Month: May 2016

Symbols and Synonyms

An LSAT logical reasoning stimulus always contains most of the major components of a logical argument. A complete argument always has at least three “statements”: one conclusion and at least two premises. All but the most trivial logical argument have at least two “terms.” After looking through a large number of published LSAT stimuli, I have yet to find any stimulus that does not include at least two of these “statements” and at least two “terms.” I have previously written about how to find a conclusion, and it is relatively simple to find a premise. In this post, I want to help students discover the terms in the argument.

We need to know what a “term” is before we can find one, however. A “term,” in a logical context, means any concept that can be precisely expressed in words. A term could be “the best third baseman in the 1960 World Series,” “the true meaning of life,” or “yellow.”

In a complete LSAT question (which consists of the stimulus as well as the correct answer), all key terms must appear at least twice.  The reason for this is that an argument must connect the concepts to form a conclusion. A term that only appears once is not really necessary to the argument, and is therefore not a “key” term.

Here are the most common argument structures in LSAT stimuli:

  • Modus Ponens: A→B, A, ∴B
  • Modus Tollens: A→B, ~B, ∴~A
  • Hypothetical Syllogism: A→B, B→C, ∴A→C
  • Disjunctive Syllogism: A v B, ~A, ∴ B
  • Conditional Counterexample: A, ~B, ∴ ~(A→B)

Note that each argument uses each term exactly twice. While it is possible to construct more elaborate arguments that use terms even more often, it is not possible to construct an argument that uses a key term less than twice. Even the simplest possible argument uses its one term twice:

  • Double Negative: A, ∴~(~A)

This means that every complete LSAT question (stimulus plus correct answer) must use each key term at least twice. Every published LSAT question has at least two key terms. Finding these terms is a foundational skill for LSAT success.

The English language makes this difficult. The LSAT writers intentionally use different words to express these key terms. They exploit negatives, double negatives, and contrapositives to make it hard to match up concepts.

Here is an example from the June 2007 PrepTest, the only test that LSAC has made freely available to all:

Although video game sales have increased steadily over
the past 3 years, we can expect a reversal of this trend
in the very near future. Historically, over three quarters
of video games sold have been purchased by people
from 13 to 16 years of age, and the number of people
in this age group is expected to decline steadily over
the next 10 years.

Which one of the following, if true, would most
seriously weaken the argument?

(E) Most of the people who have purchased video
games over the past 3 years are over the age
of 16.

The conclusion of the argument is “we can expect a reversal of this trend in the very near future.” That conclusion must contain at least one key term, but the presence of the word “this” means we are going to have to look  back  into the stimulus to figure out  what “this” refers to.

The phrases “video game sales,” “video games sold,” and “purchased video games” all refer to the same concept. The phrases “people from 13 to 16 years of age” and “people in this age group” refer to a different concept, and “over the age of 16” negates that term. The word “increase” provides us with a third term, while “decline” and “reversal of this  trend” are its negation. (Since two out of three of these instances are “decreasing,” let us call this key term “decrease” and treat “increase” as the negation.)

Since the key terms are expressed in different English words, we need some way to show that they mean the same thing. The easiest way to do this is with symbols. I will use “S” for “video game sales,” “D” for decline, and “T” for “teenager” (by which I mean the 13-16 year old customer). With these symbols, we can rewrite the argument as follows:

S→T (sales are to teenagers)
T→D (teenagers are decreasing)
∴ S→D (sales are decreasing)

The correct answer to this question is “Most of the people who have purchased video games over the past 3 years are over the age of 16.” Converting this into our symbols, we get “S→~T.” This doesn’t just weaken the above argument, it destroys it.

Learning how to spot these synonyms (and antonyms, like “increase” and “decrease”) takes work, but it is essential. If you don’t recognize the repeated terms in a stimulus, you don’t really understand the argument. If you don’t really understand the argument, you’re just guessing at the answers. If you’re guessing at the answers, your score is going to be a whole lot lower than you want it to be. So let’s start spotting synonyms!

Exploiting Antitheses

I have previously identified the “some people say” principle in an LSAT logical reasoning stimulus, and have engaged in some discussion of this phenomenon at 7Sage.com.  After some further thought and research, I have decided to revise and extend my earlier remarks.

The “some people say” formulation occurs anytime an LSAT logical reasoning stimulus refers to the words, thoughts, or beliefs of persons other than the author of the stimulus. This usually takes the form of indirect discourse (i.e., “critics insist” or “paleontologists hypothesize”) but it can be more subtle. The functional test that I apply is, “Can I put somebody else’s claim in quotes?” If so, we can label it a “some people say” claim.

In my original post, I argued that every “some people say” claim is the negation of the conclusion. That goes too far. There are a handful of cases where “some people say” something that directly supports the conclusion. I think of this as “expert testimony.” There are other cases where “some people say” what they will do in the future (campaign promises, for example). Logicians distinguish “promises” from “logical statements” (for good reason). There are some other cases where “some people say” things that are not simply the negation of the conclusion–but not many.

Instead of insisting that every “some people say” statement is the negation of the conclusion, it is more accurate and helpful to say that most “some people say” statements have this characteristic. I use the term “antithesis” for such statements, and define it as follows:

An antithesis is a claim which an LSAT logical reasoning stimulus rejects.

There are significant advantages to identifying an antithesis. The first is that it makes complex stimuli simpler. “Some people say” statements can be involved, and arguments involving them can get complicated. If you extract the antithesis, what remains is often short and simple.

The second advantage is in identifying the conclusion. While this is second nature for some students, it is very hard work for others. Since an antithesis is (by definition) a claim which the stimulus rejects, it is easy to look for a part of the stimulus that says so. If there is such a statement, that is usually the conclusion.

The third advantage is in understanding the argument as a whole. If you see a “some people say” statement and suspect that it is an antithesis, you know where the argument is going before you read it. There are only so many ways to deny a  claim, and a person who is thoroughly familiar with basic argument types can anticipate them.

Thus, for example, if the antithesis is a conditional statement (“A->B”), then the simplest argument to refute it is a counterexample (“A and ~B”). If the antithesis is an affirmation (“A”), the simplest refutation is a modus tollens (“A→B, ~B, ∴ ~A”). If the antithesis is a denial (“~B”), the simplest refutation is a modus ponens (“A→B, A, ∴ B”). While these aren’t the only ways to reject such claims, they are the most common–and a person who is actively looking for such argument patterns will be able to find them faster if they really are present.

The bottom line is this: identifying an antithesis can cut a complicated stimulus in half, expose its conclusion, and reveal the argument structure quickly and accurately. While not every “some people say” statement is an antithesis, the large majority are.