Natalie and BenDToy... so, you're basically saying everyone fend for yourself - I'm going to only take care of myself.
I think that in a healthy society, everyone does what they can to take care of others. That means that when you need less help, you give what you can, so that when you need more help, people are there to help you.
A healthy society would make sure that life is not impossible for children, people with kids, and elderly people. Does that mean that sometimes healthy adults have to work a little more? Maybe. Those same healthy adults may in a few years be disabled and need disability benefits... wouldn't you want these to be available? Or the same healthy adults may then have kids and have no family nearby to care for them when they are sick. That's a tough situation and it requires a little leniency.
It's most frequently NOT possible to cut your hours, because having kids can be very expensive (for instance, I make a good salary, but now with a kid we are really struggling to scrape by. Can I cut my hours if I need more flexibility to take care of my kid? Well, I would love to... but I can't).
Saying "why should I work harder so that someone else has time off to take care of their sick kid" is a very limited way of thinking. Okay then, why should I work harder when you're retired, just so you can get your retirement benefits? Why should I work harder when you've been injured on the job, so you can get your worker's compensation?
Katie1k, to answer your original question: I don't think you are automatically entitled to paid time off, but why don't you ask your employer about it? My boss lets me take it out of my own paid sick time.