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Fictionalism and Incompleteness

RichardWoodwardUniversity of Barcelona

Abstract

The modal fictionalist faces a problem due to the fact that her chosen story seemsto be incomplete—certain things are neither fictionally true nor fictionally false.The significance of this problem is not localized to modal fictionalism, however,since many fictionalists will face it too. By examining how the fictionalist shouldanalyze the notion of truth according to her story, and, in particular, the role thatconditionals play for the fictionalist, I develop a novel and elegant solution to theincompleteness problem.

Suppose that you are a fictionalist about some particular domain of discourse—talk of mathematical or composite objects, say. You think that the ontologicalclaims made in that discourse are misleading—there aren’t really any numbersand there aren’t really any tables. But you don’t think that we should abstain fromsaying things like “there is a unique prime number between four and six” and“there are many cheap tables at Ikea.” Rather, you think, we just need to be abit creative when understanding what’s going on when we say such things. Andthat’s where your fictionalism comes in.

Now exactly how it comes in will vary drastically depending upon the brandof fictionalism that you favour. But we can locate one core idea: the centralityof fictionalist paraphrases, so-called because they match ontologically offensivesentences up with ontological innocent surrogates, the latter regarding the con-tent of a fiction. Some examples: Field (1989) paraphrases the claim that thereare prime numbers as: it is true according to standard arithmetic that prime num-bers exist; Rosen (1990) paraphrases the claim that there are possible worlds as:it is true according to modal realism that possible worlds exist; Brock (2002)paraphrases the claim that there are fictional characters as: it is true according tofictional realism that fictional characters exist; Dorr (2005) paraphrases the claimthat there are tables as: it is true according to universalism that tables exist.

C© 2011 Wiley Periodicals, Inc.

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What role do these paraphrases play for the fictionalist? It depends who youask. Field and Dorr claim that their paraphrases play a pragmatic role; they respec-tively specify the conditions under which claims about numbers and compositeobjects are correct. In contrast, Rosen and Brock claim that their paraphrases playa semantic role; they respectively specify the truth-conditions for claims aboutpossible worlds and fictional entities. Following the latter suggestion, let’s sup-pose that the paraphrases play a semantic role: they specify the conditions underwhich the everyday claims which offend the fictionalist’s ontological scruples aretrue.

Despite the characterizations that one finds in the literature, the idea here isn’tquite that a sentence is true iff that very sentence is true according to the story.1

Rather the idea is that a sentence is true just in case its relevant regimentation istrue according to the fiction. This point is familiar from Rosen’s modal fictional-ism, which states that a modal sentence is true just in case its translation into thelanguage of counterpart-theory is fictionally true. So we can think of the semanticfictionalist as endorsing the following schema:

(F) “p” is true iff According to The Fiction, [p]T

Where ‘p’ is a sentence of the problematic discourse, ‘[p]T’ is its relevant regi-mentation, and ‘The Fiction’ is a name for the fictionalist’s chosen story.2

Having introduced semantic fictionalism, the remainder of this paper will focusupon a major problem it faces, one which first arose in connection with Rosen’smodal fictionalism.3

1 Incompleteness

Ordinary fictions are incomplete. Though we learn that Patrick Bateman has apenchant for Phil Collins and a worrying obsession with the aesthetic qualities ofbusiness cards, we do not learn what his favourite colour is or whether he has evervoted for the democratic candidate in a presidential election. American Psycho issimply silent on these points. Similarly, though we learn that Humbert Humbertwas born in 1910 in Paris, we do not learn how many other people were born inParis in that year. Lolita is simply silent on this point.

Will the fictionalist’s story be incomplete too? Seemingly so. Consider the fic-tionalist account of possible worlds bruited by Rosen, which uses a fiction basedon Lewis’s (1986) modal realism. Lewisian realism is notoriously silent aboutcertain matters—we’re not told the upper bound on the number of objects that aworld can contain, for instance. In particular, where k is a cardinal greater thanthe power-set of the actual continuum, and [p]T is there is a world containing kphysical objects, we seem to have:

• ¬According to Lewisian Realism, [p]T

• ¬According to Lewisian Realism, [¬p]T

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Similarly, consider a fictionalist account of sets, which utilizes a fiction basedon Zermelo-Frænkel set theory. This theory notoriously leaves certain claimsunsettled—the Kurepa hypothesis (ρ) is independent of its axioms, for example.4

This provides a pro tanto case for thinking that any fiction based on Zermelo-Frænkel set theory will be silent about ρ, meaning that we have:

• ¬According to Zermelo-Frænkel, ρ• ¬According to Zermelo-Frænkel, ¬ρ

These two examples illustrate what is likely to be a general phenomena: the fic-tionalist’s chosen story is likely to suffer from the same incompleteness that af-flicts everyday fictions.

