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What is the Opposite of Contrapositive?

Published in Logical Statements 3 mins read

The opposite of a contrapositive is best understood by examining the other related logical statements stemming from a conditional statement. According to the reference, the contrapositive of the conditional statement "If P then Q" is "If not Q then not P." The related statements we need to discuss are the converse and the inverse.

Conditional Statement and its Variations

Let's define a conditional statement and its variations in a tabular format:

Statement Form Example ("If it's raining, then the ground is wet.")
Conditional If P then Q If it's raining, then the ground is wet.
Converse If Q then P If the ground is wet, then it's raining.
Inverse If not P then not Q If it's not raining, then the ground is not wet.
Contrapositive If not Q then not P If the ground is not wet, then it's not raining.

Understanding the Relationship

  • The contrapositive is logically equivalent to the original conditional statement. This means that if the original statement is true, its contrapositive is also true, and vice versa.
  • The converse and inverse are not logically equivalent to the original conditional statement. They may or may not be true even if the original statement is true.

What is the Opposite?

Considering the table above, while "opposite" might be interpreted in various ways, there are two prominent "opposites" to a contrapositive. If we strictly adhere to the concept of directly negating a statement, the "opposite" of the contrapositive "If not Q then not P" would be a statement where that exact structure is negated. However, in a more practical sense of logical operations, it is more helpful to consider two distinct "opposites" by the following analysis:

  1. The Inverse:

    • The inverse statement “If not P then not Q” is logically related to, but distinct from, the contrapositive. It changes the truth of P and Q and is not logically equivalent to either the conditional or contrapositive.
    • If we use "negation" to define opposite, the inverse could be seen as an opposite of the contrapositive in that both negate P and Q, but reverse the order within the implication.
  2. The Converse:

    • The converse of the conditional, “If Q then P,” reverses the direction of the original conditional statement. While it doesn't negate anything, it provides a differing perspective by switching the roles of P and Q.
    • The converse is the closest statement to the inverse in how it relates to the conditional, and if one were to consider that the "opposite" is simply the reversing of the P and Q order, the converse is the closest.

Conclusion

Therefore, there isn't one single opposite of a contrapositive. Depending on the context, the "opposite" could refer to:

  • The inverse of the conditional, "If not P then not Q" due to the negation component.
  • The converse of the conditional, "If Q then P" due to the reversing of the implication, P then Q.

It is crucial to understand that while the inverse and converse are often confused with each other and the contrapositive, only the contrapositive is logically equivalent to the original conditional statement, as shown in the provided reference.