A Yablovian Dilemma

Thought ISSN 2161-2234

O R I G I N A L A R T I C L E

A Yablovian Dilemma

Richard Woodward

Universitat de Barcelona

Stephen Yablo (2001) argues that traditional fictionalist strategies run into trouble due to a

mismatch between the modal status of a claim like ‘2 + 3 = 5’ and the modal status of its

fictionalist paraphrase. I argue here that Yablo is best seen as confronting the fictionalist with a

dilemma, and then go on to show how this dilemma can be resolved.

Keywords fictionalism; fictionality; truth in fiction

DOI:10.1002/tht.30

Suppose that you are a mathematical fictionalist. You think that many of the claimsof pure and applied mathematics aren’t really true because there aren’t really any suchthings as numbers, sets, and functions. But you don’t think that we should stop sayingthings like ‘‘some functions are surjective’’ or ‘‘the number of cats is perfect’’ for youare well aware of the theoretical benefits that mathematics offers. You think that we justneed to be creative when understanding what’s going on when we say such things. Andthat’s where your fictionalism comes in.

Exactly how it comes in will vary depending on the species of fictionalism thatyou favour. But though various proposals have been developed in the literature, manyfictionalists seem to agree that fictionalist paraphrases play an important role. Here’sHartry Field:

The fictionalist can say that the sense in which ‘2 + 2 = 4’ is true is pretty much thesame in which ‘Oliver Twist lived in London’ is true: the latter is true only in thesense that it is true according to a certain well-known story, and the former is trueonly in that it is true according to standard mathematics. (1989, p.3)

The theory shadowing paraphrases to which Field alludes are familiar from other cases.Gideon Rosen (1990) paraphrases the claim that there are possible worlds as: it is trueaccording to modal realism that possible worlds exist, Stuart Brock (2002) paraphrases theclaim that there are fictional characters as: it is true according to fictional realism thatfictional characters exist, and Cian Dorr (2005) paraphrases the claim that there are tablesas: it is true according to universalism that tables exist.

It isn’t difficult to see why fictionalists find these paraphrases congenial. Consider anexample from fiction. Even if there aren’t really any hobbits, there is clearly something

Correspondence to: E-amil: [email protected]

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Richard Woodward A Yablovian Dilemma

right about the claim that most hobbits live in the Shire and something wrong about theclaim that most hobbits live in Mordor. And the difference is that only the first claimis true according to The Lord of the Rings. The same point applies in the mathematicalcase. Even if there aren’t really any numbers, there is clearly something right about theclaim that Six is a perfect number and clearly something wrong with the claim that Six isa prime number. And the difference, the fictionalist thinks, is that only the first claim istrue according to standard mathematics.

Though fictionalist paraphrases are a central component of many fictionalistproposals, the terminology may be misleading since speaking of ‘paraphrases’ mightbring connotations of sameness of meaning or content. The notion I have in mind ismuch weaker. Paraphrases in my sense are just mappings from sentences to sentences,and the fictionalist faces a choice when identifying the role that her paraphrases play.Going one way, she might think the paraphrases play a merely pragmatic role: they specifythe conditions under which the paraphrased sentences are correct. Going the other way,the fictionalist might think the paraphrases play a non-pragmatic role: they specify theconditions under which the paraphrased sentences are true. (The former option seemsto be preferred by Field, the latter by Rosen and Brock.) Happily, the difference betweenthese strategies won’t matter here. It is enough that proponents of each think that theparaphrased sentences are kosher insofar as their paraphrases are true.

In this paper, I shall defend these proposals from an objection due to StephenYablo (2001).

The Yablovian objection

In cases where ‘p’ is a sentence of mathematical discourse, the fictionalist offers us thefollowing paraphrase scheme:

⌈p⌉ → ⌈

According to standard mathematics, p⌉

So notice that a sentence like ‘2 + 3 = 5’ will be paraphrased by a metafictional sentence:a sentence that is not itself fictionally true but instead states that something is fictionallytrue. Yablo (2001, p.75) calls these proposals metafictionalisms, and though I think thisterminology is misleading, I’ll stick with it here.