Let’s suppose, then, that the fictionalist’s fiction is incomplete, so that there aretrue instances of the following conjunction:

(1) ¬According to The Fiction, [p]T ∧ ¬According to The Fiction, [¬p]T

The problem is now that when we plug (1) into the fictionalist’s schema (F), weget the following result:

(2) ¬(“p” is true) ∧¬(“¬p” is true)

This says that p falls into a truth-value gap—neither it nor its negation are true.But given that the truth-predicate and negation commute in the normal way, (2)entails (3):

(3) “¬p” is true ∧ “¬¬p” is true

This says that ¬p falls into a truth-value glut—both it and its negation are true.But given that the truth-predicate is disquotational, disaster follows:

(4) ¬p ∧¬¬p

So the general worry posed by incomplete fictions is that they threaten to showthat the fictionalist is committed to outright contraction.

2 Conditionals

The incompleteness problem threatens to show that fictionalism bottoms out inincoherence. What is to be done? One might think that the response is obvious:the fictionalist should simply deny that her truth-predicate behaves classically and,in particular, she should reject the rules which get us from (2) to (3). After all, it’sfamiliar that anyone who accepts the existence of truth-value gaps will need toreject the classical conception of truth. If we think that Billy’s being a borderline

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case of a tall man amounts to the fact that “Billy is tall” is neither true nor false,then we’ll have to reject the inference from its being untrue that Billy is tall to itsbeing true that Billy isn’t tall.5

Rejecting classical semantics is part and parcel of standard gap-theoretic treat-ments of, e.g., vagueness and the semantic paradoxes. But it may trouble the fic-tionalist to see the phenomenon repeated, as (Rosen, 1990, p. 343) suggests, since(i) “our ordinary ways of thinking give us no reason to expect [it]” and (ii) the onlyreason for rejecting the classical conception of truth is, to say the least, “generatedby concerns rather distant from the linguistic practice in question.” What Rosenis getting at is that the fictionalist needs an independent motivation for rejectingthe classical conception of truth. But the only motivation we have so far is to fix abug in the fictionalist’s theory. Unless something else is said, the response smacksof desperation.

Where might such independent motivations come from? Well, consider howthe fictionalist might gloss the notion of truth according to her story. In the liter-ature, many fictionalists have understood their fictional prefixes via conditionals.(Rosen, 1990, p. 344) writes:

We might begin by offering any of the following glosses: if [The Fiction] were true, pwould be true; if we suppose [The Fiction] p follows; It would be impossible for [TheFiction] to be true without p being true as well. These are not perfect paraphrases.None the less, each seems to give a fair preliminary indication of what we mean whenwe use the fictionalist’s prefix.

Similarly, Field (1989) and Divers (1999) propose to understand “according toThe Fiction” in terms of the strict conditional, and Dorr (2005) and Dorr andRosen (2002) propose to understand the prefix in terms of the counterfactualconditional.

These suggestions stand in need of an important qualification, however. For weneed to distinguish between the explicit content of the fictionalist’s story and itsimplicit content. We can think of the explicit content of the story as the conjunc-tion of the axioms that characterize whichever theory (modal realism, standardarithmetic, universalism, etc.) the fictionalist’s account is parasitic upon. The roleof the conditional is then to generate extra, implicit content. So where ‘Explicit’abbreviates the explicit content, we have:

According to The Fiction, [p]T iff Explicit� [p]T

If we take ‘�’ to be the strict conditional, we get the Field-Divers proposal; if wetake it to be the counterfactual, we get the Dorr-Rosen proposal.

