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Paths and Flow Charts

Although flow charts and paths are not, strictly speaking, ordering games, they have many of the properties found in ordering games.* Thus it is natural to analyze them here.

Flow charts and paths tend to be highly determinative. Once the chart has been constructed, the questions typically can be answered with little additional thought—often all the answers can be discerned by merely reading the chart.

The catch is that the chart may not be easy to derive. Because this type of game typically has many conditions, the chart can easily get out of control. Charting is an art. However, there are some guidelines that help:
  1. Look for a condition that starts the “flow” or that contains a lot of information.
  2. Look for an element that occurs in many conditions.
  3. Keep the chart flexible; it will probably have to evolve with the changing conditions.
Before we start, we need to address some of the hazards and symbols common to these games. Because flow charts and paths involve a “flowing” of information, the if-then symbol, —>, is the workhorse for these games. Because the information can often “flow” in both directions, the symbol “<—>“ also comes into play. A slash through a symbol indicates that information cannot flow in that direction.
For example:  means information cannot flow from A to B.

As you work through these games be alert to any opportunity to apply the contrapositive rule of logic. Often negative conditions can be expressed more clearly by rewording them in the contrapositive. For example, the statement

“if it is not sunny, then Biff is not going to the beach”

can be reworded more directly as

“if Biff is going to the beach, then it is sunny.”

It is not necessary that both parts of the if-then statement be negative for this technique to be effective.
For example, the statement “if Linda is hired, then Roland is not” can be recast as “if Roland is hired, then Linda is not.” Although in this case the contrapositive statement is no simpler than the original, it may, and often does, open up connections to other conditions.

We need to review two common fallacies associated with the contrapositive. From the statement “if A, then B” we can conclude, using the contrapositive, “if not B, then not A.” It would be fallacious, however, to conclude either “if not A, then not B” or “if B, then A.” Also note that some means “at least one and perhaps all.”

Until this point, our discussion has been out of chronological order: we have discussed how to solve flow and path games without discussing how to identify them. Path games are easy to identify; typically they involve the actual movement of an element or of information.
Some examples are
  • Four cities are connected by six roads.
  • A memo can be passed from Sara to Helen, but not from Sara to John.
  • If a litigant filed his case in federal court and lost, then he may appeal to the 4th District Court and from there to the Supreme Court.
Flow charts are harder to identify than paths. In fact, they can be quite cryptic. However, a game with many if-then conditions is often a tip-off to a flow-chart game. Unfortunately, the if-then thought is often embedded in other equivalent structures.
For example, the sentence “All A’s are B’s” can be reworded as “If x is an A, then x is a B.”
For a more subtle example take the sentence “Linda and Sara are not both hired”; it can be recast as “if Linda is hired, then Sara is not” (or “if Sara is hired, then Linda is not”).

The following drill will help you identify embedded if-then statements.

As you analyze a flow chart, look for “loops” that connect groups of elements. An example will illustrate:

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