I know they all mean to Destroy to some extent, but was told 烧毁 is specifically to use fire. What are the different contexts or situations you would use them in?
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毁灭 is the general term for 'to destroy'
销毁 also means destroy, but it is more specific.
To 销毁 something, you have to have possession of it. If you delete the data in your computer, you can claim it is 销毁数据 (or 毁灭数据), but if you hack in someone else's computer and delete data there, you can only say you are 毁灭数据.
To 销毁 something, you have to have legitimate right to do it. Destroying the opium in China's standpoint was a legal act of 销毁違法商品 (destroy illegal goods); but in the British's eyes. it was an act of 非法毁灭私有財物 (illegal destruction of private property)
To 销毁 something, that something must have physical form. You cannot 销毁 a dream, an ideal or a reputation.
销毁 has to be a human's act. We can say a natural fire 毁灭了一万棵树 (destroyed ten thousand trees) , but not 销毁了一万棵树
You can 毁灭 something that you do not have possession of. The object doesn't need to have physical form, for example: ideal, dream or enemy, and the act need not to be legal; Of course you can also legal or legally destroy (毁灭) physical things like city or building
烧毁 is the verb 烧(to burn) plus its result complement 毁 (destroy), it means "burn and destroy" or "destroy with fire"
毁灭 is used for abstract description, you can say 毁灭一座城市、毁灭一个民族 and so on.
销毁 means destroy so that no trace is remaining, for example, 销毁文件。
烧毁 means destroy with fire, therefore you can say 一场大火烧毁了这间屋子。
to destroy, usually via natural power
销 (destruct) + 毁灭 (destroy)
to destroy something by means of destruction, via human power or authorities
烧 (burn) + 毁灭 (destroy)
to destroy something on fire