Theorem 2.3.1. Modus tollens.
A conditional and its contrapositive are equivalent.
Section 2.3 Converse, inverse, and contrapositive
Related to the conditional \(p \lgccond q\) are three important variations.
- converse
- \(\displaystyle q \lgccond p\)
- inverse
- \(\displaystyle \lgcnot p \lgccond \lgcnot q\)
- contrapositive
- \(\displaystyle \lgcnot q \lgccond \lgcnot p\)
Proof.
We simply compare the truth tables.
| \(p\) | \(q\) | \(\lgcnot p\) | \(\lgcnot q\) | \(p\lgccond q\) | \(\lgcnot q \lgccond \lgcnot p\) |
| \(\lgctrue\) | \(\lgctrue\) | \(\lgcfalse\) | \(\lgcfalse\) | \(\lgctrue\) | \(\lgctrue\) |
| \(\lgctrue\) | \(\lgcfalse\) | \(\lgcfalse\) | \(\lgctrue\) | \(\lgcfalse\) | \(\lgcfalse\) |
| \(\lgcfalse\) | \(\lgctrue\) | \(\lgctrue\) | \(\lgcfalse\) | \(\lgctrue\) | \(\lgctrue\) |
| \(\lgcfalse\) | \(\lgcfalse\) | \(\lgctrue\) | \(\lgctrue\) | \(\lgctrue\) | \(\lgctrue\) |
As the two “output” columns are identical, we conclude that the statements are equivalent.
Corollary 2.3.2. Modus tollens for inverse and converse.
The inverse and converse of a conditional are equivalent.
Proof.
The inverse of the conditional \(p \lgccond q\) is \(\lgcnot p \lgccond \lgcnot q\text{.}\) The contrapositive of this new conditional is \(\lgcnot\lgcnot q \lgccond \lgcnot\lgcnot p\text{,}\) which is equivalent to \(q \lgccond p\) by double negation.
Warning 2.3.3. Common mistakes.
- Mixing up a conditional and its converse.
- Assuming that a conditional and its converse are equivalent.
Example 2.3.4. Related conditionals are not all equivalent.
- Suppose \(m\) is a fixed but unspecified whole number that is greater than \(2\text{.}\)
- conditional
- If \(m\) is a prime number, then it is an odd number.
- contrapositive
- If \(m\) is not an odd number, then it is not a prime number.
- converse
- If \(m\) is an odd number, then it is a prime number.
- inverse
- If \(m\) is not a prime number, then it is not an odd number.
Only two of these four statements are true! - Suppose \(f(x)\) is a fixed but unspecified function.
- conditional
- If \(f\) is continuous, then it is differentiable.
- contrapositive
- If \(f\) is not differentiable, then it is not continuous.
- converse
- If \(f\) is differentiable, then it is continuous.
- inverse
- If \(f\) is not continuous, then it is not differentiable.
Only two of these four statements are true!
