a model-theoretic criterion of ontology

JOHN BACON

A M O D E L - T H E O R E T I C C R I T E R I O N OF

O N T O L O G Y *

No Anglo-American philosopher in our century has attracted more attention to the problem of explicating being than has Quine. His own explication is put forward as a criterion for deciding what kinds of entity are "countenanced" in espousing a formalized theory. Thus he relativizes ontology to theories. Since Quine propounded his criterion, a fertile systematization of the theory of meaning and reference has come into its own under the name of model theory. From this point of view, a theory may be thought of as constituted by a language and a set of admissible models. Thus the relativization of ontology to theories may be further refined into a relativization to models. My basic explicandum, then, is existence in a model. The explicatum I propose is first approximated as membership in the model's domain. Since this criterion takes account not so much of what is quantified over as what is referred to, it resembles Bergmann's criterion (1954, p. 92) in some ways more than it does Quine's. For certain standard theories, however, the two criteria coincide.

Theories modeled

Let L be a language, i.e., a vocabulary with a grammar. L is interpreted by a set ML of L-models, relational structures on which truth-, denotation-, or sense-conditions are laid down for the logical constants and the meaningful expressions of L. If a sentence p is true in model m according to these conditions, we also say that m satisfies, models, or is a model of p, and similarly for a set of sentences. Such an interpretation of L, i.e., a defmition of 'L-model', denotation- or sense-conditions, and the definition of satisfaction, will also be called a modeling of L. As Quin~ restricted his criterion to standard quantificational languages, so the discussion here will be restricted to set-theoretic modelings. Since set theory is about the most powerful general framework known for the investigation of structure, the limitation is not a serious one. Nevertheless, useful insights might be forthcoming from an appraisal of the ontOlogical commitments of

Synthese 71 (1987) 1-18. © 1987 by D. Reidel Publishing Company

2 JOHN BACON

modelings formulated in ca tegory theory, A-conversion, combinatory logic, or intensional logic.

A sentence is logically true if it is true in all L-models . A sentence is T- t rue if it is true in all T-models , i.e., L-models in a subset MT of ML. The set of T- t rue sentences is a theory: T. MT will typically contain unintended models of T, e.g., numerical models, while T is perhaps intended to be about physical objects. We are thus led to recognize a further subset M~ of MT, the intended models of T, which also model a b roader " theo ry" or set of sentences I . Intuitively, I will comprise all the noncont ingent truths of the theory T as applied. A modeling of L together with a demarca t ion of MT and M~ is a modeling or interpretat ion of T. Finally, r is the actual L-model , i.e., that part icular model which corresponds to the actual state of affairs. Ideally, r ~ M~, but if T is refuted, r ¢ MT.

T h e part icular s tructure of models is left open, and standardized only as m = ( D , . . . , V). Here the first e lement D or Dm shall be the domain (empty or not) of the model m, comprising the set of basic things (if any) out o f which the model constructs meanings for cate- goremat ic expressions of L, And the final e lement V or a defined extension thereof shall, among other things, assign truth-values to the sentences of L.

1. QUINE MODELED

1.1. Ontology of models

Quine 's well known criterion of ontological commi tmen t made its debut in (1939l):

(1) It thus appears suitable to describe names simply as those constant expressions which replace variables and are replaced by variables according to the usual laws of quantification... It is to names, in this sense, that the words 'There is such an entity as' may truthfully be prefixed... We may be said to countenance such and such an entity if and only if we regard the range of our variables as including such an entity. To be is to be a value of a variable. (p. 66)

The essay in which these words appear was not actually published in toto until 1966. But Quine drew upon it in publishing the following:

(2) A word W designates if and only if existential generalization with respect to W is a valid form of inference.., names, we found, are describable simply as the constant expressions which replace these variables and are replaced by these variables according

A M O D E L - T H E O R E T i C C R I T E R I O N OF O N T O L O G Y 3

to the usual laws. In short, names are the constant substitaends of variables. . . The universe of entities is the range of values of variables. To be is to be the value of a variable. (1939d, pp. 48, 50)

We may epitomize,

(3) To be is to be the value of a bindable variable.

Now, although (2)-(3) are couched in the material mode, while (1) also speaks of performative "countenancing", there is a semantic core to Quine's doctrine here that is neither metaphysical nor pragmatic. Let R,, be the union of the variable-ranges posited in the truth-conditions for variable-binding operators of L, as applied to the L-model m. Then (3) becomes the model-relativized

(4) e exists in m iff e ~ R,,.

