What is subject matter of Logic?

One eminent philosopher has given a direct response to this question. Namely, Ludwig Wittgenstein contended that Logic is about the boundaries of reality.

Although it's not necessary, perhaps we should say a little about Wittgenstein's professor and the eminent philosopher Bertrand Russell. According to Russell, logic is the fundamental structure of all language. That all sentences have an underlying structure that followed the rules of logic, such that logic is, perhaps, about the structure of language.

Taking this idea in a much different direction (the early) Wittgenstein argues that logic structures the limits of our thinking. For Wittgenstein our thinking is shaped by our language. That is, when we think of the Cartesian doubting whether anyone else has ever thought, whether he or she is the only one who is really thinking, Wittgenstein denies this altogether. Specifically, the only reason the Cartesian can think is because they have been immersed in a language (This, with the development of psychology, linguistics, etcetera, has proved dubious). So, in essence, language shapes our thinking, but it also shapes the way we understand reality.

According to Wittgenstein language describes situations or states of affairs. That is, it gives us pictures of reality. If I say, "Tomorrow it will rain," on the early Wittgenstein's view, this is "painting" a picture of reality. What is more, language describes reality, such that it is reality. What we know about reality and how we describe reality is only expressed through language. Thus, when we want to understand reality, we want to know not just what it is at the moment, but how it might be. Here comes the punch line.

Logic tells us the limits on how the world might be or can be. For instance, it's an incontrovertible fact of logic that one thing cannot be in two places at the same time. It is true in all possible worlds that P and not-P is false. Thus, we can use this as a limit on how the world might be. For instance, I cannot be both in Tokyo and not in Tokyo (say in Los Angeles) simultaneously; it cannot be both raining and not raining in the same place at the same time (strictly speaking); and so on. In this sense, logic, according the Wittgenstein tells us what the structure and limits of reality are. Specifically, they form the boundaries of our thinking and language