Wednesday, March 14, 2007

Venn Diagrams - Part III

The standard categorical syllogism has three categorical statements or propositions: two premises — major and minor — and a conclusion. These three statements contain three terms. In the examples in the previous essay in this series, oranges, citrus fruit, apples, basketball players, people over 7 feet tall, F, and G are all terms. In a standard categorical syllogism, each term occurs in exactly two of the statements.

As an exercise, identify the terms in this syllogistic argument:

All citrus fruit is acidic,

All oranges are citrus fruit;

Therefore all oranges are acidic.

The terms are citrus fruit, acidic, and oranges. Notice that each of the terms occur in two of the statements: citrus fruit occurs in the major premise and the minor premise; acidic occurs in the major premise and the conclusion, and oranges occurs in the minor premise and the conclusion.

In a Venn diagram, the circles represent the terms (i.e., in the above argument, citrus fruit, acidic, and oranges) of the syllogistic argument. So, to diagram the citrus fruit argument, we need three circles, as follows:

The first statement says citrus fruit is acidic. As we saw in part II, in order to represent this in the diagram, we shade — remember, shading represents emptiness — the part of the citrus fruit circle that falls outside the acidic circle, thereby showing that there is no citrus fruit that is not acidic.

The second statement, oranges are citrus fruit, is shown in a similar way; we shade all of the area of the oranges circle that falls outside the citrus fruit circle.

All we need to do now is examine the diagram to see what conclusions are supported. At a glance we can see that all of the oranges circle that is not empty (not shaded) falls within the acidic things circle, and thus the conclusion all oranges are acidic is proved by the premises.

In the next installment, I will use an example from a dispute about Christian baptism to demonstrate how to spot a bad argument using Venn diagrams.

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