What are the Fundamental Logical Axioms?

In summary, there is no definitive list of logical axioms underlying all axiomatic systems, postulates, and mathematics. While attempts have been made to create a formal logic for all of mathematics, such as the Zermelo-Fraenkel axioms and Peano's axioms, none have been proven to be both correct and complete. However, there are axiom systems for logic that are both correct and complete, such as those described in the lecture notes of Lou Van den Dries.
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Atran
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Hi, I've been recently reading about logic. Is there a list of the exact logical axioms underlying all axiomatic systems, postulates and mathematics?

Thanks...
 
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  • #2
Atran said:
Hi, I've been recently reading about logic. Is there a list of the exact logical axioms underlying all axiomatic systems, postulates and mathematics?

Thanks...

The short answer is no. Russell and Whitehead tried to create (or discover) a formal logic for all of mathematics and essentially failed. The formalization of modern mathematics is based to a large extent on the Zermelo-Fraenkel axioms of Set Theory together with the Axiom of Choice (ZFC) and Peano's axioms of arithmetic.

http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
 
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  • #3
SW VandeCarr is certainly correct that there is no axiom system which is both correct and complete. This was proven by Kurt Godel in the 1930's. The mathematicians proceeded by just making up axiom system which looked correct and the started using them. They didn't care whether they are complete or not.

If you talk about logic, then the situation is quite different. There are axiom systems for logic which are both correct and complete. A good reference for this are the lecture notes of Lou Van den Dries: http://www.math.uiuc.edu/~vddries/ click on "Logic Notes" (you will need to be able to open DVI-files for this).
An axiom system for logic is described on page 38. But I don't think that it is known to be correct and complete...
 

What are the Fundamental Logical Axioms?

The Fundamental Logical Axioms, also known as the Laws of Thought, are a set of principles that form the basis of logical reasoning and deduction. They are considered to be self-evident truths and are accepted as the foundation of all logical systems.

What are the 3 main axioms of logic?

The three main axioms of logic are the Law of Identity, the Law of Non-Contradiction, and the Law of Excluded Middle. These axioms state that a statement is either true or false, that a statement cannot be both true and false at the same time, and that there is no middle ground between true and false.

Are the Fundamental Logical Axioms universally accepted?

Yes, the Fundamental Logical Axioms are universally accepted as the basis of logical reasoning. They are fundamental to all logical systems and are used in various fields such as mathematics, computer science, philosophy, and linguistics.

Can the Fundamental Logical Axioms be proven?

No, the Fundamental Logical Axioms cannot be proven as they are considered to be self-evident truths. They are accepted as the starting point for logical reasoning and cannot be deduced from any other principles.

What happens if one of the Fundamental Logical Axioms is violated?

If one of the Fundamental Logical Axioms is violated, it leads to a contradiction in logical reasoning. This can result in invalid arguments and false conclusions, making the entire logical system unreliable.

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