Is patriarchy in society a good or bad system?
I have no issue with it, regardles of being male. Like it or not, someone has to be in charge, and the members of a mammalian family, be it a wolf pack or nuclear human family, are actually happy knowing who is in charge.
I don't see any issue of men being considered the default heads of the household in many ways, but that does not mean that I think it is allowable or good to promote chauvinism. A theoretical example of a man "being in charge of things" is that relations between different households would be managed by the interaction between the two patriarchs. There would be standardization, and therefore consistency, and predictability.
Why is this beneficial in a societal context? Consider the following analogy, when my wife takes her minivan to the dealership for service, they try to bullshit her and sell her alot of unecessary services/items. When I go to the dealership, despite knowing jack shit about cars, they talk nicely to me, don't screw around with trying to B.S. me, and schedule my car for service so it can be picked up when I need it. See the difference? Our society is a chauvinist society, so in many cases, women are by default at a disadvantage in dealing with men in many social contexts.
Outside of our society being chauvinsit, my other issue with a "pure" implementation of patriarchism is that at this time, not all men accept the responsibility of being the head of the household. Many are sorry pieces of shit, frankly, and expect their wives/girlfriends/mothers to clean up after them, work, cook, take care of the kids and still have enough energy (and desire) to be sweet and sexy after Big Daddy is done drinking beer and watching football. Please. The benefits of a patriarchal society can only be relaized if all men in the society stepped up to the plate and delivered on fulfilling their patriarchal duties. In my opinion, if a man doesn't he shouldn't be entitled to patriarchal entitlements.
Would a matriarchy in society be a better or worse system?
Most likely better in some ways, and worse in others.
Would equality be the ideal goal between such systems?
Isn't equality a fundamentally desirable goal in any system? Seriously.
Analysis, discussion welcome.
This is a really good series of questions you posed.
Nowadays, despite our chauvinistic unofficial patriarchal society, there are a few realities to consider, checks and balances if you will.
First, it is the the strength of a person's character and their will that will determine if they wear the britches in their family or not. There are plenty of households nowadays where the woman of the house makes more cash than the man of the house. So, the woman's liberation movement has caused a fair amount of equality be closing the gap on the financial earnings capabilities between men and women.
Second, any Father who laments over not having one or more male children is a fool. The definition of success from a biological perspective is "did you reproduce, and pass your DNA onto the next generation?" Not "Did you reproduce and create a male offspring." I am the Father of two girls, and I am raising them the same as if they were two little boys. They are both assertive, opinionated, smart, and aware of their rights to be treated with respect and civility by everyone. As a matter of fact, both of my kids demand it, as that is the only construct of interpersoanl relations they have ever known. So, I hope I end up with two cool sons in law, but even if they are average, I don't have to worry about my grandkids, because I know that their Mothers will have their shit together, give them a stable and nurturing home, and teach them what they need to know to be independent and successful in life.
Third, despite this being a "man's world" there are certain parts of it that have been manipulated to the benefit of women. I'll guarantee you that on average, women get less speeding tickets in their lifetimes than men of their same generation. Divorce laws in some states are completely archaic. My brother had to pay out the nose to go through the steps of separating from, supporting, and ultimately divorcing his ex-wife, despite her ruining him financially, and her being unfaithful countless times.
So, despite the skew of things in all directions, the diversity of the character and will power of the men and women of the human race assists in some capacity in creating a system that is more balanced in the distribution of power between the sexes than would occur in either pure patriarchy or matriarchy, but nonethelss, our system has alot of work to do if it ever wishes to be a candidate for accomplishying the virtues of equality.
In my household, my wife and I make decisions in a very fair manner. Anything that costs more than the average price of a trip to the grocery store (~$200) we consult with each other first before impulse buying. Regarding scheduling, we split holidays between her family and mine pretty evenly. Both of us contribute all of our paychecks to the family checking account. I put my money in her account as I am not good with managing administrivia, but when it comes to strategic planning or financial management of larger resources she defers to my analysis, but even still I giver her three options, and we choose the one that both of us are most comfortable with. When it comes to interacting with the neighbors, that's my turf, as she is an INFJ and socializes on her terms.
When it comes to the kids, they have both been told that Daddy AND Mommy are both in charge. If one tells them to do something, the other will back the other parent up. We present a unified front, and that makes things alot easier to deal with when only one of us is around. But in all honesty, they give me a lot less grief than they give her. First because Daddy doesn't play, I don't give warnings, I go straight to punishment (loss of privileges, time out, penance via chores, etc.) for things they know are wrong to do, or for not doing as I command them. Second, I love hanging out with my kids, so I take them wherever I am going, and in the course of me getting my stuff done, listen to some tunes they like in the car, get a scoop of ice cream, and stop at playgrounds, so they know if they just keep their cool and behave like they know they should, that I will hook them up by letting them do things they like to do.
