Given the statement: "A || B" (where A is "take this course of action" and B is "I will quit"), if A is false (i.e., !A is true) then logically B has to be true

Given the statement: "A || B" (where A is "take this course of action" and B is "I will quit"), if A is false (i.e., !A is true) then logically B has to be true. The moment an employee says "Do this or I quit", and the company refuses to "do this", then the company is correct in assuming that the "I quit" part will apply.