Part of the fun of logic is realizing that there are multiple forms of it which change one's language and ability to express ideas. They also don't dominate one

Part of the fun of logic is realizing that there are multiple forms of it which change one's language and ability to express ideas. They also don't dominate one another—some statements' truth depends upon what logic they're formalized within. Model theory is good here in that it studies the connection between logical language and more "physical" models. Some of these models can give a relatable account for what a logic "means". For instance, Boolean logic correspond with set operations and therefore we can use that correspondence to portray a meaning for statements in Boolean-like logics. Intuitionistic logic is an interesting one if you're not familiar with it. It can be seen as the logic of working with "demonstrations", building and analyzing them. For instance, if I have an object "X" such that witnessing it immediately convinces someone of the truth of "1+1=2", and if I have a whole collection of such items, then intuitionistic logic talks about how to compose and decompose collections of these things. Another interesting operation is "not" such that "not X" is a tool for using any X you happen to have to show that the world is inconsistent. This acts as a refutation of the existence of values like X! This differs from set-like logics because, for instance, "not (not X)" is not the same as X: being able to show that if you had (a way to prove the world inconsistent if you had an X) you could show the world as inconsistent... it's a far cry away from actually having an X! Intuitionistic logic is interesting for two reasons. First, it models communication and persuasion by being an algebra of "persuasive items" like X. Second, it's "constructive" and thus works similarly to how programming languages express things through their creation.