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C00030 00004	.bb "MODAL LOGIC (part 1)"
C00035 00005	.bb PHILOSOPHICAL EXAMPLES - MOSTLY WELL KNOWN
C00040 00006	.BB EXAMPLES IN ARTIFICIAL INTELLIGENCE
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.require "memo.pub[let,jmc]" source;
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.cb CONCEPTS AS OBJECTS AND CONCEPT-VALUED FUNCTIONS

Abstract: We discuss first order theories in which ⊗concepts are
allowed as mathematical objects along with the things of which
they are the concepts.  This allows unprecedentedly straightforward
formalizations of knowledge, belief, wanting, and necessity.
Applications are given in philosophy and in artificial intelligence.



.skip to column 1
.bb INTRODUCTION

	Admitting concepts as objects - with concept-valued constants,
variables, functions and expressions - allows an ordinary first order theory
of necessity, knowledge, belief and wanting without modal operators
or quotation marks
and without
the restrictions on substituting equals for equals that either device
makes necessary.

	According to Frege (1892), the meaning of the phrase %2"Mike's
telephone number"%1 in the sentence %2"Pat knows Mike's telephone number"%1
is the concept of Mike's telephone number, whereas its meaning in the
sentence %2"Pat dialed Mike's telephone number"%1 is the number itself.
Thus if we also have %2"Mary has the same telephone number as Mike"%1, then
%2"Pat dialed Mary's telephone number"%1 follows, but %2"Pat knows Mary's
telephone number"%1 does not.

	Frege further proposed that a phrase has a ⊗sense 
which is a ⊗concept and is its ⊗meaning in ⊗oblique ⊗contexts 
like knowing and wanting, and a ⊗denotation which is its
⊗meaning in ⊗direct ⊗contexts.  
Denotations are the basis of the Tarskian semantics of first
order logic and model theory and are well understood, but sense
has given more trouble, and the modal treatment of oblique
contexts avoids the idea.  On the other hand, logicians such as
Church (1951) and Montague (1974) see a need for ⊗concepts and
have proposed formalizations, but none have been very satisfactory.

	The problem identified by Frege - of suitably limiting the
application of Leibniz's law of the substitutitivity of equals for
equals - arises in artificial intelligence as well as in philosophy
and linguistics for any system that must represent information about
beliefs, knowledge, desires, or logical necessity - regardless
of whether the representation is procedural or declarative.

	The present idea is to treat concepts as
one kind of object in a first order theory.  Thus we have an
expression whose value is Mike's telephone number and a different
though related expression whose value is the concept of Mike's
telephone number rather than having a single expression whose
denotation is the number and whose sense is a concept of it.
The relations among concepts and between concepts and other
entities are then readily expressed by first order logical
formulas.  Moreover, ordinary model theory can be used to
study what spaces of concepts satisfy various sets of axioms.

.skip 1
.BB KNOWING WHAT AND KNOWING THAT

	To assert that Pat knows Mike's telephone
number we write

!!e1:	%2true Know(Pat,Telephone Mike)%1

with the following conventions:

.item←0;
	#. Parentheses are often omitted for one argument functions and
predicates.  This  purely syntactic convention is not
important.  Another convention is to capitalize the first letter of
a constant, variable or function name when its value is a concept.

	#. ⊗Mike denotes the concept of Mike; i.e. it is the ⊗sense of
the expression %2"Mike"%1.  We will use ⊗mike when we wish to denote Mike
himself.

	#. ⊗Telephone is a function that takes the concept of a person
into the concept of his telephone number.  We will also use ⊗telephone 
which takes the person himself into the telephone number itself.

	#. If ⊗P is a person concept and ⊗X is another concept, then
⊗Know(P,X) is an assertion concept or ⊗proposition meaning that ⊗P
knows the value of ⊗X.  In {eq e1}), therefore, %2Know(Pat,Telephone Mike)%1 is
a proposition and not
a truth value.  Note that we are formalizing ⊗knowing ⊗what 
rather than ⊗knowing ⊗that or ⊗knowing ⊗how.  For AI
and for other practical purposes, ⊗knowing ⊗what 
seems to be the most useful notion of the three.

	#. ⊗true(Q) is the truth value, ⊗t or ⊗f, of the proposition ⊗Q, 
and we must write ⊗true(Q) in order to assert ⊗Q.  Later
we will consider formalisms in which ⊗true has a second argument - a
⊗situation, a ⊗story, a ⊗possible ⊗world, or even a %2partial
possible world%1 (a notion we hope to introduce).

