model minimization —an alternative to circumscription

Journal of Automated Reasoning 4 (1988) 1-13. 1 ;c~ 1988 hy Kluwer Academw Publishers.

Model Minimization- An Alternative to Circumscription

J A A K K O H I N T I K K A The Florida State Umversity, Dept. of Phdosophy, Tallahassee, FL 32306-1054, U.S.A.

(Received: January 6 1987)

Abstract. The idea underlying John McCarthy's notion of circumscription is interpreted, for formulas with finite models, as asking whether the conclusion C is true in all the mimmal finite models of the premise T. A way of modifying one of the usual proof procedures for first-order logic (the tableau method) is given which captures this idea. The result is shown to differ from the consequences of McCarthy's circumscription schema. The resulting proof procedure is extended to the case in which it is also required that the extensions of the primltwe predicates are minimal. For formulas with only infimte models, the idea on which the concept of circumscription is based is tantamount to the author's idea of restricting models to mimmal ones.

Key words: circumscription, model, minimal model, non-monotonic reasoning, tableau method, database theory.

1. Small Is Beautiful - - In Informal Reasoning

John McCarthy's notion of circumscription ~ is based on an extremely interesting insight into spontaneous human reasoning. If he is right, we human reasoners (and, McCarthy adds, intelligent computer programs) tend to assume "that the objects they can determine to have certain properties or relations are the only objects that do." In other words (my words rather than McCarthy's), human thinkers characteristically assume that the intended models of our sentences (formulas) are the minimal ones, containing only such individuals (of different kinds) as we are forced to include in them by our explicit assumptions or by the logical consequences of such assumptions. By the same token, only such kinds of individuals are instantiated in the preferred models as have to be exemplified in these models.

I am convinced that McCarthy has seen a prevalent and important feature of human reasoning, and hence of the kind of reasoning students of AI want to under- stand and to capture. McCarthy has himself presented persuasive examples to show the presence of this feature in a variety of different types of reasoning. Further evidence is not hard to come by. At least anecdotal evidence is, for instance, provided by the widespread use (or misuse, as most logicians would call it) of a definitory " i f" when "if and only if" is strictly speaking meant, in contexts like "x belongs to the set to be defined if it satisfies the condition S[x]."

Independently of McCarthy, I have been led to a similar idea in the foundations of mathematics. However, at first I applied the idea only to theories with exclusively infinite models. I shall argue elsewhere that the best way of handling the logic of

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certain important mathematical theories (with only infinite models) is to restrict by a fiat all relevant models to models which are minimal in a certain specifiable sense. 2 As

such minimal models, prime models might first seem to do a minimally satisfactory job, but on a closer look a somewhat more demanding property of models, viz., their being superprime (in a sense that can be defined) turns out to be the best explication of the minimality idea. In this way, mathematical theories (e.g., elementary number

theory) become descriptively complete (i.e., they now rule out all nonstandard models). The price one must inevitably pay is that the underlying logic becomes

incomplete. This sense of minimality applies only to infinite models, however, whereas McCarthy

is concerned also, and apparently primarily, with finite models. This prompts the question as to how the minimality idea can be carried out so as to apply also to finite models. One can also ask whether there is a more general treatment whose special cases include both the finite and the infinite case.

2. Circumscription

John McCarthy's answer to this question is his idea of circumscription. Since it is familiar from the literature, it suffices here to indicate only the main lines of this answer.

McCarthy interprets the minimality idea on which the notion of circumscription is based to mean that "the objects that can be shown to have a certain property P by reasoning from certain facts .4 are all the objects that satisfy p,,.3 This leads him to formulate circumscriptive inference by means of the following schema:

(A[~b] & (Vx)(qS[x] ~ Px)) ~ (Vx)(Px ~ ~b[x]) (1)

