2.2 扣减理由

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  Deductive Reasoning
  ::扣减理由

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  Deductive reasoning entails drawing conclusion from facts. When using deductive reasoning there are a few laws that are helpful to know.
  ::贬低推理需要从事实中得出结论。 在使用推理推理时,有一些法律有助于了解。

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  Law of Detachment: If $\begin{align*}p \rightarrow q\end{align*}$ is true, and $\begin{align*}p\end{align*}$ is true, then $\begin{align*}q\end{align*}$ is true. See the example below.
  ::分解法则:如果pQq是真实的,而p是真实的,那么q就是真实的。见下面的例子。

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  Here are two true statements:
  ::以下是两个真实说法:

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  1. If a number is odd (p), then it is the sum of an even and odd number (q).
     ::如果数字为奇数(p),则数字为偶数和奇数(q)之和。
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  2. 5 is an odd number (a specific example of p) .
     ::5 是一个奇数(p的具体例子)。

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  The conclusion must be that 5 is the sum of an even and an odd number (q).
  ::结论必须是5是偶数和奇数之和(q)。

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  Law of Contrapositive : If $\begin{align*}p \rightarrow q\end{align*}$ is true and $\begin{align*}\sim q\end{align*}$ is true, then you can conclude $\begin{align*}\sim p\end{align*}$ . See the example below.
  ::相对阳性法则:如果pQq是真实的,而QQq是真实的,那么您可以得出"p"的结论。请见下面的例子。

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  Here are two true statements:
  ::以下是两个真实说法:

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  1. If a student is in Geometry (p), then he or she has passed Algebra I (q).
     ::如果学生在几何学(p),则已通过代数I(q)。
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  2. Daniel has not passed Algebra I (a specific example of ~q).
     ::Daniel 尚未通过代数 I ( ~ q 的具体示例) 。

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  The conclusion must be that Daniel is not in Geometry (~p).
  ::结论必须是 Daniel 不在几何(~p) 中 。

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  Law of Syllogism: If $\begin{align*}p \rightarrow q\end{align*}$ and $\begin{align*}q \rightarrow r\end{align*}$ are true, then $\begin{align*}p \rightarrow r\end{align*}$ is true. See the example below.
  ::Syllogism法则:如果pq和qr是真实的,那么pär就是真实的。请见下面的例子。

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  Here are three true statements:
  ::以下是三个真实的陈述:

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  1. If Pete is late (p), Mark will be late (q).
     ::如果 Pete 迟到 (p), Mark 迟到 (q) 。
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  2. If Mark is late (q), Karl will be late (r).
     ::如果Mark迟到(q),Karl会迟到(r)。
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  3. Pete is late (p).
     ::Pete 迟到 (p) 。

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  Notice how each “then” becomes the next “if” in a chain of statements. If Pete is late, this starts a domino effect of lateness. Mark will be late and Karl will be late too. So, if Pete is late, then Karl will be late (r) , is the logical conclusion.
  ::注意每个“ 那么” 如何成为一系列语句中的下一个“ 如果 ” 。 如果 Pete 迟到, 这会引发迟到的多米诺效应。 马克会迟到, 卡尔也会迟到。 因此, 如果Pete迟到, 那么Karl就会迟到( r) , 这是合乎逻辑的结论 。

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  What if you were given a fact like "If you are late for class, you will get a detention"? What conclusions could you draw from this fact?
  ::如果你们被告知一个事实,比如“如果你上课迟到,你会被拘留吗?”你能从这个事实中得出什么结论呢?

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  Examples
  ::实例

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  Example 1
  ::例1

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  Suppose Bea makes the following statements, which are known to be true.
  ::假设比亚发表以下声明,据知这些声明是真实的。

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  If Central High School wins today, they will go to the regional tournament. Central High School won today.
  ::如果中高中今天获胜,他们将参加区域锦标赛,中高中今天获胜。

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  What is the logical conclusion?
  ::合乎逻辑的结论是什么?

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  These are true statements that we can take as facts. The conclusion is: Central High School will go to the regional tournament .
  ::这些是真实的陈述,我们可以把它们视为事实。 结论是:中央高中将参加区域锦标赛。

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  Example 2
  ::例2

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  Here are two true statements.
  ::这是两个真实的陈述。

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  If $\begin{align*}\angle A\end{align*}$ and $\begin{align*}\angle B\end{align*}$ are a linear pair , then $\begin{align*}m \angle A + m \angle B = 180^\circ\end{align*}$ .
  ::如果Aand-Bare是线性一对,那么m_A+mB=180。

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  $\begin{align*}\angle ABC\end{align*}$ and $\begin{align*}\angle CBD\end{align*}$ are a linear pair .
  ::ABC和CBD是线性一对。

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  What conclusion can you draw from this?
  ::你能从中得出什么结论?

