Circular reasoning

Definition

Circular reasoning is a logical fallacy in which the conclusion of an argument is used as a premise within the same argument, effectively going in circles without providing actual evidence. This type of reasoning can obscure the validity of an argument by assuming what it is trying to prove, making it seem valid without offering true justification. In the context of axiomatic systems, circular reasoning can jeopardize the consistency and independence of axioms by failing to establish a foundation for the axioms themselves.

5 Must Know Facts For Your Next Test

1. Circular reasoning can undermine the effort to establish a system of axioms because it does not provide independent justification for those axioms.
2. In formal logic, circular reasoning is often seen as a critical flaw that can lead to inconsistency within a theoretical framework.
3. The presence of circular reasoning can result in arguments that appear convincing on the surface but lack substantive support, which is especially problematic when discussing foundational principles.
4. To avoid circular reasoning, it's essential to ensure that premises provide independent support for conclusions rather than relying on the conclusions themselves.
5. Identifying circular reasoning can be challenging, as it may require a careful analysis of the structure of an argument to discern if it truly provides valid premises.

Review Questions

• How does circular reasoning impact the consistency of an axiomatic system?
  • Circular reasoning affects the consistency of an axiomatic system by failing to provide independent support for its axioms. When an axiom's validity relies on its own conclusion, it creates a loop that does not clarify or justify the foundational principles. This undermines the entire system because it cannot produce reliable proofs or deductions from those axioms, leading to potential contradictions within the system.
• What are some common examples of circular reasoning found in arguments related to independence of axioms?
  • Common examples of circular reasoning include arguments where one states that an axiom is true because it leads to results that are assumed to be true. For instance, saying 'Axiom A is valid because it allows us to derive theorem B, which we know is true because Axiom A asserts its truth' creates a loop. This kind of reasoning fails to establish independence because it does not provide any external justification for either the axiom or the derived theorem.
• Evaluate the implications of circular reasoning on mathematical proofs and their foundations.
  • Circular reasoning can significantly undermine mathematical proofs and their foundations by eroding trust in the results derived from those proofs. If a proof relies on assumptions that circle back on themselves, then the derived conclusions are not genuinely established. This raises concerns about the reliability and validity of mathematical systems as a whole, making it crucial for mathematicians to avoid such reasoning in order to maintain rigorous standards and ensure that each theorem rests on independently verifiable axioms.

Related terms

Axiom:

A statement or proposition that is accepted as true without proof, serving as a starting point for further reasoning or arguments.

Logical Fallacy: An error in reasoning that renders an argument invalid, often leading to faulty conclusions.

Redundancy:

The unnecessary repetition of information or ideas, which can contribute to circular reasoning by reiterating points without advancing the argument.