The Euler Diagram below represents the statement if A, then B. Consider the true mathematical statement, "if a figure is a square, then it is a rectangle," which has false converse and inverse statements. If we say "If it is raining, then I carry my umbrella," then people have some reason to assume the converse statement as well as the inverse ( if not A, then not B), "if it is not raining, then I do not carry my umbrella." If these were not the case, you might just as well have said, "I carry my umbrella all of the time!" However, the truth of the inverse and converse of a statement are logically unrelated to the truth of the initial statement. Why do we often fall into this trap (known as a converse error)? Because, everyday comments often carry an unstated second meaning. ![]() We do not need to accept the statement, "if an American city has at least one college, then it is great." Just because a premise implies a conclusion, that does not mean that the converse statement, if B, then A, must also be true. Leaving aside the truth of that conclusion, it is not a logical deduction. They hope that we will then conclude that Worcester is great. ![]() The advertisers then share with us that Worcester has ten colleges, that is, that it satisfies the conclusion of the statement. We can convert the above statement into this standard form: If an American city is great, then it has at least one college. ![]() What is the advertisement trying to suggest? Is it supplying the necessary evidence?Ī conditional statement is one that can be put in the form if A, then B where A is called the premise (or antecedent) and B is called the conclusion (or consequent). This billboard advertisement plays on the fact that people, in both daily life and within mathematics classes, tend to treat related, but logically distinct, conditional statements as equivalent. Logical Relationships Between Conditional Statements: The Converse, Inverse, and Contrapositive Every Great American City Has At Least One College.
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