LSAT Fundamentals - Conditional Logic
Philosophy majors can probably skip this one, but for the other 99% of you…if you’re not thoroughly comfortable with the logical implications of conditional statements (“If P, then Q”), then stick around…they’re important on the LSAT, and particularly…the Logic Games.
Surprisingly, there are many levels that pertain to conditionals, so, we’re going to take it from Square 1.
From the Statement Structure “If P, then Q” a few related statements can be constructed. Only one of them, however, is accurate. Let’s use actual statements with meaning, to make it easier to understand. Let’s say I have a drink. My conditional will be “If my drink is a vodka and tonic, then my drink contains alcohol.” We can readily see that this follows the “If P, then Q” structure. P = “my drink is a vodka and tonic” Q = “my drink contains alcohol.”
One possible conclusion to draw from “If P, then Q” is “If Q, then P.” This is not a valid inference. To see why, let’s just put our meaningful content into the formula. If Q, then P translates to, “If my drink contains alcohol, then my drink is a vodka and tonic.” Clearly, though, this does not follow. My drink might be a rum and coke.
Another possibility might be, “If not P, then not Q.” Again, this is not a valid inference. Let’s check it out: “If my drink is not a vodka and tonic, then my drink does not contain alcohol.” But, again, I could have a rum and coke. “Not P” would be satisfied, but “not Q” wouldn’t be true – my rum and coke WOULD contain alcohol.
That leaves “If not Q, then not P.” This is called the “contrapositive,” and it IS true. Check it out: “If my drink does not contain alcohol, then my drink is not a vodka tonic.” That one works.
Very often, a Logic Games question will offer you incorrect answer choices based on drawing one of the faulty conclusions. They want to see if you’ll jump on the wrong conclusions. But the other thing they do is offer correct answer choices based on the contrapositive, and if you don’t spot it right away and allow for it in the diagram, you’ll either miss a correct answer, or spend far too long working it out. Here’s how it plays out in LSAT land. Let’s say we have a grouping game where 8 different students take either a Math class, a Science class, or an English class. One of the clues might be:
“If Alex takes the English class, then Bob takes the Math class.” You might diagram this a number of ways, for instance: A(e) -> B(m) Or If A=e, B=m. Something quick, visual, and understandable (to you). Recognizing it as a conditional, though, you have to note that the contrapositive is true, and you have to get it diagrammed, as well. There are various ways to diagram “not.” I'll use this: (~). It’s fine to diagram it in other ways, though. The important thing, though, is to get the contrapositive in your diagram, too, in this case:
~B(m) -> ~A(e), or If B~=m, A~=e. In other words, if Bob doesn’t take the math class, then Art doesn’t take the English class. This is the logical equivalent of the given statement. The reason it’s important to write it out is that you’re guaranteed to get questions that tell you that Bob isn’t in the math class, and you have to be able to immediately rule out answer choices that put Alex in the English class. The question might tell you straight out that Bob doesn’t take English, or it might say, “If Bob is in the science class, which of the following could be true.” Then you have to make the connection: Bob in science = Bob not in math = Alex not in English. And you have to do it fast. If you can lay out the contrapositives at a glance, and reflect them in your diagrams, you’ll be in good shape on a number of questions. If you can’t, then you’re either going to come to faulty conclusions and get some answers wrong, or you’re going to use up valuable time figuring it out on the fly. So learn to:
1) recognize conditional (“if P, then Q”) statements.
2) translate to the contrapositive (“if not Q, then not P”).
3. get the contrapositive diagrammed.
Caveat: Sometimes the Q comes before the P. Just because the standard format is “if P, then Q,” don’t get lazy and assume that the first piece of information is the P, and the second is the Q. It’s the “IF” that defines which part of the sentence is the hypothesis (P), and which is the conclusion (Q). For instance:
“Carl takes Math if David takes Science.” P = “David takes Science” (the part after the “IF”). Q = “Carl takes Math.” The contrapositive is: “If Carl doesn’t take Math, then David doesn’t take science.”
Good practice Logic Game: Prep Test 33 (December 2000), section 4, game 2 (Questions 6-12). Page 177 of “The Next 10 Actual, Official LSAT PrepTests.”
Unfortunately, this post only covers the first layer of thinking you need to have about the contrapositive. Call it “Logic 101 for LSAT” 201 and 301 are more advanced, but they’re tremendously important. Coming soon to an LSAT blog near you.