I can point you to my favourite logic ;-) Here false is defined as false = (∀x. x) So in this case, it derives its meaning from whatever the meaning of the

I can point you to my favourite logic ;-) Here false is defined as false = (∀x. x) So in this case, it derives its meaning from whatever the meaning of the ∀ operator is. Its meaning is the result of applying the ∀ operator to the identity operation. Nobody is questioning that there are different possibilities for logical systems, and different possible semantics for them. The choice of logical system is up to you, and ultimately, part of your assumptions. A proof assistant uses a fixed logical system, and your mathematics needs to fit into this system to be expressible in the proof assistant. Apart from my favourite logic, there are several well-known logical systems commonly assumed to be able to represent all of mathematics more or less faithfully: first-order logic, simply-typed higher-order logic, and dependent type theory. What I like about my favourite logic is that I believe that all other logics, even quantum logics, can be expressed in it in a straightforward way. This has the advantage that once you expressed your logic in this way, you get the meaning of it for free. Of course, you still need to make sure that this is the intended meaning.