Abstract
How to handle vagueness? One way is to introduce the machinery of acceptable sharpenings, and reinterpret truth as truth-in-all-sharpenings (supervaluationism) or truth-in-some-sharpenings (subvaluationism). A major selling point has been the conservativism of the resulting systems with respect to classical theoremhood and inference. Supervaluationism and subvaluationism possess interesting formal symmetries, a fact that has been used to argue for the subvaluationist approach. However, the philosophical motivation behind each is a different matter. Subvaluationism comes with a standard story (due to Stanislaw Jaśkowski) that is difficult to sign up to. In this paper, I make use of a variant of Putnam’s well-known idea of linguistic deference, and some results in voting theory, to answer this criticism of subvaluationism. The acceptability intuitions of each member of a linguistic community amount to their voting for one or more acceptable sharpenings, with truth then characterised as truth-in-a-(contextually-determined)-sufficiency-of-sharpenings. This produces a family of logical systems that are close relations of subvaluationism, share its conservatism results, yet have stronger philosophical foundations in the workings of externalist content.
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Notes
For instance, Pablo Cobreros suggests that the advocates of paraconsistency as a way of handling vagueness (including advocates of subvaluationism) ‘seem to have spent more time arguing that paraconsistent theories are at least as good as their paracomplete counterparts, than giving positive reasons to believe on a particular paraconsistent proposal’ (Cobreros 2011, p. 211).
Similar systems supplemented by a ‘definitely’ operator, however, lack this property: the deduction theorem, and so some classical deduction rules (conditional proof, reduction, proof by cases), fail. See Williamson (1994, pp. 151–152).
Hyde (2008, p. 94). Bonini et al. (1999) presents empirical work suggesting that in borderline cases subject responses tend not to exhibit overlap, while Serchuk et al. (2011) and Ripley (2011) present further empirical work suggesting the contrary. Egré et al. (2013) also find evidence of overlap behaviour in dynamic, or forced march, sorites.
Difference here isn’t the only point of difference between the dual positions; most discussion has to do with the weak paraconsistent logic that subvaluationism makes use of, and the extent to which the failure of adjunction is or is not mirrored by failings in multiple-conclusion versions of supervaluationism (Keefe 2000b, pp. 197–198; Hyde 2008, pp. 101–103; Hyde 2009). I should note that there are also paraconsistent theories of vagueness that are not subvaluationist—for instance, David Ripley has argued that vagueness should be treated within LP (Ripley 2013).
I’m indebted to an anonymous referee for pointing out the similar motivation given in Belnap (1977) for his useful 4-valued logic, tracking the states a computer database could be in when fed possibly inconsistent or gappy information, and for noting the fact that Belnap’s account appears to find more general favour than Jaskowski’s. This may be because Belnap’s paper and its editorial introduction stress the logic’s epistemic role: ‘It is not that sentences are sometimes really neither true nor false, or both true and false; it is just that sometimes the computer is told such.’ (Dunn and Epstein 1977, p. 6).
This is a familiar point in criticism of Kuhnian incommensurability claims, made for instance by Kitcher (1993, pp. 95–105).
For similar reasons, it seems unlikely that explicit markets or other social choice mechanisms will be available here.
Computational studies of flocking have shown that highly coordinated behaviour can be driven by individual level norms of this kind; work here stems from Reynolds (1987).
Perhaps having begun at the other end of the series, though it seems that the factors inducing changes of judgement are rather complicated (Egré et al. 2013).
It also retains analogous features when adapted to the election of committees (Fishburn 1981).
Thanks to Dominic Hyde for this point.
One might perhaps then want to limit the relevant votes to, for instance, those instances of usage in which the agent is abiding by the convention of being truthful in the language that in part delimits which language is in use (Lewis 1969).
This is a point I owe to an anonymous reviewer.
I take the latter to be a version of the ‘consensus theory of truth’; the view that matters of assent and dissent are completely decisive in settling the meaning of a term. As Timothy Williamson notes, such a view would rule out the possibility of general error in our dispositions to assent or dissent (Williamson 1994, pp. 206–207).
Proof: Suppose \(\hbox {A}_{1}, {\ldots }, \hbox {A}_\mathrm{k} \; \models _{\mathrm{CL}}\) B. It follows that \( \hbox {A}_{1} \& {\ldots } \& \hbox {A}_\mathrm{k} \; \models _{\mathrm{CL}}\) B, and so all admissible interpretations of a \(\hbox {V}_\mathrm{x}\) model in which \( \hbox {A}_{1} \& {\ldots } \& \hbox {A}_\mathrm{k}\) is true are interpretations in which B is true. Suppose that \( \hbox {A}_{1} \& {\ldots } \& \hbox {A}_\mathrm{k}\) is true in model M of \(\hbox {V}_\mathrm{x}\). Then there is a decisive tally at which \( \hbox {A}_{1} \& {\ldots } \& \hbox {A}_\mathrm{k}\) is true, and so a decisive tally at which B is true. Hence \( \hbox {A}_{1} \& {\ldots } \& \hbox {A}_\mathrm{k} \; \models _{\mathrm{Vx}}\) B. Conversely, suppose it’s not the case that \(\hbox {A}_{1}, {\ldots }, \hbox {A}_\mathrm{k} \;\models _{\mathrm{CL}}\) B. Then there is at least one CL model M at which \(\hbox {A}_{1}, {\ldots }, \hbox {A}_\mathrm{k}\) are true and B is false. Obviously \( \hbox {A}_{1} \& {\ldots } \& \hbox {A}_\mathrm{k}\) is also true in M. Consider a model M* for \(\hbox {V}_\mathrm{x}\) that is just like M (and so with exhaustive and exclusive positive and negative extensions for all predicates) and with arbitrary P. All admissible interpretations for M* will be copies of M* indexed to members of P; by Compulsory Voting there will be n of these, in all of which \( \hbox {A}_{1} \& {\ldots } \& \hbox {A}_\mathrm{k}\) is true and B is false. Hence there is at least one decisive tally at which \( \hbox {A}_{1} \& {\ldots } \& \hbox {A}_\mathrm{k}\) is true and B is false, so it’s not the case that \( \hbox {A}_{1} \& {\ldots } \& \hbox {A}_\mathrm{k} \;\models _{\mathrm{Vx}}\) B.
