Abstract
It is first argued that there are strong reasons for accepting borderline identity in re. Gareth Evans’ influential argument to the effect that such identity is impossible is then rehearsed, and extensions thereof are presented that would equally well show that borderline cases in re in general and indeed borderline cases in general (whether in re or not) are impossible. The naive theory of vagueness (holding that there is no sharp boundary between positive and negative cases of application of a vague predicate) and its accompanying tolerant logics (which are not unrestrictedly transitive), developed in earlier work by the author, are then introduced. Two specific tolerant logics, basically differing on their treatment of definiteness, are offered. Generally, in both logics, Evans’ argument fails because ‘It is definite that P’ is in some sense not weaker than ‘P’. More specifically, in the first tolerant logic, ‘It is definite that x is identical with x’ is in the relevant sense not weaker than ‘x is identical with x’, and Evans’ argument fails because it illegitimately suppresses a logical truth from a valid argument; in the second tolerant logic, ‘It is definite that x is identical with y’ is in the relevant sense not weaker than ‘x is identical with y’, and Evans’ argument fails because it implicitly appeals to principles that are untenable in a naive theory adopting a tolerant logic.
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Notes
- 1.
Henceforth, this or similar qualifications on the values of ‘i’ will be implicitly understood.
- 2.
Throughout, I follow the established practice in the literature of using ‘It is definite that P’ and its like in such way that ‘It is borderline whether P’ is strongly equivalent with ‘It is neither definite that P nor definite that it is not the case that P’ (e.g. so strongly as to guarantee that they are fully intersubstitutable even in logics—which will be prominent in this paper—in which mere logical equivalence does not guarantee that).
- 3.
The text has in effect just moved from a claim of the form ‘It is not the case that, for every x, P’ to a claim of the form ‘For some x, it is not the case that P’. That move is notoriously intuitionistically invalid and admittedly less than self-evident in contexts in which reference is made to “unsurveyable” totalities like soritical series. However, for the purposes of this paper, the move will be taken as granted (see Zardini 2013c for some discussion).
- 4.
As the phrase ‘wholly present’ indicates, I’m assuming a plausible three-dimensionalist view of persistence of substances across time (according to which, roughly, substances are typically present at different times and may be so present without different parts of them existing at different times). But, contrary to what some authors seem to think (see e.g. Noonan 1982), borderline identity in re (henceforth, simply ‘borderline identity’) cannot be avoided by simply switching to a four-dimensionalist view (e.g. saying that, although it is definite that ‘Greg l ’ refers to a substance having as part the temporal slice present at t l , it is borderline whether it refers to a substance having also as part the temporal slice present at t 0 or to a substance having also as part the temporal slice present at t k ). For an analogous example could be given in modal rather than temporal terms, and few of us would be tempted by the idea—discussed e.g. by Weatherson (2013)—that a typical substance is not wholly present in all the worlds in which it is present (nor would many of us be tempted by the idea—defended e.g. by Lewis 1986—that, in a fundamental sense, a typical substance is present in only one world). In fact, given the plausible assumptions that there could be extended simples (as argued e.g. by Markosian 1998) and that these could have fuzzy boundaries, an analogous example could be given in spatial rather than modal or temporal terms (contrary to the suggestion of Williamson 1994, pp. 255–256, to the effect that fuzzy spatial boundaries—which are certainly one of the prominent ways in which objects themselves can be thought to be vague—do not imply borderline identity). Notice that the thrust of these points consists in foreclosing the existence of (temporal, modal, spatial) parts that would give rise to several equally good candidates for being the referent of the relevant expression. However, the foe of borderline identity might try to conjure up such candidates in other ways. For example, in the case of Greg0, she might claim that there are at least two three-dimensional substances, Greg′ and Greg″, such that, while Greg′ has ceased to exist at t l , Greg″ still exists at t l . She might then hold that it is borderline whether Greg is Greg′ or Greg″, so that there is nothing ‘Greg’ definitely refers to. Unfortunately for the foe of borderline identity, the metaphysics required by this non-mereological move—implying that there are indefinitely many human beings, with slightly different persistence conditions, all of which are spatially co-located at t 0 and collectively undergo a dramatic mass death as Greg’s transformation unfolds—is still in these respects wildly implausible (notice that some of the problematic features just alluded to are shared by the mereological move, which can however at least provide a ready explanation of the different persistence conditions and of the spatial co-location at a time). I’m grateful to Aurélien Darbellay, Dan López de Sa and Giovanni Merlo for urging clarifications of an earlier version of this footnote.
