---(Carnap 1956, p. 113).Variables for propositions and individuals are written in lower case, e.g. $p$ and $x$.  Variables for propositions and individual concepts are capitalized, e.g. $P$ and $X$.This talk is about expressiveness rather than for presenting a theory.\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{slide}\begin{center}	    \textcolor{red}{Concepts and propositions--2} \\        \vspace{0.3in}\end{center}We write\begin{displaymath}\begin{array}{l}    denotes(Mike,mike)\ \mbox{or when functional,}\ mike = denot(Mike) \\ \\    Telephone(Mike)\ \mbox{is the concept of Mike's telephone number}\\ \\    denot(Telephone(Mike)) = telephone(mike) \\\end{array}    \end{displaymath}\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{slide}\begin{center}	    \textcolor{red}{Knowing what and knowing that} \\\end{center}\begin{displaymath}\begin{array}{l}    knows(pat,Telephone(Mike)) \\ \\    \mbox{Suppose}\ telephone(mike) = telephone(mary) \\ \\    Telephone(Mike) \not= Telephone(Mary) \\ \\    \mbox{Possibly},\ \lnot knows(pat,Telephone(Mary)) \\ \\    \textcolor{blue}{\mbox{Truth values and propositions:}}\\    man(mike) \\    true(Man(Mike) \\    knows(pat,Man(Mike)) \mbox{ means Pat knows whether Mike is a man.} \\    \mbox{Possibly}\ knows(pat,Man(Mike)) \land \lnot man(mike) \\ \\    k(pat,Man(Mike)) \equiv true(Man(Mike)) \land knows(pat,Man(Mike)) \\\end{array}    \end{displaymath}\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{slide}\begin{center}	    \textcolor{red}{Equality and Existence } \\%        \vspace{0.3in}\end{center}%\begin{displaymath}\begin{array}{l}    true(Telephone(Mike)\ EqualsC\ Telephone(Mary)\mbox{, although}\\    Telephone(Mike) \not= Telephone(Mary) \\    telephone(denot(Mike)) = telephone(denot(Mary)) \\    telephone(mike) = telephone(mary) \\    denot(Telephone(Mike)) = denot(Telephone(Mary)) \\\end{array}    \end{displaymath}\begin{displaymath}    (\forall X)(exists(X) \equiv (\exists x)denotes(X,x)) \end{displaymath}\begin{displaymath}\begin{array}{l}   ishorseC{Pegasus} \\   Winged(Pegasus) \\   \textcolor{red}{?} true(\mbox{Winged-Horse}(Pegasus)) \\   true(\mbox{Greek mythology}, \mbox{Winged-Horse}(Pegasus)) \\   \lnot exists(Pegasus) \\\end{array}    \end{displaymath}We can have \begin{displaymath}\begin{array}{l}(\exists X)(exists(\mbox{Greek Mythology},X) \land \mbox{Winged-Horse}(X)),    \end{array}    \end{displaymath}%but most likely, there doesn't have to be a domain of Greekmythological objects.  This suggests that some of the rules ofinference of predicate logic be weakened for such theories.\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{slide}\begin{center}	    \textcolor{red}{About propositions} \\        \vspace{0.3in}\end{center}\begin{displaymath}\begin{array}{l}true(Not(P)) \equiv \lnot true(P) \\true(P\ And\ Q) \equiv true(P) \land true(Q) \\\textcolor{red}{?}\ P\ And\ Q = Q\ And\ P \\\textcolor{red}{?}\ P\ And\ (Q\ Or\ R) = (P\ And\ Q)\ Or\ (P\ And\ R) \\\end{array}    \end{displaymath}This way lies NP-completeness and even undecidablity of whether two formulas name the same proposition.\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{slide}\begin{center}	    \textcolor{red}{Functions from things to concepts} \\        \vspace{0.3in}\end{center}Numbers can have standard concepts $Concept1(n)$ is a certain standardconcept of the number $n$.  Writing $Concept1$ suggests that there mightbe another mapping $Concept2$ from numbers to concepts of them.We can have\begin{displaymath}\begin{array}{l}\lnot knew(kepler,CompositeC(Number(Planets))), \\\mbox{and also} \\knew(kepler,(CompositeC(Concept1(denot(Number(Planets)))))). \\\end{array}    \end{displaymath}\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{slide}\begin{center}	    \textcolor{red}{Functions from things to concepts--2} \\        \vspace{0.3in}\end{center}Russell's example: \emph{I thought your yacht was longer thann it is.