The verb ことわる in Japanese, written in Kanji as 断る, is notable in that it has two meanings that are nearly opposite to one another. As usual in cases where a word has more than one definition, you need to use context to tell which is being used.
The first meaning, and perhaps the more common one, is to mean “decline”, as in declining an offer. If specified, thing is that being declined or refused is prefaced by the direct object particle を (pronounced close ‘wo’).
- I declined the invitation.
If you want to be use a more politer expression, you can say the following:
- I regretfully decline your offer.
The “regretfully” part is not literally in the Japanese sentence, but I added it to try and convey the feeling of politeness.
The second meaning, which is confusingly different from the first, is when you ask someone for permission to do something.
- I’d also like to try that but we have to get permission first.
Here it is obvious that the speaker is not trying to say “we have to refuse”. You could use a similar expression like ”断らないとだめ” or “断るべき”.
There are a few other meanings of this word but the above two are the most common in my experience.