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

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.

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?
• For zero ontological commitments of math see Elaine Landry, who says axioms come from treating them as-if they were true, no ontological commitments. Russell said math and logic first principles come from abductive reasoning, which is a step closer to ontological commitments. Russell believed abductive reasoning is still important in coming up with new axioms, which we would like to have to solve things like the continuum hypothesis. He also believed math was an extension of logic. Their foundations are philosophical (abductive). Jan 5 at 3:21
• I am afraid some of your distinctions are not salient. Studies in mathematical logic do not assume ZFC, or even classical logic, nor is one necessarily bothered by "circularity" in philosophical studies of logic, it is inevitable in foundations. That logic is "prior" to numbers is very disputable, for example. The difference is not so much in the topics studied as in the methods and approaches used. Mathematicians use formalism and technical arguments, philosophers take account of technical results, but reason informally on issues where formalization is either controversial or elusive. Jan 5 at 9:02
• And interesting question, but I doubt your premise--that there is some sort of circularity. Natural numbers pre-exist logic by thousands of years, so the claim that logic is conceptually prior to natural numbers, if it means anything at all, can't mean that you need logic to use natural numbers. Furthermore, no one needed logic to be analyzed mathematically before they knew it was correct. I suspect that you are confusing analysis with justification. Logic is used to analyse natural numbers, not justify them. Mathematical logic is used to analyse logic, not justify it. Jan 5 at 14:49
• If it is accepted that Mathematics is part of Logic, and Logic is part of Philosophy, a philosophical approach to Logic is necessarily synthetic (a part interacting with other disciplines, synthesized in Philosophy), and a mathematical approach to Logic is analytic (a set of parts with multiple interactions with other parts of mathematics). Jan 6 at 11:57
• @GregoryNisbet I don't think that there is any such a thing as studying logic mathematically or philosophically. The only meaningful way you can study logic is to take into account whatever empirical facts about logic are available and reason logically. You could do that whether your are a philosopher, a mathematician or a plumber. I don't see anyone really doing this, though, so as far as I understand, nobody is really studying logic at all, except in a very limited way people working on the theory of arguments. Jan 6 at 12:09

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.