Title: Errors in Deduction

Summary: Will Sherlock ever admit he's wrong? After a tiff over the milk and an angry walk, John comes to better understand Sherlock and manages to solve the milk problem.

Rating: K

A/N: Thanks to chappysmom for giving the rough draft a read-over and offering very helpful feedback. This final version is much improved as a result, but of course, any remaining weaknesses are my own fault.

"Sherlock, why can't you ever just admit that you're wrong?" John asked, utterly exasperated and starting to get more than a bit angry. After a difficult day at the clinic, John had not been happy to discover they were out of milk (again), and that the guilty culprit had not replaced it (again). It had been the final straw that had led to the heated discussion which had been going on for several minutes.

"Of course, I can admit that I'm wrong, John. Don't be stupid. You've seen me admit that I'm factually wrong many times on a case when new information comes in. There would be no purpose to The Work if I were not willing to accept the possibility that my deductions might be in error. As for admitting that I am ethically or morally wrong," Sherlock paused for a moment, "it's unlikely."

"Well, here are some facts for you. Fact: We are out of milk. Fact: You are the one who used it up. Fact: You are perfectly capable of going to Tesco and buying milk. Conclusion: You are in the wrong here!" John's voice had risen to almost a shout by the end.

"Congratulations, John," Sherlock replied at rapid-fire pace, "you have succeeded in establishing facts that were never in dispute. However, your conclusion from those facts is only partially based in fact itself. It is also based on notions of fairness, judgments of the relative value of my time and yours, and numerous other factors that are subjective and decidedly not factual. And so," he finished bitingly, "I will not be admitting that I am wrong any time in the foreseeable future."

Very tightly, John responded, "Right then. I'm off out," and shut the door quite firmly on his way out.

John had rapidly covered the first few blocks and cataloged dozens of reasons that Sherlock was absolutely, positively wrong. As his anger dropped a couple notches from blazing mad to just royally ticked off, John began to think about what Sherlock had said.

John had long held the opinion that Sherlock refused to admit he was wrong, but Sherlock claimed that he often did. As John began to really think about it, he realized what Sherlock had said was true, and his pace slowed abruptly. He had indeed seen Sherlock form deductions and then later discard them in the light of new information, and he'd actually done it quite freely. How could his perception have been so far off?

Of course! That's all it was. John sped up, and pedestrians again began swerving out of his path. Sherlock admitted he was wrong when new information and evidence came to light. It was merely an acknowledgement that he had had incomplete information to work with. He wasn't actually ever admitting any error in his deductions. He was too arrogant for that! When had he ever admitted that someone else had made a better deduction than he had when they both had the same information to work with?

Mycroft! And John slowed again. After the explosion on Baker Street, when John came back to the flat, Sherlock had asked him about the lilo. Then Mycroft had quietly corrected his brother and said that it was the sofa. After a brief pause during which he'd looked John over and noticed who-knows-what, Sherlock had replied just as quietly, "Yes, of course."

Finally, John came to a complete halt. Sherlock had not only admitted he was wrong when someone had corrected his deduction, but it had been Mycroft, for crying out loud. And Sherlock hadn't been stroppy about it either. How had he not realized the significance of that at the time? Then John could almost hear Sherlock saying, "You see, but you do not observe."

"Oh, shut up!" John said aloud and then noticed a nearby woman looking at him strangely. "I'm sorry. Not you. I was just, you know, thinking out loud, and it just, I—Sorry." He began to walk again.

Mycroft seemed like the last person Sherlock would accept a correction from given how they sniped at each other. So Sherlock wasn't as arrogant as John had thought. He could hear a correction of his deductions, reevaluate the evidence, and admit that he was wrong and the other person right. Extraordinary!

