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Solutions to Exercifes in Chapter 1

15 June, 2015 - 15:22
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Solution to Exercise 1.1.1

The flaw is extremely hard to find. We won't actually give the solution, but here's a hint on how to go about attacking the puzzle:

Note that finding the bug in the proof is the same skill as debugging a program. A good approach is to try various degenerate inputs. In this case, there are a couple of "inputs" to the construction the length of CD is arbitrary; no matter how long or short the proof should apply equally well. Similarly, the angle 100 ◦ seems arbitrary; fiddling with inputs like these (making them very small or very large) might give you some clues as to where the bug is. A very careful drawing will clear things up.

Solution to Exercise 1.3.1

This argument is or has been commonly used for varying topics

  • marijuana,
  • alcohol,
  • all drugs,
  • handguns,
  • birth control,
  • prostitution,
  • encryption technology.

The interesting part, is that the traditional Left and Right political positions each use this argument for some of these items, while rejecting the argument when used for other items.

A more rational response is to either accept all the above, or none of the above, or to realize that the stated argument wasn't everything that there might be implicit assumptions or arguments which actually do distinguish between these cases (the different interpretations of "[X]"). Being able to articulate the differences is essential. The more refined arguments may be more nuanced, and less able to ft into a sound-bite, but lead to a better understanding of one's own values. And sometimes, upon reflection, one may realize that some of the implicit values or premises are things they actually disagree with, once they are precisely spelled out.

Solution to Exercise 1.3.2

  1. The argument isn't actually in syllogism form. For example, the following is an incorrect syllogism:


All people don’t know my file’s password

Premise (Equivalent to “Nobody knows my file’s password”, but reworded to be of the required form “All somethings have some property.”.)


All hackers are people.



Therefore, my file is secure from hackers.

Incorrect syllogism, lines 1, 2

To be a syllogism, the conclusion would have to be "all hackers don't know my file's password." The file might or might not be secure, but the above doesn't prove it.

  1. One of the two premises is wrong.


All people don’t know my file’s password.

Premise, but possibly false


All hackers are people.

Premise, but possibly false


Therefore, all hackers don’t know my file’s password.

Syllogism, line 1, 2

This proof fails, of course, if some hackers are non-people (e.g., programs), or if some people know the password. (In fact, presumably you know the password!)

Of course, even if a proof fails, the conclusion might be true for other reasons. An incorrect argument doesn't prove the conclusion's opposite!