Since in one of its uses "ontology" means the set of beings, we may also write "O(m)" for "{x: x exists in m}". Then the Quinean (4) becomes

(5) O(m) = Rm,

ontology equals bindable-variable range. In O we have a function from models to sets, i.e., a property in the parlance of modal logic (considering the models as possible worlds). What property? Exis- tence. Thus our trivial reformulation of Quine in terms of model theory enables us to give not just the extension but the intension of "exists".

1.2. Ontology of theories

On the basic doctrine (3) or (5) Quine founds his accounts of ontological commitment. He has given various versions of his criterion down through the years, committing theories and then again persons to entities and then again to kinds of entity. Of theories he once stipulated,

(6) an entity is assumed by a theory if and only if it must be counted among the values of the variables in order that the statements a]firmed in the theory be true. (1947, p. 103)

This is deceptively clear-sounding. It leaves open just how far a theory has to be interpreted in order to pack some wallop into the "must be counted". If the "theory" were uninterpreted, commitment would be

4 JOHN BACON

nil; if fully interpreted (by a unique model), then "must" will yield to "is". 1 I once suggested that Qnine's position makes the most sense if "all questions about interpretation of the statements in the theory would have to be settled except the question as to the range of the variables" (1966, p. 50). Indeed, Quine has since made this explicit: "Each of the various reinterpretations of the range (while keeping the interpretations of predicates fixed) might be compatible with the theory" (1968, p. 315). This is puzzling, however, for predicates are ordinarily interpreted on the domain or variable-range. Barring some sort of intensional or ordinary-language interpretation, how in general could we keep the interpretations of the predicates fixed while chang- ing the range? For the purposes of the present model- theoret ic r e c o n - struction of Quine, I suggest that the appropriate degree of partial interpretation is best captured by specifying Mz but not r.

Qnine has altered (6) in the second edition to read

(7) entities of a given sort are assumed by a theory . . . .

We might say that (6) defines singular commitment, (7) general commitment. That (7) does not supersede (6) for Quine is shown by his substantially reverting to singular commitment:

(8) To show that some given object is required in a theory, what we have to show is no more nor less than that the object is required, for the truth of the theory, to be among the values over which the bound variables range. (1969, p. 94)

In the same talk he relates the two:

(9) So there is more to be said of a theory, ontologically, than just saying what objects, if any, the theory requires; we can also ask what various universes would be severally sufficient. The specific objects required, if any, are the objects common to all those universes. (1969, p. 96)

More precisely,

And the theory is ontologieally eomitted to 'objects of such and such a kind,' say dogs, just in case each of those ranges contains some dog or other. (1968, p. 315)

Now, although singular commitment is the simpler notion, it is less useful than general commitment. For one thing, there might easily be no particular object a theory was committed to, as Quine implies by the 'if any' in (9). 2 Furthermore, it turns out that singular commitment can be taken as a special case of general commitment. The latter,

A M O D E L - T H E O R E T I C C R I T E R I O N O F O N T O L O G Y 5

then, may be defined thus:

(10) Fs exist according to Ti f f F [q O(m) ¢ A for all m ~ Mz.3

The invocation of "all models m" here echoes the "must" in (6), the "required" in (8)-(9), and the "has to" in (13). As the general ontological commitment of T we may take

(11) O(T) ={F: Fs exist according to T}.

Notice that, loosely speaking, O(T) will be simpler the larger its member classes are. T would have minimal ontological commitment if

O ( T ) = U O(m). m E M I

At the opposite extreme are unit classes, which yield singular commitment:

(12) e exists according to Ti f f {e} ~ O(T).

