Something else to realize is that your mom may not be "all there" - so to speak (for any number of reasons, old age being just one). My own mother was able to hold it all together for far longer than my wife and I realized from over-the-phone conversations (my mother and I lived in different states - her in California, me in Arizona); it wasn't until she fell and broke her hip that we realized just how much help she really needed. It wasn't until later that we realized that she had been fooling us quite adroitly about her own deterioration (indeed, she probably didn't realize it herself - a person generally can't be objective about their own brain state, unfortunately).
Now, maybe (hopefully), your mother isn't in this same situation - but you may want to get her evaluated as soon as possible. You may also want to discuss things about wills, power of attorney (medical and general), living trusts, etc - right now, before your mother deteriorates any further. My own mother was adamant about us (my wife and I) "not being in her business" - until she fell, and we found that she was hording 3 years of uncashed/undeposited social security checks in her purse. Her arguing with you about the "logical order of things" may be nothing - or it may be an indicator of something greater.
We found out later in the hospital that my mom had a form of aphasia (she couldn't use any "A" words - show her an apple, she knew what it was, but couldn't name it properly - over the rest of her brief time with us, her aphasia progressed - sometimes to a point where we had to wrack our brains trying to figure out what she was talking about - she would say things that didn't make any sense at all; complete nonsense - until you unraveled them in an "aphasic" manner - then it made complete sense).
Do what you can while you still have time, while your mom still have some sense of awareness about her - because if your mom is declared incompetent by the State, and you don't have your legal p's and q's in order, things will get really difficult, really quick - not to mention expensive (you'll be in for a shock if you ever have to price good nursing home care for dementia/Alzheimer's patients).
Good luck, and I hope it works out well for both of you - I really do.