Reflexive fictionalisms

Reflexive fictionalisms DANIEL NOLAN & JOHN O’LEARY-HAWTHORNE

1. Introduction Modal Fictionalism and the unusual problems it faces have been widely discussed of late. Realizing the scope for fictionalism about a variety of domains, many have wondered whether recent insights into the problems for modal fictionalism have any further implications. We will argue that one of the main problems facing modal fictionalism is merely a special case of a general problem that faces a genus of fictionalisms of which modal fictionalism is merely a species. We will suggest that one particular diagnosis of this problem becomes more obviously attractive when viewed in this broader context, arguing that not only modal fictionalism, but other inter- esting fictionalisms (including fictionalism about mathematics and fictionalism about universals) cannot provide the ultimate grounding for the areas of discourse in question.

To introduce this class, we will begin by recapping the ‘story so far’ in the case of modal fictionalism. The form of modal fictionalism presented by Gideon Rosen (1990) will be our focus for simplicity. It is the thesis that while claims made using modal operators are often literally true (for instance, the claim ‘possibly swans are blue’ is perfectly in order), it nevertheless is false that there are any non-actual possible worlds, concrete or otherwise, and so the claim ‘there is a possible world where swans are blue’ is literally false. However, since talk of possible worlds is at the very least quite useful when discussing modal issues, and perhaps even indispensable, the modal fictionalist wishes to rescue the usual ‘Possible Worlds translations’ of modal claims, and does so by translating claims involving modal operators into claims about what is true according to a certain fiction about possible worlds. Talk of possible worlds is then useful in talking about modality because there is a procedure for moving back and forth from the fiction about possible worlds to workaday claims involving modal operators. According to Rosen’s fictionalist, the connection between ‘possibly swans are blue’ and ‘there is a possible world where swans are blue’ is that possibly swans are blue iff, according to the modal fiction, there is a world containing blue swans. In general

Possibly p iff according to the modal fiction, at some world p , and Necessarily p iff according to the modal fiction, at all worlds p . Using the device of a story about possible worlds and the above ‘trans-

lation procedure’, Rosen claims that we may be able to have the benefits of

ANALYSIS 56.1, January 1996, pp. 23-32. 0 Daniel Nolan &John O’Leary-Hawthorne

24 DANIEL NOLAN 8r JOHN O’LEARY-HAWTHORNE

(i) having ordinary modal talk come out as literally true most of the time, and (ii) having recourse to the elegance of a possible worlds framework for modal thinking, but without the ontological cost of non-actual concrete possible worlds and possibiliu. The modal fiction Rosen chooses to rely on is very nearly the one presented as non-fiction by David Lewis (1986) - with a few minor alterations.

A serious problem has been identified for this approach in recent litera- ture (Rosen 1993, Brock 1993), which can be illustrated by considering the following claim:

(1) Necessarily, there exist many worlds. Treat the scope of the implicit quantifier in (1) as quantifying over ubso- lutely everything - (1) is to be read in the widest possible way. The modal fictionalist does not want to be forced to endorse (l), since at best the fictionalist will wish to remain agnostic about whether or not there are many worlds, and at worst the fictionalist will want to reject outright the claim that many worlds exist, understood literally.

Should it be necessarily true by the fictionalist’s own lights that there are many worlds, the fictionalist will seemingly be led to embrace modal realism after all, since necessarily p entails p . Unfortunately, it is hard for the fictionalist to resist such a conclusion. Consider the fictionalist’s translation scheme. (1) will be true iff

(2) According to the modal fiction, at all worlds, there exist many worlds.

and Lewis’s fiction does say that the modal universe contains many existing worlds, and the ‘at all worlds’ is just a redundant quantifier when ‘there exist many worlds’ is taken to have a very wide scope. ‘At all worlds, there exist many worlds’, when interpreted in the intended way, is true according to Lewis 1986, so it is true according to the modal fiction. And since (2) is true iff (1) is, by the fictionalist’s lights, then (1) must be true by the fictionalist’s lights. That there are in fact many worlds follows from (l), and so the modal fictionalist is committed to the existence of the many entities that fictionalism was introduced to avoid commitment to. Rosen’s initial modal fictionalism in the form proposed is unsuccessful.* A broader understanding of the problem is afforded, we believe, by recognizing that the problem isn’t peculiar to modal fictionalism. Moreover, we will be offering

We have slightly reworked the BrocWRosen objection both for expository purposes and also because the present formulation of the argument avoids the solution offered in section 3 of Menzies and Pettit 1994.