This would hold only if all intended models overlapped, a very strong condition

1.3. Personal commitment

Theoretical commitment as defined in (10)-(12) is transparent: it doesn't allow a theory to commit us to something that in fact doesn't exist 4 (unless, embracing "noneism", we can bring off the trick of forming nonempty classes of nonexistents). This reflects exegeticaUy the transparent occurrence of "an entity" in (6) and "some given object" in (8). (7) admits less clearly of a transparent reading. It seems perhaps to allow a theory affirming "3x(x is a unicorn)" to commit us to unicorns, even though no variable-range of any model of such a theory could contain unicorns. Ontological commitment in this in- tentional sense becomes intelligible if we relativize it to subjects, as Quine does in this passage:

(13) we are convicted of a particular ontological presupposition if; and only if, the alleged presuppositum has to be reckoned among the entities over which our variables range in order to render one of our affirmations true. (1948, p. 13)

The relativization here to "our variables" and "our affirmations" personalizes the theory espousal under consideration. Roughly speak-

6 J O H N B A C O N

ing, what seems to be going on here is that a subject, say S, espouses a theory commiting him to an "alleged presupposi tum' . Reformulating this in terms of general commitment, we might seem to get something like

3 T : S is commit ted to T, and F~ O(T).

But this entails that there are Fs, whereas S's commitment is compatible with there being no Fs. What we must do is move the second conjunct inside the scope of S's commitment:

(14) T commits S to Fs iff S is commit ted to claim of T that: T is true and F c O( T).

And S is ontologically commit ted to Fs iff there is some theory that so commits him. (I have suppressed the obvious time parameter.) Notice that (14) is formulated so that " F " occurs opaquely and " T " trans- parently, opening the latter to comparatively unproblematic existential generalization.

1.4. Existence statements in the object language

From (4) it is an easy matter to make an ontological descent 5 into the object language L. What we want are statement forms whose truth- conditions will be

V(a) exists in m, i.e., V(a) ~ Rm

and

V(P) f70(m) ~ A

for singular and general existence respectively. Let L be standard, with a single sort of bindable variable x, only unrestricted quantifiers as variable-binders, and an identity sign. For standard L, the desired statement forms are

(15) 3x(x = a) (16) 3xPx,

as Quine has pointed out:

Now if the theory affirms the existentially quantified identity "(3x)(x = a), " . . . "a" is being used to name an object... We may indeed take "(3x)(x = a)" as explicating "a exists." (1969, p. 94)


A general existence statement, e . g . , . . . "There are horses" . . . says that there is at least one entity satisfying a certain condition. In logical symbols, the whole appears as an existential quant i f ica t ion: . . . (3x)(x is a horse) . . . . (1939d, p. 44f)

For many-sorted theories a disjunction would do it, with as many disjoined copies of (15)/(16) as there are variable-sorts (subject to type restrictions, of course).

The availability of (15)-(16) does not mean that the semantic ascent to model theory was all for nought, however. As Jackson has argued in a rather different reconstruction of the notion of ontological commitment, it is not simply the acceptance of a sentence as such but the acceptance of a particular semantic interpretation of it that is ontological committal (1980).

1.5. Sentential commitment

The account of commitment, as opposed to existence, developed thus far is metalogical. It is easily extended to sentences by taking them as unit theories, i.e., theories with a single nonlogical postulate, the sentence in question. The theoretical commitment of a sentence is not quite what one might at first expect, however. For example, (15) says that a exists, but it does not precisely commit one ontologically to a. For " a " may designate nonrigidly over the intended models of (15): it may be that V(a) ~p V'(a) for two intended models m, m' c MI. If so, barring other restrictions on Ms, the theoretical ontological commitment of (15) may be nil. (I assume that ML includes models with empty domains.) Similar remarks apply to (16), although it is more likely to commit. It doesn't necessarily commit us to Ps, for there may be intended models with valuation functions V, V' such that V(P) ~ V'(P) or even V(P)f3 V'(P)= A. The most plausible way to beef up the ontological commitment of sentences so that it coincides with what they say is to identify Mz with {r}. For if we have just one model, rigid designation is assured.

1.6. Beyond the Ouinean model

This whole part has simply been a gloss on Quine, sympathetic I hope, even though he would probably not be pleased with every detail. While the introduction of elementary model-theoretic notation may seem gratuitous, I believe it brings out certain points in Quine's theory

8 J O H N B A C O N

of ontology more sharply than they would otherwise appear. More im- portantly, however, the battery of concepts developed with Quine's help will stand us in good stead even as we part from Quine's basic criterion of ontology.

2. A N A L T E R N A T I V E C R I T E R I O N

2.1. Extension to the domain

As it stands, I contend, Quine's criterion provides a sufficient but not a necessary condition for existence in a model of a first-order theory. For higher-order theories, it is not even a sufficient condition. In this part I propose to extend Quine's ontological criterion in three stages. The first extension I suggest is to include in the ontology of a model not just the union of its variable-ranges but its whole domain. Thus (5) becomes

(17) O(m) = D m

With " O " thus redefined for models, the subsequent chain of definitions (10)-(14) may stand.

