# What's there to do in logic? [closed]

I recently asked a similar question about category theory (it got more interesting answers on MO though, I hope for this one I'll get answers here too), but here I'm going to reduce the scope a little for this one, since I'm interested in something more specific.

In what follows, when I say "logic" I don't mean something as wide as what's on the arXiv for instance, where it seems like anything not considered mainstream mathematics can end up in the "Logic" category (I may be wrong about this last point); I actually mean somethin very restricting that doesn't even encompass set theory. In what follows, "logic" will be the (mathematical, i.e. I'm not talking about foundations either, or philosophy which are definitey linked) study of proofs, so the associated proof systems, the semantics for these (whether it be set-grounded semantics or categorical semantics or anything), the syntax-semantics interplay etc. -this includes Gödel type theorems, but also completeness theorems, or Lindström's theorem, the study of non classical logics (intuitionistic, linear, modal) etc.

If it's not clear what I mean by "logic" I can try to clarify some more but I believe (hope) it's pretty clear by now (of course this is not necessarily what I mean by logic in general, but for this question, it is)

Now that this has been said, here's my question : is there still some research in logic nowadays ? What is it, what do logicians do ? What are some paradigmatic questions, some (relatively recent) interesting results, ideas ?

Is the research in logic mostly "pure mathematics" or applied (in computer science especially I believe) ?

Is there a lot of category theory going on in there ?

An answer by a logician saying what he or she does in his or her work would be great for instance !

EDIT : I'm seeing in the comments that some of what I said has been misinterpreted (perhaps by my own fault and lack of clarity); so I'll try in this bit to clarify some things :

1- This restriction of what "logic" means is not what I think logic is, it's only for the purposes of this question that I redefined the term, so as to have a simple word to refer to what I'm referring to. In particular, I'm not saying that, e.g. set theory, or the study of foundations is not logic in general. It's simply that, for this question I would like them not to be taken into account (in a reasonable sense of course). I'm sorry if anyone doing research in those areas felt offended by my lack of consideration for these (note that I'm actually interested in these areas, just not for this question)

2- What I said about the arXiv was more of an impression and something that annoys me; in that any paper which uses some bits and pieces of tools that are usually reserved to logicians will end up in the "logic" category (group theory papers that have nothing to do with logic besides the fact that they use the word "ultrafilter"; any paper that uses $k=\{0,...,k-1\}$ -of course I"m exaggerating) : it seems like there's a fear from mathematicians of logic and so anything that touches "logic" will be sent there. But that's just a feeling, perhaps it's only where I'm from that logicians are disrespected (and unfortunately, they are).. In any case I'm sorry if anyone felt insulted by this comment about the arXiv, I certainly didn't intend to.

3- Model theory, and anything that touches the syntax/semantics interface, falls into the scope of this question. Of course not all of model theory does, for instance the more algebraic side of it doesn't but, e.g. the completeness theorem, or the theorems relating properties of the class of models of $T$ with the syntactic form of $T$ do.

## closed as too broad by Andrés E. Caicedo, Namaste, Batominovski, Xander Henderson, JonMark PerryJun 29 '18 at 11:26

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• All you've "clarified" is what logic is not, according to You. It is very difficult to carve out and define a field, as you see it, in negative terms. That's like me telling you I'd like to know the ingredients of a pie, but the pie isn't mainstream blueberry, nor raspberry, nor blackberry, nor is it made of peach, nor apple. That that this has been said: here's my question: are their recipes for my pie; what are some pragmatic questions, some relatively new pies. What is it pie makers do? Is most of the research about pies done by bakers, or perhaps CS. specialists? blah, blah, blah – Namaste Jun 28 '18 at 21:49
• @amWhy : I did write : "study of proofs, so the associated proof systems, the semantics for these (whether it be set-grounded semantics or categorical semantics or anything), the syntax-semantics interplay etc. -this includes Gödel type theorems, but also completeness theorems, or Lindström's theorem, the study of non classical logics (intuitionistic, linear, modal) etc."; I hoped it would be enough. Should I expand on that ? – Max Jun 28 '18 at 21:57
• You write "it seems like anything not considered mainstream mathematics can end up in the "Logic" category". I follow the math.LO category of the arXiv closely, and at least 90% of what appears there is serious research in well-developed subfields in mathematical logic. Your question suggests that these areas are both "not mainstream mathematics" and also "not logic". Which assessment is more insulting probably depends on the logician... – Alex Kruckman Jun 29 '18 at 3:57
• If you're really uninterested in model theory, set theory, and most of recursion theory, you should probably retitle the question "what's there to do in proof theory / foundations?" – Alex Kruckman Jun 29 '18 at 3:58
• Does finite model theory fall into your interests? – Larry B. Jun 29 '18 at 4:10