What Is the Inverse of a Statement?

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In propositional logic, a conditional statement is an "if-then" sentence construction that links together an independent (p) and dependent (q) proposition. For any conditional statement, there are three possible permutations of it: the inverse, the converse and the contrapositive. The inverse has the same syntax as the original statement, except that both the p and q propositions are negated.

Proposition

  • From a logician's standpoint, a "proposition" is a statement that defines an idea or concept. This statement can refer to an idea that is tangible (e.g. the dog is hungry) or abstract (e.g. the dog is noble). The only requirement is that the proposition must be either "true" or "not true." Propositional logic does not allow for an ambiguous third value.

Conditional Statement

  • A conditional statement consists of two propositions: a hypothesis and a conclusion. The syntax of the statement establishes a specific relationship between the propositions; namely, which proposition is the hypothesis and which is the conclusion. Generally, the syntax of a conditional statement uses an "if-then" construction, i.e. "If [hypothesis], then [conclusion]."

    Translated, this statement is claiming that whenever the hypothesis has a certain value (e.g., "true"), the conclusion must have a particular value (e.g. "true"). For example, if the proposition "it is raining" is true, then the proposition "the ground is wet" must be true.

The Hypothesis, Or "P"

  • In textbooks, logicians use the letter "P" to represent the hypothesis of a conditional statement. Within this context, the hypothesis can be considered "independent" because its value is not predicated on the value of the other proposition.

The Conclusion, Or "Q"

  • Meanwhile, the letter "Q" is reserved for the conclusion of a conditional statement. The conclusion is dependent on the hypothesis, but only for certain values of the latter. For example, if "it is raining" is true, then "the ground is wet" must also be true. However, if "it is raining" is false (i.e. there's no water falling from the sky), then "the ground is wet" could be either true or false (i.e. maybe a dam broke and flooded the area). The original conditional statement only defines the relationship for one value of the hypothesis.

Inverse Statement

  • In the inverse of a conditional statement, the values of both the hypothesis and conclusion are inverted. For example, if the original statement was "if it is raining, then the ground is wet," the inverse of that statement would be "if it is not raining, then the ground will not be dry."

    Note: Constructing the inverse isn't about switching all of the propositions' values to "false." Rather, the essence of the inverse statement is "inversion," which meaning switching "true" values to "false" as well as "false" values to "true." For example, the inverse of the statement, "If I do not have a coat, then I will get hypothermia," would be "If I do have a coat, then I will not get hypothermia."

References

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