(17) has the seeming defect of permitting gratuitous inflations of m's ontology. For if ( D , . . . ) is a model of T, so is (D tAX, . . . ) , provided that bound variables are restricted to part of D. X is then unused in (D U X , . . . ) . It seems silly to affirm, by (17), that O ( D U X . . . . ) = D U X. My answer is that this inflation is innocuous, as it gets washed out again when we proceed to the ontology of T (10) with its restriction to intended models.

Another consequence of (17) that gives us pause is that the usual designation-condition for names,

V(a) ~ Din,

will guarantee that every name designates an existent. In this respect, the new criterion is in the same boat as Quine's, however, for the only way his (15) can come out false in a standard language is in the empty domain. As Quine once said, "If the word 'Pegasus' designates something then there is such a thing as Pegasus" (1939d, p. 45).


2.2. Shades of free logic

The alternative criteria (5) and (17) may be contrasted in their application to models for free logic. Two different modelings of free logic have been put forward: outer-domain semantics and supervaluational semantics. In the outer-domain semantics, the compre- hensive domain D of a model m is allowed to be wider than the range R of the bound variables. But the outer domain D - R is not seman- tically gratuitous like X in the preceding section. D - R provides the empty names with referents: call them "shades". Now, it must be noted in fairness to Quine that he never envisaged the application of his criterion to a theory based on free logic: cf. (2), which implies pretty clearly that D = R. It is natural all the same for free-logicians to claim Quine's paternity for their treatment of quantification (e.g., Hintikka, 1966, pp. 73ff). For on their interpretation only R comprises "existents": shades don't "exist". Thus they agree with Quine in accepting (5). By (17), in contrast, shades do exist in outer-domain models for free logic.

This may seem a misunderstanding of free logic. Certainly it ac- cords neither with the origins of free logic (Leonard, 1957) nor with the intentions of free-logicians. I reply, first, that nothing in the axiomatics or model theory of free logic prevents shades from existing (cf. my 1969). And R might be any class whatsoever, representing any property whatsoever. Thus the restriction of existents to R is a limitation on the models admitted, i.e., a theory. Second, I invite you to consider how you would react if you learned of a new advance in physical theory distinguishing quarks along a new line: "existent" and "nonexistent" quarks. Would you be capable of taking the "noneist" physicist at his word, and excluding the "nonexistent" quarks from his ontology?

On the supervaluational semantics for free logic, on the other hand, no outer domain is posited: Rm = D,, for every supervaluational model m. Now, for such a model the Quinean ontology coincides with that advocated here, as indeed with that intended by the free-logicians. (My objection to supervaluational semantics for free logic is that it evaluates simple sentences with empty subjects in an ad hoc manner. Its treatment of simple sentences is a cross between the approaches taken in the objectual and the substitutional interpretations of quantification. On the latter, see Section 3.5 below.)

10 J O H N B A C O N

2.3. Contingent domains

It is good semantic form to include in the domain(s) of a model all the set-theoretically simple nonlinguistic objects (other than A) that will be used as members o f sets, terms of relations, or arguments and values of functions that figure in the model on its application. More precisely, the objects included should be those that are not stipulated to be linguistic or set-theoretically compound in the definition of the modeling in question. (Special models might then be constructed out of linguistic materials, as in a Henkin proof.) This is not always done, however. For example, a system of modal logic may be interpreted by models (W, E, R, g, V), where W is a set ("possible worlds"), E is a function assigning to each w e W a domain Ew, and so on (cf. e.g., Thomason, 1970, p. 70, building on Kripke, 1963, pp. 65ff). At first it might look as though W exhausted the ontology of such a model. But the domain function E takes as values sets of new items not in W and not constructed out of materials available on the basis of W. It is clearly in the spirit of (17) to reckon not just W but

wu U w E W

as the ontology of the model. Such examples point the way to a second extension of our ontological criterion.

2.4. Nonlinguistic atoms

The point is that, besides the domain, other elements involved in the model may introduce new simple or unanalyzed individuals. Provided these are nonlinguistic, 6 they belong in the ontology of the model. In order to make this idea precise, I first define recursively the set A(X) of "atoms" of an arbitrary item X defined by M to be involved with an L-model m as such. Functions and relations are here taken as sets.