	#. The formulas are in a sorted first order logic with functions
and equality.  In the present informal treatement, we will not be
explicit about sorts, but we will try to be typographically consistent.

	The reader may be nervous about what is meant by ⊗concept. 
He will have to remain nervous; no final commitment will be made
in this paper.  The formalism is compatible with a variety
of possibilities, and these can be compared using the
models of their first order theories.
However, if {eq e1}) is to be reasonable, it must not
follow from {eq e1}) and the fact that Mary's telephone number is the
same as Mike's, that Pat knows Mary's telephone number.

	The proposition that Joe knows ⊗whether Pat knows Mike's telephone number,
is written

!!ee1:	%2Know(Joe,Know(Pat,Telephone Mike))%1,

and asserting it requires writing

!!ee4:	%2true Know(Joe,Know(Pat,Telephone Mike))%1,

while the proposition that Joe knows ⊗that Pat knows Mike's tephone number
is written

!!ee2:	%2K(Joe,Know(Pat,Telephone Mike))%1,

where ⊗K(P,Q) is the proposition that ⊗P knows that ⊗Q. 

	Consider the function ⊗denot and the statement

!!e2:	%2∀P1 P2.(denot P1 = denot P2 ⊃ denot Telephone P1 = denot Telephone P2)%1.

Here %2denot X%1 is the ⊗denotation of the concept ⊗X, and {eq e2})
asserts that the denotation of the concept of %2X%1's telephone
number depends only on the denotation of concept %2X%1.
The variables in {eq e2}) range over concepts of persons, and
we regard {eq e2}) as asserting
that ⊗Telephone is ⊗extensional with respect to ⊗denot. 
Note that our ⊗denot operates on concepts rather than on expressions, but
a theory of expressions will require a denotation function for them also.

	⊗Know is extensional with respect to ⊗denot in its first argument,
and this expressed by

!!e3:	%2∀P1 P2 X.(denot P1 = denot P2 ⊃ denot Know(P1,X) = denot Know(P2,X))%1,

but it is not extensional in its second argument.
(Note that all these predicates and functions are entirely extensional
in the underlying logic, and the notion of extensionality presented
here is relative to the function ⊗denot.)

	The predicate ⊗true and the function ⊗denot are related by

!!e4:	%2∀Q.(true Q ≡ (denot Q = t))%1

provided truth values are in the range of ⊗denot, and ⊗denot may
also be provided with a %2(partial) possible world%1 argument.

	If we want a system in which not all concepts have denotations,
then we should use a predicate ⊗denotes(X,x) instead of a function.
The extensionality of ⊗Telephone would then be written

!!e41:	%2∀P1 P2 x u.(denotes(P1,x)∧denotes(P2,x)∧denotes(Telephone P1,u) ⊃
denotes(Telephone P2,u))%1.

	In order to combine concepts propositionally, we need analogs
of the propositional operators such as ⊗And, which we shall use as an infix,
and axiomatize by

!!e5:	%2∀Q1 Q2.(true (Q1 And Q2) ≡ true Q1 ∧ true Q2)%1, etc.

Assume that the corresponding formulas for ⊗Or, ⊗Not, ⊗Implies, and
⊗Equiv have been written.

	The equality symbol "=" is used with its usual logical meaning
of identity,
so that %2X = Y%1 asserts that ⊗X and ⊗Y are the same concept.
To discuss concepts of particular equalities, we introduce ⊗Equal, and
⊗Equal(X,Y) is the proposition that ⊗X and ⊗Y have equal denotations.
Thus we have

!!ee51:	%2∀X Y.(true Equal(X,Y) ≡ denot X = denot Y)%1.

	Propositions formed by quantification present more of a problem.
We will want
a function ⊗All(var,exp), where ⊗var is a "variable" and ⊗exp is
some kind of "concept-valued expression".
We will need objects called ⊗vars and variables
ranging over them as well as variables
ranging over "concept-valued expressions".
In any case, the basic fact about quantifiers is something like

!!e6:	%2true All(x,E) ≡ ∀x'.(true Subst(x',x,E))%1,

where ⊗subst(x,y,z) is a suitable analog of the LISP ⊗subst. 

	This will be elaborated subsequently using the notion of
extensional form.  Note that variables ranging over concepts
and quantifying over concepts
require no new formalism; problems arise only when the concept
itself includes quantification.