Here A[P] is a complex formula containing the one-place predicate P and A[~b] the same formula with a complex expression ~b (with "x " as its only free variable)

replacing P. This schema can be obviously generalized so as to apply to many-placed predicates

and furthermore to several predicates at the same time. However, the concept of circumscription does not seem to capture the intended idea

optimally. There are at least two kinds of problems about it. First, in some cases it

seems to yield too strong implications. Second, it does not suffice to restrict models

to minimal ones. As to the former kind of problem, several different difficulties will be discussed later

in this paper. Suffice it here to indicate the general trend of these difficulties by means of a not altogether serious example. Let the circumspection schema deal with Alice's " theory" of eating and perceiving, as Lewis Carroll had programmed her. Indeed, let ~b(x) stand for "Alice eats x" and Px "Alice sees x". Then by means of the circumscription schema it can be proved that, in the immortal words of the Mad Hatter rebuking Alice, "you might as well s a y . . , that 'I see what I eat' is the same thing as 'I eat what I see'. ''4 This conclusion is, of course, quite as absurd as Lewis Carroll intended it to be.

MODEL MINIMIZATION 3

As to the infinite case, McCarthy shows how his circumspection schema leads to the schema of mathematical induction. It is known, however, that the schema for mathematical induction does not suffice to restrict the models of an axiomatic theory of elementary arithmetic to the minimal model, which in this case is the intended structure N of natural numbers. In this case, the circumscription schema thus does some of the work that is needed here, but it does not do the whole job.

Further objections to the circumscription idea will be presented later.

3. Minimization Implemented

Can we do better? The idea of model minimization easily guides us to a more satisfactory treatment in the finite case. In order to see what is involved, let us assume that we are trying to prove a conclusion C from a finite set of premises T in a first-order language, say, by means of the Beth tableau method) Normally, the tableau construction begins with T on the top of the left column and C in the right column. The construction proceeds step by step by means of familiar rules, and the aim is to close the tableau by closing all its subtableaux. These rules are but the usual Gentzen- type rules of logical inference read upside down, with formulas in the left column playing the role of formulas in front of the sequent sign and with formulas in the right column playing the role of formulas that follow the sequent sign.

The model minimization idea can now be taken into account by initially modifying the tableau procedure as follows:

(1) The tableau construction rules are modified in such an (obvious) way that no formulas are transferred from one column to another.

(2) A subtableau is closed also when S and ~r,,~S, for any formula S, both occur in the same column, not just (as Beth stipulates) when S ocurs in both columns.

(3) Those entries in the left column which are independent of the presence of C in the right column are said to form the designated part of the column. Clearly, (3) does not affect the tableau construction as such at all.

(4) The only rule of tableau construction that actually has to be modified is the rule for existential instantiation in the left column (EIL). What it normally says is the following:

ElL: If (3x)S[x] occurs in the left column o fa subtableau while no formula of the form S[a] occurs in it, then one may add S[b] to the same column, where 'b' is a new constant.

Even this rule technically need not be modified. What has to be done is to amplify it by stipulating that our application of EIL creates a number of additional subtableau construction lines, alternative to the one initiated by ElL in its usual form (see above). They will be called the ghosts of the normal subtableau, which we shall call the primary construction (primary subtableau). They are not related to each other or to the primary subtableau disjunctively, as the alternatives created by an application of the

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disjunction rule (in the left column) are, but in a more complex way soon to be explained.

Each ghost of a primary subtableau shares the initial part with the primary one. It differs from the primary one, created by an application of EIL, by the fact that instead of the new formula S[b] it initially contains a formula of the form S[a,], where a, is one of the constants occurring in the designated part of the subtableau. For each such a, we have a different ghost initiated by S[a,].

Ghost tableaux are closed in the same way as primary ones. After a ghost subtableau is closed, it is henceforth disregarded completely, and said to be dead.

Each ghost subtableau is constructed precisely in the same way as the corresponding primary subtableau. Each formula in a ghost has a counterpart in the primary subtableau and vice versa. This is easily seen to be always possible, except in one case, this is when in the primary subtableau EIL is applied. Assuming that the corresponding existentially quantified sentence in the ghost construction is (3y)S[ y, a,], it may then happen that there already is a formula of the form S[cj, a,] in the left column of the ghost.

Then the ghost construction is continued simply by considering S[cj, a,] (for each cj) as the counterpart to the new formula S[d, a,] introduced by ElL in the primary construction. Thus the analogy between a primary construction and its ghosts can always be maintained.

