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Logic-II (Diagramming)

Most arguments are based on some variation of an if-then statement. However, the if-then statement is often embedded in other equivalent structures. Diagramming brings out the superstructure and the underlying simplicity of arguments.

If the premise of an if-then statement is true, then the conclusion must be true as well. This is the defining characteristic of a conditional statement; it can be illustrated as follows:

This diagram displays the if-then statement “A—>B,” the affirmed premise “A,” and the necessary conclusion “B.” Such a diagram can be very helpful in showing the logical structure of an argument.
There are three statements that can be derived from the implication “if A, then B”; two are invalid, and one is valid.
From “if A, then B” you cannot conclude “if B, then A.”
For example, if it is cloudy, you cannot conclude that it is raining. From experience, this example is obviously true; it seems silly that anyone could commit such an error. However, when the implication is unfamiliar to us, this fallacy can be tempting.

Another, and not as obvious, fallacy derived from “if A, then B” is to conclude “if not A, then not B.” Again, consider the weather example. If it is not raining, you cannot conclude that it is not cloudy—it may still be overcast. This fallacy is popular with students.

Finally, there is one statement that is logically equivalent to “if A, then B.” Namely, “if not B, then not A.” This is called the contrapositive.

To show the contrapositive’s validity, we once again appeal to our weather example. If it is not cloudy, then from experience we know that it cannot possibly be raining.

We now know two things about the implication “if A, then B”:
  1. If A is true, then B must be true.
  2. If B is false, then A must be false.
If you assume no more than these two facts about an implication, then you will not fall for the fallacies that trap many students.
Example: (If-then)

If Jane does not study for the GMAT, then she will not score well. Jane, in fact, did not study for the GMAT; therefore she scored poorly on the test.

When symbolizing arguments we may let a letter stand for an element, a phrase, a clause, or even an entire sentence.  The clause “Jane does not study for the GMAT” can be symbolized as ~S, *and the clause “she will not score well” can be symbolized as ~W.  Substituting these symbols into the argument yields the following diagram:



This diagram shows that the argument has a valid if-then structure. A conditional statement is presented, ~S—>~W; its premise affirmed, ~S; and then the conclusion that necessarily follows, ~W, is stated.


Most of the arguments that you will have to diagram are more complex than this one—but not much more. In fact, once you master diagramming, you will find these arguments rather routine.


Note: Where the tilde symbol, ~, means “not.”

Embedded If-Then Statements

Usually, arguments involve an if-then statement. Unfortunately, the if-then thought is often embedded in other equivalent structures. In this section, we study how to spot these structures.
Example: (Embedded If-then)

John and Ken cannot both go to the party.

At first glance, this sentence does not appear to contain an if-then statement. But it essentially says:


“if John goes to the party, then Ken does not.”


Note, the statement “if Ken goes to the party, then John does not” expresses the same thing. So we don’t need to state both.

Example: (Embedded If-then)

Danielle will be accepted to business school only if she does well on the GMAT.

Given this statement, we know that if Danielle is accepted to business school, then she must have done well on the GMAT.


Note: Students often wrongly interpret this statement to mean:


“If Danielle does well on the GMAT, then she will be accepted to business school.”


There is no such guarantee. The only guarantee is that if she does not do well on the GMAT, then she will not be accepted to business school.


“A only if B” is logically equivalent to “if A, then B.”

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