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Logic-II (Diagramming)

We thoroughly covered diagramming in the game section. Diagramming is also useful with arguments. However, the diagrams won’t be as elaborate as those used with games. In fact, in these cases, the term “diagramming ” is somewhat of a misnomer. Rarely will we actually draw a diagram; instead we will symbolize the arguments, much as we did the conditions of the games.

Diagramming is very helpful with arguments that ask you to select the statement that is most similar in structure to the original. The first step with these arguments is to decide whether the original statement is valid. If it is, the answer must be valid as well. If it is invalid, then the answer must be invalid. Some common questions for these types of arguments are
  • The logical structure of the argument above is most similar to which one of the following?
  • Which one of the following arguments contains a flaw that is most similar to one in the argument above?
Typically these arguments use some variation of an if-then statement, often the contrapositive.  Before we begin diagramming, we need to review some of the logical connectives that were introduced earlier and discuss in more detail those connectives only briefly covered.

As stated in the introduction to this section, most logical-structure arguments are based on some variation of an if-then statement. However, the if-then statement is often embedded in other equivalent structures. We already studied embedded if-then statements in the chapter on flow charts. Still, we need to further develop the ability to recognize these structures.

By now you should be well aware that if the premise of an if-then statement is true then the conclusion must be true as well. his 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.
Example: (If-then)

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

When symbolizing games, we let a letter stand for an element. When symbolizing arguments, however, 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 LSAT” 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.

At first, many students get hopelessly lost with logical-structure arguments because they develop tunnel vision—analyzing the meaning of each word. For these arguments, you should step back and take a bird’s-eye view. Diagramming brings out the superstructure and the underlying simplicity of these arguments.

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 law school, then she must have done well on the LSAT.


Note: Students often wrongly interpret this statement to mean “if Danielle does well on the LSAT, then she will be accepted to law school.” There is no such guarantee. The only guarantee is that if she does not do well on the LSAT, then she will not be accepted to law school.

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

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