We shall restrict the ghost generation to applications of EIL in the designated part of the overall tableau construction.

Notice that ghost tableaux can form a branching hierarchy: there may be ghosts of ghosts of ghosts . . . .

4. Mini-Consequence Defined

It is not very hard to see how the intuitive idea of model minimization can be implemented by means of the ghost constructions. For the purpose, let us see what can happen in a tableau construction which starts with the premise T and the conclusion C. Let's assume that we are considering one subtableau s.

Let's first assume that the designated part ds of s is completed, i.e., that it reaches a stage at which no further applications of tableau rules to ds are possible while the

designated part ds of the tableau is still open, i.e., while no contradictions are present. (In other words, in ds no formula occurs in both columns and for no F do both F and ~,~F occur in the same column.) Then it is seen that ds describes a finite model M~ of T. The interesting question is whether M~ is minimal.

In order to determine whether it is, we can examine all the ghosts of ds. Clearly, no tableau rules can be applied to them, either. Hence there are two possibilities concern- ing one of the ghost constructions g: (i) it is closed; (ii) it is completed.

If (i), g can be disregarded. If (ii), g defines a smaller finite model of T. In order to determine whether it is minimal, we have to examine the ghosts of g. If one of them is completed, we have found a still smaller model of T. Eventually this process will


come to an end with a minimal completed tableau construction for T which defines

a minimal model M0 of T.

This is the story of what happens if s is completed. I f s is closed, then so clearly are

all its ghosts, and s therefore yields no model for T.

I f s goes on to infinity without being closed or completed, then it does not yield a finite model. But even then, one of the ghosts of s, say g, can sometimes be completed

after a finite number of tableau construction steps. Then we can find a minimal finite

model by treating g in the same way as s was treated above in the case when it was

completed.

I f neither s nor any of its ghosts is ever completed, we do not obtain a minimal

model from the modified tableau construction. ( C f Section 10 below.)

Consider now one of the cases where the moified tableau construction has produced a minimal model, i.e., where we have a construction which has come to an end with

a completed open subtableau or, rather, completed designated part of the construc-

tion, where no ghost of that construction line is similarly completed. Let the indi-

vidual constants occurring in the completed construction be a~, a 2 . . . . . a k. The

minimality assumption can then be brought to play by continuing the subtableau as follows:

(i) Add to the left column

(Vx)(x = al v x = a2 v ' ' . v x = ak).

Alternatively, for first-order logic without identity, we must replace any formula or subformula in the subtableau of the form (3x)S[x] by

Sial] v S[a2] v ' ' ' v S[ak]

and every formula or subformula of the form (Vx)S[x] by

S[a,] & S[a2] & " ' " & S[ak].

(ii) Continue the subtableau normally.

It is of course understood that only the designated part of the left column of the

subtableau in question comes to play in the continuation of the tableau construction.

I shall call this the minimization rule. It is calculated to capture the same idea as others have tried to capture by means of the schema of circumscription.

If in the entire tableau beginning with Tin the left column and C in the right column every stage of each subtableau has at least one continuation which can be closed by means of the minimization rule or else can be closed altogether, then I shall say that C is a mini-consequence of T. Of course, in each subtableau, the minimization rule is to

be used at most once. The notion of mini-consequence is the alternative I am offering to trying to capture the model minimization idea by means of the procedure of

circumscription.

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5. Examples and Observations

Several different observations are in order here. First, the notion of mini-consequence yields in many cases the same results as

reasoning by circumscription. For instance, consider McCarthy's sample theory 6

consisting (in a slightly different notation) of the conjunction

Isblock(a) & Isblock(b) & Isblock(c). (2)

If this is used as my T, and if the conjunction rule is applied to (2), the tableau

construction will come to an end with (2), Isblock(a), Isblock(b), and Isblock(c) in the left column. All these formulas will be designated ones. Then the minimization rule

enables us to add to the left column

(Vx)(x = a v x = b v x = c). (3)

But this means that (3) is a mini-consequence of (2), just as it can be obtained from (2) by circumscriptive inference.