(18) 1. A(A) = A. 2. If M does not stipulate X to be a set, then A(X) = {X}. 3. If X ~ Y then A(X) ~_ A(Y).

X is linguistic iff A(X) contains some expressions of L. Using " L " ambiguously for the set of L's expressions, I now define the ontology


of X relative to M as its nonlinguistic atoms:

(19) O ( X ) = A ( X ) - L

It is a consequence of (19) that

(20) O(m) = U o(x) X ~ m

In particular, O(Dm) = Dm will be included. So far in this section, I have defined O only for the general form of

L-models as defined by M. Particular models might, of course, have further structure not stipulated by the definition; for example, the elements of D might be sets of expressions. Such particulars are here treated as ontologically irrelevant. The model has its ontology in virtue of its defined form.

As before (10)-(14) extend O beyond models to theories and persons.

2.5. Composite existence

According to this criterion, we shall never be committed ontologically to sets. For the special case of a theory of sets, see section 3.6. But what about the sets or other abstract entities that the use of predicates or higher-order quantifiers might be held to commit us to? Have we not simply defined away that important problem? If so, let us face the problem again by a diplomatic extension of our terminology. Let us call the mode of existence defined thus far by O "subsistence", and let us consider what composite entities might be said to exist in a broader sense of the term, or to "consist". The rough idea is that a composite entity exists in a model if it plays an essential semantic role in the application of the model. This is the third extension of our ontological criterion.

More precisely, relative to a modeling M,

(21) X consists in m iff 1. O(X) ~ A, 2. X is nonlinguistic, and 3. M ineliminably mentions or quantifies over X in

defining or applying m.

Clause (1) rules out purely set-theoretic items, such as the two


truth-values T ={A} and F = A, since O(T)= O(F)= A. Clause (2) would rule out the valuation or denotation function V used in many models, for A(V) contains expressions of L, making V linguistic (cf. (20)). Clause (3) presents problems to be taken up in the next section. Assuming that it might be clarified, I proceed to define

C(m) = {X: X consists in m} E(m) = O(m) U C(m).

E(m) is the ontology of m in the broad sense, embracing composite as well as basic entities. We might distinguish O(m) as its "subontology". Once again, C and E may be extended as in (10)-(14), writing " C " or " E " for " 0 " .

2.6. Eliminability

(21) is unfortunately vague. Clause (2) should be qualified somehow to prevent the contrivance of ad hoc linguistic functions as entity shel- ters. For example, the ontology evader could add an inessential linguistic argument to every element of the model. On the other hand, I don't want to go so far as to count designation or denotation relations as entities in the ontology: hence stipulation (2). What is intuitively wanted is that X's linguisticness, if any, be eliminable by trivial reformulation of M. Which brings us to the "ineliminably" in clause (3). I will not give a precise definition of this term but only some examples. They should suffice to make clear the sort of thing I have in mind. 7

3. APPLICATIONS

3.1. Predicate Classes

First, a well known example from Quine. It is common practice in the semantics of predicate logic to assign classes to predicates as their extensions. Thus, for m = (D, V), the denotation-condition for one- place predicates P is often given as

(22) V(P) c_ D.

Then the truth-condition for a simple sentence with subject a is

V(Pa) = T iff V(a)~ V(P).


Now, V(P) cannot be ruled out of consistence by (21.1). But, as Quine has often pointed out (e.g., 1972, p. 80), there is no need here to extend V to predicates. We can use instead a true-of or denotation relation Den such that

P Den x iff x ~ V(P). (

V as applied to predicates is then definable as Den'. All this is clearly just a rearrangement of deck chairs, no material alteration. Accord- ingly, mention of V(P) is eliminable: standard first-order objectual quantification theory does not commit us to the consistence of subsets of D. On the other hand, it does commit us to the consistence of D itself, for " D " is not eliminable from the kind of modeling considered here.

