Contrary to popular opinion, there is no reason human society is intrinsically misogynistic. If, as you say, your society makes a conscious decision to attempt to avoid that fate, they’re likely to succeed.
Let’s run through some particulars.
Physical Force and Equity
Societies do not, as a rule, run on physical force. If you have a society that is working relatively smoothly, especially if the society has generally agreed that peaceful coexistence is preferable to continual violence and danger, then you’re not going to need a great deal of violence to get things done. Indeed, violence will quickly come to be indicative of serious problems to be dealt with.
The crucial point is that the society must hold the respect of the large majority of its members. Because each of us is individually associated with the collective, crimes against individuals appear, from the perspective of each individual, to be crimes against society as much as against individuals. This nullifies the “not my problem” effect that is so crippling in, for example, much of today’s United States.
In essence, the notion of obedience to law—don’t kill, don’t steal, don’t rape, etc.—must rest on mutual social respect rather than on fear of punishment. In real life, we don’t just attack others and steal their stuff, nor does it usually occur to us to do so. This is because we respect the society of which we are members, and we recognize that the same respect is therefore due to other members. The notion that intensive, violent policing is absolutely necessary to prevent people turning into psychopaths is a sick fantasy, albeit a popular one. Most tribal societies do not have police systems at all, for instance.
So we can set one point aside: although women will tend, all things being equal, to lose physical fights with men, this has very, very little to do with whether they will be treated as second-class citizens (or worse).
But What About Farming?
Look up rice, the world’s most popular crop. Who plants and harvests it? Who maintains it in the paddies? Why… women and children? Huh. Apparently the notion that farming is automatically men’s work is a chauvinistic fantasy too.
Now rice is not an ideal crop in all circumstances, but it’s a very good one in many. This is because while the total labor put into it is perhaps higher than with some crops, that labor can be stretched out over long periods of time. In addition, it tolerates a fairly wide climatic range.
An interesting effect of a constant-labor crop like this is that children become remarkably valuable in the short term. You can certainly put an 8-year-old to work in the paddies without feeling you’re being cruel. The obvious example is water-cycling. You have a simple pedal mechanism (you can surely scrounge that from the wreckage of your apocalypse, or just build your own like the Chinese have for many, many centuries) with paddles or a water-screw. Put the bottom end in the lower paddy and the top above the upper one. Now pedal. The idea is to move water from the lower to the upper, slowly, in a controlled way, thus mitigating against the constant effects of seepage. Since it turns out that you get better crops the more precisely and constantly you do this, your children become valuable members of your farming society very soon.
One big concern that tends to come up in these things is an evaluation of what any given person can or cannot do on his or her own. Certain kinds of back-breaking labor seem to require men, because only men have the upper body strength to do it themselves. But this is a nonsense.
To begin with farming. Rice, once again, points to another essential issue for your society: collectivism. Everyone gets rice equally. It’s not my paddy, my rice, but our paddies, our rice.
But this point expands greatly.
In our little town, Frieda makes furniture, Warren does weaving, Delilah is great at dying, and Charles makes cheese. When we sit down to a meal, do we celebrate Charles because he’s the most useful? Well, only if we’re crawling on the floor naked. Otherwise, everyone has contributed. And the same goes for Betty the baker, Sam the sausage-maker, and so on and so forth.
This should be obvious. If you have a balanced, planned society, you draw on the skills and talents of each individual. All those abstractions about upper-body-strength and whatnot fail in the face of real life. Charles, for instance, is built like a gorilla… but there was that childhood accident, and he can’t walk without crutches. So do we send him out to do farming because he’s got a penis, or kill him because he’s not “whole,” or do we enjoy the cheese?
In fact, almost nothing that has to happen in the village is intrinsically sex-divided. Interestingly, however, some tasks are likely to become so. The trick is to avoid this being seen as a matter of superiority on one side or another.
Take weaving, let’s say. Anyone can do it—there’s no special strength or whatever involved. But in many communities through the ages, weaving became a female activity. Why? All kinds of local reasons, few if any now recoverable. But once you’ve established a situation like this, it tends to perpetuate itself.
In essence, what happens is that a whole bunch of women get together to talk while they work. And they talk about what interests them, of course. And, as everyone knows perfectly well, if you get a bunch of women together to talk about whatever they feel like talking about, the conversation will tend over time to become woman-oriented. I don’t mean that women talk about mothering and menstruation and so forth all the time, only that women and men do tend to have socially-distinct conversation patterns in any given society. From a male perspective, the women may be gossiping or whining or whatever, but then again, from a female perspective the analogous male conversation is a bunch of grunting, fart jokes, and penis-waving. Neither is true—but these conversations do tend to be sex-restrictive.
Now once weaving becomes a female activity in this sense, you’re going to see men somewhat disinclined to join, because they feel unwelcome and uncomfortable. (As a side note, I wonder whether your postapocalyptic equal society will be open about those gay men who do feel more comfortable in such social circles, and have them join the weaving.)
