I'm not sure what you mean. All formalisations of non-classical logic operate in the framework of first-order logic, in the sense that the informal meta-language in which the non-classical logics are explained in is traditional first-order logic, see e.g. [1].
I don't know what "dialectical logic" logic is.
[1] Graham Priest, An Introduction to Non-Classical Logic
> All formalisations of non-classical logic operate in the framework of first-order logic, in the sense that the informal meta-language in which the non-classical logics are explained in is traditional first-order logic
What gave you this idea? It is frequently useful to work over different base logics to obtain models of non-classical logics. E.g., when you work in categorical logic you usually work over an intuitionistic logic which can then be interpreted in an arbitrary topos. This gives you the ability to internalize any constructions you do "on the outside".
There is a large community of mathematicians fixated on classical first-order logic, but this is just because of tradition. For models of classical first-order set theory it just doesn't make much of a difference. This is not true of "all formalizations of non-classical logic".
That said, a lot of the development of category theory that I'm aware of, is taking place with set theory. Certainly the two books [1, 2] that I learnt category theory from do. But that was a while back, and it might be the case that this is different now, I have not kept up with this research direction. For example is there a full, unrestricted formalisation of category theory in HoTT?
[1] J. Adamek, H. Herrlich, G. E. Strecker, Abstract and Concrete Categories: The Joy of Cats.
[2] S. Mac Lane, Categories for the Working Mathematician.
Textbooks are usually supposed to be accessible to a wide audience so it makes sense when discussing foundations to start from a (hopefully) familiar set theory. It's usually a trade-off, since you end up repeating yourself when it comes to "internalized" constructions. "Sketches of an Elephant" is a good example of a textbook that pretty much presents everything twice. Once in an ambient set theory and once internally.
What I meant specifically is work such as the following:
Internal Universes in Models of Homotopy Type Theory - https://arxiv.org/abs/1801.07664
which explicitly works in an extensional type theory with some axioms to simplify and generalize a complicated model construction.
> For example is there a full, unrestricted formalisation of category theory in HoTT?
You can formalize category theory in HoTT as presented in the book. This has some advantages over a presentation in extensional type theory or set theory (being able to work up-to equivalence) and some disadvantages (universes in categories have to be constructed explicitly, since the ambient universe is not truncated). In my opinion, it's not the case that one is strictly superior - in the end it always depends on what you want to do.
Dialectical logic is a name given to describe the non-formal method apparent in Hegel's work, in which two opposing poles confront each other and progress towards a positive outcome.
I'd question whether the people who talk about logic in analytic philosophy departments and mathematics have a monopoly on the usage of an ancient term and concept which has found immense value in non-formal contexts too.
They do not have monopoly, but have a right to a technical usage, which in this case is just short for formal logic.
As far as I know there is no formal equivalent of Hegel's logic. This is not a criticism (my favourite thinker is Jung) but is still a relevant distinction.
Of course, though my point was about the term "logic" in general rather than "formal logic". I am happy to say that Hegel's logic isn't a formal logic, but it's not exactly what we know to be "informal logic" either (i.e arguments and fallacies). Funnily enough, a Boolean opposition between formal and informal logic doesn't seem to include the full range of possible values...
Ultimately meaning closely tracks use. But if you want to maximise the probability of being understood, then it's wise to use a term in the way that the audience you are talking to is using it. And on a comp-sci centric site like Hacker News, that means to understand "logic" in the sense of formally valid truth-preserving inferences. All the more so since the post I was replying to, and my own clearly were arguing about formal logic.
As an aside, there have been attempts at formalising logical approaches a little more in the tradition of Hegel [1], maybe started in earnest by Paul Lorenzen. But this only got real traction in the 1990s with Girard's linear logic [2] game semantics of logic [3]. The tradition of game semantics is firmly within the Fregean tradition of logic as formal logic.
I've made myself understood using "logic" (as in a logic) a few times on Hacker News, for instance "the logic of capital" - nobody objected to this kind of usage. You made a link between understanding and the lack of the need to "learn" logic later in life - which ties in very well with what you might not have expected - Hegel's logic, which aims to deal with the intersection between logic and the movement of thought very closely. Implicitly there was the idea that classical logic represents how we use logic internally, which is a notion also firmly in the camp of continental logic theorists. So it's not relevant that you were talking about formal logic - after all, this site is for interesting things, not formal things.
I also think it's worth asking whether the formalization of Hegel's logic (or one similar to his, even superficially) can preserve the meaning. Hegel was famously against what he saw as the rise of mathematical formalism and its precursor, Greek and Latin terms, in new philosophical texts - because he didn't view our consciousness as being "at home" in such concepts. This is of course entangled with the critique of positivism and the reduction of concepts and scientific knowledge to formal logically linked facts. Already this definition of "logic" which you use to strictly only include Ferge and Russell's descendants shifts the conversation to a world in which the totality of things can be explained by this family of logics - after all, anything outside that would be "illogical". I'm not saying it's a deliberate move.
If the browsers of Hacker News are only able to understand a very limited definition of "logic" then perhaps it is good to broaden their minds.
Thank you for the first link in particular, even though it seems dialogical logic doesn't seem to capture some key Hegelian features, such as the uniqueness of the negation of the negation.
There is no logic of capital. The context "of capital" disambiguates the phrase
and makes it clear that the discussion is about a politics, and has
about as much to do with logic as the holy roman empire was holy,
roman, or an empire.
> Hacker News are only able to understand a very limited definition of "logic"
Why underestimate HN? Why not ask: how is using a word in non-standard meaning helping people's understanding?
As an aside, there's quite a few working logicians on HN, including
two how have published on features of Hegel's conception of (what he
called) logic, albeit filtered through Sellars and Brandom's Hegel
reading.
I don't know what "dialectical logic" logic is.
[1] Graham Priest, An Introduction to Non-Classical Logic