Onto the objection. Yablo (2001, p.76) writes:

It ordinarily strikes us that 2 + 3 is necessarily 5; it could not have been otherwise.But it could (perhaps) have been otherwise that 2 + 3 = 5 according to standardmath. For standard math could (perhaps) have been different. Certainly it is not apriori that standard math turned out the way it did. It is not a priori then thataccording to standard math, there are everywhere continuous functions that arenowhere differentiable. That there are functions like that is the kind of thingmathematicians take themselves to know a priori.

Yablo raises two problems in this passage: one about modal status (necessary vs.contingent) and one about epistemic status (a priori vs. a posteriori). In what follows,

2 Thought (2013) © 2013 Wiley Periodicals, Inc and the Northern Institute of Philosophy


I focus on the modal problem for the sake of brevity. (And the proposed solution to themodal problem can be transposed to solve the epistemic problem too.) The Yablovianallegation is, then, that fictionalists are in trouble due to a mismatch between the modalstatus of a mathematical claim and the modal status of its fictionalist paraphrase.

Faced with this objection, the metafictionalist has at least two defensive strategies.On the one hand, she might attempt to make the case that there is, despite initialappearances, no mismatch between the modal status of a mathematical claim and themodal status of its fictionalist paraphrase. On the other hand, she might grant that thereis indeed a modal mismatch but attempt to make the case that this feature is not itselftroubling.

Now, perhaps the metafictionalist will be tempted to pursue the second strategy.After all, perhaps all she wants is a license to move back and forth between claimsconcerning numbers and claims concerning fictions and we have no reason yet to thinkthat this requires more than the material equivalence of ‘2 + 3 = 5’ and its fictionalistparaphrase (i.e., that the apparent truth of ‘2 + 3 = 5’ is reflected in the genuine truth ofits paraphrase). But two considerations suggest that the metafictionalist will be on saferground if she can successfully implement the first strategy outlined above and show thatthe modal mismatch that Yablo outlines is not genuine.

Firstly, the retreat to material equivalence is unavailable to certain fictionalists. Inoted earlier that some metafictionalists appeal to fictionalist paraphrases to specify theconditions under which the claims of the target discourse are true. But if her paraphrasespreserve truth-conditional content, it would seem that the metafictionalist must hold thatmathematical claims are not only materially but modally equivalent to their fictionalistparaphrases. So the modal mismatch that Yablo outlines would, if genuine, be fatal to atleast some articulations of metafictionalism.

Secondly, the general fictionalist project is not simply to earn the right to say certainthings but also to enjoy the benefits that are often associated with doing so. It is familiar,e.g., that talking about numbers and possible worlds is beneficial insofar as it improvesour inferential lot. And it may well be that the metafictionalist needs more than materialequivalence to establish that she can enjoy these inferential benefits. For instance, thestrategy that John Divers (1999) develops to show that the inferential benefits associatedwith talking about possible worlds are available to modal fictionalists—a strategy gener-alized beyond the modal case in Woodward (2010)—explicitly requires that fictionalistparaphrases preserve not only alethic status but modal status.1 So the modal mismatchthat Yablo outlines would close this route to the benefits the fictionalist aims to deliver.

In light of the these considerations, I submit that metafictionalists will be in a strongerposition if they can show that there is, appearances notwithstanding, no mismatch inmodal status. It is this strategy that I shall pursue on their behalf.

The Yablovian dilemma

As Yablo presents it, his problem seems to admit of an obvious solution. The claimthat 2 + 3 = 5 is true according to standard mathematics admits of two readings.2 On



the former, shifty reading, the expression ‘‘standard mathematics’’ picks out (at time tand world w) whichever mathematical theory happens to be standardly accepted at thattime and world. On the latter, rigid reading, the expression ‘‘standard mathematics’’picks out (at time t and a world w) whichever theory happens to be standardly acceptedat the present time in the actual world. And Yablo clearly requires the shifty readingbecause there is no modal mismatch if ‘‘standard mathematics’’ is read rigidly. If M is thestandard mathematical theory at our time and world, it will be true at every world that2 + 3 = 5 is true according to standard mathematics given that 2 + 3 = 5 is true accordingto M. So shouldn’t the fictionalist simply insist that the rigid reading be enforced?

