对三进制逻辑简短的介绍

A Brief Introduction to Ternary Logic对三个的逻辑简短的介绍Jorge Pedraza Arpasi _Jorge Pedraza Arpasi ＿7th November 20032003 年十一月7 日1 Introduction1 介绍Based on the work of Ivan Guzm´an de Rojas [1] and some basic notions of binary logic and 基于伊凡Guzm ′的工作de Rojas ［1] 和二进位逻辑的一些基本观念和algebra of Boole, we will introduce and show some elementary properties of ternary logic. Boole 的代数学，我们将会介绍并且表示三个逻辑的一些初步财产。

These properties can be resumed as follows: a) Ternary logic is a generalization of binary这些财产能依下列各项被重新开始：一）三个的逻辑是一个二进的一般化logic, b) it has not a structure to be a Boolean algebra, c) it is based on more than three逻辑，b）它不有结构成为布尔数学体系的代数学，c）它以超过三为基础basic operations, and d) its tautologies and contradictions are more complicated for find基本操作、和d）它的同义字之反复和矛盾对发现更复杂out.在外。

The generalization is in the sense that if one proposition p is true(false) under the rules一般化在感觉中，如果一个建议p 在规则之下是真实的（错误的）of binary logic then it is true(false) under the ternary logic.当时二进位逻辑，它在三个的逻辑之下是真实的（错误的）。

The lack of Boolean structure,缺乏布尔数学体系结构，in ternary logic, is compensated by powerful tools for inferential analysis [1].在三个的逻辑中，为推理的分析［1] 被有力的工具偿还。

In this note, for the binary case, we will use the values {0, 1} which means true=1 and在这笔记中，为二进位的情形，我们将会使用意谓的价值｛0,1} 真实的＝1 和false=0. Whereas that for ternary case we will use the values {0, 1, 2} which means true=1, 错误的＝0. 然而为三个的情形我们将会使用意谓的价值｛0,1,2} 真实的＝1,false=2, and “perhaps true perhaps false”=0. This notation is the most adequate for a later错误的＝2, 和" 也许真实的也许错误的"=0. 这一个记号法是最适当的为一稍后algebraic analysis.代数的分析。

By x we will mean a simple statement or proposition whereas by p or f被x ，我们将会意指一份简单的陈述或建议然而藉着p 或fwe will mean a composed proposition which depends on other simple propositions, hence we 我们将会意指一个取决于其他简单的建议的镇静的建议，因此我们also refer they as functions.也提到他们为功能。

_The author is maintainer of the website Aymara Uta Internet: .＿作家是网站Aymara Uta 英特网的保养者：.112 Binary Logic(Classical Logic)2 二进位的逻辑（古典的逻辑）We can see the binary logic as a system L whose elements called propositions or statements 我们能视二进位的逻辑为一个元素呼叫建议的系统L 或陈述are valued on the set {0, 1} [2].在固定的｛0,1}[2] 上被评价。

This set {0, 1} we denote as being Z2. Thus, if x is我们当做作为Z2 指示的这一组｛0,1}. 因此，如果x 是a proposition, the value of x can be seen as a mapping _ : L !一个建议，x 的价值能被视为一映射＿: L ！{0, 1} such that｛0,1} 以致于_(x) =＿(x）=8<8<:：1; if x is true1; 如果x 是真实的0; if x is false0; 如果x 是错误的It is standard, in almost all the logical literature, to ignore the mapping _.在几乎所有合乎逻辑的文学中，它是标准，不理睬这映射＿.Therefore,因此，for the sack of practical purposes, it is assumed that _(x) = x and from this, x = 1 means 对于实际目的的麻布袋，一般假定＿(x）= x 和从这，x ＝1个方法x is true and x = 0 means x is false.x 是真实的和x ＝0 方法x 是错误的。

Over L are defined the following basic operations上面的L 被定义下列的基本操作•The negation ¬ (unary operation “not”)？否定？（一元的操作" 不")•The disjunction _ (binary operation “or”)？分离＿(二元运算" 或")•The conjunction ^ (binary operation “and”)？连接词＾(二元运算" 和")The system L is closed under any of these three operations, in the sense that if x, y 2 L, 系统L 在这些之中的任何一个三操作之下被关闭，在感觉中哪一如果x，y 2 L，then both ¬x 2 L, x _ y 2 L, and x ^ y 2 L.然后两者的？x 2 L，x ＿y 2 L 、和x ＾y 2 L。

From these three operations we can derive从这些三操作，我们能源自16 binary operations, among them the implication Imp(x, y) = x !16个二元运算，在他们之中涵义小鬼（x，y）= x ！y and the equivalencey 和同等Equiv(x, y) = x $ y.Equiv （x，y）＝x ＄y。

The value of these basic operations and any other composed propositions这些基本操作和任何其他的镇静建议的价值depend on the value of each component x and y as can be seen in the true table for取决于每个成份x 和y 的价值因为能在真实的桌子中被见到these operations shown in Table 1在表1 显示的这些操作Another way to describe propositions based on the above basic operations is by considering 另外的一个方法描述以上述的基本操作为基础的建议是藉由考虑them as functions.他们当做功能。

In this way, the unary operator “negation” is a function f : Z2 !这样，一元的操作员" 否定" 是一个功能f ：Z2!Z2,Z2,and a binary operation such as the “disjunction” is a functi on f : Z22而且一个二元运算，像是" 分离" ，是一个功能f ：Z22!！Z2. In general, byZ2. 大体上，被combining the three basic operations we can define binary logic functions as mappings结合我们能把二进位的逻辑功能定义为映射的三基本的操作f : Zn2f ：Zn2!！Z2.Z2.When n = 1 we have one-variable functions f(x), and there are 221 = 4 of these kind当n ＝ 1 我们有单一变数功能f（x）, 而且有221= 这些类型中的4个of functions which are: the Identity or Affirmation id(x), the Negation N(x), the Tautology 功能是：身份或肯定遗传素质（x），否定N（x）, 同义字之反复22Neg(x) Disj(x, y) Conj(x, y) Imp(x, y) Equiv(x, y)Neg （x）Disj （x，y）Conj （x，y）小鬼（x，y）Equiv （x，y）x y ¬x x _ y x ^ y x !x y ？x x ＿y x ＾y x ！y x $ yy x ＄y1 1 0 1 1 1 111011111 0 1 0 0 01010000 1 1 1 0 1 001110100 0 0 0 0 1000001Table 1: Basic operations or functions in binary logic表1: 二进位的逻辑中的基本操作或功能_ (x) and the Contradiction (x).＿(x）而且矛盾（x）.All these four functions are also called modal functions这些四个功能也叫做型的功能of x and they are shown in the Table 2x 和他们在表2 中被显示x id(x) N(x) _ (x) (x)x 遗传素质（x）N （x）＿(x）(x）1 1 0 1 0110100 0 1 1 000110Table 2: All the one-variable binary functions表2:所有的单一变数二进动作When n = 2 we have two-variable functions f(x, y), and there are 222 = 16 of these当n ＝ 2 我们有两变数的功能f（x，y）, 而且有222=16 这些kind of functions which are shown in the Table 3. Notice, in that Table, that f2(x, y) is在表3 中被显示的功能的类型. 注意，在那一张桌子中，f 2(x，y）是the disjunction, f8(x, y) is the conjunction, f5(x, y) is the implication, f7 is the equivalence, 分离，f 8(x，y）是连接词，f 5(x，y）是涵义， f 7 是同等，whereas f1, f16 are the tautology and the contradiction functions, respectively.然而f 1, f 16 是同义字之反复，而且矛盾动作，分别地。