自然演繹法系統之比較

A Critical Comparison of Natural Deduction Systems

自然演繹法（natural deduction）是大多數基礎邏輯課程中所教的證明系統，然而不同的教科書所採用的自然演繹法系統不盡相同，造成了教學與學習上的困擾。特別是在量詞推論規則上，我們有兩套完全不同的系統。其中一個系統（本文稱之為「Gentzen系統」）使用了一條看起來較為複雜的「存在個例化規則」（existential instantiation），而另一個系統（本文稱之為「Copi-Kahane系統」），則是用了一條看起來較為簡單的存在個例化規則。雖然目前大部分中文邏輯教科書都採用了「Copi-Kahane系統」的規則，然而也有少數中英文基礎邏輯或中階邏輯教科書是採用「Gentzen系統」。這造成了教學上的一些問題，例如它也許會造成基礎邏輯與中階邏輯課程銜接上的一些難度。在本文中，我將從邏輯教學的觀點來比較「Gentzen系統」與「Copi-Kahane系統」的優劣，並對於我們應該採用哪一個系統來教學提出我的建議。 Natural deduction is the logical system most commonly used in teaching elementary logic. However, different textbooks may adopt different versions of the natural deduction system, which can be quite annoying to many teachers and students of logic. More precisely, with regard to the inference rules for quantifiers, there are two completely different systems of natural deduction. One system (which I shall call the ＂Gentzen system＂) has a rule of ＂existential instantiation＂ that looks pretty complicated, whilst the other system (which I shall call the ＂Copi-Kahane system＂) has a much simpler rule for existential instantiation. Although most of the Chinese textbooks for elementary logic use the rules of the Copi-Kahane system, there are still some textbooks for elementary logic or intermediate logic which use the Gentzen system. This leads to some problems especially in teaching logic, as students may find it somehow difficult to learn intermediate logic (which usually use the Gentzen system) if what they learn in elementary logic is the Copi-Kahane system. In this paper, I shall critically compare the Gentzen system with the Copi-Kahane system to judge which one is better for teaching elementary logic.