If only things were so simple. There is a serious modal difficulty facing the fictionalist,but Yablo misidentifies the problem. It’s worth remembering that fictionalists want tosave not only pure mathematics but applied mathematics too. But consider the contrastbetween (1) and (2):

(1) 2 + 3 = 5(2) The number of planets is eight

These two claims intuitively differ modally: it ordinarily strikes us that it is necessary that2 + 3 = 5 but contingent that the number of planets is eight. However, consider theirrespective fictionalist paraphrases:

(3) According to standard mathematics, 2 + 3 = 5(4) According to standard mathematics, the number of planets is eight

Now, (4) will sound rather odd. We don’t usually think of standard mathematics asconcerning itself with planets and other concrete whatnots. But note that the fictionalistneeds her fiction to say things about planets to save sentences like (2). And even if shecan ensure that (3) and (4) both hold, the challenge the fictionalist then faces is to ensurethat there is no modal mismatch between (1) and (3) on the one hand and (2) and (4)on the other. The necessity of (1) should be reflected in the necessity of (3) and thecontingency of (2) should be reflected in the contingency of (4).

Enforcing the rigid reading of ‘‘standard mathematics’’ means that the necessityof (1) is reflected in the necessity of (3). But if the rigid reading is enforced, there isa strong case for thinking that (4) is necessary. Given that it is presently true at theactual world that there are exactly eight planets, it should also be true according toour (present, actual) standard theory that the number of planets is eight. But the samereasons that previously allowed us to say that (3) is necessary now seem to force usto say that (4) is necessary too. The rigidifying strategy thus gives the right result inthe pure mathematical case but the wrong result in the applied mathematical case. Ofcourse, if the shifty reading were enforced in the applied case, the contingency of (2)would be reflected in the contingency of (4). But then we’d be back to the drawing boardwith (1).

Though Yablo did not present it in this way, his worry is best seen as a dilemma, andYablo’s error is that he exclusively focuses on only one horn. The dilemma is that either‘‘standard mathematics’’ is given a shifty reading or a rigid one. Grasping the first horn



seems to commit the metafictionalist to too many contingencies. Grasping the secondhorn seems to commit her to too few.

Having identified the real problem that lies behind Yablo’s worry, I shall now arguethat the problem can be solved. Solving the problem, however, will require us to get clearon the role that fictionality plays within the context of metafictionalism.

Fictionalism and fictionality

A seductive thought is that to say that p is true according to a fiction is to say that thereis a conditional connection between the relevant fiction on the one hand and p on theother. When Rosen (1990, p.344) considers the question of how we might explicate theconcept of fictionality to someone who did not already understand fictional operators,he writes:

We might begin by offering any of the following glosses: if THE FICTION were truethen p would be true; if we suppose THE FICTION, p follows; it would be impossible forTHE FICTION 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 meanwhen we use the fictionalist’s prefix.

As the relevant literature shows, many other fictionalists have agreed with thesesuggestions: Field (1989) and Divers (1999) gloss ‘‘according to THE FICTION’’ interms of the strict conditional, and Dorr (2005) and Dorr and Rosen (2002) gloss theprefix in terms of the counterfactual conditional.

To fix ideas, suppose the fictionalist explicates her prefix in terms of a strictconditional. It is crucial to this strategy that we distinguish between the explicit contentof the fictionalist’s story and its implicit content. Think of the story’s explicit content asthe conjunction of the axioms that characterize the theory upon which the fictionalist’saccount is parasitic. Then the role of the conditional is to generate extra, implicit content.Where ‘EXPLICIT’ abbreviates the explicit content, we have:

According to THE FICTION, p iff EXPLICIT ⊃ p

Again, there are questions about whether this constitutes an analysis of the conceptualcontent expressed by the fictionalist’s prefix or a weaker kind of explication, but set thesequestions aside. The simple fact is that the proposal seems obviously inadequate.

Suppose for the moment that we’re just worried about arithmetic, so that the explicitcontent of our fiction can be taken as the conjunction of the Dedekind-Peano axioms.Then in the pure mathematical case, the present proposal seems to give the right results:2 + 3 = 5 is necessitated by the Dedekind-Peano axioms, after all. But in the appliedmathematical case, the proposal seems to deliver obviously wrong results. How on earthdo the Dedekind-Peano axioms necessitate that the number of planets is eight?

