# The Subject Matter of Logic: Explaining what logic is about

**Logics are formal systems with many different applications. The boundary between logics and other formal systems, like mathematics, is unclear. One way of clarifying this boundary is by appealing to the subject matter; defining their purpose, for instance, showing how the truth of the premises guarantees the conclusion, demonstrating reasoning, or modelling an argument.
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There are two contemporary philosophical debates where the subject matter of logic is relevant. The first is that of logical pluralism, which needs a way to determine whether a logic is correct. The second is the argument that the normativity of logic supplies a mechanism for determining whether a logic is correct. In these debates, the subject matter of logic is relevant, but not discussed. And it does not need to be because the participants implicitly agree that the answer is validity. They also agree that this does not advance the debate.

Validity is a dead-end because it only transforms the question. Logicians clarify validity by giving a definition. But it is a well-established fact that there is more than one way of defining a relation of logical consequence. So, how does one determine whether a given definition is correct? Or to put it another way, how does one know whether that definition captures validity? On this point, opinions are deeply divided. Moreover, there is no clear strategy for resolving the difference of opinion. MacFarlane comments that appeals to intuitions about validity are prevalent in these debates. Furthermore, he remarks, these intuitions are a product of education.

In many parts of the advanced literature, logicians agree that the subject matter of logic is validity. But this explanation won't do as an introduction for a beginner. As Newton-Smith says, this 'has the fault of explaining the obscure in terms of the equally obscure.' Instead, teachers must give what Mates calls 'an informal and intuitive account of the matters with which logic is primarily concerned.'

The constraints on introductory explanations provide an opportunity to investigate the subject matter of logic in a novel way and perhaps reveal the source of intuitions about validity, which might, in turn, shed light on questions of pluralism and normativity. My thesis examines the way that teachers introduce the subject matter of logic to beginners.

I begin by exploring a slice of history and a tradition of logic instruction. I argue that a reliance on this tradition leads to flawed teaching in the modern context. After that, I examine modern introductory texts and the strategies they use to present the subject matter of logic. I draw several lessons from this examination.

I conduct three interviews with Gillian Russell, Dave Ripley, and Johan van Benthem in which I ask philosophic and pedagogic questions directed by each of their interests. These dialogues are illuminating on their own. But brought together they show the interaction between the teacher's theory of the subject matter of logic and the practice of introductory logic teaching.

After this investigation into modern logic pedagogy, I present a framework for a solution to the pedagogic problem. It is a framework because there is no single best way to introduce logic to a beginner. But it is possible to develop a guiding structure. The solution which I present relies on a widely accepted 'trivial' form of pluralism and a kind of relativism which I introduce, and argue for, in this thesis. This discussion also includes practical suggestions for developing introductory logic courses.