Understanding the Promise
When A promises to give consideration to the police for finding his lost child, it raises important questions about the enforceability of such promises under contract law.

Legal Principles of Consideration
- **Definition of Consideration**: Consideration refers to something of value that is exchanged between parties in a contract. For a promise to be enforceable, there must be a valid consideration.
- **Past Consideration**: If A's promise to reward the police comes after the act of finding the child, it is considered past consideration and is generally not enforceable.

Public Duty and Legal Obligations
- **Public Duty**: The police have a legal obligation to perform their duties, which include searching for missing persons. Thus, their actions in finding the child do not constitute consideration in a contractual sense.
- **Implication of Promises**: A's promise might be seen as a moral obligation rather than a legal one, as the police cannot be legally bound to act based on A's promise.

Conclusion on Enforceability
- **Lack of Consideration**: Since there was no exchange of consideration at the time the promise was made, and the police were already obligated to find the child, A's promise is not enforceable.
- **Moral vs. Legal Obligation**: A may feel an ethical responsibility to reward the police, but this does not create a binding legal contract.
In summary, without valid consideration and given the existing obligations of the police, A's promise to reward them for finding his child is unenforceable by law.