Such conditional analyses of the fictionalist’s prefix interact with the incom-pleteness objection to semantic fictionalism. For given a conditional analysis, thefirst premise of the objection is equivalent to the following:

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¬(Explicit� [p]T) ∧ ¬(Explicit� [¬p]T)

The thing to note is that this directly conflicts with conditional excluded middle(CEM), which states that every instance of the following schema is logically true:

(p� q) ∨ (p� ¬q)

Some conditionals obey (CEM); others do not. It is common ground that the mate-rial conditional supports (CEM) and that the strict conditional does not. The caseof the counterfactual is vexed: Stalnaker (1968, 1981, 1984) and Williams (2009)both defend (CEM) whilst Lewis (1973) and Bennett (2003) both argue against it.

For the moment, suppose that the fictionalist analyzed her prefix in terms ofa conditional that does not support (CEM). She now faces the prospect that herfiction is incomplete in the sense that some things are neither true according to thefiction nor false according to the fiction. But the very coherence of her fictionalismis thereby threatened, meaning that it is likely that she will be forced to reject heroriginal schema (F). This isn’t as bad as it might seem, however. For as Nolan(2008) has pointed out, (F) will still hold in exactly those cases over which thefiction delivers a verdict. So whilst (F) doesn’t hold in full generality, we stillhave:

(F1) (According to The Fiction, [p]T ∨ According to The Fiction, [¬p]T) iff (“p” istrue iff According to The Fiction, [p]T)

By retreating in this manner, the fictionalist can block the original argument fromincompleteness to contradiction. For the argument required that (F) held toutcourt, whereas the present fix tells us that it doesn’t hold when we’re dealingwith fictional incompleteness. The fix is effective, but somewhat inelegant. Wecan do better.

3 Indeterminacy

Instead of glossing her prefix in terms of the strict conditional, the fictionalistmight appeal to the counterfactual conditional: to say that [p]T is true accordingto The Fiction is to say that [p]T would have been true had its explicit contentbeen true.

As I noted above, whether the counterfactual conditional supports (CEM) is avexed question, but let’s join the Stalnaker-Williams axis and accept (CEM) forthe counterfactual. So every instance of the following holds:

(p� q) ∨ (p� ¬q)

(Where ‘�’ is the counterfactual.) Every instance of the following is therebysemantically guarantee to hold too:

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(Explicit� [p]T) ∨ (Explicit� [¬p]T)

But given our counterfactual analysis of “according to,” we now know that thefictionalist’s story is complete with respect to p, because we have:

(According to The Fiction, [p]T) ∨ (According to The Fiction, [¬p]T)

The question emerges: what is left of the incompleteness problem? One mightthink that the fictionalist is let off the hook completely since (1) is now semanti-cally guaranteed to be false. But matters are not quite so simple.

In the fall of 2008, the Spanish footballer Cesc Fabregas was appointed cap-tain of Arsenal, succeeding his French teammate William Gallas. Now considerthe following pair of counterfactuals:

• If Fabregas and Gallas were compatriots, both would be French• If Fabregas and Gallas were compatriots, both would be Spanish

Both of these counterfactuals seem unacceptable, but their disjunction is aninstance of (CEM). So the proponent of (CEM) faces a challenge: she mustreconcile (CEM) with the intuitive data. How can the disjunction of thesecounterfactuals be a logical truth despite the fact that neither counterfactual isacceptable?

Now, the Stalnaker-Lewis approach to counterfactuals boils down to the ideathat a counterfactual is true iff the closest worlds at which the antecedent is true areall worlds at which the consequent is true. But, contra Lewis, Stalnaker assumesthat there is always a unique closest p-world (at least when p is possible). Bothagree that there are two worlds: w (where both Fabregas and Gallas are French)and v (where both are Spanish). Why should one rather the other be the closestworld to actuality? There seems to be no grounds for ‘selecting’ w rather than vwhen we are evaluating these counterfactuals. The demand that there are neverties for closeness seems to imply that Stalnaker must arbitrarily favour one worldover the other.

Stalnaker responds by claiming that whilst one of the worlds is closer, it is in-determinate which world is the closest one. In cases of apparent ties for closenessthere are many admissible selection functions (in this case, there is one that selectsw as closest and one that selects v). We can then say that a counterfactual ‘p�q’ is supertrue iff q holds no matter which admissible selection function we pick,superfalse iff ¬q holds no matter which admissible selection function we pickand indeterminate iff q holds relative to some admissible selection functions and¬q holds relative to others. The point is that the semantic content expressed bythe counterfactual conditional is, for Stalnaker, indeterminate. So both of the pre-vious counterfactuals are indeterminate, e.g., not because there is any (relevant)indeterminacy in their antecedents or consequents, but because the counterfac-tual conditional is itself semantically indeterminate. But every instance of (CEM)

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holds since each instance is true relative to every way of making the counterfactualprecise. In this way, Stalnaker aims to reconcile (CEM) with the intuitive data.