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  This is an example of the Law of Detachment, therefore:
  ::这是《分遣队法》的一个例子,因此:

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  $$\begin{align*}m\angle ABC + m \angle CBD = 180^\circ\end{align*}$$
  ::* mABC+mCBD=180

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  Example 3
  ::例3

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  Determine the conclusion from the true statements below.
  ::确定以下真实语句的结论。

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  Babies wear diapers.
  ::婴儿穿尿布。

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  My little brother does not wear diapers.
  ::我弟弟不穿尿布

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  The second statement is the equivalent of $\begin{align*}\sim q\end{align*}$ . Therefore, the conclusion is $\begin{align*}\sim p\end{align*}$ , or: My little brother is not a baby .
  ::第二个声明相当于QQ。 因此,结论是_p,或者:我的小弟弟不是婴儿。

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  Example 4
  ::例4

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  Here are two true statements.
  ::这是两个真实的陈述。

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  If $\begin{align*}\angle A\end{align*}$ and $\begin{align*}\angle B\end{align*}$ are a linear pair, then $\begin{align*}m \angle A + m \angle B = 180^\circ\end{align*}$ .
  ::如果Aand-Bare是线性一对,那么m_A+mB=180。

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  $\begin{align*}m\angle 1 = 90^\circ\end{align*}$ and $\begin{align*}m\angle 2 = 90^\circ\end{align*}$ .
  ::m1=90和m2=90。

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  What conclusion can you draw from these two statements?
  ::你能从这两个发言中得出什么结论?

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  Here there is NO conclusion. These statements are in the form:
  ::以下没有结论。这些声明的形式如下:

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  $$\begin{align*}& p \rightarrow q\\ & q\end{align*}$$
  ::pqqq

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  We cannot conclude that $\begin{align*}\angle 1\end{align*}$ and $\begin{align*}\angle 2\end{align*}$ are a linear pair.
  ::我们无法得出1和2是线性对子的结论。

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  Here are two counterexamples:
  ::以下是两个反例:

  

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  Example 5
  ::例5

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  Determine the conclusion from the true statements below.
  ::确定以下真实语句的结论。

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  If you are not in Chicago, then you can’t be on the $\begin{align*}L\end{align*}$ .
  ::若你不在芝加哥,

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  Sally is on the $\begin{align*}L\end{align*}$ .
  ::莎莉在L线上

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  If we were to rewrite this symbolically, it would look like:
  ::如果我们用象征性的文字重写它, 它看起来像:

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  $$\begin{align*}& \sim p \rightarrow \sim q\\ & q\end{align*}$$
  ::{\fn方正黑体简体\fs18\b1\bord1\shad1\3cH2F2F2F

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  Even though it looks a little different, this is an example of the Law of Contrapositive. Therefore, the logical conclusion is: Sally is in Chicago .
  ::尽管看起来略有不同,但这是《阴阳性法》的一个例子。因此,合乎逻辑的结论是:莎莉在芝加哥。

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  Review
  ::回顾

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  Assuming the premises are true, determine the logical conclusion and state which law you used (Law of Detachment, Law of Contrapositive, or Law of Syllogism). If no conclusion can be drawn, write “no conclusion.”
  ::假设这些前提是真实的,确定逻辑结论以及你所使用的法律(《分遣队法 》 、 《 抵触法 》 、 或 《 赛洛格主义法 》 ) 。 如果无法得出结论,请写“无结论 ” 。

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  1. People who vote for Jane Wannabe are smart people. I voted for Jane Wannabe.
     ::投票给Jane Wangbe的人都是聪明人 我投票给Jane Wangbe
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  2. If Rae is the driver today then Maria is the driver tomorrow. Ann is the driver today.
     ::如果雷是今天的司机 那么玛利亚是明天的司机 安是今天的司机
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  3. All equiangular triangles are equilateral. $\begin{align*}\triangle ABC\end{align*}$ is equiangular.
     ::所有等角三角都是等角的。 ABC 是等角 。
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  4. If North wins, then West wins. If West wins, then East loses.
     ::北方赢,西方赢,西方赢,西方赢,东方输
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  5. If $\begin{align*}z > 5\end{align*}$ , then $\begin{align*}x > 3\end{align*}$ . If $\begin{align*}x > 3\end{align*}$ , then $\begin{align*}y > 7\end{align*}$ .
     ::如果 z> 5, 那么 x> 3. 如果 x> 3, 那么y> 7 。
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  6. If I am cold, then I wear a jacket. I am not wearing a jacket.
     ::如果我冷,那我就穿夹克 我不穿夹克
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  7. If it is raining outside, then I need an umbrella. It is not raining outside.
     ::如果外面下雨,我需要伞,外面没有下雨
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  8. If a shape is a circle, then it never ends. If it never ends, then it never starts. If it never starts, then it doesn’t exist. If it doesn’t exist, then we don’t need to study it.
     ::如果形状是一个圆圈,那么它就永远不会结束。如果它永远不会结束,它就永远不会开始。如果它从来没有开始,它就不存在。如果它不存在,我们就不需要研究它。
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  9. If you text while driving, then you are unsafe. You are a safe driver.
     ::如果您在驾驶时发短信, 您将不安全。 您是一个安全的驾驶员 。
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  10. If you wear sunglasses, then it is sunny outside. You are wearing sunglasses.
      ::如果你戴太阳镜,外面阳光明媚,你戴着太阳镜
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  11. If you wear sunglasses, then it is sunny outside. It is cloudy.
      ::如果你戴太阳镜,外面阳光明媚,多云。
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  12. I will clean my room if my mom asks me to. I am not cleaning my room.
      ::如果我妈妈让我打扫房间,我会打扫房间 我不是在打扫房间
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  13. Write the symbolic representation of #8. Include your conclusion. Does this argument make sense?
      ::写入 8 的符号表示 。 包含您的结论 。 此论点合理吗 ?
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  14. Write the symbolic representation of #10. Include your conclusion.
      ::写下 #10 的象征性表示。 包括您的结论 。
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  15. Write the symbolic representation of #11. Include your conclusion.
      ::写下 #11 的象征性表示。 包括您的结论 。

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

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  Resources
  ::资源