	The conceptual functions can be related to ordinary
extensional functions.  Thus

!!e7:	%2∀P.(denot Telephone P = telephone denot P)%1,

and ⊗telephone can be used in any purely extensional context,
e.g. in the following "law" expressing the effects of dialing
someone's number in the notation of (McCarthy and Hayes 1970):

!!e8:	%2∀p1 p2 s.(speaking(p1,p2,result(p1,dial telephone p2,s)))%1

which asserts that a situation in which ⊗p1 and ⊗p2 are speaking
results from ⊗p1 dialing %2p2%1's telephone number.

	If we were to adopt the convention that a proposition
appearing at the outer level of a sentence is asserted and were to
regard the denotation-valued function as standing for the sense-valued
function when it appears as the second argument of ⊗Know, we would have
a notation that looks like ordinary language in
handling obliquity entirely by context.  There is no guarantee that
general statements could be expressed unambiguously without
circumlocution, but we take
the possibility as an additional sign that we are moving toward
the expressiveness of natural language.

	As mentioned before, %2"Pat knows Mike's telephone number"%1
is written

!!e9:	%2true Know(Pat,Telephone Mike)%1.

We can write %2"Pat knows Mike's telephone number is 333-3333"%1

!!e10:	%2true K(Pat,Equal(Telephone Mike,Concept1 "333-3333")%1

where ⊗K(P,Q) is the proposition that ⊗denot(P) knows the proposition ⊗Q 
and %2Concept1("333-3333")%1 is
some standard concept of that telephone number.

	The two ways of expressing knowledge are somewhat interdefinable,
since we can write

!!e11:	%2K(P,Q) = And(Q,Know(P,Q))%1,

and

!!e12:	%2true Know(P,X) ≡ ∃A.(constant A ∧ true K(P,Equal(X,A)))%1.

Here %2constant A%1 asserts that ⊗A is a constant, i.e. a concept
such that we are willing to say that ⊗P knows ⊗X if he knows it is
equal to some particular constant.  This is clear enough for some
domains like integers, but it is not obvious how to treat knowing
a person.  Another possibility is to introduce for the elements
of certain domains a function ⊗Concept that gives a sort of ⊗standard 
⊗concept of an element of the domain.
Then we can rewrite {eq e12}) as

!!e13:	%2true Know(P,X) ≡ ∃a.true K(P,Equal(X,Concept a))%1.

	{eq e12}) and {eq e13}) expresses a ⊗denotational definition
of ⊗Know in terms of ⊗K.  A ⊗conceptual definition seems to require
something like

!!e14:	%2Know(P,X) = Exist(aA,K(P,Equal(X,Concept aA))%1,

where ⊗aA is a "variable", but we will postpone a
discussion of the interpretation of {eq e14}).

.bb "MODAL LOGIC (part 1)"

	We will divide our treatment of necessity and possibility into
two parts.  In %2unquantified modal logic%1, the arguments of the
modal functions will not involve quantification although quantification
occurs in the logic.

	%2Nec Q%1 is the proposition that the proposition ⊗Q is necessary,
and %2Poss Q%1 is the proposition that it is possible.  To assert
necessity or possibility we must write %2true Nec Q%1 or %2true Poss Q%1.
This can be abbreviated by defining %2nec Q ≡ true Nec Q%1 and %2poss Q%1
correspondingly, but these are predicates in the logic with ⊗t and ⊗f 
as values so that ⊗nec ⊗Q cannot be an argument of ⊗nec or ⊗Nec. 

	Before we even get to modal logic proper we have a decision to
make - shall %2Not Not Q%1 be considered the same proposition as ⊗Q, or
is it merely extensionally equivalent?  The first is written

!!e41a:	%2∀Q. Not Not Q = Q%1,

and the second

!!e42:	%2 ∀Q.true Not Not Q ≡ true Q%1.

The second follows from the first by substitution of equals for equals.

	If we choose the first alternative, then we may go on to identify
any two propositions that can be transformed into each other by Boolean
identities.  This can be assured by a small collection of propositional
identities like {eq e41}) including associative and distributive laws for
conjunction and disjunction, De Morgan's law, and the laws governing the
propositions ⊗T and ⊗F.  In the second alternative we will want the
extensional forms of the same laws.
When we get to quantification a similar choice will arise, but if we
choose the first alternative, it will be undecideable whether two
expressions denote the same concept.  I doubt that considerations of linguistic
usage or usefulness in AI
will unequivocally recommend one alternative, so both will have
to be studied.

