How Often is First-Order and Modal Logic Used in Math?

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• kyphysics
In summary, the conversation discusses the usefulness and necessity of modal logic, with one participant expressing their interest in taking a modal logic class and another participant discussing the potential benefits of learning modal logic for philosophy. The conversation also touches on the overlap between modal logic and computer science and the potential for modal logic to improve reasoning abilities. Ultimately, it is concluded that while modal logic may not have practical applications in science and math, it can be enjoyable and useful for certain philosophical areas.
kyphysics
I'm a philosophy minor, who took first-order logic (philosophy department, not the math side) and I have a chance to take modal logic next semester (also under philosophy). It's going to be a much smaller class and will likely be harder. But it's also essential to doing work in and/or understanding some areas of philosophy (for example, philosophy of religion's ontological argument for God's existence is based on an argument in modal logic). In trying to gauge whether it would be worth my time, I'm wondering how often modal logic (and/or first-order) is used in math subjects?

Thanks.

I have never come across any discipline in maths in which modal logic is useful, let alone necessary. I have yet to be convinced that there are any statements or deductions in modal logic that cannot be formulated in Classical First-Order Predicate Logic together with ZFC set theory.

My understanding is that the main use of modal logic is to enable one to express concisely certain metaphysical hypotheses involving counterfactuals. They can be expressed without it, but the expression is less concise. The semantics of modal logic is also quite interesting. Since modal logic is the most-used lingo in certain philosophical disciplines, in particular the metaphysics of identity, free will and Kripkean philosophy of language, it is necessary to have at least a rudimentary understanding of it in order to make sense of readings in those areas.

So, from a science and maths point of view, there is no benefit to learning modal logic. From a philosophy point of view, it depends what areas you want to focus on. In the more practical areas like ethics, politics and philosophy of life, it is not generally used.

Having said that, modal logic is kind of fun so, if you don't have to give up on something useful in order to do it, why not?

fresh_42
Well, there are people who specialize in mathematical logic, but I think they are sadly, increasingly rare, or kept as pets in computer science departments. :) Seriously though, there is some nice overlap in CS.

Also as @andrewkirk pointed out, it's fun (for people like us) just on its own terms.

-Dave K

andrewkirk said:
I have never come across any discipline in maths in which modal logic is useful, let alone necessary. I have yet to be convinced that there are any statements or deductions in modal logic that cannot be formulated in Classical First-Order Predicate Logic together with ZFC set theory.

My understanding is that the main use of modal logic is to enable one to express concisely certain metaphysical hypotheses involving counterfactuals. They can be expressed without it, but the expression is less concise. The semantics of modal logic is also quite interesting. Since modal logic is the most-used lingo in certain philosophical disciplines, in particular the metaphysics of identity, free will and Kripkean philosophy of language, it is necessary to have at least a rudimentary understanding of it in order to make sense of readings in those areas.

So, from a science and maths point of view, there is no benefit to learning modal logic. From a philosophy point of view, it depends what areas you want to focus on. In the more practical areas like ethics, politics and philosophy of life, it is not generally used.

Having said that, modal logic is kind of fun so, if you don't have to give up on something useful in order to do it, why not?

Yeah, that's really a bummer. I was hoping there was some transferability of those skills, because it's supposed to be a harder class. I'm not sure if I'll take it now - although, I'm still leaning towards it.

I actually probably will and drop a different "hard" class mainly because I'm sick of not being able to read and analyze philosophy arguments using modal logic.

Even the mind/soul-body argument has a modal structure.

I totally agree that modal logic can be done for fun's sake. Honestly, that's one of the less practical reasons I'm considering it so highly! I'm sure my parents would be thrilled that I'm using credits for personal enjoyment and curiosity.

dkotschessaa said:
Well, there are people who specialize in mathematical logic, but I think they are sadly, increasingly rare, or kept as pets in computer science departments. :) Seriously though, there is some nice overlap in CS.

Also as @andrewkirk pointed out, it's fun (for people like us) just on its own terms.

-Dave K

That's cool. Too bad I'm not doing anything CS related (Econ. and History major with minors in Philosophy and Math).