}can be treated similarly, although it requires a function going from theconcept $Length(Youryacht)$ to what I thought its value was.\begin{displaymath}\begin{array}{l}denot(I,Length(Youryacht)) > length(youryacht)\end{array}    \end{displaymath}\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{slide}\begin{center}	    \textcolor{red}{Functions from things to concepts--3} \\        \vspace{0.3in}\end{center}        We may also want a map from things to concepts of them inorder to formalize a sentence like, {\it ``Lassie knows the location ofall her puppies''}.  We write this%\begin{displaymath}(\forall x)(ispuppy(x,lassie) \supset knowsd(lassie,LocationdC(Conceptd(x))).\end{displaymath}%$Conceptd$ takes a puppy into a dog's concept of it, and $Locationd$ takes a dog's concept of a puppy into a dog's concept of its location.The axioms satisfied by $knowsd$, $Locationd$ and $Conceptd$ can be tailoredto our ideas of what dogs know.\begin{displaymath}\begin{array}{l}(\exists n2)(k(pat,Concept2(n2)\ EqualsC\ Telephone(Mike)))\\\quad\quad \equiv knows(pat,Telephone(Mike))\\\mbox{or}\\knows(pat,Telephone(Mike)) \\ \quad\quad \equiv denot(pat,Telephone(Mike)) =telephone(mike).\end{array}    \end{displaymath}\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{slide}\begin{center}	    \textcolor{red}{Concepts as approximate entities} \\        \vspace{0.3in}\end{center}$\bullet$ Approximate entities occur in human common sense reasoning. They don't have if-and-only-if definitions, e.g. the rock and iceconstituting Mount Everest.$\bullet$ The set of individual concepts of Greek mythology is anotherapproximate entity.  Few of them have denotations.$\bullet$ The logical way of handling approximate entities is toaxiomatize them weakly.  \textcolor{blue}{Did Pegasus have a mother?}$\bullet$ $exists(\mbox{Greek Mythology},Pegasus)$,\\$\lnot exists(\mbox{Greek Mythology},Thor)$,\\$\lnot exists(\mbox{Greek Mythology},\mbox{George Bush})$,\\$exists(\mbox{Greek Mythology},Mother(Pegasus))?$\\\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{slide}\begin{center}	    \textcolor{red}{Mr.~S and Mr.~P} \\        \vspace{0.3in}\end{center}Two numbers $m$ and $n$ are chosen such that $2 \leq  m \leq  n \leq  99$.Mr.~S is told their sum and Mr.~P is told their product.  The followingdialogue ensues:Mr.~P: I don't know the numbers.\\Mr.~S:  I knew you didn't know.  I don't know either.\\Mr.~P:  Now I know the numbers.\\Mr~ S:  Now I know them too.\medskipIn view of the dialogue, what are the numbers? ``Two puzzles involving knowledge''\\www-formal.stanford.edu/jmc/puzzles.html\\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\begin{slide}\begin{center}	    \textcolor{red}{Formalizing Mr.~S and Mr.~P} \\        \vspace{0.3in}\end{center}\begin{displaymath}\begin{array}{l} knows(person,pair,time), k(person,Proposition,time) \\ \mbox{persons: } s,\ p,\ sp \\ \lnot knows(p,Pair0,0) \\ knows(s,Sum(Pair0),0) \\ knows(p,Product(Pair0),0) \\ (\forall pair)(sum(pair) = sum(pair0) \\ \quad\quad \rightarrow \lnot k(s,Not(Pair0\ Equal\ Concept1(pair)),0))\\ k(sp,\ldots,0)\\ \end{array}    \end{displaymath}In the paper $A(w1,w2,person,time)$ means that in world $w1$, world $w2$is possible for $person$ at $time$. ``Two puzzles involving knowledge''\\www-formal.stanford.edu/jmc/puzzles.html\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{document}\begin{slide}\begin{center}	    \textcolor{red}{} \\        \vspace{0.3in}\end{center}\begin{displaymath}\begin{array}{l}    \end{array}    \end{displaymath}\vfill\end{slide}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%A concept is a member of a concept space characterized by axioms.  Thus a group element is a member of a groupEvade paradoxes by weakening.