- 5.
For which we could also adduce the additional consideration that, extremely plausibly, there is something (i.e. Greg) that is definitely Greg, from which it follows that there is something that ‘Greg’ definitely refers to (since it is definite that being Greg implies being referred to by ‘Greg’), and so that there is something that ‘Greg0’ definitely refers to (since it is definite that being referred to by ‘Greg’ implies being referred to by ‘Greg0’).
- 6.
It is common to think that such candidates are easily available. For example, in the case of ‘baldness0’, it is common to think that such candidates could be: the property of having at most l hairs on one’s scalp, the property of having at most l + 1 hairs on one’s scalp, the property of having at most l + 2 hairs on one’s scalp etc. Unfortunately, this common thought forgets that the stuff about hairs on one’s scalp is just a useful oversimplification of what baldness consists in: it is definite that a man with l hairs on his scalp that are however much thicker than normal and uniformly distributed so as to cover the whole of his scalp with a bushy mane is not bald, and so, after all, the simple property of having at most l hairs on one’s scalp is not a good candidate for being the referent of ‘baldness0’. And, as in so many other cases of conceptual analysis, the prospects of coming up with a series of more complex precise properties strong enough as to rule out all definite cases of non-baldness and at the same time weak enough as to rule in all definite cases of baldness are bleak. Even setting this aside, points similar to those raised at the end of Footnote 4 apply here: it is metaphysically wildly implausible to think that there are indefinitely many trichological disorders, with slightly different exemplification conditions, all of which conspire to affect poor b 0.
- 7.
For which we could also adduce the additional consideration that, extremely plausibly, there is something (i.e. baldness) that is definitely baldness, from which it follows that there is something that ‘baldness’ definitely refers to (since it is definite that being baldness implies being referred to by ‘baldness’), and so that there is something that ‘baldness0’ definitely refers to (since it is definite that being referred to by ‘baldness’ implies being referred to by ‘baldness0’).
- 8.
A related argument is offered in Salmon (2005), pp. 243–246 (whose first edition is from 1982), which however also requires some basic principles from the theory of ordered pairs. Mutatis mutandis, all the discussion to follow applies equally well to Salmon’s argument.
- 9.
Contrary to other discussions of the same topic, ours is conducted having in view as paradigmatic candidate cases of borderline identity mainly cases of cross-temporal and cross-modal identity (see Sect. 16.1 and in particular Footnote 4). But, as is well-known, in such cases indiscernibility of identicals has to be handled with some care: for example, we don’t want to conclude from the fact that in a world Greg is German and in another world Greg is not German that Greg (in the former world) is distinct from Greg (in the latter world). The properties discernibility across which entails cross-temporal and cross-modal distinctness are naturally thought of as properties exemplified in an atemporal and amodal way, like the property of being human, the property of being concrete and the property of being identical with Greg. We do want to conclude from the fact that Greg is human and a dumpling is not human that Greg (at least in any world in which he exists) is distinct from the dumpling (at least in any world in which it exists), no matter precisely in which worlds they happen to exemplify these properties (e.g. Greg may not be human in worlds in which he does not exist). Since the property of being definitely identical with Greg0 is arguably one of the properties that are exemplifiable in an atemporal and amodal way, the application of indiscernibility of identicals required by argument E in cases of borderline cross-temporal or cross-modal identity is legitimate. I’m grateful to Aurélien Darbellay for raising this issue.
- 10.
Henceforth, ‘location’ and its relatives will be understood as ‘partial location’ and its relatives.
- 11.
To elaborate a bit, if x is part of y, y is identical with y + x. But, if it is definite that x is at p, surely it is definite that y + x is at p. That is not only intuitively compelling; if the logic of definiteness is as strong as K, it follows from the uncontroversial fact that it is definite that, if x is at p, y + x is at p. An application of indiscernibility of identicals that is as legitimate as the one made in argument E (since it only involves expressions—‘y’ and ‘y + x’—such that there is something they definitely refer to) yields then the desired conclusion.
- 12.
It’s worth mentioning that there is a relative of argument F which even more closely resembles argument E and which is thereby more general (in that it does away with talk of location). That relative can be got from argument E by replacing the formulas x = x and \(x = y\) with \(x \trianglelefteq x\) and \(x \trianglelefteq y\) respectively and by replacing the principles of reflexivity of identity and indiscernibility of identicals with reflexivity of parthood and monotonicity of definite parthood over parthood (if x is part of y and it is definite that z is part of x, then it is definite that z is part of y) respectively. Notice that the latter principle, although it may initially come across as slightly less plausible than (MDPLP), can be supported in a way analogous to how (MDPLP) has been supported in Footnote 11.
