3.0 Prelude to Symbolic Logic and Proofs 符号逻辑与证明·导言

本页译自 LibreTexts · Discrete Mathematics (Levin) 第 3.0 章。公式经 MathJax 渲染，自定义宏已注入。

Logic is the study of consequence. Given a few mathematical statements or facts, we would like to be able to draw some conclusions. For example, if I told you that a particular real-valued function was continuous on the interval $[0,1]$, and $f(0) = -1$ and $f(1) = 5$, can we conclude that there is some point between $[0,1]$ where the graph of the function crosses the $x$-axis? Yes, we can, thanks to the Intermediate Value Theorem from Calculus. Can we conclude that there is exactly one point? No. Whenever we find an “answer” in math, we really have a (perhaps hidden) argument. Mathematics is really about proving general statements (like the Intermediate Value Theorem), and this too is done via an argument, usually called a proof. We start with some given conditions, the premises of our argument, and from these we find a consequence of interest, our conclusion.

逻辑学研究的是推论。给定若干数学命题或事实，我们希望能够得出一些结论。例如，如果我告诉你，某个实值函数在区间 $[0,1]$ 上连续，且 $f(0) = -1$、$f(1) = 5$，我们能否断定在该区间内存在某点，使函数图像穿过 $x$ 轴？能，借助微积分中的介值定理即可。我们能否断定恰有一个这样的点？不能。在数学中，每当我们找到一个「答案」，实际上都（或许隐含地）伴随着一段论证。数学真正要做的，是证明一般性命题（如介值定理），而这同样通过论证完成，通常称为证明。我们从若干给定的条件——即论证的前提——出发，由此找到我们关注的推论，也就是结论。

The problem is, as you no doubt know from arguing with friends, not all arguments are good arguments. A “bad” argument is one in which the conclusion does not follow from the premises, i.e., the conclusion is not a consequence of the premises. Logic is the study of what makes an argument good or bad. In other words, logic aims to determine in which cases a conclusion is, or is not, a consequence of a set of premises.

问题在于，正如你与朋友争论时必定体会到的那样，并非所有论证都是「好」论证。一个「坏」论证，是指其结论并非由前提推出，即结论并非前提的推论。逻辑学研究的是：是什么使一个论证为好或为坏。换言之，逻辑学的目标在于判定：在哪些情形下，某个结论是（或不是）一组前提的推论。

By the way, “argument” is actually a technical term in math (and philosophy, another discipline which studies logic):

顺便说一句，「论证」其实是数学（以及哲学——另一门研究逻辑的学科）中的一个专门术语：

Definition: Arguments

定义：论证

An argument is a set of statements, one of which is called the conclusion and the rest of which are called premises. An argument is said to be valid if the conclusion must be true whenever the premises are all true. An argument is invalid if it is not valid; it is possible for all the premises to be true and the conclusion to be false.

论证（argument）是一组命题（statements），其中一条称为结论（conclusion），其余称为前提（premises）。当所有前提为真时结论必定为真，则称该论证有效（valid）。若论证不是有效的，则称其无效（invalid）；也就是说，完全可能所有前提都为真而结论为假。

For example, consider the following two arguments:

例如，考虑下面两个论证：

	If Edith eats her vegetables, then she can have a cookie.
	Edith eats her vegetables.
$\therefore$	Edith gets a cookie.

（中文说明：第一个论证——前提：若伊迪丝吃蔬菜，则她可以吃一块饼干；伊迪丝吃了蔬菜。结论（$\therefore$ 因此）：伊迪丝得到一块饼干。）

	Florence must eat her vegetables in order to get a cookie.
	Florence eats her vegetables.
$\therefore$	Florence gets a cookie.

（中文说明：第二个论证——前提：弗洛伦斯必须吃蔬菜才能吃到饼干；弗洛伦斯吃了蔬菜。结论（$\therefore$ 因此）：弗洛伦斯得到一块饼干。）

(The symbol “$\therefore$” means “therefore”.)

（符号「$\therefore$」表示「因此」。）

Are these arguments valid? Hopefully you agree that the first one is but the second one is not. Logic tells us why by analyzing the structure of the statements in the argument. Notice the two arguments above look almost identical. Edith and Florence both eat their vegetables. In both cases, there is a connection between the eating of vegetables and cookies. But we claim that it is valid to conclude that Edith gets a cookie, but not that Florence does. The difference must be in the connection between eating vegetables and getting cookies. We need to be skilled at reading and comprehending these sentences. Do the two sentences mean the same thing? Unfortunately, in everyday language we are often sloppy, and you might be tempted to say they are equivalent. But notice that just because Florence must eat her vegetables, we have not said that doing so would be enough (she might also need to clean her room, for example). In everyday (non-mathematical) practice, you might be tempted to say this “other direction” is implied. In mathematics, we never get that luxury.

这两个论证有效吗？希望你认同：第一个有效，第二个无效。逻辑学通过分析论证中命题的结构来告诉我们原因。注意，上面两个论证看起来几乎一模一样。伊迪丝和弗洛伦斯都吃了蔬菜；两种情形下，吃蔬菜与吃饼干之间都存在联系。但我们主张：由前者可以推出伊迪丝得到饼干，却推不出弗洛伦斯得到饼干。差别必定在于「吃蔬菜」与「得到饼干」之间的联系。我们需要善于阅读并理解这些句子。这两句话意思相同吗？遗憾的是，在日常语言中我们常常不严谨，你或许会倾向于说它们是等价的。但请注意：仅仅因为弗洛伦斯「必须」吃蔬菜，我们并没有说这样做就「足够」（例如，她可能还需要整理房间）。在日常（非数学）语境中，你或许会倾向于认为这「另一个方向」也成立。在数学中，我们从来没有这种便利。

Before proceeding, it might be a good idea to quickly review Section 0.2 where we first encountered statements and the various forms they can take. The goal now is to see what mathematical tools we can develop to better analyze these, and then to see how this helps read and write proofs

在继续之前，不妨快速回顾一下第 0.2 节，我们正是在那里首次接触到命题及其各种形式。现在的目标是：看我们能发展出哪些数学工具来更好地分析这些命题，进而看清这些工具有助于我们阅读和撰写证明。

	If Edith eats her vegetables, then she can have a cookie.
	Edith eats her vegetables.
\(\therefore\)	Edith gets a cookie.