I agree that fun is always a factor and when I consider that I took a class on world music for an elective and will likely never use it, then I don't feel so bad taking modal logic. :)

kyphysics said:
That's cool. Too bad I'm not doing anything CS related (Econ. and History major with minors in Philosophy and Math).

That actually sounds awesome. I have a friend who has a masters in Econ, is a history buff, and who is working on his PhD. in math right now. I stay out of arguments with him because I always lose.

I agree that fun is always a factor and when I consider that I took a class on world music for an elective and will likely never use it, then I don't feel so bad taking modal logic. :)

Well, here are 90 pages about the history of mathematical modal logic. So at least one major combined with two of your minors. :)

-Dave K

kyphysics said:
I totally agree that modal logic can be done for fun's sake. Honestly, that's one of the less practical reasons I'm considering it so highly! I'm sure my parents would be thrilled that I'm using credits for personal enjoyment and curiosity.

Of course studying ANY kind of logic or math helps your reasoning abilities.

-Dave K

andrewkirk said:
I have never come across any discipline in maths in which modal logic is useful, let alone necessary.
I would have signed this without hesitation, but then I've found this on Wikipedia (by chance and on a not quite serious search for "topological logic": "In abstract algebra and mathematical logic, topological Boolean algebra is one of the many names that have been used for an interior algebra in the literature."
The mathematical structure of modal logic, namely Boolean algebras augmented with unary operations (often called modal algebras), began to emerge with J. C. C. McKinsey's 1941 proof that S2 and S4 are decidable,[15] and reached full flower in the work of Alfred Tarski and his student Bjarni Jónsson (Jónsson and Tarski 1951–52). This work revealed that S4 and S5 are models of interior algebra, a proper extension of Boolean algebra originally designed to capture the properties of the interior and closure operators of topology. Texts on modal logic typically do little more than mention its connections with the study of Boolean algebras and topology. For a thorough survey of the history of formal modal logic and of the associated mathematics, see Robert Goldblatt (2006).
https://en.wikipedia.org/wiki/Modal_logic#Development_of_modal_logic

I'm sure modal logic isn't necessary to deal with Boolean algebras, so this remark is meant to be an interesting addition rather than a point of criticism.

I double majored in philosophy in undergrad, with a focus on logic. Modern modal logic is every bit as rigorous as any other first order predicate logic you'll run across. If it's taught from that perspective (rigor, as opposed to a historical perspective), it can actually be extremely valuable to further mathematical studies. Kripke semantics provides insight into model theory, completeness and incompleteness, etc., plus my classes got into issues that foundational mathematics classes don't normally get into. This includes technical things like intensionality versus extensionality and how that affects the identity relation, but also things with a distinctly philosophical flavor--think Lewis's metaphysics of modal realism. But at least the way I was taught, my modal logic classes had much more of a mathematical feel than a philosophical one.
Caveat: I'm American, and the recent Anglo-American philosophical tradition is heavily influenced by the logical positivists (like Russell and early Wittgenstein), and so tends to appear a lot more mathematical than the trends in Continental European philosophy, which is heavier on the existentialists (like Sartre and Heidegger).

dkotschessaa

1. How is first-order logic used in math?

First-order logic is used in math to formally represent mathematical statements and proofs. It allows for precise and unambiguous statements about mathematical objects and their relationships, which is essential in rigorous mathematical reasoning.

2. What are some examples of modal logic in math?

Modal logic is used in math to reason about possibility and necessity. For example, it can be used to reason about theorems that hold in some but not all cases, or to prove statements about infinite sets.

3. How often is first-order logic used in math compared to other logics?

First-order logic is the most commonly used logic in math. It is the foundation of many mathematical theories, including set theory, number theory, and calculus. Other logics, such as modal logic, are often used in conjunction with first-order logic to reason about specific mathematical concepts.

4. Are there any limitations to using modal logic in math?

While modal logic can be a useful tool in math, it does have its limitations. It can be difficult to formalize certain mathematical concepts and properties using modal logic, and it may not always provide intuitive or useful results.

5. How important is it for mathematicians to have a strong understanding of logic?

A strong understanding of logic is essential for mathematicians. Logical reasoning is a fundamental part of mathematical thinking and is necessary for constructing and evaluating mathematical arguments. Without a solid understanding of logic, it is difficult to make meaningful and accurate contributions to mathematics.

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