# Solving a riddle using Boolean logic

1. Feb 16, 2014

### spaghetti3451

1. The problem statement, all variables and given/known data

A famous lawyer takes on an apprentice on one condition: the lawyer will train the apprentice on the business, and the apprentice will have to pay the lawyer only after she wins her first case. Right after the end of the apprenticeship, the lawyer sues his own apprentice for the amount owed.

The lawyer argues that if he wins the case, then he will be paid his due. If the apprentice wins the case, he will still get paid because she agreed to pay if she wins the first case. The apprentice on the other hand claims that if she wins then by the court's order she is no longer required to pay. On the other hand if she loses the case, then according to the original contract she is no longer obliged to pay.

2. Relevant equations

3. The attempt at a solution

Let W be the proposition "The apprentice wins her first case" and let P be the proposition "The apprentice pays her lawyer."

The apprentice will have to pay the lawyer only after she wins her first case, i.e., W$\leftrightarrow$P.

Lawyer's arguments:

1. If the lawyer wins the case, then he will be paid his due, i.e., $\neg$W→P. This does not follow from the premise.

(The court rules do not apply since the premise excludes court rules.)

2. If the apprentice wins the case, the lawyer will still get paid (because she agreed to pay if she wins the first case), i.e., W→P. This is the premise itself.

(The original agreement applies since the premise is the original agreement itself.)

Apprentice's arguments:

1. If the apprentice wins, then (by the court's order) she is no longer required to pay, i.e., W→$\neg$P. This does not follow from the premise.

(The court rules do not apply since the premise excludes court rules.)

2. If the apprentice loses the case, then (according to the original contract) she is no longer obliged to pay, i.e., $\neg$W→$\neg$P. ???

2. Feb 16, 2014

### maajdl

The lawyer will lose and will be paid.
When the court judges the case, the apprentice hasn't yet won any case and the lawyer loses.
However, as soon as the court has judged the case, apprentice must pay according to the contract.
Otherwise, the lawyer might sue a second time an win the case then.

3. Feb 16, 2014

### spaghetti3451

I am not sure if your analysis follows the rules of discrete maths.

I am wondering if we should base our analysis of the problem only on the original payment contract or if the lawyer and his apprentice are also bound by the court rules.

In practical scenarios, informal contracts such as those agreed upon by the lawyer and his apprentice carry little weight. Court decisions overrule all such informal agreements.
However, this is a hypothetical problem in discrete maths where personal contracts could hold sway over the outcome of events.
In this regard, I am not sure if I should treat the court rules as a red herring.

I understand that the heart of the problem lies with the conflict between the personal agreement drawn out by the lawyer and her apprentice and the opposing court rules.
I simply need a clue to finish up the problem.

4. Feb 17, 2014

### maajdl

At the time the lawyer goes to court, the apprentice hasn't won any case yet.
Therefore, the decision of the court, in this precise case is clear: the lawyer lose the case, the apprentice wins.
After the court has decided so, then the apprentice has won his first.
From that time, the lawyer has the right to demand his due again.
He would then win if he was going to court for a second time.
Therefore, the apprentice will then have to pay.

Time is the aspect that needs to be formalized if you absolutely want to follows formal rules.
But formal rules are of no help if you miss the importance of time in this question.

Last edited: Feb 17, 2014