A different style of argument for the conclusion that modal fictionalists cannot deny the existence of worlds is provided by Hale 1995. (But see Rosen 1995 for a reply.) We shall not be explicitly discussing Hale’s objection here.

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a strategy for dealing with each manifestation of the problem, one that appears rather compelling from the broader perspective.

2 . The analogy with mathematical fictionalism

A similar problem to the problem that arises for the modal fictionalist will arise for a certain sort of mathematical fictionalist.2 Suppose that one thought that there really were no such things as numbers, but that talk about numbers was an extremely useful device for performing the sorts of tasks that we use mathematics for. One might then wish to justify talking loosely as if there were numbers, and fictionalism seems to offer such a justification. Suppose, then, that the mathematical fictionalist proposes the following translation scheme in order to explain the connection between true (mathematical) claims about respectable entities and claims made about the merely fictional numbers:

(3) For any sort of entity ‘p, There are tz ‘ps iff according to the mathematical fiction, the number N numbers the q s

where numbering is a fictional relation which holds between numbers and groups of objects (the ‘how many’ relation, if you like), n is a place holder for a numerical adjective (such as ‘three’, for example), and N is the name of the number corresponding to the numerical adjective (e.g. ‘Three’). So, to take a concrete example, the following is an example of the schema (3):

(3’) There are two moons of Mars iff according to the mathematical

The mathematical fictionalist can take over some fairly standard Platon- istic treatment of numbers as the work to be treated as the mathematical fiction, and thus, supposedly, can enjoy all the benefits of Platonist mathematics without the ontological cost. The mathematical fictionalist can then reason away using the mathematical fiction, and apply the results where needed by back-translation.

A similar problem will arise for this sort of mathematical fictionalist which arises for the fictionalist about possible worlds. This problem arises when one considers the very entities the mathematical fictionalist wishes to

fiction, the number Two numbers the moons of Mars.

Certain rescue attempts for modal fictionalism aimed at avoiding the BrocWRosen objections have been proposed in recent issues of this journal. See Menzies and Pettit 1994, section 5 and Noonan 1994 (endorsed by Rosen 1995). We identify difficulties for those rescue attempt in a separate piece.

2 Strictly speaking, the position that will he outlined might better be described as numerical fictionalism, as the fictionalist treatment will only he discussed in relation to numbers. A properly developed mathematical fictionalism would also owe us a story about geometry, groups, functions and much else besides.

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be fictionalist about. For consider the following claim:

(4) There are (at least) three numbers. According to the translation scheme, this is true iff

( 5 ) According to the mathematical fiction, the number Three numbers the numbers (or at least numbers some subset of the numbers).

Now it is true according to Platonists that there are at least three numbers (i.e. the number Three numbers a subset of the numbers). So with the obvious candidate fiction (the standard mathematical story), it follows from the translation mechanism that it is literally true that there are at least three numbers (and in fact that there are infinitely many). So the mathematical fictionalist is committed to realism about numbers after all.

3. What the two cases have in common Notice that both modal fictionalism and mathematical fictionalism are cases where a fictional domain of objects (worlds or numbers) are set up to assist in reasoning involving certain operators (modal operators or the ‘number’ operators such as ‘there are three ...’). And notice that in each case the difficulty arose from sentences comprised of those operators being applied to the objects postulated by the fiction (modal statements about worlds in the first case, statements about the number of numbers in the second). Finally, notice that in each case that some applications of the operators involved automatically bring ontological commitment (necessarily, rps exist entails that rps exist, and there are n p entails that there are 40s).

These three commonalities tempt us to generalize the statement of the problem that we have noticed in the cases of modal and mathematical fictionalisms. We can delineate a class of fictionalisms with the above characteristics: firstly, the fictionalism treats as fictional a domain of objects which realists in that area take to be the objects in virtue of which certain operators or predicates are applicable; secondly, that the operator in question is capable of entailing existential commitment in virtue of at least some of its usages; and finally, that the type of operator or predicate in question can be intelligibly applied to objects in the domain which the fictionalist desires to be fictionalist about. Let us call fictionalisms in this class reflexive fictionalisms - reflexive because it is possible to apply the relevant operator or predicate in claims about the relevant fictional objects.

The problem for some reflexive fictionalisms is that there will be an embarrassment derivable from their theory when the theory relies on a simple biconditional that tells us of some operator that it applies iff according to the fiction involved, the relevant entities have the appropriate

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characteristics, or alternatively have the relevant relations to the non- fictional entities which appear on the left-hand-side of the biconditional. The embarrassment will be that some operator which entails ontological commitment will be demonstrably applicable to the very objects whose existence the fictionalist wishes to deny (or be agnostic about).

