I am not interested in the philosophical part of this question :-)

When I look at mathematics, I see that lots of different logics are used : classical, intuitionistic, linear, modal ones and weirder ones ...

For someone new to the field, it is not easy to really see what they have in common for justifying the use of the word "logic". Is it just because of a filiation with classical logic ?

I have attempted to find an answer in the literature. Some papers are telling me that a logic is a pre-order. It is not a satisfactory answer to me. I imagined that it may be related to the use of some specific connectors : but linear logic is telling me it is not so simple. I imagined that it may be related to some symmetry properties of the rules of the system : but it is dependent on how the logic is formalized. Then, I had the crazy idea (after discovering the Curry-Howard isomorphism) that it may be related to the computational content of the system. But, it is obviously wrong.

So, I have not progressed and I am still wondering if there may be a point of view allowing to see what all these systems have in common.

I have avoided the use of the word "truth" in this question. I am expecting a mathematical answer if there is one. There are too many philosophical problems related to the notion of truth.

But, perhaps my question is a naive one ...