	The same question arises in modal logic.  Namely we must choose
between the ⊗conceptual ⊗identity 

!!e43:	%2∀Q.(Poss Q = Not Nec Not Q)%1,

and the weaker extensional axiom

!!e44:	%2∀Q.(true Poss Q ≡ true Not Nec Not Q)%1.

We will write the rest of our modal axioms as conceptual identities,
but their translation into extensional form is easy.

	We have

!!e45:	%2∀Q.(Nec(Q) Implies Q) = T)%1,

and

!!e46:	%2∀Q1 Q2.(Nec(Q1) And Nec(Q1 Implies Q2) Implies Nec(Q2)) = T)%1.

yielding a system equivalent to T.

	S4 is given by

!!e47:	∀Q.(%2Nec Q = Nec Nec Q)%1,

and S5 by

!!e48:	%2∀Q.(Poss Q = Nec Poss Q)%1.

	Actually, there may be no need to commit ourselves to a particular
modal system.  We can simultaneously have the functions ⊗NecT, ⊗Nec4 and ⊗Nec5, 
related by axioms such as

!!e49:	%2∀Q.((Nec4 Q Implies Nec5 Q) = T)%1

which would seem plausible if we regard S4 as corresponding to provability
in some system and S5 as truth in the intended model of the system.

	Presumably we shall want to relate necessity and equality by
the axiom

!!e50:	%2∀X.nec Equal(X,X)%1,

and we may want the stronger relation

!!e51: %2∀X Y.(X=Y ≡ nec Equal(X,Y))%1

which asserts that two concepts are the same if and only if the equality
of what they may denote is necessary.

.bb PHILOSOPHICAL EXAMPLES - MOSTLY WELL KNOWN


	Some sentences that recur as examples in the philosophical literature
will be expressed in our notation so the treatments can be
compared.

	First we have %2"The number of planets = 9"%1 and %2"Necessarily
9 = 9"%1 from one doesn't want to deduce %2"Necessarily the number
of planets = 9"%1.  This example is discussed in (Kaplan 1969).
Consider the sentences

!!e20:	%2¬nec Equal(Number Planets, Concept 9)%1

and

!!e21:	%2nec Equal(Concept number planets,Concept 9)%1.

Both are true.  {eq e20}) asserts that it is not necessary that the
number of planets be 9, and {eq e21} asserts that the number of
planets, once determined, is a number that is necessarily equal to 9.
It is a major virtue of our formalism that both meanings can be
expressed and are readily distinguished.
Leibniz's law of the replacement of equals by equals causes no
trouble, because %2"The number of planets = 9"%1
may be written

!!e22:	%2number(planets) = 9%1

or, using concepts, as

!!e23:	%2true Equal(Number Planets, Concept 9)%1,

and %2"Necessarily 9=9"%1 is written

!!e24:	%2nec Equal(Concept 9,Concept 9)%1,

and these don't yield the unwanted conclusion.

	The following sentence attributed to Russell is is discussed
by Kaplan: %2"I thought that your yacht was longer than it is"%1.  We
can write it

!!e52:	%2true Believed(I,Greater(Length YourYacht,Concept denot Length YourYacht))%1

where we are not analyzing the pronouns or the tense, but are using
⊗denot to get the real length of the yacht and ⊗Concept to get back
a concept of this true length so as to end up with a proposition
that the length of the yacht is greater than that number.
This looks problematical, but if it is consistent, it is probably useful,
and I think it is consistent.

	The function ⊗Concept used in the above examples merits further
study.  It seems useful to provide such a function mapping integers
into standard concepts of integers, and we used a similar function
for mapping telephone numbers regarded as strings of digits into
concepts of them.  This can be extended to other domains, and there
is no need to look for a unique preferred map from objects to their
concepts.  Any maps that are found useful can be used.

	Ryle used the sentences %2"Baldwin is a statesman"%1 and
%2"Pickwick is a fiction"%1 to illustrate that parallel sentence
construction does not always give parallel sense.  We would render
the first as %2true Statesman Baldwin%1 or %2statesman denot Baldwin%1
or %2statesman baldwin%1, while the second can only be rendered
as %2true Fiction Pickwick%1 or %2fiction Pickwick%1.
.BB EXAMPLES IN ARTIFICIAL INTELLIGENCE

	A computer program with general intelligence must be able to
represent facts about what information it lacks and where and how it
is to be obtained.  The example problem I have been considering is
that of representing what a traveler knows about the information airline clerks,
travel agents, and reservation computers, and airline guides have
relevant to a proposed trip.
This is still rather difficult, but the following considerations
have emerged:

.item←0
	#. Unless we formalize ⊗knowing ⊗what, we add to our heuristic
difficulties, because the theorem prover or other reasoner has an extra
existential quantifier to deal with.