Proof: Suppose \(\models _{\mathrm{CL}}\) B. Then B is true at all classical interpretations, and so at all admissible interpretations in all models for \(\hbox {V}_\mathrm{x}\). In each model, by Compulsory Voting, there is at least one complete tally, and so at least one decisive tally at which B is true. So \(\models _{\mathrm{Vx}}\) B.
Conversely, suppose it’s not the case that \(\models _{\mathrm{CL}}\) B. Then there is at least one CL model M at which B is false. Consider a model M* for \(\hbox {V}_\mathrm{x}\) that is just like M (and so with exhaustive and exclusive positive and negative extensions for all predicates) and with arbitrary P. All admissible interpretations for M* will be copies of M* indexed to members of P; by Compulsory Voting there will be n of these. Hence there is at least one decisive tally at which B is false, so B is false in M* and it’s not the case that \(\models _{\mathrm{Vx}}\) B.
Proof: Suppose it’s not the case that \(\hbox {A}_{1}, {\ldots }, \hbox {A}_\mathrm{k} \;\models _{\mathrm{SbV}}\) B. Then there is at least one SbV model M at which \(\hbox {A}_{1} ,{\ldots }, \hbox {A}_\mathrm{k}\) are true and B is not. So there is at least one admissible interpretation of M at which \(\hbox {A}_{1}\) is true, ..., at least one admissible interpretation of M at which \(\hbox {A}_\mathrm{k}\) is true, and at none of these is B true (because B is true at no interpretation in M). Consider a model M* for \(\hbox {V}_\mathrm{x}\) that is just like M, with arbitrary P, but that admits just a copy of each of these interpretations indexed to each member of P. Hence no matter what x is, there will be a decisive tally at which \(\hbox {A}_{1}\) is true, ..., a decisive tally at which \(\hbox {A}_\mathrm{k}\) is true, and no decisive tally at which B is true, so it’s not the case that \(\hbox {A}_{1}, {\ldots }, \hbox {A}_\mathrm{k} \;\models _{\mathrm{Vx}}\) B. Conversely, suppose it’s not the case that \(\hbox {A}_{1}, {\ldots }, \hbox {A}_\mathrm{k} \;\models _{\mathrm{Vx}}\) B. Then there is at least one \(\hbox {V}_\mathrm{x}\) model M at which \(\hbox {A}_{1} ,{\ldots }, \hbox {A}_\mathrm{k}\) are true and B is not; hence there is a decisive tally at which \(\hbox {A}_{1}\) is true, ..., a decisive tally at which \(\hbox {A}_\mathrm{k}\) is true, and no decisive tally at which B is true. But then there must be at least one admissible interpretation at which \(\hbox {A}_{1}\) is true and B is not, ..., and at least one admissible interpretation at which \(\hbox {A}_\mathrm{k}\) is true and B is not. Consider a model M* for SbV that is just like M without P and with indices dropped, that admits just these interpretations. It will be a model where there is at least one admissible interpretation of M at which \(\hbox {A}_{1}\) is true, ..., at least one admissible interpretation of M at which \(\hbox {A}_\mathrm{k}\) is true, and no interpretation at which B is true, and so it’s not the case that \(\hbox {A}_{1}, {\ldots }, \hbox {A}_\mathrm{k} \; \models _{\mathrm{SbV}}\) B.
My thanks to an anonymous reviewer for making clear the need for this point here.
Other modifications to the system seem to me to have more plausibility. For instance, one might wish to weight each vote by the intensity of the acceptability intuition held by the agent (an idea suggested by Julian Lamont). This is akin to making use of preferential approval voting, a method in which each agent can distribute percentages of their vote among the candidates as they wish (79 % to A, 3 % to B, 18 % to C, say). This modification doesn’t change the conclusions drawn above; the result is still a set of systems closely akin to subvaluationism. Another way in which we could adjust matters is by getting rid of the condition of Compulsory Voting, and allowing for language users who find no sharpening of some term acceptable. If person k has a ‘null’ vote, this can be modelled by indexing k to every admissible interpretation, and then renormalising the proportions needed for a winning vote accordingly. (In approval voting, voting for everyone is much the same as voting for none.) Since the results above hold for the whole range of values of x, they will hold for the renormalised system. This is messier, but in principle it can be done if we have reason to be concerned about that limit. Finally, my examples have involved cases where it’s plausible there are countably many votes, but (as an anonymous reviewer has suggested) there seems to be no objection to extending the system to allow for uncountable cases, and so for continuous cases of vagueness [for which see Weber and Colyvan (2010)].
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Acknowledgments
I’m indebted to Mark Colyvan, Dominic Hyde and audiences at the \(4{\mathrm{th}}\) World Congress of Paraconsistency and the University of Queensland and University of Tasmania philosophy seminar series for discussion of this paper. Research on this paper was funded by an Australian Research Council Discovery Grant to Mark Colyvan and Dominic Hyde (Grant Number DP0666020).
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Chase, J.K. Voting and vagueness. Synthese 193, 2453–2468 (2016). https://doi.org/10.1007/s11229-015-0859-1
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DOI: https://doi.org/10.1007/s11229-015-0859-1