However, in the case of the following formula its mini-consequences differ from its

circumscriptive consequences:

Isblock(a) v Isblock(b). (4)

By starting from (4) in the left column we arrive quickly at a situation in which the tableau construction has generated two subtableaux each with the following formula inserted in virtue of the minimization rule into their left column:

(Vx)(x = a v x = b). (5)

This of course means that (5) is a mini-consequence of (4). In contrast to this, the circumscription schema yields the conclusion

(Vx) (x = a) v (Vx) (x = b) (6)

which is not equivalent with (5). Which conclusion is "right", the mini-consequence (5) or the circumscriptive one

(6)? Even though no absolute decision is possible here, it seems that the mini-

consequence (5) is much more natural than the circumscriptive consequence (6). For suppose that someone actually puts forward (4). Clearly the propounder is envisaging a situation in which both a and b are present. He or she might very well be reluctant to introduce any other individuals, but most of us innocent bystanders would surely think that the speaker would be welshing on his or her existential presuppositions if he or she tried to retract an existential commitment either to a or to b. Hence (5) rather than (6) presumably is the right consequence.

It can also be seen, by means of suitable examples, that a logic based on the notion of mini-consequence is not monotonic. In this respect, too, it is like circumscription.

In order to see this, consider three sentences, T~, T2, T3, which say, intuitively speaking,


T~ = there exist at least n individuals with the property P;

= there exist at least n - 1 individuals with the property P;

T~ = there exist precisely n - 1 individuals with the property P.

For instance, if n = 2, we have:

Ti = ( 3 x ) ( 3 y ) ( P x & P y & x :/: y)

T2 = (3x )Px

T~ = ( 3 x ) ( P x & (Vy)(Py ~ x = y))

Let us now consider what the minimal models of T~ - T3 are like, in a sense of "minimal model" which is intuitively obvious in the present application and which

will be discussed further in Section 6 below. In the minimal model of T~ there are

precisely two P's. Hence ~ is true in it, i.e., ~ is a mini-consequence of T~. The

minimal model of T2 has precisely one P, wherefore T3 is true in it. Hence /'3 is a

mini-consequence of ~ . Ye t / ' 3 is not a mini-consequence of T~. As an additional example, consider the theory T defined by the following axioms:

(Vx)(3y) Rxy

(3y) (Vx) ~,~Rxy

(Vx) (Vy) (Vz) ( R x y & R x z ~ y = z)

(Vx ) (Vy ) (Rxy ~ ~cx~Ryx)

(7)

(8)

(9)

(10)

For vividness, let us say that y is a successor to x whenever Rxy . (The converse of

this relation will naturally be called precedence.) Then (7)-(10) say, intuitively, that

everyone has a successor, that someone is not preceded by anyone, that no one has

more than one successor, and that successor relation is antisymmetric. Then a sentence C is a mini-consequence of T i f f it is true in the model of the following

structure:

o , o , o (11)

No/ If a requirement is added to T saying that there must be at least 5 individuals in its

models, then the C is a mini-consequence of T i f f it is true in both of the following

structures:

(a ) O , �9 , �9 , o ( 1 2 )

",,,o/ (b) o

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Notice that while T allows there to be more than one individual without pre- decessors, in the model that serves to characterize its mini-consequences in this way, there is only one such individual.

6. Towards a Model Theory of Mini-Consequence

A model theory can be developed for the notion of mini-consequence. For the purpose, we have to define a few concepts. Assume, for the sake of argument, that we are dealing with a first-order language L where the only nonlogical concept is a two-place relation R. (Our definitions are easily generalized from this case to an unrestricted first-order language.) Assume also that a model M of L and a mapping f o f the domain of individuals do(M) of M into itself is given. Then it is said t h a t f defines an m-automorphism iff the following conditions are satisfied.

(i) Iffa, b e do(M) and iff(a) -r f (b) , then Rf(a) f (b) iffthere are a', b' ~ do(M) such

that f (a ) = f(a') , f (b) = f(b') , Ra'b'. (ii) If f (a) = f (b) -- c, then it must be the case that Raa iff Rbb and that Rcc iff Raa

and Rbb.