- 13.
(MDPLE) can be supported in a way analogous to how (MDPLP) has been supported in Footnote 11. Letting pla(x) be the sum of the places at which x is located (and assuming that there is something ‘pla’ definitely refers to), if x exemplifies y, pla(y) is identical with \(\mathrm{pla}(y) +\mathrm{ pla}(x)\). But, if it is definite that p is part of pla(x) (which is tantamount to its being definite that x is at p), surely it is definite that p is part of pla(y) + pla(x). That is not only intuitively compelling; if the logic of definiteness is as strong as K, it follows from the uncontroversial fact that it is definite that, if p is part of pla(x), p is part of pla(y) + pla(x). An application of indiscernibility of identicals that is as legitimate as the one made in argument E (since it only involves expressions—‘pla(y)’ and ‘pla(y) + pla(x)’—such that there is something they definitely refer to) yields then that it is definite that p is part of pla(y) (which is tantamount to the desired conclusion).
- 14.
It’s worth mentioning that there is a relative of the argument considered which even more closely resembles argument E and which is thereby more general (in that it does away with talk of location). That relative employs plural talk and assumes that there are some things ‘the bald people’ definitely refers to. (Notice that this is a fair assumption given that, at this point in the text, we’re working under the hypothesis that there are borderline cases in re: if there is something ‘baldness’ definitely refers to, the things exemplifying it surely are the things ‘the bald people’ definitely refers to. Notice also that, independently of these issues, the arguments to follow can be reworked as arguments targeting at least borderline being-some-of in re.) Letting ⊑ and bb express being-some-of and the bald people respectively, the argument in question can be got from argument E by replacing the formulas x = x and x = y with xx ⊑ xx and xx ⊑ bb respectively and by replacing the principles of reflexivity of identity and indiscernibility of identicals with reflexivity of being-some-of and monotonicity of definite being-some-of over being-some-of (if the xx are some of the yy and it is definite that the zz are some of the xx, then it is definite that the zz are some of the yy) respectively. Notice that the latter principle, although it may initially come across as slightly less plausible than (MDPLP) (or (MDPLE)), can be supported in a way analogous to how (MDPLP) has been supported in Footnote 11. Indeed, letting \(\doteqdot \) and ll express plural identity and the plurality formed by the bald people together with b l respectively, taking argument E, replacing the formulas x = x and x = y with \(ll \doteqdot ll\) and \(ll \doteqdot bb\) respectively and replacing the principles of reflexivity of identity and indiscernibility of identicals with reflexivity of plural identity and indiscernibility of identical pluralities respectively, we obtain \(\neg \mathcal{D}(ll \doteqdot bb) \vdash \neg (ll \doteqdot bb)\), which is tantamount to \(\neg \mathcal{D}Bb_{l} \vdash \neg Bb_{l}\) (since \(ll \doteqdot bb\) is tantamount to Bb l ).
- 15.
Throughout, I use square brackets to disambiguate constituent structure.
- 16.
\(\mathrm{val}_{\mathfrak{M},\mathrm{ass}}\) is the model- and assignment-relative valuation function that will be defined below in the text in terms of \(\mathfrak{M}\) (and in particular in terms of \(\mathrm{id_{\mathfrak{M}}}\) ). The circularity is not vicious as we’re here only constraining rather than explicitly defining \(\mathrm{id_{\mathfrak{M}}}\). It is of course not guaranteed that such constraint is satisfiable. Models in which it is satisfied will be sketched in Theorems 5 and 6.
- 17.
In turn, we can assume that disjunction, implication and particular quantification are defined in the usual way.
- 18.
(N\({}^{\neg,\mathcal{D}}\) ) was more informally labelled ‘simple definitisation’ in Sect. 16.2.
- 19.
Notice that, with classical logic as background logic for the modal logic, the list of modal principles given in the text is multiply redundant. But this need no longer be so if a non-classical logic is used as background logic for the modal logic, in particular if the non-classical logic in question is in the relevant respects so weak as V 0 is. Notice also that, with the exception of (N \({}^{\neg,\mathcal{D}}\) ), the listed principles will not be directly concerned by our discussion. However, the validity of principles (C) and (N \({}^{\neg,\mathcal{D}}\) ) serves to mark the extent to which definiteness is still closed under logical consequence in V 0 and its like (see the discussion in Sect. 16.6 ), while the validity of principles (D)–(5) serves to show that Evans’ argument fails in V 0 and its like even under very strong assumptions about definiteness (setting aside that, as I’ve noted in Sect. 16.2 , at least (5) is widely rejected because of higher-order vagueness).