3.2. World-lines

As another example, take the use of world-lines in the semantics of quantified modal logic. In Thomason's system Q2 (1969, p. 136(6)) and my Q4 (1980, p. 196, IVl), these are quantified over unrestrictedly, both in the object language and in the truth-conditions for the quantifiers. World-lines are accordingly ineliminable from the models used: they consist in those models. On the other hand, in Thomason's Q3 (1970, p. 70(2)) and my Q5 (1975, p. 350), only constant terms may express arbitrary world-lines. Variables are restricted to "straight" world-lines (constant functions from worlds to individuals) in my semantics for these systems (1980, p. 197, Ivarl). But clearly a constant function is a needless complication (which I adopted for architectonic reasons). Thomason just uses the constant individual values of my constant functions as his variable-range (1970, p. 70(1)). 8 As for nonrigid names, although we might state their sense-conditions a s

Int(a) ~ D w or Int(a): W---~ D,

reminiscent of (22), all we really need is

Vw(a) E D,

whereupon Int is definable as AxAwVw(x). I conclude that use of the quantified modal logics Q3 and Q5 does not presuppose the existence of world-lines, whatever else it may presuppose.


3.3. Inner domain of free logic

The above examples should help to explain eliminability, although they don't cover all bases. To revert to free logic (section 2.2) for a moment, the inner domain R there would ordinarily be included in a model. Accordingly, it would seem to consist. But there are ways of seemingly eliminating it. First, we might replace countable R in a model by the sequence of its members. This seems artificial. Yet, if a model were stipulated to include a unit class {x}, we should unhesitat- ingly welcome its replacement by x. The only significant difference between the two cases would seem to be infinity versus finitude. A finitude constraint would rule out important models, however. On this point, I am content to leave the criterion vague.

A more interesting way of eliminating the inner domain from the semantics of free logic is to take the free-logicians' "E" or "E!" as a primitive predicate and then let the quantifiers range over V(E). This is eliminable in favor of " ' E ' Den" as in section 3.1:

V((x)p) = Tiff V'(p)= T for all V' like V except that 'E ' Den V'(x) and maybe V'(x) ~ V(x).

Quantification over assignment functions V' is in turn eliminable in favor of a (linguistic) semantic-substitution function on assignments.

3.4. Possible worlds

As it is usually interpreted, with a set W of basic items "which we think of as possible worlds", modal logic commits us ontologically to ("putative") possible worlds. However, for many purposes the so- called "possible worlds" can be construed linguistically, as state des- criptions or classical models. On such a modeling, they wouldn't even consist; nor would their accessibility relation, being likewise linguistic. These reflections suffice to clear propositional modal logic of ontological commitment to possible worlds. And I hold that propositional modal logic is enough to give meaning to the modal operators.

When we add quantifiers, we may be obliged ultimately to admit nonlinguistic possible worlds to our ontology. (There may not be enough classical models or maximal consistent sets to go around.) These may be understood by analogy to the linguistic possible worlds.


The real conceptual problems about modal logic - what accessibility is, and how we can learn transworld individuating functions (seeing as they both involve worlds beyond our reach) - are not ontological but metaphysical and epistemological.

3.5. Substitutional quantification

The ontological criteria developed here do not work for substitutional interpretations of the quantifiers. They would yield a result of nil commitment for theories so formulated. Such a result is implausible, for applied substitutional quantification will still involve names desig- nating individuals. It's just that the individuals don' t get mentioned in the semantics. Substitutional models are accordingly excluded from consideration here. (For an extension of a Quinean criterion to substitutional quantification, see Gottlieb [1980].)

3.6. Set theory

A special problem arises for our definition (18) when the theory we are considering is a set theory with no individuals or nonsets, for then E(m) = A for all intended models m of the theory. Yet intuitively we should like to say that such a theory committed us to an ontology of sets. In this particular case, it seems more reasonable to replace (18.1) by

A(A) = {A}

Thus, for set theory without individuals, O(m)= {A}, and E(m) will comprise all the nonlinguistic set-theoretic constructs out of A used in m .

3.7. Ad hoc vs. intended models

We have already deplored in section 2.6 the subterfuge of entity evasion by linguization. There are many other ploys for introducing spurious luxuriance or austerity into modelings. For example, take a model m = (D . . . . ) with a large domain D = O(m). Now "identify" the members of D with set-theoretically constructed numbers. Poof! no ontology: O(m) = E(m) = A; or, if we use the modified criterion of section 3.6, O(m)= {A}. Of course, this artificial model might not be

16 JOHN BACON

an intended one, in which case its existence would be innocuous so far as theoretical commitment was concerned. This brings out our heavy reliance on the demarcation of Mz, the intended models of a theory. Without it, we should be obliged to come up with a model-theoretic reconstruction of Ockham's Razor, a criterion for ordering models according to their natural parsimony. That is a task for another time; here I leave ontological shearing to the intentions of the community of theoreticians. For rough guidelines they can do no better than to turn to Quine:

