Vagueness, Identity and Leibniz's Law
Williamson (Timothy)
Source: Bottani - Individuals, Essence and Identity, 2002
Paper - Abstract

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Author’s Introduction

  1. When we reflect on the individuation1 of objects, we soon encounter hard cases. We are presented with pairs of objects the grounds for classifying which as identical look no better and no worse than the grounds for classifying them as distinct; we have no idea how to decide the question. For instance, the diachronic identify of the ship of Theseus2 was already a matter of philosophical debate when Plutarch wrote, and has still not been settled:
      The thirty-oared galley in which Theseus3 sailed with the youths and returned safely was preserved by the Athenians down to the time of Demetrius of Phalerum. At intervals, they removed the old timbers and replaced them with sound ones, so that the ship became a classic illustration for the philosophers of the disputed question of growth and change, some of them arguing that it remained the same, and others that it became a different vessel. (Plutarch, 'Life of Theseus' §23, Scott Kilvert (trans.) [I960])
    If the name 'S1' is conferred by ostension at the time of Theseus4 on the ship then present, and the name 'S2' is conferred by ostension at the time of Demetrius of Phalerum on the ship present at that later time, then it seems unclear whether S1 = S2; S1 and S2 are apparently neither clearly identical nor clearly distinct. Examples of this kind abound.
  2. Someone might describe the case thus:
      There are objects of which it is vague whether they are identical. Such vagueness concerns the objects themselves, not their names. When the name 'S1' was introduced, it was clear which ship was being pointed at; likewise when the name 'S2' was introduced. Thus there are objects (the ships S1 and S2) and a relation (identity) such that it is vague whether those objects stand in that relation. That amounts to vagueness in reality itself, not merely in our mental or linguistic representations of reality.
    Such a description contradicts the usual understanding of vagueness as located only in representations.
  3. We might doubt whether ships are perfectly representation-independent objects, for they are artefacts, individuated at least partly in accord with the intentions of their makers or users; intentions involve mental representations. But similar identity puzzles arise for natural objects, such as planets, many of which (idealists notwithstanding) would have had the same existence and nature even if beings capable of forming representations had never evolved.
  4. Cases of vague identity5 have also been claimed to occur in quantum mechanics6 (Lowe [1994]). However, no theory of vagueness has made any serious contribution to the understanding of problems in quantum mechanics7. It is not as though we understand what is physically happening in such cases but need a theory of vagueness to describe it clearly. Quantum mechanics8 in its present mystifying state has no obvious morals for the theory of vagueness.
  5. Many philosophers conceive vagueness as a kind of indeterminacy. On such a view, if it is vague whether objects are identical then it is indeterminate whether they are identical, in roughly the sense that there is no right answer to the question 'Are they identical?'. Thus reality itself would be indeterminate. That would explain our prolonged failure to resolve philosophical disputes about identity. Yet Gareth Evans [1978] and Nathan Salmon [1982] (pp.243-5), have argued formally that it cannot be indeterminate whether objects are identical. Very roughly: if it is indeterminate whether x is identical with y, then x has a property that y lacks, because it is determinate that x is self-identical, and therefore determinate whether x is identical with x, thus x is distinct from y, by Leibniz's Law9 of the indiscernibility of identicals10, so it is after all not indeterminate whether they are identical. The claim of vague identity11 is self-refuting. In response, numerous attempts have been made to construct a coherent account of vague identity12 within a framework of many-valued logic.


See "Edgington (Dorothy) - Williamson on Vagueness, Identity and Leibniz's Law" for a response.

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