The Parallelism Between Evans's Vague Object Argument and Kripke's Theorem of the Necessity of Identity


  • Ken Akiba  ''Vagueness as a Modality,'' in: The Philosophical Quarterly, vol. 50, no. 200, 2000,
  • Ken Akiba  ''Identity Is Simple,'' in: American Philosophical Quarterly, vol. 37, no. 4, 2000,
  • Ken Akiba  ''Vagueness in the World,'' in: Noûs, vol. 38, no. 3, 2004,
  • Ken Akiba  ''Introduction,'' in Ken Akiba and Ali Abasnezhad eds., Vague Objects and Vague Identity: New Essays on Ontic Vagueness, Springer, Logic, Epistemology, and the Unity of Science, vol. 33, 2014,
  • Brian Garrett  ''Some Comments on Evans's Proof,'' in Ken Akiba and Ali Abasnezhad eds., Vague Objects and Vague Identity: New Essays on Ontic Vagueness, Springer, Logic, Epistemology, and the Unity of Science, vol. 33, 2014.


さて、私が今まで気が付かなかったその意外なこととは、Gareth Evans さんによる vague objects の存在を否定する証明の一部が、Saul Kripke さんによる同一性の必然性を述べる有名な定理 (∀x∀y( x = y ⊃ □( x = y ))) の対偶になっている、ということです。そして、実際その通り対偶になっているならば、かつ (Quine さんの反論に逆らって) Kripke さんのその定理が本当に成り立っているとするならば、その時、Evans さんの証明も、反駁されるべきものでなく、妥当だと認められることになる、ということのようです。これは個人的に意外です。

まず、Evans さんの、その証明を掲げます。典拠先は、

  • Gareth Evans  ''Can There Be Vague Objects?'' in: Analysis, vol. 38, no. 4, 1978, p. 208, reprinted in his Collected Papers, Oxford University Press, 1985, p. 176, and in Rosanna Keefe and Peter Smith eds., Vagueness: A Reader, The MIT Press, 1996, p. 317,

です。Analysis から原文の一部と、その逐語的な試訳/私訳を提示致します。Evans さんの文章における註はすべて引用者、訳者によるものです。誤訳しておりましたら、誠にすみません。あくまで英語原文を参照していただき、もしもわかりにくいところがございましたら、その時に限り和訳を参照ください。誤訳が含まれている可能性が大ですので、絶対に参考程度で済ませてくださいますようよろしくお願い申し上げます。

 Let 'a' and 'b' be singular terms such that the sentence 'a = b' is of indeterminate truth value, and let us allow for the expression of the idea of indeterminacy by the sentential operator '▽'.
Then we have:

   (1)  ▽( a = b )

(1) reports a fact about b which we may express by ascribing to it the property 'x[▽( x = a )]'*1:

   (2)  x[▽( x = a )]b.

But we have:

   (3)  〜▽( a = a )

and hence:

   (4)  〜x[▽( x = a )]a.

But by Leibniz's Law, we may derive from (2) and (4):

   (5)  〜( a = b )

contradicting the assumption, with which we began, that the identity statemtent 'a = b' is of indeterminate truth value.

 'a' と 'b' を、文 'a = b' が不確定な真理値を持つような単称名とし、そして文演算子 '▽' によって、不確定性という観念の表現を想定することにしよう。

     (1)  ▽( a = b ).

(1) は b に関するある事実を伝えており、b に性質 'x[▽( x = a )]'*2 を帰属させることによって、この事実を伝える。つまり

     (2)  x[▽( x = a )]b.


     (3)  〜▽( a = a )


     (4)  〜x[▽( x = a )]a.

