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I've looked at several resources to learn about logic and metalogic, and they first present syntactic consequence and semantic consequence as separate things and then try to show how each implies the other in a sound and complete system.

But how did humans develop formal systems and the notion of syntactic consequence in the first place? Wouldn't they have had to develop such systems based on what semantic consequences they agreed with?

I'm confused by the process of just arbitrarily setting up a formal system and saying 'if a formula follows the rules of inference then there is a proof,' and then trying to prove if it agrees with some semantic model.

I'm sure there is a good reason, but I would like to get a good intuitive sense of why, and these kinds of "soft" issues are usually glossed over in learning materials about logic that I've come across.

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