Meanwhile, there are going to be analogous “male” activities: baking, let’s say. If in our society baking becomes a very masculine, even macho activity, what will happen? Pretty soon, they’ve probably got this whole thing going about how bread is a woman who has to be pressed into shape (or a breast to be fondled, etc.), and sweat, and heat, and columnar phallic ovens, and whatever.
Now I choose baking for the simple reason that it does require considerable upper-body strength, it’s hard work, and in many societies the bakers—especially if part of a collective that includes threshing and milling—are male. But also because lots of societies deem any form of cooking intrinsically female.
The point should be clear: sex-division with respect to labor is not about intrinsic abilities or the nature of any given task, but about social construction. Provided you want to end up with equity across the sexes, you need to make sure that people value cloth as much as they do bread (and so forth).
Equity and Law
I suggest that you start with the assumption that law, in the strong sense, will be a minimal issue in your society. Not only is constant policing not necessary most of the time, but all the complexities of contracts and so forth are irrelevant to your postapocalyptic society. Furthermore, you have a problem working out what sort of law to depend on in the first place.
Common law, a la Britain and the US, depends on precedent, which means a vast library of case law. But in your postapocalyptic society, has such a library been preserved well? Who’s maintaining it? Do the people generally trust the lawyers to do this? They don’t trust lawyers now, so why would they in the future?
Inquisitorial law, as in France, has the advantage that at least it’s answerable to truth instead of precedent and opinion. And yet, the principal force preventing rampant abuse is very strong hierarchical submission: this court doesn’t abuse its power because the next higher court will crush it brutally if it does. We cannot count on a system like this in our broken world.
There are other possibilities, of course, but my point is that legalism simply isn’t a good place to begin planning.
So what does this mean for women?
It means that women must be equal citizens in actuality and not merely in law. It means, to put that another way, that you can’t build your society by simply trying to legislate the morality you want: you have to build a society that works.
How It Really Works
On the Farm
Women can carry babies on their backs while doing an awful lot of farm work, from planting and harvesting rice to mucking out or feeding livestock. Children over about 6 months do not need continuous breast-feeding: they can eat “solid food” and be topped up with breast-milk whenever convenient. Children who don’t have to breast-feed constantly can be carried or watched by men just as well as by women.
As social planners, you’ll want to focus on crops that are long-term sustainable in a number of senses. You don’t want to destroy your fields, certainly, but you also want crops that work well collectively. Rice is an obvious example, but there are many others. Corn is an interesting case: the planting is a brutal couple of days where everyone pitches in, and then there’s harvesting, drying, storage, and so on. But there’s also this weird thing that corn crosses very fast, and the only way to prevent this (or to do it on purpose) in order to keep the best corn for the future is to hand-pollinate. People have been doing this in Mesoamerica for thousands of years. And no, it doesn’t require men: it requires patience, intelligence, and care. Ultimately, your farms will center around collective labor.
In the Village
If you’re planning a society, you must insist that pure individualism is in fact selfishness. Everyone must cooperate in the face of the apocalypse, or all will suffer and die.
Tasks can be sex-divided, as noted above, but their worth cannot be constituted in this way. When you find that hunting is considered more valuable than cooking, and hunting is done by men and cooking by women, this tells you that the society is sexist, not that there is any intrinsic connection between these tasks and these sexes, nor that there is any intrinsic superiority of one task over the other.
I suggest that you emphasize the village collective is the centerpiece of your society, because it allows for a great deal of planning while granting enormous latitude for local particularities.
Since any and every task can be apportioned however a society wishes, there is no reason that home life has to be different from this. The first thing to realize is that “home” and “family” are social constructions as much as anything else. There is no reason whatsoever to think that humans naturally live in married couples. Marriage, or anyway preferential bonding, does seem to be enormously more common than other arrangements, but that does not mean that husbands and wives must live together in private spaces.
So you have a spectrum.
At one end, have the women live together in one kind of longhouse and the men together in another. Couples have private communal spaces they can use for intimacy, physical and otherwise, at will. Life centers around the village, and there are certain spaces in the village that are especially central to communal existence.
At the other end, have couples live together in private spaces whose boundaries are impermeable. Your mother-in-law may never, ever come to visit: after the first day, when she comes to inspect to be sure her son isn’t marrying some dreadful hussy, she’s forbidden to enter the marital space. Communal and private life are sharply divided.
Humans have existed happily everywhere along this spectrum. No part of it requires or assumes that women are inferior to men.
One of the remarkable things about human beings is that they do come up with an amazing variety of different ways of doing things. Another remarkable thing is that they have a strange habit of claiming that their way is the only way because of nature.
Pretty much everything you hear about how men or women have to this or that is of this kind: it’s social construction. Ideology, to put it another way. Women get paid less in US society because the society is inequitable, period. There is no other justification. (This doesn’t mean it’s easy to fix, of course.) Women do the cooking in most traditional societies because that worked out to be rather convenient when somebody has to stay home with the kids and somebody has to go out hunting. But this isn’t the way nature is, it’s just how things tend to fall out. If you look at nomadic societies, for instance, you’ll find that this sexual division doesn’t always work thus. And now that the members of your society understand what is and isn’t actually required by physiology, they’re also not bound by traditional assumptions.