Initiative is a core part of a good life. It means doing things instead of passively waiting for stuff to happen. Initiative is self-leadership.
Initiative involves willingness to make choices. Don't just go on Facebook/Twitter/Reddit/etc and scroll through your feed every day. Don't just turn on the TV and watch whatever's on. Then you are making one choice (read feed, watch TV) and that's about it.
If you always finish books you read, and read them in order from start to finish, then you are only making one choice per book – to read it. You aren't making the choice to continue after each chapter, and you aren't making choices about which parts to read and why those. (Reading out of order is frequently a good idea with non-fiction books, like the kind of book with an index, and sometimes with fiction too.)
The more decision points you have in your life, and the more you actively think about them and make a choice, then the more control you have over your life. Whereas if you set things up to mostly coast along, then it's more like life is this thing which is out of your control and just happens to you.
Initiative is kinda about wanting some things and then making them happen. You have to think about what you want first. If you aren't introspective, you'll be kinda aimless and lost. When you have some ideas, keep looking for different ways to approach them until something works. Be persistent. Be willing to try many small steps, only some of which work, until you make some progress. Don't give up.
Initiative is about choosing goals yourself and pursuing them, rather than passively waiting for stuff in life to come to you. It's about selecting things – choosing one over another – instead of just doing whatever's in front of you. It's about (proudly) making choices instead of coasting. It's about leadership – leading yourself – rather than following some other leader.
Responsibility is something you can only achieve when you control your life. You are responsible for outcomes which you make choices to control. Passivity is irresponsibility.
Passivity lets your life be controlled by static memes, by other people, by society, by your culture, and by chance. If you aren't making decisions in accordance with your values, they'll get made in another way. If you don't have some knowledge of how you want your life to be, and make it happen, then some other knowledge (from memes, other people, whatever) will take charge.
Looking for ways to be responsible for more stuff can help. Like some CEOs are like, "well i couldn't help it, my employee fucked that one up". "i delegated it. it's not my fault". But a better CEO goes "no matter what, if something goes wrong, that's my responsibility. i should have found a way to make it turn out better." A good CEO looks to take responsibility for everything he can, rather than looking for excuses.
Self-leadership is required to be responsible. And so is an attitude of not making excuses. Instead of saying why something was impossible or out of your control, figure out a way to change it. Keep looking for more you can do, rather than reasons you can't. Have some disrespect for limitations.
Do you want to live? Do you really? Do you want to decide on things to do, and do them, and take responsibility for the outcome? Do you want to initiate this yourself, with no one telling you to do it, and no one else to blame if you make a mistake?
People say they want an active, chosen life. But most people don't act like this very fully or consistently. Many people don't really even know what to do with themselves if they get some free time.
Many people consider responsibility a duty, a chore, a hassle to dodge. They see potential for work and sacrifice. They fear being pressured by obligations. They think of trying to do what they are supposed to. (Supposed to – in service to whom?) Many feel similarly about morality.
But if you like and want your life, wouldn't you want to be responsible for what happens in it? Wouldn't you want to make the decisions and determine the outcome? Wouldn't you want knowledge to help you make good life decisions (moral knowledge)? Why wouldn't you want to run your own life? Who or what else would you want to be in control, if not you? Who would you expect to take responsibility for making your life go well, if you don't?
How can a passive (low-initiative) person change? What do they need to know to change? How can they change given that learning/changing takes initiative (it doesn't just happen passively), which they don't have?
If a person is irresponsible, won't they be irresponsible about the project of becoming a responsible person?
People are mixed. They aren't completely passive or irresponsible. You control some of your life, you make some decisions. You can expand on that. You can make step by step increases. It may seem daunting at first, but it will get easier as you go along, because you'll be a better, more effective person after each step.
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Acknowledgments: some ideas presented are modified from, or inspired by, ideas from Karl Popper, Ayn Rand, David Deutsch and William Godwin. Thank you!