We shall say that a finite model M is minimal iff there is no m-automorphism which maps do(M) into a proper subset of do(M). The following is the basic result that can then be established in this direction:

Completeness for mini-consequences: C is a mini-consequence of Ti f f it is true in

each minimal model of T.

For instance, the reason why (12) (a)-(b) are non minimal models of the theory defined by (7)-(10) is that they each admit of an m-automorphism which identifies the individuals represented by solid dots.

The result just stated is essentially a completeness result for my modified tableau rules, including the mini-consequence rule. In fact, this completeness is an almost direct consequence of the way the concept of mini-consequence was defined above. It is easily seen that, in order to prove the desired result, it suffices to prove the following:

LEMMA. 1: Assume that M is a minimal model of T. Then the applications of our

modified tableau rules can always be chosen so that in at least one subtableau, or in at least one of its ghosts, all the designated formulas are true in M.

This can be proved by induction on the length of the subtableau construction considering the different tableau rules. The only tricky case is EIL (the left-handed existential instantation rule, modified as indicated). By the inductive hypothesis the formula (3x)S[x] to be instantiated is true in M. Now assume that none of the individual constants a, already in the designated part of the tableau can make S[a,] true in M. Then some other individual b in the domain of M (not yet named by the constants in the tableau) will do so. In applying EIL to (3x)S[x] so as to yield a new formula of the form S[d], we can take " d " to be the name of b.


Again, assume that S[x] is satisfied in M by some individual at whose name already occurs in the designated part of some subtableau. Then adding a formula S[b] with a

new symbol "b" to the left column will not necessarily preserve the truth of all

the formulas of the designated part in M. However, they are all true in M as far as the ghost subtableau is concerned which is generated by the addition of S[a,] to the tableau instead of S[b].

It is easily seen that this tableau construction comes to an end in a finite number of moves in the sense that the designated part of the left column cannot be extended. Then it is seen that this distinguished part is a model set with the same individuals as M and with all its members true in M, in brief, it characterizes completely the

model M. Since each minimal model M is obtainable in the way specified by Lemma 1, C is

a mini-consequence of T i f f it is true in all of them, as claimed.

7. Mini-Consequences Differ from Inferences by Circumscription

As a further example, consider a primitive " theory" T of biological family relation- ships within the human race which guarantees that the relation "ancestor" of (the ancestral of the relation of being a parent of) is irreflexive, asymmetrical and tran- sitive. Let us further add an axiom which says that Adam and Eve are the only people without parents. It is easily seen that then it is a mini-consequence of Ttha t everybody is a descendant of Adam and Eve. For in trying to construct a model for T you must introduce parents for each human being (different from Adam and Eve), their parents, etc., without being able to return to earlier members of the sequence. Hence the only way in which the model to be constructed can be finite is for each such sequence of parents to come to an end with Adam and Eve.

An interesting thing about this conclusion is that the circumscription scheme does not validate it. For fairly obviously, all that you can prove by means of the circumscription schema is that each human being either has a father and a mother or else is Adam and Eve. From this it does not follow, of course, that there cannot be infinite chains of earlier and earlier ancestors.

Thus the notion of mini-consequence is stronger than that of inference by circum-

scription, in the sense of validating more inferences. This example seems to show the superiority of mini-inference over circumscription.

For clearly the same idea of not multiplying entities without necessity which motivates the schema of circumscription also motivates the mini-consequence that all human beings are descendants of Adam and Eve. A model in which this statement is true is more economical than a model in which it is false; furthermore, there is no necessity of assuming infinitely long chains of ancestors. Hence by the same heuristic principles as guide inferences by circumscription, the biblical mini-consequence ought to be valid. It is, but it is not validated by the schema of circumscription.

In this example, the mini-consequence in question cannot be established in my approach by a finite argument. However, this does not spoil the counter-example. For

10 JAAKKO HINTIKKA

instance, I can add the assumption that every descendant of Adam or Eve is separated from him or her by fewer than n generations. Then the mini-consequence in question can actually be proved by a finite argument (tableau construction) without destroying its role as a counter-example.

