【大問４ 読解問題】
Read the texts in Sections A and B, and answer the questions. Section A: Choose the best option from a - d for questions 1 - 5.
In the analysis of any argument, questions and answers to questions (assertions) should never be isolated from each other. In other words, every argument is really a dialogue, and should be evaluated as such.
Every argument has two sides. It is the obligation of an answerer in reasonable dialogue to give an informative and relevant direct answer to a reasonable question if he/she can. If an answerer truly does not know whether the proposition queried is true or false, he/she should have the option, in reasonable dialogue, of replying 'I don't know' or ‘no commitment one way or the other.' In other words, the ignorant answerer should be able to admit his/her ignorance.
For, as Socrates reminded us, the beginning of wisdom is to admit your ignorance if you really don't know the answer to a question. Hence, any structure of dialogue that does not allow an answerer the ( A ), in replying to questions, would not be tolerant of wisdom.
The idea that an answerer should concede that he/she doesn't know the answer, if he/she really doesn't, is reflected in a traditional fallacy called the ad ignorantiam fallacy. Consider the following dialogue:
Elliot: How do you know that ghosts don't exist?
Zelda: Well, nobody has ever proved that ghosts do exist, have they?
Here, Elliot asks Zelda to give justification for her commitment to the proposition that ghosts do not exist. Zelda answers by shifting the burden of proof back onto Elliot to prove that ghosts do exist. This reply is said to commit the fallacy of arguing from ignorance (argumentum ad ignorantiam); just because a proposition has never been proved true, that does not mean that it is false. You cannot argue from ignorance.
Fermat's Last Theorem in mathematics can be a good illustration of this point. The theorem, written in 1637 stating that it is impossible to separate any power higher than the second into two like powers (no three positive integers a, b, and c satisfy the equation $a^n+b^n = c^n$ for any integer value of n greater than 2), had never been proved true until 1994, when Andrew Wiles and Richard Taylor worked out a proof based on methods developed by other mathematicians.
Prior to 1994, it was ( B ) whether it can be proved that Fermat's Last Theorem is unprovable.
Douglas Walton Informal Logic A PragmaticApproach
1. Which of the following best fits in blank A?
a. no-commitment option
b. negative-answer option
c. positive-response option
d. response with wisdom
2. Which of the following arguments meets the definition of the argumentum ad ignorantiam?
[I] Proposition A is not known to be true; therefore, A is false.
[II] Proposition A is not known to be false; therefore, A is true.
a. I only
b. II only
c. both I and II
d. neither I nor II
3. Which of the following is NOT true of the argument below?
[Some philosophers have tried to prove God does not exist, but they have failed. Therefore, God exists.]
a. The argument is not consistent with Socrates' sense of wisdom.
b. The conclusion is similar to Zelda's response to Elliot's question in the text.
c. The replacement of the conclusion with “Therefore, God does not exist” makes the argument sound.
d. The argument is a case of argumentum ad ignorantiam.
4. Which of the following best fills in the blank labeled B?
a. an appropriate proposition
b. an open question
c. a disclosed problem
d. a challenging issue
5. What could we say about Fermat's Last Theorem before 1994?
a. It had not been proved because it could not be proved.
b. All that was known was that it might just be very difficult to prove.
c. Mathematics does not allow argument from ignorance.
d. Whether a proposition has been proved is analogous to whether it can be proved.
Section B: Choose the best option from a - d for questions 6 - 10.
I. Statements about the natural numbers can often be regarded as sequences of statements P, for all n (n=1, 2, 3...).
II. The principle of mathematical induction states: Given any statement about the natural numbers $P_n$ , if the following conditions hold:
1. $P_1$ is true.
2. Whenever $P_k$ is true, $P_{k+1}$ is true.
Then $P_n$ , is true for all n.
III. We can apply the principle of mathematical induction to a sequence of statements P, as follows:
1. Write out the statements $P_1$, $P_k$, and $P_{k+1}$
2. Show that $P_1$, is true.
3. Assume that $P_k$ is true. From this assumption (it is never ( A ) to prove $P_k$ explicitly), show that the truth of $P_{k+1}$ follows. This proof is often called the induction step.
4. Conclude that P, holds for all n.
IV. We now prove the statement
$P_n : 1+2+2^2+...+2^{n-1}=2^n-1$
by mathematical induction using the steps outlined in III.
$P_1$, is the statement $1 = 2^1-1$,
$P_k$ is the statement
$1+2+2^2+...+2^{k-1}=2^k–1$,
$P_{k+1}$ is the statement
$1+2^2+...+2^{k+1-1}=2^{k+1}–1$,
which can be rewritten as
$1+2+2^2+...+2^{k-1}+2^k=2^{k+1}–1$
Now $P_1$, is true,
since $1=2^1-1 2-1=1$ is true.
Assume the truth of $P_k$ and, comparing it to $P_{k+1}$,
note that the left side of $P_{k+1}$ differs from the left side of $P_k$ only by the single additional term ( B ).
Hence, starting with $P_k$, add $2^k$ to both sides.
$1 +2 +2^2+...+2^{k-1}=2^k–1$
$1+2+2^2+...+2^{k-1}+2^k =2^k+2^k-1$
Simplifying the right side yields:
$2^k+2^k-1=2•2^k–1=2^1•2^k–1$
$=2^{k+1}–1$,
thus
$1+2+2^2+...+2^{k-1}+2^k=2^{k+1}-1$ holds.
But this is ( C ) the statement $P_{k+1}$. Thus the truth of $P_{k+1}$ ( D ) from the truth of Pk.
Thus, by the principle of mathematical induction, $P_n$ holds for all n.
Fred Safier. Schaum's Outline of Precalculus.
6. Which of the following is closest in meaning to "induction” in the text?
a. a method of discovering general rules and principles from particular facts and examples
b. the process of using information you have in order to find the answer to a problem
c. the act of bringing something into use or existence for the first time
d. a means of reaching specific facts and examples from general statements and guidelines
7. Which of the following best fits in blank A?
a. possible
b. necessary
c. easy
d. conditional
8. Which of the following best fits in blank B?
a. $2$
b. $2^{k-1}$
c. $2^{k}$
d. $2^{k+1}$
9. Which of the following best fits in blank C?
a. eventually
b. precisely
c. accurately
d. properly
10. Which of the following best fits in blank D?
a. follows
b. induces
c. deduces
d. precedes
【大問４ 読解問題 解答】
Section A
1. a
2. c
3. c
4. b
5. b
Section B
6. a
7. b
8. c
9. b
10. a
【大問４ 読解問題 解説】
説明文。長文を読み進めながら適語補充と文章の順序を解答します。早稲田理工学部では、英語での数学能力が求められます。本年度は、フェルマーの最終定理と、数学的帰納法が題材です。どちらも高校数学を終えていれば、理解できる内容ですが、英語ではどのように数式を立てるのか、対策しておくとよいでしょう。
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