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Exhibit 5.3 – The Logic of “If … then” Statements

Another way of understanding “If … then” statements is through an understanding of necessary versus sufficient conditions. A necessary condition must be present for an event to occur but will not, by itself, cause the given event to occur. A sufficient condition is enough, by itself, to ensure that the event will occur. In more technical parlance, a necessary condition is a condition which, if absent, will not allow the event to occur. A sufficient condition is a condition which, if present, will cause the event to occur.

When a person argues “If A, then B” and then argues “If B, then A,” he or she erroneously reverses the conditional statement. The reason that a conditional statement cannot be reversed is that the original “If … then” statement functions as a necessary condition. When it is reversed, an “If … then” statement erroneously turns into a sufficient condition. In the example above, being “green” is a necessary but not a sufficient condition in order for something to be considered U.S. money. Obviously, other factors besides “green coloring” need be present, including special watermarked paper, unique insignia, and precise size. By reversing the “If … then” statement we erroneously suggest that being green-colored is enough of a criterion for something to be considered U.S. money.

Try one more example: “I gave my pet hamster water every day and he still died.” Giving your pet hamster water each day is a necessary condition for keeping him alive, but it is not a sufficient condition. Obviously, a hamster needs many other things besides water, one of which is food.

“NO-SOME-MOST-ALL” STATEMENTS

Many errors are committed in drawing inferences because ordinary speech is inherently ambiguous. For example, take the four statements below:

I.  No As are Bs

II. All As are Bs

III. Some As are Bs

IV. Most As are Bs

To study the meaning of these four statements, refer to Exhibit 5.4. We can see that statement I corresponds to either diagram (1) or (2), but usually to diagram (1). Statement II could represent either diagram (3) or (4), although usually diagram (4). Statement III could typify any one of diagrams (5), (6), or (7), although usually (6). Statement IV could refer to either diagram (8) or (9), but typically (9). This is evidence that ordinary speech can hamper clear thinking, and that it is often necessary to use non-verbal symbols to reinforce clear thinking.

There are two major differences between “most” and “some” statements. First, it is assumed that “most” implies majority (greater than half), while “some” implies minority (less than half). Second, whereas “some statements” automatically imply reciprocality, “most statements” do not necessarily imply reciprocality. For example, the statement “some doctors are wealthy people” implies that some wealthy people are also doctors. But the statement “most doctors are wealthy people” does not necessarily mean that most wealthy people are doctors.

The diagrams in Exhibit 5.4 summarize the concepts of mutual inclusivity, mutual exclusivity, and overlap. Either circles are embedded inside one another, or circles are completely separated, or circles overlap with one another. Basically, there are eight possibilities.

Exhibit 5.5 provides a summary of logical equivalency statements. With a better understanding of the visual representations of these relationships, the next step is to be able to combine these visuals with verbal logic statements expressed in English. This is a translation exercise. For example, within the area of inclusion, we must be able to see that all of the following are equivalent forms: “All cats are mammals”; “Every cat is a mammal”; “If it is a cat, then it is a mammal”; “Only mammals are cats”; and “No cat is not a mammal.”

MUTUAL INCLUSIVITY AND EXCLUSIVITY

Exhibit 5.4 – Overlap and Non-Overlap Scenarios

Statement I — No As are Bs — refers