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The author asks why && and || need different precedence (Unless I misunderstood).

This is because `a && (b || c)` has a different truth table than `(a && b) || c`

&& having higher precedence than || makes sense because && is boolean multiplication and || is boolean addition.

This makes so much sense but I never thought of it this way before so explicitly. Neat!

Has anyone ever found a link between boolean (F2) and "normal" base (F[n], or N) arithmetics ?

That doesn't explain why they would need a different precedence. `a + b - c` is not the same as `a - b + c` even through the precedence's of + and - are the same (in most syntaxes).

Granted, the intersection of precedence and associativity is a great source of confusion (and any number of annoying Facebook memes)

I'm not a formally trained mathematician, but one way to rationalize + and - having the same precedence, is to consider them as inverses of each other, so your two examples can be written as

a + -b + c

And

a + b + -c

At which point precedence and associativity no longer matter. The same can be done with multiplication and division

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