What, if anything, is the difference between studying logic mathematically and studying it philosophically?

Question

There seems to be a distinction between studying logic mathematically and studying it philosophically and, in practice, it is reasonably clear which framework one is using when one studies logic.

I've used this distinction in the past to try to defend using very strong theories (classical FOL + ZFC) to analyze simple ones (that they arguably depend on) like propositional calculus by claiming that the apparent conceptual circularity is a philosophical concern and is not a mathematical concern. Now I'm wondering whether this position is wrong or naive.

I'm wondering whether this position (i.e. there is an important/meaningful distinction in practice between studying logic mathematically and studying logic philosophically) has a name and whether any philosophers have described, defended, or rebutted it specifically.

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When you study logic mathematically, you assume ordinary mathematical reasoning in the background (which in practice usually means classical logic and ZFC or NBG or another mainstream set theory), and aren't particularly concerned about the meaning of the deductive system you're working with. You're just analyzing it as a mathematical object. Also, more to the point, when studying logic mathematically one is not bothered by conceptual circularity, such as using natural numbers at the meta-level even though logic is conceptually prior to natural numbers.

Studying logic philosophically would include questions that are not specifically about analyzing the behavior of a logical system, using a possibly-formalized background theory like ZFC or ordinary mathematics.

Here are some questions that I think illustrate the concept of investigating a logic philosophically. Many of them are extremely naive, but that's just because I made them up instead of grabbing them from somewhere.

• How well does some deductive system capture inferences that are valid in natural language?
• What are the ontological commitments of a deductive system (such as plural logic or second-order logic)?
• What's the right way to measure how "good" a logic system is?
• Is there one true logic?
• Is logical pluralism a coherent position?
• What, if anything, is the relationship between logic and truth?

Answer 1

To contrast "studying logic mathematically" with "studying logic philosophically" is perhaps not looking at the issue quite straight. Logic overlaps with mathematics in a two-way fashion. We can use mathematical methods to make logic more rigorous, and we can use logic to study the foundations of mathematics. These activities are inevitably circular, though that doesn't make them useless. But equally, it doesn't create a dichotomy in how one studies logic, or in how one uses logic. Logic is logic, however mathematical or not one chooses to make it.

In the same way that there is a branch of philosophy concerned with the philosophy of mathematics, there is also a branch of philosophy concerned with the philosophy of logic. Many of your bulleted questions are fundamental questions in this domain. What is logic all about? If there are logical truths, what are they true of? What exactly is logical consequence and validity? How do we have knowledge of logic? Is logic in need of some kind of justification or foundation? If so, how can we provide one? Why are there lots of logics? Is only one correct, and if so, which one? Is logic normative, and if so in what way? How does logic connect to related subjects such as computation, reasoning, linguistics and epistemology?

There is a substantial literature on the philosophy of logic, and answering your questions would require a lengthy essay for each. To address a few points:

How a formal system relates to and captures valid inferences in natural language, and how well it does so, is always a compromise between a number of conflicting desiderata. We want a logic to be simple, consistent, accurate, expressive, general, practical, obvious, computable, and to make a good fit with other disciplines where it is put to use. It is impossible to do all these things, so all our systems of logic are trade-offs. The fact that first order classical logic is so ubiquitous is an indicator that it does a pretty good job, but it still has many limitations.

Logical pluralism has strong and weak versions. In its weak form, we may allow that different logics can coexist because they are concerned with a different modality, or a different semantic property. E.g. that classical validity is concerned with perserving truth, while intuitionistic logic is concerned with warranted assertability, or relevance logic with channels of information. In its stronger form, pluralists accept that distinct logics can coexist for a single domain, modality, or semantics.

Fundamentally, logic is concerned with the relation of logical consequence. But logicians cannot agree on what this is. There are about a dozen competing accounts of logical consequence, or of what it is that makes a valid argument valid.

Here is a list of some books on the philosophy of logic:

• Beall and Restall. Logical Pluralism.
• Cohnitz and Estrada-Gonzalez. An Introduction to the Philosophy of Logic.
• John Etchemendy. The Concept of Logical Consequence.
• Susan Haack. Philosophy of Logics.
• Matthew McKeon. The Concept of Logical Consequence.
• W.V. Quine. Philosophy of Logic.
• Stephen Read. Thinking about Logic.
• Ian Rumfitt. Boundary Stones of Thought.
• Penelope Rush. The Metaphysics of Logic.
• Graham Priest. Doubt Truth to be a Liar.
• Stewart Shapiro. The Oxford Handbook of Philosophy of Mathematics and Logic