One person may not unilaterally alter a contract, such as a contract for the provision of goods or services or a promissory note. This includes oral, or spoken, agreements as well as written ones. Instead, changing the terms, such as to reduce the amount owed, requires the agreement of both parties.
The person writing the check may attempt to argue that by writing “paid in full,” he offered to settle the debt for that amount, and that by cashing the check the other party accepted the offer, forming a new agreement to accept that amount as full payment. However, a contract requires consideration; one party must pay or provide something to the other party. Furthermore, the consideration must be something that the party is not already obligated to pay or provide.
In this case, person A owes money to person B. If person A offers person B some portion of that, that money will not constitute consideration since A had to pay that much—and more!—any way. The preexisting obligation negates the check’s value as consideration, and without consideration, no agreement or contract would be formed. Without a new agreement or contract being formed, the first person, person A, cannot unilaterally (on his own) alter the terms of the existing obligation. When you are owed money by a person, you can cash a check for less than the full amount due to you without writing off the balance or accepting that check as payment in full. You’d have to credit it againtst the balance owed, but that’s the only obligation. Of course, you can decided to accept less than full payment, if you’re in a charitable mood, or think that’s the most you’re going to get, or think that the relationship with the client is sufficiently important. But that must be your choice—it can’t be unilaterally forced on you.