4. Diagnosis Let us return to mathematical fictionalism. It is fairly clear that the paradox that arises for the fictionalist can be avoided by adopting a more complicated algorithm for moving back and forth from talk that explicitly deploys a fiction operator to talk that does not. In particulal; the fictionalist might say that the inference from ‘There are three F’s’ to ‘According to the mathematical fiction, Three numbers the F’s’ is appropriate only for a limited range of F’s, including a exclusion clause that, inter alza, denies the propriety of the move when ‘number’ is the substituend of ‘F’. Is this an acceptable way of dealing with the paradox? It is not. Tinkering with the inference rules in the way we have just entertained imposes a semantic divide that is intuitively ad hoc. It is intuitively clear that the literal meaning of ‘There are at least three . . .’ in the case of ‘There are at least three numbers’ is the same as the literal meaning of ‘There are at least three ...’ in the case of ‘There are at least three dogs’. When the fictionalist who understands English dissents from the latter and assents to the former it is intuitively clear that she understands ‘There are three’ in just the same way. From the point of view of these semantic platitudes, our strategy for the fictionalist seems ad hoc, a complication introduced simply because the theory has got into trouble.

Does this mean that no form of fictionalism can be defended? Of course not. Only one form of fictionalism about numbers is touched by our main problem, a fictionalism that says that the literal content of, e.g., number operators is to be explained by (or reduced to) propositions about what’s true according to a fiction. Call the type of fictionalism which purports to provide such a reduction or explanation ‘strong fictionalism’. There is another variety of fictionalism; let us call this other variety ‘timid fictionalism’. ‘Timid’ fictionalism says that claims involving number operators are often strictly and literally true but that they are not true in virtue of any facts about what is true according to a story that involves an ontology of numbers. Our fictionalism will then merely concern ontological claims about numbers, and will say that claims implying ontological commitment to numbers are all, strictly speaking, false. We shall thus not bring in fictions at all to evaluate the strict and literal use of numerical language. Uses of that language, in their strict and literal use, that are ontologically

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committed to numbers, will all be taken at face value and reckoned strictly and literally false. Claims involving number operators that are not ontologically committed to numbers, will be given a strict and literal assessment that does not involve recourse to fictions and which will, inevitably, reckon some such claims true, others false. How is there room now for being fictionalist in any respect? Well, with all this in place, the timid fictionalist will elucidate a way of making fruitful use of talk about numbers by graft- ing a fiction about such objects onto claims involving number operators that are independently understood. The timid fictionalist recognizes the usefulness of talk about numbers even though there aren’t any, and earns the right to carry on using that way of talking by appealing to a fiction- based account. In this spirit, she will cook up inference rules that allow us to pass back and forth from number operator talk to talk about what is true according to some fiction about a domain of numbers. We shall use our independent understanding and evaluation of claims involving number operators as the test of adequacy for such rules. Thus she will say that

‘There are two moons of Mars’ is strictly speaking true iff it is true according to the Number Fiction that the number of moons of Mars is identical to Two.

In itself, this inference rule has the same style as that of strong fictionalism. However, unlike strong fictionalism, this ‘timid fictionalism’ does not use such biconditionals as a way of elucidating the literal meaning of some uses of number operators; rather, they are a way of introducing a loose way of using talk about numbers that is not ontologically committing. With the above sort of inference rule in place, the timid fictionalist can say: ‘When I am speaking loosely, were I to say “The number of moons of Mars is identical to Two”, take that as equivalent to “According to the fiction, the number of moons of Mars is identical to Two”, a claim which in turn obeys the inference rule above.’

In this way, the timid fictionalist utilizes the translation rules from operator talk to fiction talk merely as a way of assigning a loose use to ontological number talk, not as a way of elucidating the meaning of number operators. And for this reason, the danger of semantic ad hocery that arose with strong fictionalism does not arise here. So long as the inference rules give talk about numbers a useful role to play, they will serve their purpose. In this context, it is perfectly acceptable to tailor the inference rules in such a way as to provide the most convenient way of using talk about numbers. The timid fictionalist has a fiction-independent understanding of both ‘There are at least three numbers’ and ‘There are at least three kangaroos’, in their strict and literal usage, reckoning the former false and the latter true. Given this understanding, she will clearly wish to

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tailor the inference rules from fiction talk and back in such a way as to preclude endorsing

