哥德爾本體論證明

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哥德爾本體論證明是數學家库尔特·哥德尔11世紀意大利僧侶聖安瑟倫對於神存在性的本體論論點整理並改進後所作的數學表達方式。聖安瑟倫後曾有17世紀莱布尼茨提出了另一個較複雜的宇宙論證版本,而這個就是哥德爾所研究並嘗試用其本體論邏輯論點去澄清的版本。

雖然哥德爾有宗教信仰,他從未發表這個證明。他在1970年代絕食而死的前幾年不斷將這個論點向身邊的朋友們展示,他去世九年後,即1987年,這論點才被出版。

哥德爾的論證證明用上了由他本人及克里普克20世紀邏輯學家所發展的模态逻辑,分開了必需的真與偶然的真。 表示必然性,而 表示可能性。證明的關鍵在於利用「神可能存在」(定理2)及神的極致性(定義1)去推導出「神必然存在」(定理4)。在S5模態邏輯系統的框架下,這項結論可謂全然有效,因此相當驚人。然而,若使用相同的邏輯推論去假設極致偉大的存有不存在,也同樣沒有任何自相予盾之處。

證明[编辑]

聖安瑟倫的論點[编辑]

11世紀意大利僧侶聖安瑟倫,其論點用最簡潔的表達如下:「God, by definition, is that than which a greater cannot be thought(i.e.). God exists in the understanding(i.e.). If God exists in the understanding, we could imagine Him to be greater by existing in reality. Therefore, God must exist.」。也就是:

:神是我們所能相像得到最偉大的存有。

:實際存在的物體比想像中的物體更偉大。

推論:神存在。

哥德爾的證明[编辑]

哥德爾的證明若以符號表達,則如下:

 解「像神特性」,為任一特性, 解 「 為正(也可作「善」或「偉大」)特性」,  解「 x 擁有  特性」, 解「必需存在」,  解 「 是 x 的本質(essence)」, 表示「必然性」,而  表示「可能性」:

語譯[编辑]

定理 1至4的證明[编辑]

證明中用到的公設[编辑]

哥德爾證明中的公設有5項:

公設 0: 在所有特性中挑出特性是可能的。哥德爾定義正特性頗不清晰:「正解作在道德美學上為正(independently of the accidental structure of the world)......It may also mean pure attribution as opposed to privation (or containing privation)." (Gödel 1995)

然後我們假設對於所有正特性,以下幾個條件正確(可被總結為「那些正特性們形成了一個超滤子」):

公設 1: 假如 φ 為正特性且 φ 導出 ψ,那 ψ 也是正。
公設 2: 假如 φ 是一個特性,那要麼φ與其邏輯非——「非φ」,有一個且只有一個為正特性。
公設 3: 「像神特性」G乃是正特性
公設 4: 若φ為正特性,則其必需為正特性。
公設 5: 必需存在性E是一個正特性。This mirrors the key assumption in Anselm's argument.

批評[编辑]

哥德爾本體論證明的大部份批評,皆在於其公設部份。正如任何邏輯系統,假如其所依賴的公設備受懷疑,則結論也會受到懷疑。此情況特別適用於哥德爾的證明,因為其所依賴的5條公設,全部也是可以質疑的。此證明並不表示其結論正確,但假如你接受了那些公設,結論就是正確的。

很多哲學家質疑這些公設。第一層的攻擊,在於指出沒有任何理據技持為何這些公設為正確。第二層則是這些公設帶來一個不受歡迎的結論。This line of thought was argued by Sobel,[1] showing that if the axioms are accepted, they lead to a modal collapse where every statement that is true is necessarily true.

There are suggested amendments to the proof, presented by C. A. Anderson,[2] but argued to be refutable by C. A. Anderson and Michael Gettings.[3] Sobel's proof of modal collapse has been questioned by Koons,[4] but a counter-defence by Sobel has been given.[來源請求]

The proof has also been questioned by Oppy,[5] asking whether lots of other almost-gods would also be "proven" by Gödel's axioms. This counter-argument has been questioned by Gettings,[6] who agrees that the axioms might be questioned, but disagrees that Oppy's particular counter-example can be shown from Gödel's axioms.

There are many more criticisms, most focusing on the philosophically interesting question of whether these axioms must be rejected to avoid odd conclusions. The broader criticism is that even if the axioms cannot be shown to be false, that does not mean that they are true. Hilbert's famous remark about interchangeability of the primitives' names applies to those in Gödel's ontological axioms ("positive", "god-like", "essence") as well as to those in Hilbert's geometry axioms ("point", "line", "plane"). According to Fuhrmann (2005) it remains to show that the dazzling notion prescribed by traditions and often believed to be essentially mysterious satisfies Gödel's axioms. This is not a mathematical, but merely a theological task.[7]:364–366 It is this task which decides which religion's god has been proven to exist.

参见[编辑]

參考文獻[编辑]

  • C. Anthony Anderson, "Some Emendations of Gödel's Ontological Proof", Faith and Philosophy, Vol. 7, No 3, pp. 291–303, July 1990
  • Kurt Gödel (1995). "Ontological Proof". Collected Works: Unpublished Essays & Lectures, Volume III. pp. 403–404. Oxford University Press. ISBN 0195147227
  • A. P. Hazen, "On Gödel's Ontological Proof", Australasian Journal of Philosophy, Vol. 76, No 3, pp. 361–377, September 1998
  • Jordan Howard Sobel, "Gödel's Ontological Proof" in On Being and Saying. Essays for Richard Cartwright, ed. Judith Jarvis Thomson (MIT press, 1987)
  • Melvin Fitting, "Types, Tableaus, and Godel's God" Publisher: Dordrecht Kluwer Academic ©2002, ISBN 1402006047 9781402006043

外部連結[编辑]

  • ^ Jordan Howard Sobel. Gödel's ontological proof. (编) Judith Jarvis Thomson. On Being and Saying: Essays for Richard Cartwright. Cambridge/MA & London, England: MIT Press. Nov 1987: 241–261. ISBN 978-0262200639. 
  • ^ C. A. Anderson. Some emendations of Gödel's ontological argument. Faith and Philosophy, 7:291–303, 1990.
  • ^ Gόdel's Ontological Proof Revisited | C. Anthony Anderson and Michael Gettings, Gödel's Ontological Proof Revisited
  • ^ Koons, Robert C. "Sobel on Gödel’s ontological proof." (2005)
  • ^ Oppy, Graham. "Gödelian ontological arguments." Analysis 56.4 (1996): 226–230.
  • ^ Gettings, Michael. "Gödel's ontological argument: a reply to Oppy." Analysis 59.264 (1999): 309–313.
  • ^ André Fuhrmann. Existenz und Notwendigkeit — Kurt Gödels axiomatische Theologie [Existence and Necessity — Kurt Gödel's Axiomatic Theology] (PDF). (编) W. Spohn. Logik in der Philosophie [Logic in Philosophy]. Heidelberg: Synchron. 2005: 349—374 (German).