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Point of formal semantics?

A formal semantics is invariably a mapping from one formal system to 

logical language + inference rules -> sets and relations
computer language + operations -> sets and relations

Certainly proving that something gives you the same result in two different 
formal systems might give you a little encouragement that you're doing the 
right thing.

But why does anyone claim that this gives you any clue as to the "meaning" 
of the formal system on the left hand side?

Why are some people so adamant that this sort of thing must be done?


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