‘There are at least three numbers’ is strictly and literally true iff according to the fiction, there are at least three numbers,

while allowing ‘There are at least three kangaroos’ is strictly and literally true iff according to the fiction, Three numbers some of the kangaroos

But she will not thereby be forced to say that ‘There are three’ means one thing in the case of ‘There are three numbers’, another in the case of ‘There are three kangaroos’. For the timid. fictionalist never intended the fiction- theoretic biconditionals to elucidate the meaning of any use of ‘There are three’, only to graft a loose use of number ontology onto some of those uses. The danger of semantic ad hocery only arises when fictions are used to give a semantics of some but not all uses of an operator where, intuitively, the uses given the fiction-theoretic semantics appear semantically homogeneous with sundry uses that are not. This is the strong fictionalist’s game, not the timid ficti~nalist’s.~?~

Timid mathematical fictionalism is thus better placed than strong fictionalism to handle the paradox is an acceptable way. We note in passing that in any case, strong fictionalism about mathematics is an utterly bizarre view - we would not ordinarily think that the fact that there are two moons of Mars is an artefact of some complicated story-telling on our part - rather, the claim that there are two moons of Mars is made true by big lumps of rock in outer space. This becomes especially obvious when it is remembered that claims about how many objects of a certain sort there are translate into first-order logical sentences expressed just in terms of quan- tificatiori and non-identity. It seems crazy to suppose that ‘Deimos exists and Phobos exists and Deimos is not identical to Phobos’ is true in virtue of a fiction, unless one was going to be anti-realist enough to suppose that all truths (or nearly all truths) are true in virtue of stories we tell. Unlike strong fictionalist about mathematics, timid fictionalists can, of course, be suitably realist about the moons of Mars, including facts concerning their quantity.

3 There is another style of fictionalism that is relatively untouched by the paradox, one which says that all use of the discourse - whether of the operator or ontological variety - is strictly speaking false or truthvalueless, and which then sets out inference rules involving fictions as giving a use for that discourse and according to which, loose1y speaking, various sentences of that discourse come out as true (or at least useful). Nolan (forthcoming) calls this ‘broad fictionalism’.

Note that the most important recent effort at fictionalism about numbers - Hartry Field’s 1980 - is not a strong fictionalism.

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Very similar points apply to modal fictionalism, we suggest.s If one attempts to use fictions to provide an account of the content of modal operators and tries to rtLut the paradox by tinkering with the basic inference rules, then one will invite the charge of semantic ad hocery. If, on the other hand, all that is wanted is a way to employ devices which apparently quantify over possible worlds (and thus gain the pragmatic benefits of thinking extensionally in modal situations) without ontological commitment to such entities, then presumably any translation device that gets the job done will be okay, provided that it functions straightforwardly and intuitively in most of the everyday cases (and modal claims about possible worlds are certainly do not constitute most of those cases).‘That is, the fact that the translation device is somewhat gerrymandered is only a worry for the strong variety of modal fictionalism (that variety which seeks to provide a fiction-theoretic analysis of modal operators). It is not a worry for the timid modal fictionalist (the modal fictionalist who is not seeking a metaphysical analysis of modality, but simply an ontologically-light excuse for quantification over possible worlds in modal reasoning).6 Further, even leaving aside the paradox, strong modal fictionalism is strange in its own right (as it was in the mathematical case). It is strange in its own right to suppose that ‘If I had dropped the pen, it would have fallen’ and ‘Neces- sarily, there are no true contradictions’ are made true by stories we tell.

The thing to learn from the observation that this paradox affects a range of reflexive fictionalisms and can be avoided by timid fictionalisms but not by strong ones is, we suggest, that reflexive fictionalisms should not be strong.

5. Applying the lesson It might also be wondered whether our observations about reflexive fictionalisms are of more general use than considerations of modal and mathematical fictionalisms. A final example might help to highlight the more general application. Consider Fictionalism about universals: the view that there really are no properties or relations, but that it is convenient to talk as if there are for various purposes (making law-like generalizations, making general observations about similarities, constructing colour spec- tra for the purpose of choosing how things are to be coloured, and so on). Thus we can claim that roses are red while denying that, literally speaking, roses have Redness, but nevertheless allow that we can meaningfully and

Of course, a full defence of our diagnosis, as it applies to modal fictionalism, would require a critical comparison with other, rather different rescue attempts that have been offered to resolve to BrocWRosen difficulty. Space precludes our undertaking such a task in this paper.

