2007/10/07

南轅北轍

《Gödel, Escher, Bach》p.94 如是說:

[...] Consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it.

而 Max 曾強調

Repeat after me: Consistency is a property of a deductive system; it has nothing to do with semantics!

兩者說法南轅北轍。不過我想這是唯名定義的問題:Hofstadter 這裡用的 consistency 應該是 Max 的 soundness。下面 Hofstadter 就定義

But now let us say exactly what is meant by consistency of a formal system (together with an interpretation): that every theorem, when interpreted, becomes a true statement.

不過 Hofstadter 又另外定義 "internal" consistency 為

A system would be internally inconsistent if it contained two or more theorems whose interpretations were incompatible with one another, and internally consistent if all interpreted theorems were compatible with one another.

這似乎就比較接近一般講的 consistency,但也不太一樣。

再多看幾頁,就會覺得其實 Hofstadter 賦予 formal systems & semantics 的內涵比 mathematical logic 考慮的還要多,所以兩者間大概沒辦法做完全直接的對應。

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買了一桶蛋捲回來吃 XD。

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