## Sunday, September 09, 2007

### How to solve the logic problems

On Friday, I took the How Logical Are You? quiz. I thought it might be useful to discuss how to answer each of the questions (I won't give the actual answers, though).

Question 1: There's an easy way and a hard way to solve this problem. The easy way is to examine each answer in turn and see if it satisfies the conditions in the question: If Ralph is 60, is he four times as old as Frank is at 15? If so, add 20 years to each age: When Ralph is 80, will he be twice as old as Frank at 35?

There's an additional shortcut. In a question like this, typically all the answers will satisfy the simpler (usually first) condition, but only one will satisfy the more complicated. Check the more complicated condition first for each answer; check the simpler condition only if the answer satisfies the first condition.

You can solve questions like this the hard way by solving simultaneous algebraic equations. Note: You should learn basic algebra, it's very useful.

There are two variables, Ralph's and Frank's present ages, so there have to be two conditions (There are an infinite number of ways for two variables to satisfy one condition). Write down the conditions algebraically (R means Ralph's present age, F means Frank's):
1. R = 4 * F
2. (R + 20) = 2 * (F + 20)
Solve both equations for either R or F (since (1) is already solved for R, we only have to solve the other equation):
1. R = 4 * F
2. R = (2 * (F + 20)) - 20 = 2 * F + 2 * 20 - 20 = 2 * F + 20
Substitute one solution for R into the other solution; now we have one equation with one variable:
1. 4 * F = 2 * F + 20
Solve this equation using ordinary algebra and you have Frank's present age; since you know Ralph is four times older than Frank, you can quickly get Ralph's present age.

You can also solve this problem using graphs. Each condition specifies a line; the answer is the point where the lines intersect. I've left the numbers off the graph so as to not spoil the question. Question 2: This question tests your understanding of material implication. The key words are "if ... then ..." and the absence of the key word "only". (See question 4). What follows the "if" is usually called the antecedent (often labeled as p), and what follows the "then" is called the consequent (q); we can express a material implication in logical notation as p -> q. Sometimes the test taker will try to trip you up by reversing the order in the question (see question 4).

There is only one valid way of transforming a material implication by changing the order of the antecedent and consequent and/or adding "not" to either or both: The contrapositive: reverse the antecedent and consequent, and add not to both: if not q then not p (~q -> ~p). All the other combinations are invalid. The two most common invalid combinations have special names: The converse: q -> p and the inverse: ~p -> ~q. All the other combinations (such as ~p -> q) are invalid, but don't have names.

Question 3: This is a fairly easy question to solve; I think most people will get this one right by intuition. It's interesting, though, because it's self-referential: All the possible answers refer to themselves: Each statement refers to a set of statements, and the set includes each statement. To devotees of Russell and Whitehead, this question is not, strictly speaking, meaningful. If you allow this sort of self-reference, directly or indirectly, you can create paradoxes: statements that are both valid and invalid. The problem with Russell and Whitehead's strict prohibition of self reference is, as Kurt Gödel showed, you can end up with statements that are neither true nor false, thus casting philosophical doubt on law of the excluded middle.

Question 4: This is a very tricky question! It looks like the "if" makes "Neko drives" the antecedent and "Neko goes to the movies" the consequent (see Question 1). But "only if" (as opposed to "if and only if") is a tricky grammatical way to introduce the consequent. The question should be read as, "If Neko goes to the movies, then she drives." You can then apply the same sort of analysis as described in question 1: Only the contrapositive is valid; the inverse and converse are not valid.

Question 5: This question is meant to distract you with irrelevant information, so I'll just give you a couple of hints: The total number of socks and shoes is irrelevant; the answer depends only the number of different colors. Second, since the question asks for the fewest number of each to guarantee pair, you need only consider the worst case scenario; it's irrelevant that even fewer choices might give a pair, or that the correct answer might yield more than one pair.

Two of the next three questions are tricky and get into quantified logic and sets. I'll address them in another post.

1. Question 2 is also an example (whether deliberate or not) of misdirection. Here are some intelligent people trying to solve it using their knowledge of the rules of baseball.

2. Indeed. As I note in the second post, a key characteristic of multiple-choice logic problems is that many of the answer choices will be indeterminate according only to the premises offered in the question.

3. Fantastic breakdown! Big thanks thanks thanks!!!!!

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