6 For a fuller discussion of the stronghimid divide, see Nolan (forthcoming).

REFLEXIVE FICTIONALISMS 3 I

usefully talk about Redness and its similarity to Orangeness, its relation- ship to Scarletness, and so on. This is because, according to a fictionalist about Universals, there is a story about Redness and other universals, and a procedure exists for translating claims using predicates such as ‘is red’ and claims about what is true of Redness in the fiction of universals. The obvious biconditional to take advantage of in the case of properties would be a schema of the form:

( 6 ) X is cp iff According to the fiction, pness is instantiated by X (and there would be a similar translation scheme for relations)

This scheme could be used for nearly all predicates, and so, the fictionalist might claim, we can have the advantages of realism about universals without the ontological cost, This form of fictionalism is a reflexive one, however, as it makes sense to predicate things of universals (the objects intended to be merely fictional). Can an instance of the paradox for reflexive fiaionalisms be found which makes the simple biconditional unsatisfactory?

An example of how the reflexive paradox arises: Suppose we think that any literally true subject-predicate sentence implies ontological commitment to the thing denoted by the subject term. Now universals themselves, according to the fiction, have properties and stand in relations. Redne~s,~ for example, will, according to the fiction, instantiate Monadicity (i.e. according to the fiction, Redness will be monadic, that is, Redness is a property rather than some sort of relation). So the following will be true:

(7) According to the fiction of universals, Redness instantiates Monadicity

and by the translation procedure, therefore, (8) Redness is monadic

will be literally true. So the fictionalist about universals will be committed to the literal existence of Redness after all, given the criterion for existence mentioned above. Similar moves can be made in virtue of the relations in which universals are supposed to stand according to the fiction. Because cricket balls are red, it can be derived that Redness is instantiated by cricket balls by means of translations involving the relation of Instantiation and its converse (assuming that according to the fiction Universals are related to their Instances by an Instantiation relation).* Once again, our diagnosis will run along the same lines: fictionalism about universals shouldn’t be strong.

7 Redness might not even appear in the fiction if the fiction is one of ‘sparse’ universals only (e.g. if the fiction is similar to Armstrong 1978). Readers with such qualms are invited to substitute a suitably sparse property in its stead for purposes of the example.

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6. Conclusion The problem for modal fictionalism which we discussed is not an isolated problem, generated solely by the peculiarities of modality or a localized quirk in the procedure of employing the specific fictionalist biconditional offered by Rosen. There is a quite general category of fictionalisms, and it is in virtue of the combination of features which go to make up this category that modal fictionalism faces the problem it does. We have argued that the solution to the general problem is to avoid strong fictionalism in these cases. That is, in the cases of reflexive fictionalisms at least, we should see the fictions as useful for helping us in our thinking and re-asoning, but not as suitable for providing the material for an analysis or a reduction of the discourses in question. This, in turn, indicates that the explanatory role that fictions can play in nominalist strategies of avoiding ontological or ideological commitments in these areas can only be limited. The serious metaphysical explanations of modality, mathematics, and predication must be sought el~ewhere.~

RSSS, Australian National University Canberra ACT 0200, Australia [email protected]

hawthorn@coom bs.anu. edu.au

References Armstrong, D. 1978. A Theory of Universals. Cambridge: Cambridge University Press. Brock, S. 1993. Modal fictionalism: a response to Rosen. Mind 102: 147-50. Field, H. 1980. Science Without Numbers. Princeton: Princeton University Press. Hale, B. 1995. Modal fictionalism: a simple dilemma. Analysis 55: 63-67. Lewis, D. 1986. On the Plurality of Worlds. Oxford: Basil Blackwell. Menzies, P. and P. Pettit. 1994. In defence of fictionalism about possible worlds. Analy-

Nolan, D. Forthcoming. Three problems for ‘strong’ modal fictionalism. Noonan, H. 1994. In defence of the letter of fictionalism. Analysis 54: 133-39. Rosen, G. 1990. Modal fictionalism. Mind 99: 327-54. Rosen, G. 1993. A problem for fictionalism about possible worlds. Analysis 53: 71-81. Rosen, G. 1995. Modal fictionalism fixed. Analysis 55: 67-73.

is 54: 27-36.

It almost goes without saying that, as in the modal and mathematical cases, strong fictionalism about universals is an extremely odd view even when one leaves the paradox to one side. Only someone of an extremely anti-realist bent could find any prima facie plausibility in the view that ordinary true predications are made true in virtue of some fact about human story-telling. (Note also the regress problem.)

We are grateful to Peter Menzies, Graham Oppy and Philip Pettit for their comments on a draft of this piece.

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Reflexive fictionalisms