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What is 3rd Order Logic?

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> What exactly is 3rd Order logic?

In short (and just a tad bit simplified), it's all about what kind of quantifications are you allowed to do:

First order: quantification over individual variables, i.e., variables representing individual objects the theory is talking about.

Second order: quantification over set variables, i.e., variables representing sets of objects the theory is talking about

Third order: quantification over variables representing sets of sets of objects the theory is talking about.

And so on.

> And what is going on with its relationship to elementary topoi?

I don't know. I have not looked too much into category theory.

> Previously I had believed that higher-order logics (above 2nd) require a semester of Model Theory.

Depends entirely on what you want to do with the logic. You will need knowledge of model theory for many topics related to first-order logic too (or any logic for that matter).

> Upon deeper investigation, to my surprise, I found that Model Theory is not used here.

Again, that entirely depends on what are you trying to do. You will not need model theory in every situation.

>Instead category theory and "elementary topoi" are used instead.

I guess whatever you're reading is building foundations starting from category theory, instead of the more classical approach of building on top of set theory.

> What the heck is going on here? I feel like I'm taking crazy pills.

What is going on where exactly? Seems like you got yourself confused or jumped to some conclusions.
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Just as in first-order logic we have quantifiers that range over objects, second-order logic has quantifiers ranging over sets of objects, third-order logic has quantifiers ranging over sets of sets of objects, and so on. The exact definitions used can vary, but the general principle of quantifiers ranging over sets (or, alternatively, propositions) is pretty consistent.

I'm not sure what the rest of your comment is referring to. Third-order or higher logic is not especially qualitatively different than second-order logic. You can certainly still look at it from a model-theoretic perspective, though some of the results can be messier in higher-order logic. And while I'm sure there are people who have looked at these structures from a category-theoretic perspective, elementary topoi aren't really any more closely connected to third-order logic than to nth-order logic for any other value of n. Is there some specific book or paper you are talking about here?

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