In response to this question, the mathematical fictionalist should concede that theDedekind-Peano axioms do not necessitate that the number of planets is eight. But the



subsequent point is that the strict conditional account of the fictionalist’s prefix canbe rescued by enriching the explicit content of the fiction. Suppose that we add to thefiction an encyclopædia: a list of all the nominalistic facts about our world. Doing soensures that it is true according to our fiction that there are exactly eight planets. Butintuitively the axioms of our theory say that there are exactly n Fs iff the number of Fsis n. Put everything together and we can ensure that it is fictionally true that the numberof planets is eight. That the number of planets is eight is entailed by the explicit contentand the nominalistic encyclopædia.3

(Objection: the proposal overgenerates, since it entails that it is fictionally true thatwater is H2O and that London is the host city for the 2012 Olympics. But surely suchthings are not true according to standard mathematics.

Reply: the proposal ensures that these things are fictionally true and so the contentof THE FICTION outstrips the content that we might intuitively associate with standardmathematics. But insofar as the fictionalist needed to ensure that it is fictionally true thatthe number of planets is eight, she already needed a fiction that is richer than standardmathematics. When push comes to shove, the fictionalist should either change the initialterminology or stress that the fiction she works with is not standard mathematics in anyintuitive sense.)

In place of the original account of the fictionalist’s prefix, then, we have:

According to THE FICTION, p iff EXPLICIT ∧ ENCYCLOPÆDIA ⊃ p

Though this approach is superior to its predecessor, we need to tread carefully.Formulating the encyclopædia requires care. For suppose that there are only finitelymany concrete objects: exactly n. Then we don’t want to pack into the encyclopædia theclaim that there are exactly n things. For then the axioms of mathematics, together withthe encyclopædia, will entail a contradiction. To avoid this problem, the fictionalist willneed to restrict the quantifiers of the sentences in the encyclopædia by the predicate ‘isconcrete’. The encyclopædia will thus not say that there are exactly n things, but will sayinstead that there are exactly n concrete things (if that is indeed the case). The threat ofcontradiction will thereby be avoided.4

With our account of fictionality in place, we can now ask: how does all of this helpto solve the dilemma with which Yablo confronts the fictionalist?

The Yablovian dilemma resolved

Recall that Yablo’s dilemma is that the fictionalist must either give ‘THE FICTION’ a shiftyreading or a rigid reading. Grasping the first horn of this dilemma is alleged to beproblematic because it commits the fictionalist to far too many contingencies. Graspingthe second horn is alleged to be problematic because it commits her to far too few.

We now know, however, that the content of THE FICTION is determined by twofactors: its explicit content, conceived of as the conjunction of the mathematical axioms,and its encyclopædia, conceived of as a list of nominalistic truths about our world.



Accordingly, the dilemma is relocated as the key questions do not so much concern thebehaviour of the expression ‘THE FICTION’ as they concern the expressions ‘EXPLICIT’ and‘ENCYCLOPÆDIA’.

Syntactically, both ‘EXPLICIT’ and ‘ENCYCLOPÆDIA’ abbreviate sentences: the formerabbreviates the conjunction of mathematical axioms and the latter abbreviates theconjunction of nominalistic truths. There is, in this setting, a sense in which we cannotask whether these expressions are rigid designators or shifty ones: they aren’t designatorsafter all. But suppose we ask whether, at an arbitrary world w, p is true according to THE

FICTION. Evidently, the fictionalist does not want to say that this question is answeredonce we have figured out whether p is entailed by the conjunction of the axioms of ourmathematical theories and the nominalistic truths that hold at our world. In particular,there is no guarantee that the nominalistic truths that hold that our world also hold atw: for all that’s been said so far, w might be world where there are exactly seven planets.What the fictionalist wants to say, then, is that ‘ENCYCLOPÆDIA’ would have abbreviateda different conjunction of claims had the nominalistic truths been different, and thecontent of THE FICTION would have been different as a result. So rather than import intoTHE FICTION the nominalistic truths that hold at our world, we should instead importthe nominalistic truths that hold at w itself.

Where ‘ENCYCLOPÆDIA-(w)’ abbreviates the nominalistic truths that hold at w, thefictionalist should therefore hold that, at a world w, it is true according to THE FICTION

that p iff p is entailed by the conjunction of EXPLICIT and ENCYCLOPÆDIA-(w). And noticethat this immediately underwrites the contingency of claims like (4). If v is a world wherethere are less than eight planets, it will not be true at v that, according to THE FICTION,there are exactly eight planets for this latter claim will not be entailed by the conjunctionof the axioms of our mathematical theory and the nominalistic truths that hold at v. Inthis setting, there is no modal mismatch between (2) and (4): (2) is intuitively contingentand this is reflected in the contingency of its fictional paraphrase.

