The existence of logic seems somewhat mysterious. It's this thing that seems to exist, but unlike other things that exist, it doesn't seem to exist anywhere in specific or in any tangible form. Further, while it is easy to mock Plato for mysticism when he posits perfect forms existing in some kind of mysterious Platonic Realm, that's actually uncomfortably close to a description of what logic is often seen as.
I think this is an interesting question. I know some friends who know a lot more about philosophy than I do on social media. A lot of people who aren't as well-read in philosophy only come across notions of what the ontology of what things like logic and mathematics might be through Platonism. I'm not as familiar with them myself, yet I'm aware there is a much wider variety of options for what constitutes the ontology of logic to explore beyond Platonism. I haven't read Plato, and say I couldn't rightly say what if anything is wrong with it. Yet learning about things like the ontology of logic has led me to think more recent and obscure options to explain such things than the Platonic realm are better. I don't think they're the kind of views someone with only a cursory understanding of academic philosophy would have heard of. I've been saying 'things like the ontology of logic', because I've actually thought more specifically about the ontology of mathematics. I've also talked to some friends who know much more than me about maths, logic, and philosophy. I would suggest looking into the following fields for a much greener and greater garden of potential answers to your question:
I will also ask my friends what answers they would give to this question, and then I will report them back here.
https://www.lesswrong.com/posts/mHNzpX38HkZQrdYn3/philosophy-of-numbers-part-2
Basically, we have a mental model of logic the same way we have a mental model of geography. It's useful to say that logical facts have referents for the same internal reason it's useful to say that geographical facts have referents. But if you looked at a human from outside, the causal story behind logical facts vs. geographical facts would be different.
Is there some issue about the status of logic that doesn't apply to maths, or language, or games?
There are degrees of existence, in a way. The more repeatable, ubiquitous and predictive/predictable something (a concept, an observation, an algorithm) appears, the more it feels to be imbued with the magic of "existence". Logic, if you refer to a mathematical concept, has a high degree of existence. It's not universal, a lot of people don't relate to it, but it feels like a real if not necessarily a tangible thing. Logic doesn't exist as much as, say, a chair, but probably more than, say, fairies. it may indeed be interesting to ask the where question. Where do fairies exist? Where do chairs exist? Where does logic exist? Well, fairies certainly exist in fairy tales, so the next question is where do fairy tales exist? Maybe in the minds of those telling and retelling the stories. Maybe, like many concepts, they emerge once the complexity of the underlying substrate reaches a certain threshold. I suspect logic can be treated similarly. Actually, so can chairs.
It might be the case that "existence" always means the same thing, but comes in degrees. Or it might be the case that "existence" has literal and metaphorical senses. Or something else.
It would be really helpful to list the things that seem to be pointing to logic existing. One of the answers would be that establishing that there is reason to think that logic exists will fail thus there is no need to think of the mode of something that is not.
A lot of language that at one level seems to be about existence can be turned into forms where it's not neccesarily so. "There exists a" means a value can be picked to satisfy a condition. For example unicorn satisfies "being horned" but that doesn't make unicorns exist (ie it's perfectly resonable to assert both that unicorns are horned and that there are no unicorns).
There is also the interesting question whether logic could be any different. Could 2+2 equal 5? (note the danger how 5 would be just 4 by another name.) Could logic turn out to be different or be created differently? One issue for example that if you imagine that number times number could be -1 that just points to another entity (imaginary number) rather than change in existing entities. One pecular possible property of their mode of existence pointed by this would be that there is no state to point in their existence. You can look at a ball and there claims about it's position etc but when you "look" at numbers what you say can't be evidence in the same way that ball reports would be correspondences to world state. There is no ambiguity on the state of logic and it's questionable whether a "different state of logic" could even make sense. (All this kind of wackiness can be implied as perfectly expected and reasonable in such concepts such as "aprior")
Appliability of logic in physical world is sort of a theorem based on the laws of physics (mostly more metaphysical and less technical like the persistence of objects, that themselves as theorems of the basic laws of physics) and the laws governing the process of formulating atomic statements based on the observations.
At the same time we need to be careful as we can easily fall into the trap of unfalsifiability -- when the predictions of logic fail, we're used to say that the problem was with our atomic statements.
That's just the sketch of the full explanation of the topic, which would require at least a chapter.
I'm not sure what "exist" means in this context. IMO, logic is a somewhat ambiguous word, but the common use is for a particularly powerful base model (that is, a meta-model that can be extended in many directions to make predictions). It doesn't exist any more (nor any less) than "thermodynamics" or "love" exists - these are concepts that can be used to communicate and predict states of the universe, but don't directly correspond to a state of the universe.
Actually, for the question of existence, the power and applicability of the model doesn't matter. it exists in the same sense as bad models exist, too. It's an idea, or pattern of processing inside a brain.
https://www.lesswrong.com/posts/mHNzpX38HkZQrdYn3/philosophy-of-numbers-part-2
Basically, we have a mental model of logic the same way we have a mental model of geography. It's useful to say that logical facts have referents for the same internal reason it's useful to say that geographical facts have referents. But if you looked at a human from outside, the causal story behind logical facts vs. geographical facts would be different.