00100		THE "ORDINARILY" PROBLEM
00200	
00300		The "ordinarily" problem is part of the problem of
00400	expressing formally the information expressed in ordinary
00500	language.  For artificial intelligence purposes this must be
00600	done in such a way that the fact that a certain strategy will
00700	achieve a goal follows from the description of a
00800	situation.  We shall introduce the problem in connection with
00900	an example.
01000	
01100		Consider the well known puzzle of the farmer with a
01200	wolf, goat, and cabbage that is usually expressed in
01300	English approximately as follows:
01350	
01400		"A farmer wishes to cross a river with a wolf, a goat,
01500	and a cabbage.  There is a boat that can hold himself and one
01600	of his charges.  If he leaves the wolf alone with the goat, then
01700	the goat will be eaten, and if he leaves the goat alone with the 
01800	cabbage, then the cabbage will be eaten. How can he get
01900	his charges across the river?"
02000	
02100		The expected solution is something like "The farmer should
02200	row the goat across, then come back for the wolf or cabbage, row
02300	it across, take the goat back with him and leave the goat
02400	while he takes the remaining other animal across, and finally go
02500	back and get the goat."
02600	
02700		We would like to have a computer program that when
02800	given the problem expressed in English as above would come up
02900	with the above solution.  After all, a reasonably clever person will
03000	come up with that solution, and almost anyone will accept it.
03100	Our first step toward making computers solve 
03200	such problems is to try to devise a formal language in into which
03300	they can be translated
03400	and in terms of which the correctness of the proposed solution
03500	follows formally from the description of the problem.  Two
03600	problems arise.
03700	
03800		The first problem is that the human uses additional
03900	facts in solving the problem beyond those given in its statement.
04000	Moreover, the necessary facts about time, causality, the uses of
04100	boats, the facts that the wolf, goat, and cabbage are material
04200	objects which have certain properties thereby, the fact that
04300	being eaten is destructive, etc. are not all ordinarily expressed
04400	in English in a readily formalizable way.  In fact, formalizing
04500	these facts depends on problems of philosophical logic which
04600	is not very far advanced.  Some approaches to these problems
04700	are described in (McCarthy 1960), (McCarthy 1963), and (McCarthy
04800	and Hayes 1969).  But there is another source of difficulty that
04900	is the main subject matter of this paper.
05000	
05100		Suppose we add to the above description of the problem
05200	the two sentences "The river is too swift for boating.  One
05300	hundred yards upstream there is a bridge across the river."
05400	Notice that these two sentences do not contradict the previous
05500	set, i.e. the enlarged description of the situation is still
05600	consistent.  Nevertheless, the common sense solution
05700	to the altered problem is completely changed.  Now the solution
05800	is "The farmer should walk with his charges to the bridge
05900	and cross the bridge."
06000	
06100		In order to discuss this peculiar matter we shall
06200	introduce some symbols:
06300		A  denotes the original statement of the problem except
06400	for the question "How  can he get his charges across the river?"
06500		B denotes the additional sentences about the swiftness
06600	of the river and the bridge.
06700		C denotes common knowledge expressed in sentences.
06800		S1  denotes the usual solution to the problem.
06900		S2  denotes the solution of crossing the bridge.
07000	
07100		If we suppose the sentences to be formalized in mathematical
07200	logic the following difficulty arises.  If the original statement
07300	of the problem is correct we would expect that  A∪C entails
07400	S1 and if the revised statement of the problem is correct
07500	then  A∪B∪C  entails  S2 but not S1.  But in ordinary logic
07600	whenever a set X of sentences entails a sentence S,
07700	any superset Y of X also entails S.  We shall call this property
07800	of a logic the inclusion property.
07900	
08000		The common sense way out of the difficulty is to say
08100	that implicit in the original statement of the 
08200	problem is the assertion that the facts given are all the
08300	specific facts relevant to the problem.    The problem is to 
08400	formalize this assertion and to devise a form of logic that does
08500	not have the inclusion property.  Such a logic would have to
08600	be a literature in the sense of (McCarthy and Hayes 1969).
08700	
08800		Another approach to the problem is through the idea of a
08900	minimal model of a set of sentences of first order logic which
09000	we shall introduce as follows: 
09100	
09200		Let L be a many sorted first order logic, S a set of 
09300	sentences of L and M1 and M2 two models of S.  We shall say
09400	that M1 is contained in M2 (written M1≤M2) if
09500	domain(M1) ⊂ domain(M2) and the predicate letters of M1 are
09600	included in those of M2 and for any x,y,...,z ε domain(M1)
09700	and predicate letter P of M1, P(x,y,...,z) is true in M1)
09800	if and only if it is true in M2.
09900	
10000		A model M of a set S of sentences is called minimal
10100	with respect to collection G of sorts, if it does not 
10200	contain a model M1 such that the domain(M1)∩g ⊂ domain(M)∩g
10300	properly for some g ε G.
10400	
10500		A set S of sentences minimally entails a sentence p
10600	if p is true in every minimal model of S.
10700	
10800		We propose to apply the idea of minimal entailment
10900	to the farmer's problem by considering a sort of material
11000	objects and a sort of strange properties of rivers.  In the
11100	minimal models of the original problem A∪C there are no
11200	bridges and no peculiarities with the river.  Therefore,
11300	if we have a rule in common knowledge C that a boat can 
11400	be rowed across a river if there is nothing wrong with
11500	the river or the boat, the boat will be rowable in minimal
11600	models of A∪C; however, there will be no bridge
11700	in these models.  On the other hand, in the minimal models
11800	of AUBUC there will be a bridge but the river won't be rowable.
11900	
12000		In its present form the idea seems a bit far fetched since
12100	the introduction of the sort of peculiarities of rivers is
12200	quite arbitrary.  Nevertheless, I think something along these
12300	lines can be made to work.