we have paid out to this contractor over half the amount due and he has not finished the work
Your primary legal relief will be through suing the contractor for breach of contract. However, the court will determine whether the contractor breached his contract with you based on the whether he materially complied with the terms of your agreement. As such, it is important that this contract specifies terms that the contractor breached. For instance, if the contract requires the contractor to pay specific damages in the event of his breach, then you could certainly recover that amount in a contract suit. However, if the contract requires the contractor to complete work before receiving the second half of the contract price and he did not finish the work, then you could hire another contractor to finish the work and sue the contractor in breach for an amount equal to the difference between what you paid to have the work completed by another contractor and the lower amount you would have paid under the initial contract.
Also, I will stress again that you carefully consider what your contract specifically requires from both the contractor and you. You do not want to withhold payment from the contractor due to an alleged breach of contract only so that a court will then decide that the contractor was either not in default or in material default under the terms of the contract and it is actually you now who is in breach by withholding payment.
Lastly, in addition to basic breach of contract, each state has specific state statutes that may address particular industries (such as construction) and various consumer rights. These state specific statutes may very well be additional avenues of relief for you.