# Logic-II (Diagramming)

*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-Then**

*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:

*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.

*cannot*conclude “if B, then A.”

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”:

- If A is true, then B must be true.
- If B is false, then A must be false.

*(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

*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.

*(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.

*(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.”**