	#. Similarly in treating belief we need something like
%2denot(Telephone Mike,Pat,s)%1 standing for what Pat believes Mike's
telephone number to be in the situation ⊗s. 
Neither is formalized in the philosophical literature.

	#. Modal logic offers difficulties especially as we need
often need multiple modalitieπ like %2"believes he wants to know"%1
in a single sentence, and this makes the Kripke possible worlds
semantics for modal logic almost impossibly cumbersome.
Modal logic is especially troublesome if oblique contexts are
only a small part of the problem.

	#. For this reason, the most useful of the earlier treatments seemed
to involve making the argument of knowledge or belief a sentence
or term and weakening the Montague and Kaplan (1963) knowledge axioms
suitably to avoid their paradox.  However, it is not easy to implement
a reasoning program that goes into quoted phrases.

	Consider the following easier example:

	Joe wants to know Mike's telephone number.  He knows that
Pat knows it and that Pat likes Joe.  We want the program to decide
on Joe's behalf to ask Pat for Mike's telephone number.

.skip 1
.bb PHILOSOPHICAL ATTITUDES


	My motivation for introducing concepts as objects comes from
artificial intelligence.  Namely, I want computer programs that can
reason intelligently about who wants what or who knows what.
This leads to considering examples like that of the previous section
and seems to have the following philosophical consequences:
.item←0

	#. Since we cannot hope to make a program capable of understanding
the whole world, we try to formalize knowledge etc. in a way that
enables the program to act intelligently in a limited domain.

	#. We are not especially attached to the usages of natural
language except in so far as they suggest useful formalizations.

	#. There is no harm in introducing lots of abstract entities
like concepts and no inclination to restrict ourselves to entities
that can be defined finitistically.  This is because we aren't interested
in making our own knowledge more secure (as philosophers often are) but
rather want to make a computer program act effectively even at the cost
of having it reason naively.  In designing such programs, we take for
granted our own common sense views of the world.

	I must confess, however, to finding this attitude philosophically
attractive, i.e. first find a formal system that reproduces common
sense reasoning - naively if necessary, and then try to make it
secure.
.skip 1
REFERENCES

Church, Alonzo (1951), The Need for Abstract Entities in Semantic Analysis,
in %2Contributions to the Analysis and Synthesis of Knowledge%1,
Proceedings of the American Academy of Arts and Sciences, %380%1, No. 1
(July 1951), 100-112.  Reprinted in %2The Structure of Language%1, edited
by Jerry A. Fodor and Jerrold Katz, Prentice-Hall 1964

Frege, Gottlob (1892), Uber Sinn und Bedeutung. %2Zeitschrift fur
Philosophie und Philosophische Kritik%1 100:25-50.  Translated by
H. Feigl under the title "On Sense and Nominatum" in H. Feigl and
W. Sellars (eds.) %2Readings in Philosophical Analysis%1, New York
1949.  Translated by M. Black under the title "On Sense and Reference"
in P. Geach and M. Black, %2Translations from the Philosophical
Writings of Gottlob Frege%1, Oxford, 1952.

Kaplan, David (1969), Quantifying In, from %2Words and Objections:
Essays on the Work of W.V. Quine%1, edited by D. Davidson and J.
Hintikka, (Dordrecht-Holland: D. Reidel Publishing Co.), pp. 178-214.
Reprinted in (Linsky 1971).

Linsky, Leonard, ed.(1971) %2Reference and Modality%1, Oxford Readings in
Philosophy, Oxford University Press.

McCarthy, J. and Hayes, P.J. (1969) Some Philosophical Problems from
the Standpoint of Artificial Intelligence. %2Machine Intelligence 4%1,
pp. 463-502 (eds Meltzer, B. and Michie, D.). Edinburgh: Edinburgh
University Press.

Montague, Richard (1974), %2Formal Philosophy%1, Yale University Press

.SKIP 2

.begin verbatim
John McCarthy
Stanford Artificial Intelligence Laboratory
Stanford University
Stanford, California 94305
.end

%7This draft of CONCEP[S76,JMC] PUBbed at {time} on {date}.%1