It has long been recognized that negation in Aristotle’s term logic differs syntactically from negation in classical logic: modern external negation attaches to propositions fully formed, whereas Aristotelian internal negation forms propositions from sentential constituents. Still, modern external negation is used to render Aristotelian internal negation, as may be seen in formalizations of Aristotle’s semantic principles of non-contradiction and of excluded middle. These principles govern the distribution of truth values among pairs of contradictory propositions, and Aristotelian contradictories always consist of an affirmation and a denial. So how should we formalize a false denial? In the literature, we find that a false denial is formalized by means of two negation signs attached to a one-place predicate. However, it can be shown that this rendering leads to an incorrect picture of Aristotle’s principles. In this paper, I propose a solution to this technical problem by devising a formal notation especially for Aristotelian propositions in which internal negation is differentiated from external negation. I will also analyze both principles, each of which has two logically equivalent forms, a positive and a negative one. The fact that Aristotle’s principles are distinct and complementary is reflected in my new formalizations.

中文翻译：

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内在否定与亚里士多德的不矛盾原则和排中原则

人们早就认识到，亚里士多德术语逻辑中的否定在句法上不同于古典逻辑中的否定：现代外部否定依附于完全形成的命题，而亚里士多德的内部否定则从句子成分形成命题。尽管如此，现代外在否定仍被用来呈现亚里士多德的内在否定，正如在亚里士多德的非矛盾和排中的语义原则的形式化中可以看到的那样。这些原则支配着成对矛盾命题之间真值的分配，而亚里士多德的矛盾总是由肯定和否定组成。那么我们应该如何将虚假否认正式化呢？在文献中，我们发现虚假否认是通过附加到一个位谓词的两个否定符号来形式化的。然而，可以证明这种渲染导致了对亚里士多德原理的错误描述。在本文中，我通过设计一种形式符号来解决这个技术问题，特别是为亚里士多德的命题，其中内部否定与外部否定不同。我还将分析这两个原则，每个原则都有两种逻辑上等价的形式，一种是肯定的，一种是否定的。亚里士多德的原则是独特的和互补的这一事实反映在我的新形式化中。每一种都有两种逻辑上等价的形式，一种是正的，一种是负的。亚里士多德的原则是独特的和互补的这一事实反映在我的新形式化中。每一种都有两种逻辑上等价的形式，一种是正的，一种是负的。亚里士多德的原则是独特的和互补的这一事实反映在我的新形式化中。