Literally any law a government creates is an extension of its morality. If a law is something you don't believe in that doesn't make it any less a moral. Why is this question even worded this way? Wouldn't it be a whole lot more productive to express which value's we find moral?
Anything the government does with its police powers should be based in morals. Otherwise you have amoral hedonistic chaos, And that would be tyranny because its only foundation would be in the whims of the enforcers.
Greco - Roman - American Empire much?
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Of course the Government does moral policing. Every policy discussion is a moral discussion. Most we take for granted but they are technically moral decisions. We make a moral decision that murder is wrong and therefore it is outlawed. We make a moral decision that not fulfilling contracts is wrong and therefore we enforce contracts. We make a moral decision that theft is wrong and therefore we police it. And on and on and on.
The question isn't whether or not we should use policy to enforce morals. That's largely what policy does after all. The question is which things are suitable for government to enforce.
Everyone has an idea of what is moral and not moral. I grantee that you will meet someone with different morals than you, Unless you live under a rock. To say the government is going to police it's citizen's morals means it's going to make everyone believe the same thing in the same way. I can't agree with that.
One example is a women's right to vote. If the government did moral policing than, At the time, It would have been illegal to say that a women can vote because under the law women weren't allowed to vote. That entire movement would have never happened.
Also, Who decides what's moral or not? Is believing in god moral? Is abortion moral (that's the big moral issue right now)? Is hating dogs moral?
Yes, I know, Most people who would say yes won't bring it down to liking dogs or not, But you see my point right? There is no moral right or wrong. Just opinions.
Why should the government tell people exactly how to live their life? As long as they don't hurt anybody there should be no problem with their morals and beliefs. If they are doing an immoral action that causes harm to another individual that can be quantified, Then it would be necessary to step in. Government shouldn't tell us how to live.