Now, the current proposal is that the fictionalist should solve the incomplete-ness problem by analyzing ‘According to The Fiction’ in terms of the counterfac-tual and endorsing (CEM). If the fictionalist goes this way, then she will reject thefirst premise of the incompleteness argument, since (CEM) entails its negation.As our discussion of Stalnaker’s semantics makes vivid, however, the fictionalistwill also be forced to accept that a disjunction can be true without either of itsdisjuncts being true. (At least, she’ll need to accept this if she wants to explain theintuitive data.) The fiction is determinately complete in the sense it is definitelythe case that either [p]T is fictionally true or [¬p]T is fictionally true. But there isno determinate way in which her story is complete. This is because the content ofThe Fiction is determined by the counterfactual implications of its explicit con-tent. And whenever it is indeterminate whether Explicit counterfactually implies[p]T, it will be indeterminate whether [p]T is true according to The Fiction.

Furthermore, recall that the name ‘The Fiction’ was initially introduced as aname for the set of sentences (or propositions) that form the content of the fic-tionalist’s story. On the present proposal, however, it is indeterminate whethercertain sentences are fictionally true. This is because the name ‘The Fiction’ isreferentially indeterminate: there are a variety of sets of sentences, each of whichare equally eligible candidates to be its denotation. Let’s call each candidate set acompletion of The Fiction. Each completion is complete: for any sentence, eitherthat sentence or its negation is a member of a completion. But whether a sentenceis part of the content of The Fiction is a matter of whether it is a counterfactualimplication of the story’s explicit content. By (CEM), either Explicit counterfac-tually implies [p]T or it counterfactually implies [¬p]T. So the set of sentencesthat are the counterfactual implications of Explicit is complete. Each completionis a candidate to be the set of sentences that are the counterfactual implications ofthe explicit content. But it is indeterminate which completion ‘The Fiction’ refersto.

With this idea in place, we can provide a supervaluational reconstruction ofsemantic fictionalism. We can say, that is, that a sentence p is true just in caseits translation [p]T is a member of every completion. But notice that [p]T is amember of every completion just in case the counterfactual ‘Explicit� [p]T’ isdeterminately true. This, given our counterfactual gloss of the fictionalist’s prefix,yields the following fictionalist schema:

(F2) “p” is true iff �(According to The Fiction, [p]T)

(Where ‘�’ is read ‘it is determinately the case that’.)Retreating from (F) to (F2) doesn’t itself avoid the incompleteness problem: we

can just re-run the argument by reformulating (1) as the claim that [p]T is neitherdeterminately true according to the fiction nor determinately false according tothe fiction. But we now have independent reason to reject the classical conception

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of truth; in particular, we have good reason for rejecting the inference from the un-truth of p to the truth of ¬p. For since our fictionalist proposal is supervaluationalin character, it comes as no surprise to learn that the truth-predicate doesn’t be-have classically. After all, standard supervaluational treatments of anything entailthat the truth-predicate behaves non-classically. But we’re not rejecting classicalsemantics just to fix the incompleteness bug. Rather, our independently motivatedanalysis of the fictionalist’s prefix predicts the non-classical behavior of the truth-predicate. Our solution to the incompleteness problem flows from the role thatcounterfactuals play within the fictionalist’s account.

It might be thought troubling that my solution to the incompleteness problemrequires the fictionalist to modify her initial proposal. But consider the followingfictionalist schema:

(F3) p iff According to The Fiction, [p]T

Initially, there was little point in distinguishing (F) and (F3) since they are clas-sically equivalent. And it’s noticable that the standard formulations of semanticfictionalism are in line with (F3) rather than its predecessor. (Rosen, 1990, p. 332),e.g., tells us that “the fictionalist’s parasitic proposal is. . . the schema: p iff accord-ing to [The Fiction], [pT].” The point is then that the supervaluational fictionalistcan accept that (F3) holds in full generality: this schema holds in a classical set-ting and in our supervaluational setting. But it’s characteristic of supervaluation-ism that “p” is not equivalent to “ ‘p’ is true”, so it comes as no surprise to learnthat (F) comes apart from (F3). Moreover, whilst the fictionalist needs to modifyher original metalinguistic schema, one central aspect of her original proposal—(F3)—is retained.

