My take on this problem is that it is a classic tension between unjust enrichment and contract terms. You contracted for only 2 mows and he delivered 4 mows accidently. If you pay only for two mows, you got the benefit of two extra mows (the unjust enrichment), but are paying for something that you didn't ask for (the contract terms). If you pay for all 4, you are paying for a service you didn't want. In my opinion, you would probably end up winning in a small claims type of action because you specifically requested a less frequent mowing, and the lawn service accepted your terms. But this is far from a done deal, for instance if you knew about the extra mowings, but didn't stop them, (or they can paint that picture regardless of your real state of mind) it looks like you tried to get free mowings without paying for it, which might lower your chances of winning.
Practically, I would concentrate on the negotiations rather than the legal stance because no amount of legal correctness will probably convince the business owner, and it probably isn't a worth his time to sue you to recover. You could send a check with a letter explaining that you had only requested two mowings, and that you are only willing to pay for 2. Write "Paid in Full" on the check in the memo area, and hope they cash it. If so, you are probably done because you had a legitimate dispute about the amount owed, and you offered to settle for the amount of the check. Cashing the check probably counts as acceptance of your offer. The other choice the business owner has is to send you to collections, where they will probably recover less than your 50% payment, so the dollar-wise move on their part is to just cash the check.
You can see more about "paid in full" checks here: http://www.weisslawstl.com/2005/12/01/accord-satisfaction-how-notation-check-sometimes-unwittingly-resolve-payment-dispute
It may vary according to the jurisdiction you live in.
I had a similar dispute with an electrician. I had warned him before he came out that I had isolated the problem to a long line of buried cable that required special equipment to diagnose. I asked them to show up with the right equipment to diagnose, including boots and machetes to follow the run of cable. They showed up with neither, failed to diagnose the problem, then sent me a bill for the callout. I explained my position and that I was happy to either go to court if they wanted to sue, or go to court if they turned me over to collections, and that they should just walk away and I wouldn't complain about the time they wasted showing up unprepared to diagnose my problem. They did walk away after much huffing-and-puffing about it.