What is Logic?

It seems to me that what we mean, when we say that some reasoning is logical, is that it is the kind of reasoning that would always, if we started with nothing but some truths, take us to nothing but truths. It is, for example, logical for me to deduce that you are a human from the (presumed) truth that everyone reading this is human, because if all the objects of a certain kind have a certain property, then any particular object of that kind will have that property. Why is that the case? Well, it just seems to be self-evident. I suppose that I could show the self-evidence by drawing a lot of red dots: given those dots, each of them is, of course, red (and each of them is a dot). Such reasoning is self-evident because it gives us just some of what we already had. Now, it seems to me that first-order logic is a mathematical model of such logical reasoning, which is about things (subjects) and their properties (predicates). In short, I take predicate logic to be linguistic, and first-order logic to be mathematical. But I am beginning to wonder if I have got this all wrong, because I found the following description of predicate logic on a paper in a book in the library:

Predicate logic (first-order logic) is a formal system that extends propositional logic by breaking sentences down into subjects, predicates (properties), and quantifiers. It enables precise reasoning about objects by defining properties and relationships, allowing for statements like "all dogs are blue", which propositional logic cannot express.