That my proposal retains (F3) also highlights that the central element in myresponse to the incompleteness problem is the idea that the semantic fictionalistshould analyze claims of incompleteness in terms of indeterminacy. That is, myrecommendation is that the fictionalist should reject the initial premise of the ar-gument from incompleteness to inconsistency. It is only if the fictionalist couplesthis idea with a standard, gap-theoretic treatment of indeterminacy that she be-comes forced to reject both her original schema (F) and the classical conceptionof truth. So my response isn’t based upon a non-classical theory of truth, but itentails it given a gap-theoretic treatment of indeterminacy.

There is, then, a final twist in the tale. Even spotting (CEM), the counterfac-tual analysis of the fictionalist’s prefix predicts the non-classical behavior of thetruth-predicate only if accommodating indeterminacy requires us to jettison clas-sical semantics. For the fictionalist is committed to truth-value gaps only if weare working with a gap-theoretic treatment of indeterminacy. This is Stalnaker’spreferred option, and arguably the standard approach to indeterminacy that canbe found in the literature. But it isn’t mandatory. It’s well known, for instance,that we can uphold classical semantics if we take indeterminacy to be rooted inour epistemic limitations.6 Moreover, there are theories of indeterminacy which

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reject epistemicism but maintain classical semantics.7 My proposed solution tothe incompleteness problem is thus available to both classically-minded fiction-alists and those more tolerant of non-classical semantics. The essential elementsin my solution are (i) that the fictionalist’s story is complete—given (CEM)—and(ii) that it is indeterminate what is true according to the story. If the fictionalistreckons that indeterminacy requires a non-classical semantics, she will need toboth reject the classical conception of truth and retreat from her original schemato my (F2). But if she held that indeterminacy doesn’t require a shift to the non-classical left, she can maintain classical semantics and her original schema (F).The incompleteness objection will be avoided either way, of course. So whilst mysolution to the problem was developed in a non-classical setting, this is not anessential feature of the solution: classically-minded fictionalists can accept it too.

4 Conclusion

The incompleteness problem threatens to show that semantic fictionalism bottomsout in incoherence. I developed a novel response to this problem, based on a pop-ular way of understanding the notion of truth according to the fictionalist’s story.This solution is familiar in the sense that it rejects the classical conception of truth.But what’s striking is that our counterfactual analysis of the fictionalist’s pre-fix predicts—given (CEM) and a gap-theoretic treatment of indeterminacy—thatthe truth-predicate will behave non-classically. The benefits of this novel solutionare that it avoids the threat of contradiction, models indeterminacy in a familiarfashion and does not require the inelegant restrictions on the fictionalist’s schemasuggested by Nolan. I therefore offer the solution as a principled and motivatedsolution to the incompleteness problem.8

Notes1 See Sider (2008) and Woodward (2010).2 One might worry that no self-respecting fictionalist can be a semantic fictionalist. For the se-

mantic fictionalist says that it’s true that are numbers and true that there are tables. But isn’t the fic-tionalist’s whole schtick that such claims are strictly and literally false? I defend semantic fictionalismfrom this worry in Woodward (2008).

3 See (Rosen, 1990, pp. 341–344). (Chihara, 1998, pp. 175–177), (Sider, 2002, p. 314), and (Fine,2003, p. 117) reiterate the problem.

4 The generalized continuum hypothesis and the axiom of choice are in the same boat.5 We’ll also have to reject the T-schema. In for a penny, in for a pound.6 Williamson (1994).7 McGee and McLaughlin (1995, 2004) argue that semantic indeterminacy is consistent with clas-

sical semantics. See Williamson (2004) for criticism. Barnes and Williams (2010) maintain a classicalsemantics and take indeterminacy to be a metaphysical phenomenon.

8 Thanks to Elizabeth Barnes, Ross Cameron, John Divers, Dominic Gregory, Bob Hale, DanielNolan, Tatjana von Solodkoff, Jason Turner, Robbie Williams, and an anonymous reference for Nous.The ideas developed in this paper were presented at the Universities of Leeds and Geneva, and at thesecond Phloxshop conference at Humboldt University, Berlin. Many thanks to my audiences on thoseoccasions.

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