Even though I have formulated this example in somewhat frivolous-looking terms, the underlying point is a serious one. It has, for instance, been argued that Aristotle in fact assumed a principle of philosophical reasoning which amounts in effect to the kind of application of the idea of mini-consequence which is at issue in my Adam-and- Eve example. 7 More precisely, it has been argued that Aristotle assumed that in a partly ordered set all chains must be finite, which comes close to minimality require- ments of the kind we are dealing with here.

8. Mini-consequence and database theory

It seems to me that a concept like mini-consequence or circumscription is destined to play an important role in database theory. 8 My reasons for this belief can be explained

as follows:

(i) Database theory should be developed in such a way that the data in question do

not include only atomic (particular) propositions (negated or unnegated), but can also include general propositions, more generally, arbitrary first-order propositions. This is necessary if database-driven reasoning is to be successfully appli- cable to scientific or clinical reasoning.

(ii) If this is done, it will be awkward to define a database logically by means of closure axioms, i.e., axioms which include a sentence of the form

(Vx)(x = al v x = a 2 v " . . v x = ak) (13)

where a~, a2 . . . . . a~ are all the individuals named in the database.

For one thing, this procedure can easily turn a perfectly acceptable database into an inconsistent one. It also makes it difficult to develop a natural theory of operations on databases, such as adding new data to it or dropping some of the data.

(iii) For these reasons, it seems to me that the closure of a database should be accomplished by means of an optional extra assumption which can be used only when there is some reason to think that the database is supposed to be complete.

(iv) Even then, the closure requirement should not restrict the model that the database defines to individuals mentioned in the database. It should also contain at least those individuals whose existence is logically implied by the database.

The notion of mini-consequences is intended to be a way of reaching these ends. It seems to realize very well the idea that each database has to be considered as telling us everything that needs to be told about its subject matter. At least it realizes this idea better than closure axioms.


9. Minimizing Predicate Extensions

The ideas so far developed can be extended in different directions. One of them is the

following: It can be argued, along the same lines as are followed above and are also

followed by McCarthy, that in ordinary reasoning human thinkers often assume, not only that the intended models are as frugal as they can be with respect to the existence of individuals, but also that the extensions of certain given predicates, or maybe just the extension of a single given predicate P, can only be as large as they must be.

This idea can easily be captured by the same method as was used to formulate the notion of mini-consequence. The modified tableau-building procedure outlined above leads (where it comes to an end with completed constructions) to a specification of a number of alternative minimal models of T in which C must be true in order to be a mini-consequence of T. These models are not fully specified by the construction, however. Consider, for instance, the case of a one-place predicate P occurring in T. In the description of a model (model set) reached by the procedure, it is specified that

certain individuals Cl, c2 . . . . must have P and that certain others dl, ~ . . . . must not have it, in other words, that the following must be true in the model:

Pci, Pc2 . . . . . ~',~Pdl, -~P~ . . . .

However, the extension of P is otherwise left open.

Now clearly we can implement the idea that the extension of P is assumed to be minimal by requiring that only cl, c2 . . . . have this predicate in the minimal model we have reached. This means changing the minimization rule in such a way that all the formulas v, JPe~, c'~Pe~_ . . . . are also added to the left column, where the e, of all the individuals in the minimal model different both from all the c, and from all the d,, when the construction of the designated part of a subtableau comes to a completion.

Whenever C can be derived from T by the latest modified tableau method, C is true in all the minimal finite models of T in which the extension of P is as small as possible.

This is easily generalized so as to implement the minimization of any finite number of predicate extensions. Thus the idea of minimizing predicate extension is also easily

captured by our technique.

10. Infinite Case

The idea of model minimization can thus be implemented fully and without difficulties whenever the relevant minimal models can be finite. It appears that in most actual applications we are in fact dealing with the finite case. But what happens when we have a set of first-order premises (assumptions) which have only infinite models? A case in point would be a Peano-type axiomatization of elementary number theory. Does the idea of model minimization have interesting applications to such cases?

