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Transitive Property

These arguments are rarely difficult, provided you step back and take a bird’s-eye view. It may be helpful to view this structure as an inequality in mathematics.
For example, 5 > 4 and 4 > 3, so 5 > 3.

Notice that the conclusion in the transitive property is also an if-then statement. So we don’t know that C is true unless we know that A is true. However, if we add the premise “A is true” to the diagram, then we can conclude that C is true:

As you may have anticipated, the contrapositive can be generalized to the transitive property:

Example: (Transitive Property)


If you work hard, you will be successful in America. If you are successful in America, you can lead a life of leisure. So if you work hard in America, you can live a life of leisure.

Let W stand for “you work hard,” S stand for “you will be successful in America,” and L stand for “you can lead a life of leisure.” Now the first sentence translates as W—>S, the second sentence as S—>L, and the conclusion as W—>L. Combining these symbol statements yields the following diagram:



The diagram clearly displays the transitive property.

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