Answer: No, Paul has not breached a contract.
Explanation: To answer this, we must first we must define what a contract is.
A contract is an agreement between two or more people that is legally binding, and which guides or governs the actions or conducts of the parties involved.
A quality that makes a contract legally binding is that it is enforceable by law.
In the scenario given in the question above, Paul has not breached any contract because there isn't one. The promise to buy dinner has not been legally bound, therefore, it is not enforceable by law, in essence, it is not qualified to be called a contract.