Moreover, though ‘ENCYCLOPÆDIA-(w)’ is shifty insofar as it abbreviates a differentconjunction of nominalistic truths at different worlds, ‘EXPLICIT’ is instead rigid insofaras it abbreviates the same conjunction of axioms at all worlds. And we also know thatpure mathematical cases like (1) are distinctive because the axioms alone entail that2 + 3 = 5. So it doesn’t matter which nominalistic claims we import into the fiction: atevery world, EXPLICIT will entail that 2 + 3 = 5 and so it will be necessary that, accordingto THE FICTION, 2 + 3 = 5. There is thus no modal mismatch between (1) and (3): (1) isintuitively necessary and this is reflected in the necessity of its fictionalist paraphrase.

We can see that the approach to fictionality presented earlier avoids the modalproblem that Yablo outlined. But which horn of the Yablovian dilemma has beengrasped? Well, it is natural to think of ‘THE FICTION’ as a name which refers to a fiction.And we have seen that at worlds where there are exactly seven planets, ‘THE FICTION’refers to a story according to which the number of planets is seven, even though at ourworld, ‘THE FICTION’ refers to a story according to which the number of planets is eight.But fictions are intuitively individuated in terms of their content, meaning that ‘THE

FICTION’ refers to distinct fictions at different worlds. So it is clear that my approach



treats ‘THE FICTION’ as a shifty designator rather than a rigid one. However, the problemthat Yablo outlined is avoided since it does not follow that it is contingent that it istrue according to THE FICTION that 2 + 3 = 5. At every world, ‘THE FICTION’ refers to afiction according to which 2 + 3 = 5. In this sense, the shifty behaviour of ‘THE FICTION’is localized and only manifests itself in applied mathematical cases.

I thus submit that once the role that fictionality plays within the context of fictionalismis understood, Yablo’s dilemma can be dissolved. But though my solution is natural andelegant, it would be disappointing if it required the fictionalist to explicate her prefixin terms of the strict conditional. After all, there are good reasons to think that thefictionalist is better off explicating her prefix in terms of a counterfactual conditional.5

Thankfully, the strategy generalizes.

The solution generalized

Suppose that the fictionalist explicates her prefix via a counterfactual conditional, asfollows:

According to the FICTION, p iff EXPLICIT →p

Let ‘p’ be the claim that the number of planets of eight. Then if we assume the standardapproach to counterfactuals, the truth of the right-hand side of this biconditionaldepends on whether or not all of the closest worlds to actuality where the axioms ofthe mathematical fiction are true are worlds where the number of planets is eight. Andthis indeed seems to be the case. But when we ask whether the right-hand side of thisbiconditional is necessary, the question is whether, for any world w, all of the closestworlds to w where EXPLICIT is true are worlds where the number of planets is eight. Andthis seems to be false. For let v be a world where there are exactly seven planets. Then allof the closest worlds to v where maths is true are worlds where the number of planetsis not eight, but seven. In this setting, the counterfactual explication of the fictionalist’sprefix predicts that it is not necessary that it is fictionally true that the number of planetsis eight.

Let ‘p’ be the claim that 2 + 3 = 5. Then the question of whether the right-hand sideof our principle is necessary turns on whether, for any world w, all of the closest worldsto w where the axioms of standard mathematics are true are worlds where 2 + 3 = 5.And this seems to be true. For the axioms entail that 2 + 3 = 5. But we have:

�(p ⊃ q

) |= �(p �→q

)

In this setting, the fact that the axioms of standard mathematics necessitate that 2 + 3 = 5entails that it is necessary that 2 + 3 = 5 would have been true had those axioms beentrue. The counterfactual explication of the fictionalist’s prefix thus predicts that it isnecessary that 2 + 3 = 5 is fictionally true.6

The fictionalist does not need to explicate her prefix in terms of a strict conditional todissolve the dilemma that Yablo outlines. Those who prefer to work with counterfactualscan solve the problem too. The choice the fictionalist faces is whether to build a shifty



element into her fiction directly by including an encyclopædia or whether to build ina shifty element indirectly by explicating her prefix via a counterfactual. Either way,however, there will be no modal mismatch between the modal statuses of her fictionalistparaphrases and the modal statuses of the paraphrased claims.