だが、ライプニッツの法則により、我々は (2) と (4) から次を引き出すことができて、すなわち、

     (5)  〜( a = b )

これは我々が最初に論証を始めた際の前提、同一性言明 'a = b' は不確定な真理値を持つ、ということに矛盾する。


 What people have become aware of in the last dozen years or so, however, is that the argument (1) through (5) need not be blocked at all. We can accept the argument - or, more specifically, the argument from one implicit disjunct ¬Δ( a = b ) of (1) to (5) - as a valid argument, essentially the contrapositive of Kripke's (1971)[''Identity and Necessity''] argument for the necessity of identity, a = b → □( a = b ), where □ is replaced with Δ. The upshot is simply that identical things are determinately identical, or that there are no identical things that are only indeterminately identical. Evans somehow thought that this is contradictory, but it simply is not. Accepting it does not affect the main theses of onticism at all.*3


なお、この引用文の末尾にある言葉 'onticism' について、秋葉先生の説明を上げておきます。

[O]nticism […] holds that there are objectively vague facts (or states of affairs) in the world. That John is a tall man is such a state of affairs. The vague property being a tall man exists in reality, and Michael (definitely) possesses that property, while John neither definitely does nor definitely does not possess it.
 As is already apparent from the above brief charaterizations, onticism […] embraces a larger ontology that includes vague things (vague facts, states of affairs, propositions, properties, individuals, etc.) along with crisp things. Onticism holds that the property being a tall man exists as a vague property, whereas semanticism [which is an opponent of onticism] maintains that it does not exist, […]*4

One apparent species of ontic vagueness [which is accepted by onticists] is the existence of vague objects. Vague objects, as we use the term here, are individual physical objects whose spatiotemporal boundaries are vague. Many - indeed, probably most - ordinary, so-called ''medium-sized,'' physical objects around us, including rather small objects such as cells and molecules and rather large objects such as stars and planets, are vague objects in this sense.*5

Evans さんの論証が Kripke さんの同一性の必然性定理の対偶になっているという秋葉先生のご指摘を、もう少し追加しておきます。そして、そのように対偶になっていることのいみとは何なのかについて、秋葉先生の言葉を記しておきます。

 Many advocates of vague objects […] furthermore contend that the identity of vague objects can itself be vague. More precisely, they contend that the identity statement ''a is identical to b'' can be indeterminate in truth value if ''a'' or ''b'' is a name of a vague object.
 Certainly one can be sympathetic to the view that there are vague objects in the world, but disagree with the claim that the identity of such objects can be vague or indeterninate. A version of Kripke's proof shows that the latter claim [i.e., the claim that the identity of vague objects can be vague] cannot be correct. Just repalce ''necessarily'' in Kripke's original proof with ''determinately.'' Suppose that a is identical to b. a is determinately identical to a. So, by Leibniz's Law, b also must be determinately identical to a. Thus the identity between a and b cannot be indeterminate.*6

Evans 論証が Kripke さんによる同一性の必然性定理の対偶になっていることのいみの一つ目は、同一性自身は曖昧ではあり得ない、ということのようです。

 As the title of Evans's paper ''Can there be vague objects?'' suggests, Evans took his argument to show that vague or indeterminate objects are impossible, but it does not show that; it only shows that indeterminate identity is impossible. Note that Evans's argument is in fact just the contrapositive of the Kripkean proof discussed earlier. Kripke's proof is completely general and does not concern what kinds of objects there are; consequently, it does not demonstrate the impossibility of vague objects. Neither can Evans's proof show that vague objects are impossible.*7

そして、Evans 論証が Kripke さんによる同一性の必然性定理の対偶になっていることのいみの二つ目は、Kripke さんの問題の定理の変項が取る対象は、曖昧な対象であれ、曖昧でない対象であれ、何であれ、特に制限は設けられていないので、この定理からは曖昧な対象の非存在を立証できない、ということのようです。



*1:For convenience, we replace the original accent circonflexe attached over x by overline. The same holds for the following similar cases.

*2:ここで最初の 'x' の上に横棒がありますが、英語原文では '^' が置かれています。便宜上、ここでは横棒で代用します。以下同様です。

*3:Akiba, ''Introduction,'' p. 9.

*4:Akiba, ''Introduction,'' p. 3.

*5:Akiba, ''Introduction,'' p. 5.

*6:Akiba, ''Identity Is Simple,'' p. 395.

*7:Akiba, ''Identity Is Simple,'' pp. 395-396.