Why the emphasis on first-order logic?

In summary: True, the "inaccessible cardinal axiom" is ambiguous, as it could be mixed up with the axiom that says that there exists an inaccessible cardinal, quite a different kettle of fish.In summary, according to the first blog, second-order logic has some advantages over first-order logic, such as the ability to do away with messy schemata, but it has some drawbacks as well. Second-order logic is not complete, meaning that it does not have a proof procedure for all its theorems. Second-order logic may become useless if
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nomadreid
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It seems to me that, despite several systems developed for higher-order logics, almost all the attention in Logic is devoted to first-order. I understand that higher-order logics have some drawbacks, such as the compactness theorem and the Löwenheim-Skolem theorems and other such not holding in second-order. However, there are some advantages as well, for example doing away with a lot of messy schemata. I would understand if the whole Logic community was composed of Intuitionists or Constructivists, since sets/classes using quantification over sets or relations are harder to construct, or if the Logic community was worried about programming issues, but since neither of these is in fact the case, I am puzzled by the preponderance of attention to first-order results. I know this is a broad question, so any responses will be appreciated.
 
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In First-Order Predicate Logic we can axiomatise ZFC set theory, and once we have that we can do almost all interesting mathematics. So there is no need to tiptoe amongst the minefields that arise in higher order logics.

The only well-known bit I'm not sure about is Category Theory, which seems to need something more than just set theory. It's on my to-do list to one day do enough Category Theory to understand what platform it requires to give it sufficient rigour.

Perhaps something about transfinite ordinals as well - another item on my to-do list.
 
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Thanks, andrewkirk. That makes sense. Actually, almost all the theory on transfinite ordinals that I have seen concentrates on a first-order base as well. But it seems so nice to be able to finitely axiomatise ZFC, possible in second-order but not in first-order. It would also make some theories of truth simpler, perhaps.
I think Category Theory can be done mostly in second-order, but I am not strong on Category Theory, so I won't go into that.
 
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andrewkirk said:
The only well-known bit I'm not sure about is Category Theory, which seems to need something more than just set theory. It's on my to-do list to one day do enough Category Theory to understand what platform it requires to give it sufficient rigour.
Much of category theory can be done in ZFC as well. For some constructions however, you need the universe axiom, which is equivalent to the postulate that for every cardinal, there exists a larger inaccessible cardinal.
 
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rubi said:
Much of category theory can be done in ZFC as well. For some constructions however, you need the universe axiom, which is equivalent to the postulate that for every cardinal, there exists a larger inaccessible cardinal.
Indeed, and at first glance the universe axiom (aka the inaccessible cardinal axiom) cannot be stated within first-order logic -- but I am ready to be corrected on this. Anyone? Nice example, rubi.
 
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nomadreid said:
Indeed, and at first glance the universe axiom (aka the inaccessible cardinal axiom) cannot be stated within first-order logic -- but I am ready to be corrected on this. Anyone? Nice example, rubi.
I wouldn't want to call the universe axiom the inaccessible cardinal axiom, because there are several ways to postulate an inaccessible cardinal and the universe axiom is just one of them. The universe axiom can also be stated in first-order logic, but it is quite complicated. You can find it on Metamath: http://us.metamath.org/mpeuni/ax-groth.html
 
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The Metamath version is impressive, thanks.
True, the "inaccessible cardinal axiom" is ambiguous, as it could be mixed up with the axiom that says that there exists an inaccessible cardinal, quite a different kettle of fish.
 
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  • #8
nomadreid said:
It seems to me that, despite several systems developed for higher-order logics, almost all the attention in Logic is devoted to first-order. I understand that higher-order logics have some drawbacks, such as the compactness theorem and the Löwenheim-Skolem theorems and other such not holding in second-order. However, there are some advantages as well, for example doing away with a lot of messy schemata. I would understand if the whole Logic community was composed of Intuitionists or Constructivists, since sets/classes using quantification over sets or relations are harder to construct, or if the Logic community was worried about programming issues, but since neither of these is in fact the case, I am puzzled by the preponderance of attention to first-order results. I know this is a broad question, so any responses will be appreciated.
This may be enlightening:
https://www.lesswrong.com/posts/MLq...ss-incompleteness-and-what-it-all-means-first
https://www.lesswrong.com/posts/SWn4rqdycu83ikfBa/second-order-logic-the-controversy
 
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Demystifier: Thanks, I enjoy the "less wrong" blogs. The essence of the first one (mostly introductory) with respect to my question is the quote
"The lack of completeness means that the truths of second order logic cannot be enumerated - it has no complete proof procedure."
And in the second, he repeats that point before then going into an interesting discussion that I will have to go over again (I've bookmarked it) about the relationships between first order, second order, set theories, and the physical universe. On a first reading I am not actually sure that the author comes to a conclusion in terms of physicists' interest in second-order, but I will be re-reading it. Thanks again.
 
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I've done some AI programming using FOPL, and also looked into higher-order logic from the viewpoint of developing AI software. AFAIK there is no practical alternative right now to using FOPL for the type of application I was working on. One benefit is the availability of the resolution method of theorem proving. For a reference see https://www.amazon.com/dp/0137117485/?tag=pfamazon01-20.
 
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1. Why is first-order logic important in scientific research?

First-order logic is a formal system of reasoning that is widely used in scientific research because it allows for clear and precise definitions of terms and logical deductions to be made. This makes it an essential tool for modeling complex systems and making predictions based on evidence.

2. What are the advantages of using first-order logic in scientific investigations?

First-order logic provides a solid foundation for logical reasoning and allows for the formulation of precise hypotheses and theories. It also enables scientists to identify and analyze the logical structure of arguments and avoid common fallacies. Additionally, using first-order logic allows for the application of mathematical methods and tools to analyze complex systems.

3. Can first-order logic be applied to all areas of scientific research?

Yes, first-order logic can be applied to various fields of scientific research, including physics, biology, computer science, and linguistics. Its flexibility and versatility make it a useful tool for analyzing and modeling complex systems in different domains.

4. How does first-order logic differ from other types of logic?

First-order logic is a type of formal logic that differs from other types, such as propositional logic, in that it allows for quantification over individuals and predicates. This means that it can handle statements about specific objects and their properties, making it more suitable for modeling real-world phenomena and making inferences about them.

5. What are some limitations of first-order logic in scientific research?

While first-order logic is a powerful tool for reasoning and making predictions, it does have its limitations. For example, it cannot handle uncertain or probabilistic information, and it may not be able to capture the full complexity of certain systems. In these cases, alternative logics or approaches may be needed to supplement or replace first-order logic.

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