- 20.
(EXC) and (EXH) were more neutrally labelled ‘¬-L’ and ‘¬-R’ respectively in Sect. 16.2.
- 21.
A property is non-composite iff it is neither conjunctive nor disjunctive. Arguably, a composite property is exemplified in virtue of some non-composite properties being exemplified and of logical facts about conjunction and disjunction. That in turn plausibly suggests that, if identicals are indiscernible across composite properties, they are such in virtue of their being indiscernible across non-composite properties and of logical facts about conjunction and disjunction. But, for reasons I’ll partially adumbrate in Sect. 16.6 , the required logical facts about conjunction and disjunction do not obtain in V 0 and its like and in fact cannot possibly obtain in any naive theory of vagueness adopting as tolerant logic V 0 or one of its like. Therefore, (II) is arguably the proper formulation of the principle of indiscernibility of identicals in V 0 and its like. Even more strongly, for reasons I cannot go into in this paper, identicals cannot possibly be indiscernible across composite properties in any naive theory adopting as tolerant logic V 0 or one of its like, and, on a natural way of making metaphysical sense of any such theory, this is so because indiscernibility across composite properties may concern the same object in different circumstances (and it is uncontroversial that the same object may exemplify different properties in different circumstances, see Footnote 9 ).
- 22.
With the usual proviso that τ be free for σ in \(\varphi\) .
- 23.
(R) and (II) were more informally labelled ‘reflexivity’ and ‘indiscernibility of identicals’ respectively in Sect. 16.2 .
- 24.
Many theories of borderline identity block Evans’ argument by rejecting (DD) while preserving (II), claiming that (DD) is only a degenerated principle resulting from the compelling (II) plus questionable assumptions about negation like (EXC) and (EXH) (see e.g. Parsons 1987, pp. 9–11). However, contrary to such assessment, (DD) strikes me as equally compelling as (II): if x and y are discernible with respect to a property, how could they fail to be distinct? Moreover, the move in question does not apply at least to the semantic-equivalence extension of Evans’ argument developed in Sect. 16.2 .
- 25.
\((\mathrm{{TR}}^{\vdash })\) was more informally labelled ‘transitivity’ in Sect. 16.2 .
- 26.
\((\mathrm{{TR}}^{\vdash })\) is a version of transitivity for logical consequence with a somewhat intermediate strength. It is strong in that, for example, it allows one to apply transitivity in the presence of side premises and conclusions (contrast with the principle saying that, if \(\varphi \vdash \psi\) and \(\psi \vdash \chi\) hold, then \(\varphi \vdash \chi\) holds). It is weak in that, for example, it does not allow one to apply transitivity in order to dispense with intermediate premises taken distributively rather than collectively (contrast with the principle saying that, if \(\Xi \vdash \Lambda,\Theta \) holds and, for every \(\varphi \in \Theta \) , \(\Delta,\varphi \vdash \Gamma \) holds, then \(\Delta,\Xi \vdash \Lambda,\Gamma \) holds). In fact, the naive theory of vagueness requires failures of transitivity even in the absence of side premises and conclusions and, unsurprisingly, even that weak version of transitivity fails in V 0 and its like (see Weir 2005 for a different family of non-transitive logics in which such weak version of transitivity is preserved). However, argument E requires transitivity in the presence of side premises , and so we focus on \((\mathrm{{TR}}^{\vdash })\) .
- 27.
Notice that the law of excluded middle does hold in V 0 and its like, being a straightforward consequence of (EXH).
- 28.
Henceforth, ‘definitelyi’ or its like is the result of concatenating the empty string with i occurrences of ‘definitely’ or its like.
- 29.