Our acceptance of an ontology is, I think, similar in principle to our acceptance of a scientific theory, say a system of physics: we adopt, at least insofar as we are reasonable, the simplest conceptual scheme into which the disordered fragments of raw experience can be fitted and arranged. Our ontology is determined once we have fixed upon the over-all conceptual scheme which is to accommodate science in the broadest sense . . . . (1948, p. 16f)

4. CONCLUSION

My aim has been to adapt Quine's criterion of the ontological commitment of theories couched in standard quantificational idiom to a much broader class of theories by focusing on the set-theoretic structure of the models of those theories. For standard first-order theories, the two criteria coincide on simple entities. Divergences appear as they are applied to higher-order theories and as composite entities are taken into account. In support of the extended criterion, I appeal to its fruits in treating the various examples considered above and to the healthY intuitions of the non-noneists among us. Don ' t O(m) and E(m) comprise just the things we should have thought existed according to a part icular interpretation m of a language or a theory? Whatever the answer (and it will hardly be unanimous), I hope to have pointed the way towards a recognition of ontology as a worthwhile branch of modern theory.

NOTES

* Earlier versions of parts of this paper were read at New York University, at the Australasian Association of Philosophy, and at the University of Sydney. I am very grateful to William Barrett, Keith Campbell, Gregory Currie, Kenneth Gemes, Toomas Karmo, and Stephen Read for comments and criticisms. t Bacon (1966, p. 49f) and Chihara (1973, p. 100) have taxed Quine here with

A M O D E L - T H E O R E T I C C R I T E R I O N OF O N T O L O G Y 17

surreptitious recourse to modality or the theory of meaning. However, in (1966, p. 3) Quine says he only meant "necessary condition", i.e., (presumably) formal implication of some sort. 2 I am grateful to Kenneth Gemes for calling this point to my attention. 3 The account of theories given in the introduction may well be oversimplified: cf. Stegmiiller (1978, pp. 46tt). (10) can be extended to a StegmOller theory-element (Mp, Mpp, M, C, I) by letting Mr be the theoretical supplementation of the partial possible models in I in accordance with the constraints C. 4 This was pointed out to me by Keith Campbell, giving rise to the present section. Cf. Chihara (1973, p. 96f).

In (1969, p. 311) I spoke of "semantic descent", but Yehoshua Bar-Hillel rightly pointed out that "ontological descent" is more appropriate (cf. "celestial ascent", "infernal descent"). 6 Here I part ways with Jackson, who includes linguistic entities among the ontological commitments of a sentence on an interpretation (1980, pp. 310f). 7 It might be more satisfactory to delete the "ineliminably" from (21.3) and try to find a model-theoretic definition of "M' is the result of eliminating something from M", or "M' is just like M except for some eliminations". Something like the present (21) could then be got as the intersection of the ontologies of corresponding models in similar modelings. I have had no success with this approach so far. 8 This exposition is anachronistic: I learned about these systems from Thomason.

R E F E R E N C E S

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Bacon, John: 1969, 'Ontological Commitment and Free Logic', Monist 53, 310-19. Bacon, John: 1980, 'Substance and First-order Quantification Over Ifidividual-

concepts', Journal of Symbolic Logic 45, 193-203. Bacon, John: 1975, Basic Logic, Extensional and Modal (York College, New York.

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and Existence, University of Wisconsin, Madison Press, 1960, 91-105. Chihara, Charles: 1973, Ontology and the Vicious-Orcle Principle, CorneU University

Press, Ithaca, N.Y. Gottlieb, Dale: 1980, Ontological Economy: Substitutional Quantification and Mathe-

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18 JOHN B A C O N

(eds.), Readings in Philosophical Analysis, Appleton-Century-Crofts, New York, 1949, 44-151.

Quine, Willard V.: 1939l, 'A Logistical Approach to the Ontological Problem', reprinted in The Ways of Paradox and Other Essays, Random House, New York, 1966, 64-69.

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Calculi', in Lambert (ed.), Philosophical Problems in Logic: Some Recent Develop- ments, Reidel, Dordrecht, 56-76.

Department of Traditional and Modern Philosophy University of Sydney Sydney, New South Wales 2006 Australia

a model-theoretic criterion of ontology

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