Modus tollens in logic
In logic, there is an argument style called modus tollens:

If  H0 then R
Not R
therefore
Not H0

This is a valid argument.

Modus tollens misapplied to p-values
Some people mis-apply this to p-values, saying:

If H0 then probably not R
Not R
therefore
Probably not H0

This is not valid.

Example of proper and improper modus tollens

Proper:

If a person is an American citizen, then that person is not a member of parliament
This person is a member of parliament
therefore
This person is not an American citizen

Improper

If a person is an American citizen, then that person is probably not a senator
This person is a senator
therefore
This person is not an American citizen