It is not unusual to admit that the analogue of this can happen in a non-deductive case. Consider, for example, a non-deductive consequence relation \(\Vvdash \) relativised to a biographer’s evidence (where \(\Gamma \Vvdash \varphi\) holds iff, given the biographer’s evidence, the state of information represented by \(\Gamma \) lends credibility to \(\varphi\) ), with the biographer’s evidence being good enough as to lend credibility both to ‘On 25/12/1915, Kafka had ham for dinner’ and to ‘On 26/12/1915, Kafka had sausages for dinner’. Then, while we have that both \(\varnothing \Vvdash \) ‘On 25/12/1915, Kafka had ham for dinner’ and \(\varnothing \Vvdash \) ‘On 26/12/1915, Kafka had sausages for dinner’ hold, and presumably also that ‘On 25/12/1915, Kafka had ham for dinner’, ‘On 26/12/1915, Kafka had sausages for dinner’ \(\Vvdash \) ‘On 25/12/1915, Kafka had ham for dinner and, on 26/12/1915, Kafka had sausages for dinner’ holds, it is a familiar point that ‘On 25/12/1915, Kafka had ham for dinner’ and ‘On 26/12/1915, Kafka had sausages for dinner’ cannot be suppressed in the last argument on pain of giving rise to the preface paradox (see Makinson 1965). What is distinctive of the position under discussion is to maintain that the analogue of this can happen in the deductive case (I explore further the illuminating similarities between tolerant logics and certain non-deductive consequence relations in Zardini 2013b).
- 30.
\((\mathrm{{T}}^{\vdash })\) was more informally labelled ‘fact entails definite fact’ in Sect. 16.2. The proper converse of \((\mathrm{{T}}^{\subset })\) is of course:
- \((\mathrm{{T}}^{\supset })\) :
-
\(\varnothing \vdash \varphi \supset \mathcal{D}\varphi\) holds,
which is typically rejected even by logics of definiteness that accept \((\mathrm{{T}}^{\vdash })\) (as, by contraposition on material implication and a version of modus ponens, it yields the contrapositive of \((\mathrm{{T}}^{\vdash })\) ). Given the usual definition of implication, in \(\mathbf{V_{0}}\) and its like \((\mathrm{{T}}^{\supset })\) follows from \((\mathrm{{T}}^{\vdash })\) by (EXH) and the properties of disjunction in V 0 and its like. I think that this is as it should it be, as (T ⊃ ) no less than \((\mathrm{{T}}^{\vdash })\) is arguably integral to the appealing conception of definiteness that I’m about to introduce in the text. As for the alleged bad consequence of \((\mathrm{{T}}^{\supset })\) (\(\neg \mathcal{D}\varphi \vdash \neg \varphi\) ), I’m going to argue in this section that, in a naive theory of vagueness adopting a tolerant logic, it is not such.
- 31.
I’m well aware that (EXH) is rejected by many non-classical logics of vagueness (and that even (EXC) is rejected by some of them). But, setting aside the question of the meaning and logic of the negative constructions ordinarily used in natural language, I think that we clearly do have a notion of a sentence \(\varphi\) failing to hold that, on the face of it, is exclusive and exhaustive with respect to \(\varphi\). V 0 is a theory, among other things, of that arguably theoretically fundamental notion and of the borderline cases that arise with respect to it.
- 32.
I’ve briefly argued in the text that, for no i, one has non-inferential reasons for accepting \(\mathcal{B}(g_{0} = g_{i})\). If one does not have inferential reasons either for accepting it, that is obviously sufficient for undercutting the letter of the second reason against the conclusion of argument E, quite independently of all the previous stuff in the text about (CCTLC). But, even assuming that one does not have inferential reasons either for accepting \(\mathcal{B}(g_{0} = g_{i})\), the richness of articulation afforded by the normative theory which accompanies a naive theory of vagueness adopting a tolerant logic and which leads to the rejection of (CCTLC) is needed in order to undercut subtler renderings of the spirit of the second reason against the conclusion of argument E. For example, given that (CCTLC) has deliberately been formulated in terms of the broader notion of thinking (which encompasses also attitudes other than acceptance), the foe of the conclusion of argument E could observe that, in V 0 and its like, given the obvious additional constraints on the domain \(\exists x\mathcal{B}(g_{0} = x) \vdash \mathcal{B}(g_{0} = g_{0}),\mathcal{B}(g_{0} = g_{1}),\mathcal{B}(g_{0} = g_{2})\ldots,\mathcal{B}(g_{0} = g_{999,999})\) holds and that there should be no objection in a naive theory adopting a tolerant logic to the normative principle concerning multiple-conclusion arguments saying that, if one has non-inferential reasons to accept all the premises of a valid argument, one has inferential reasons for conditionally accepting each of its conclusions (i.e. conditionally on the rejection for non-inferential reasons of all the other conclusions). Letting conditional acceptance be the relevant mode of thinking, the foe of the conclusion of argument E could then use the conclusion of argument E and (CCTLC) to infer that, for every i, one is committed to accepting conditionally g 0 ≠ g i (i.e., conditionally, for every other \(j\), on the rejection for non-inferential reasons of \(\mathcal{B}(g_{0} = g_{j})\) ). These are already in themselves rebarbative commitments. And, if the logic of definiteness also yields \(x\neq y,\neg \mathcal{D}(x\neq y) \vdash \varnothing \) (see Sect. 16.2), the foe of the conclusion of argument E could turn those commitments into downright unacceptable ones by inferring that, for every i, one is committed to accepting conditionally the jointly inconsistent g 0 ≠ g i and \(\neg \mathcal{D}(g_{0}\neq g_{i})\) (again, conditionally, for every other j, on the rejection for non-inferential reasons of \(\mathcal{B}(g_{0} = g_{j})\) ), from which it follows that, unacceptably, all options open to one are unacceptable options on which one is committed to accepting two jointly inconsistent sentences. This more sophisticated argument is effectively blocked by rejecting (CCTLC) on the grounds adduced in the text.