But Sherlock could have done that and still been snippy, made some comment about a stopped clock being right even twice a day or some such thing. But, he hadn't. Sherlock's reaction to Mycroft's correction suggested that he had immediately assumed that Mycroft must be right. Since when did Sherlock ever admit that Mycroft was right, even about something as simple and obvious as the time of day? But Mycroft with the same evidence had out-deduced Sherlock, and effortlessly, barely even looking at John. Could Mycroft actually be better at deducing than Sherlock, and Sherlock actually be willing to admit that, at least to himself, and to some degree to Mycroft? Would wonders never cease?

And with that, John realized he wasn't nearly as angry at Sherlock anymore. But, the milk problem still needed solved, or he'd be angry again soon enough. John was still convinced that he was right about that, even if it wasn't factual enough for Sherlock. What to do about it?

John remembered something his therapist had said once when he'd been very angry at Harry, more than he usually would have been. His therapist asked him to think about what was really upsetting him. Was all of that anger a result of Harry or was some of it coming from other sources but being directed toward her?

So, why was he angry about the milk? It wasn't because of money. A few weeks after they became flatmates, John had initiated an awkward one-sided conversation about how to divvy up expenses for groceries and household items. After looking at him for a moment, Sherlock handed him a credit card and said, "Here, use this when you think it appropriate."

Startled, John asked, "Wouldn't you rather keep it and I can ask for it when I need it and tell you what for?"

"Boring. You will make every effort to be scrupulously fair. Take it." As John slowly took the card, Sherlock said with a smirk, "Besides, if you ever were to misuse the privilege, you would be kidnapped in short order."

And so, John wasn't quite sure whose money was helping to pay for the household expenses, Sherlock's or Mycroft's, but he did use it as he thought appropriate.

So was he angry because he didn't want to always be the one to do the shopping? Well, there definitely was a part of him that did think it was more than a bit unfair. But, did he want Sherlock to do the shopping? John actually couldn't picture Sherlock managing to go to Tesco, get everything he was supposed to without permanently ticking off the workers, get it all home and put away in the proper places. It probably was best that John did the shopping. But to find out it needed done after he'd already gotten home from a horrible day…. Ahh! There was the problem.

He was ready to go back to the flat, after he made a quick stop for milk and a few other things. And not only would he return with a solution to the milk problem, but he felt that he understood Sherlock better than he had before. After a long, horrible day, and a really bad fight, it seemed like the evening might end on a more positive note.

When John entered the flat, Sherlock looked up and started to open his mouth, but John spoke first. "Right. So, I figured out what the problem really is. I don't mind too much always doing the shopping. What really peeves me is when I discover we're unexpectedly out of something and I have to do without or go to Tesco right then, especially after a horrible day. So, I have a solution. When you use up milk or anything else, you text me so that I know and can add it to the shopping list."

Sherlock raised his eyebrows. "You don't want me to add it to the list myself?"

"No, because if you'd added it while I was at the clinic, I'd have still come home and been surprised to find no milk when I could have make a quick stop on the way. Which would have been more efficient, and less annoying. I'd ask you to mention it and add it when I'm here and only text me when I'm out, but since you don't always notice when I've left, that seems doomed to fail. Just text when you use something up, and I'll take care of it. Think you can do that?"

"Yes. Good. Quite logical. John…. I appreciate that you're willing to do the shopping. I—regret that I didn't think to text you about the milk."

"Yeah, well, thanks." John took a deep breath. "What I said was unfair, about you never admitting you're wrong. You're right. You do admit when you're in error in your deductions. I've even see you do it when it's your 'arch-enemy' pointing it out," with a rueful grin. "So, I'm sorry."

"John," Sherlock began and paused, uncertain what to say. Then as a sort of peace offering, he said "Who do you think taught me to deduce?"

"Well, I—" John was actually a bit gobsmacked, unsure of what to say. And then he found his footing. "Actually, I thought you were born deducing. Figured you gave the obstetrician the shock of his life when you told him his life's story before he'd finished cutting the cord."

"Hardly, John," Sherlock said with a wry smile.

And it was all good again. John knew there would be other arguments. He may have succeeded in putting an end to The Great Milk Wars, but there'd be plenty of other things to bicker about. For now though, there was peace.