The answer is an emphatic yes. In fact, I have argued elsewhere that the most promising approach to theories like Peano arithmetic is precisely to restrict the models to be considered once and for all to minimal ones. 9 This prompts of course

12 JAAKKO HINTIKKA

the question: minimal in what sense? The cardinality of the domain of a model no longer serves as an index of its "size" in the sense which is relevant to its minimality, for clearly, e.g., one countable model can be richer than another countable model. Hence, some sort of qualitative minimality is what counts here. Now the finest qualitative distinctions among different individuals in the domain of a model are apparently those effected by the so-called types (in the sense of model theory), l~ Minimality will then presumably mean being what model theorists call primacy (being a prime model). In fact, it turns out that the concepts of type and of primacy have to

be generalized. ~ This gives rise to a notion stronger than primacy, which I have called superprimacy. 12

It can be shown that an overall restriction on models to superprime ones enables ut to formulate a complete (in the sense of categorical) theory of elementary number theory. ~3 In this sense, the idea of model minimization turns out to be extremely important in the foundations of mathematics. This seems to me to vindicate in a most impressive way John McCarthy's original intuition.

The other side of the G6delian coin is that we cannot of course obtain a complete axiomatization in a sense that involves a complete proof procedure. ~4 Hence the

modified proof procedure sketched in this paper for mini-consequences cannot be generalized to the infinite case.

The operative question becomes, rather, the problem of looking for such stronger rules of proof as would capture partly (but increasingly extensively) the effects of the overall restriction of models to superprime ones. Various schemata of induction can be thought of as (partial) ways of doing so. This intriguing problem lies beyond the scope of the present paper, however.

Acknowledgement

The work reported here was made possible by NSF Grant # IST - 8310936 (Information Science and Technology), Principal Investigators Jaakko Hintikka and C. J. B. Macmillan.

Notes

t See McCarthy, John, 'Circumscription - a form of non-monotomc reasoning', Artificial Intelligence 13, 27-39 (1980). 2 See 'Extremality assumptions in the foundations of mathematics ' , in PSA 1986, vol. 2 (eds. A. Fine and M. Forbes), Philosophy of Science Association, East Lansing, MI 1987; and 'Is there life in the foundations of mathematics after G6del?' (forthcoming).

Op. Cit., p. 27. 4 See The Annotated Ahce (ed. Martin Gardner), Penguin, New York (1965), p. 95. 5 Beth, Evert W., 'Semantic entailment and formal derivablhty', Mededelingen van de Koninkloke Neder- landse Akademie van Wetenschappen N.R. 18, no. 13, Amsterdam, 309-342 (1955).

Op. Ctt., p. 32. 7 See Beth, Evert W., Foundations of Mathematics, Harper and Row, New York (1966), pp. 9 12. (Actu- ally, Beth formulates his principle in a somewhat different way. Wha t is given here seems to be me to be the g~st of the principle he has in mind.)


Cfi, e.g., Reiter, Raymond, "Towards a logical reconstruction of relational database theory', m On Conceptual Modelling (eds. M. L. Brodie, J. Mylopoulos and J. W. Schmidt), Springer, New York (1984), pp. 191 238. (See especially pp. 191,218 and 227 229 for the "closed world assumption".) 9 0 p cir., (note 2 above). to See, e.g., Chang, C. C., and J. J. Keisler, Model Theory, North-Holland, Amsterdam (1973), pp. 77 78. t t Op cir. (note 2 above). ~20p. ca. (note 2 above). 13 Op. cit (note 2 above). It is important here to keep apart the several senses ofcompleteness~ all the more so because they are frequently confused w~th each other. What is at msue here is the power of a first-order axiom system (theory) to restrict its models to the intended (standard) ones. Thls has to be distinguished from the completeness of a formalization of logic, which means recursive enumerability of all (and only) valid formulas, and from the deductive completeness of a formalized system, which means the formal derivability of C or coC for each C in the language of the system. 14 It is only in this sence that G6del proved the incompleteness of elementary arithmetic.

model minimization —an alternative to circumscription

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