- 33.
(M) may be rescued from the problems discussed in the text if one were to go tolerant in reasoning about \(\vdash \) itself. But, on a natural way of implementing it, this move would equally block as tolerantly invalid the reasoning based on the conclusion of argument E and (M) to the effect that nothing is a definite borderline case of identity.
- 34.
All this implies that not only does the conclusion of argument E hold in V 1 but also argument E itself is valid.
- 35.
Thus, generally, both V 0 and V 1 think that the problem with Evans’ argument consists in the fact that \(\mathcal{D}\varphi\) is not weaker** than \(\varphi\). The particular instance on which V 0 focusses is \(\mathcal{D}(x = x)\) not being weaker** than x = x while, contrary to what V 0 thinks happens in other instances, being weaker* than it (with the consequence that the former cannot be suppressed in a valid argument although it is a logical truth). On the contrary, V 1 thinks that \(\mathcal{D}(x = x)\) is not only weaker* but also weaker** than x = x (with the consequence that the former can be suppressed in a valid argument) and focusses instead on \(\mathcal{D}(x = y)\) not being weaker** than x = y while being weaker* than it. Thanks to Krzysztof Posłajko for questions that led to this observation.
- 36.
Nor does it imply a sort of “supervaluationist thingy” in which, although it is asserted that it is definite that something among a finite number n of things is thus and so, it is also asserted of each of these things that it is not definite that it is thus and so (a pattern that, under minimal assumptions, (T\({}^{\vdash }\) ), (EXC) and (EXH), would lead to a straightforward inconsistency). For example, as we’ve seen, the position under discussion asserts not only \(\mathcal{D}\exists x\mathcal{B}Hx\) but also \(\mathcal{D}\exists x\mathcal{D}\mathcal{B}Hx\), and, for many is, it does not assert \(\neg \mathcal{D}\mathcal{B}Hg_{i}\) (although for no i does it assert \(\mathcal{D}\mathcal{B}Hg_{i}\) ).
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Acknowledgements
An earlier version of the material in this paper has been presented in 2013 at the PERSP Metaphysics Seminar (University of Barcelona). I’m grateful to the members of the seminar and to an anonymous referee for very stimulating discussions and comments. I’m also grateful to the editors Ken Akiba and Ali Abasnezhad for inviting me to contribute to this volume and for their support and patience throughout the process. During the writing of the paper, I have benefitted from the FP7 Marie Curie Intra-European Research Fellowship 301493, as well as from partial funds from the project FFI2011-25626 of the Spanish Ministry of Science and Innovation on Reference, Self-Reference and Empirical Data, from the project FFI2012-35026 of the Spanish Ministry of Economy and Competition on The Makings of Truth: Nature, Extent, and Applications of Truthmaking, from the project CONSOLIDER-INGENIO 2010 CSD2009-00056 of the Spanish Ministry of Science and Innovation on Philosophy of Perspectival Thoughts and Facts (PERSP) and from the FP7 Marie Curie Initial Training Network 238128 on Perspectival Thoughts and Facts (PETAF).
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Zardini, E. (2014). Evans Tolerated. In: Akiba, K., Abasnezhad, A. (eds) Vague Objects and Vague Identity. Logic, Epistemology, and the Unity of Science, vol 33. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7978-5_16
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DOI: https://doi.org/10.1007/978-94-007-7978-5_16
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