Conclusion

Yablo alleges that standard fictionalist proposals are in trouble due a mismatch betweenthe modal status of a mathematical claim and the modal status of its fictionalistparaphrase. We have seen that the fictionalist faces a dilemma when solving thisproblem, one which concerns the question of whether ‘THE FICTION’ receives a rigidinterpretation or a shifty one. But we have also seen that this question can be addressedonce we properly understand the role that fictionality plays within the context offictionalist proposals. In particular, it is crucial to remember that the content of thefictionalist’s fiction is determined by two factors: what is explicitly says and the way theworld is. And whether the fictionalist appeals to strict conditionals or counterfactualsin order to flesh this out, she can ensure that the necessity of a claim like ‘2 + 3 = 5’ isreflected in the necessity of its fictionalist paraphrase, and also that the contingency of aclaim like ‘the number of planets is eight’ is reflected in the contingency of its fictionalistparaphrase. Whichever way she goes, Yablo’s dilemma is thus resolved.

Acknowledgements

The author thanks John Divers, Anthony Everett, Daniel Nolan, Tatjana von Solodkoff,Robbie Williams, and an audience at the fourth VPMMM workshop in Barcelona.This article was written during a visit to the University of Hamburg in connectionwith the ANR-DFG Nominalizations project (SCHN 1137/2-1): many thanks to theDAAD for supporting my visit. My research on this paper was also supported by myinvolvement in the Nature of Assertion: Consequences for Relativism and Fictionalismproject (FFI2010-169049), the Vagueness and Physics, Metaphysics, and MetaMetaphysicsproject (FFI2008-06153), and the PERSP—Philosophy of Perspectival Thoughts andFacts project (CSD2009-00056). Many thanks to the DGI, MICINN, and the SpanishGovernment for supporting these projects.

Notes1 This is an implicit. Divers’ strategy requires that the fictionalist’s schema—p iff According to

the fiction, p—is necessary. Given K, it follows that its left-hand side is necessary

(contingent) iff its right-hand side is necessary (contingent) too.

2 cf. Divers (1999, p.338).

3 The appeal to an encyclopaedia is familiar from the literature on modal fictionalism: see

Rosen (1990, p.335).

4 cf. Williams (2010, pp.118–20).

5 See Woodward (2011).



6 A wrinkle. If the fictionalist takes her fiction to be impossible, she will need to reject the above

principle. (Sketch of an argument: if p is impossible, then p entails any arbitrary sentence q.

So the fictionalist who embraces the counterfactual explication of her prefix must reject the

principle on pain of accepting that her fiction is trivial.) But these issues are familiar and the

fictionalist should think very carefully before claiming that her fiction is impossible: see

Woodward (2010) for discussion.

ReferencesBrock, Stuart. ‘‘Fictionalism about Fictional Characters.’’ Nous, 36 (2002): 1–21.

Divers, John. ‘‘A Modal Fictionalist Result.’’ Nous, 33 (1999): 317–46.

Dorr, Cian. ‘‘What We Disagree About When We Disagree About Ontology,’’ in Fictionalism in

Metaphysics, edited by Mark Kalderon. Oxford: Oxford University Press, 2005, 234–86.

Dorr, Cian and Gideon Rosen. ‘‘Composition as a Fiction,’’ in The Blackwell Guide to Metaphysics,

edited by Richard M. Gale. Oxford: Basil Blackwell, 2002, 151–74.

Field, Hartry. Realism, Mathematics and Modality. Oxford: Blackwell, 1989.

Rosen, Gideon. ‘‘Modal Fictionalism.’’ Mind, 99 (1990): 327–54.

Williams, J. Robert G. ‘‘Fundamental and Derivative Truths.’’ Mind 119 (2010): 103–41.

Woodward, Richard. ‘‘Fictionalism and Inferential Safety.’’ Analysis 70 (2010): 409–17.

Woodward, Richard. 2011. ‘‘Fictionalism and Incompleteness.’’ Nous 46 (2012): 781–90.

Yablo, Stephen. ‘‘Go Figure: A Path Through Fictionalism,’’ in Midwest Studies in Philosophy

XXV: Figurative Language, edited by Peter French and Howard K. Wettstein. Oxford: Basil

Blackwell, 2001, 72–102.


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