Final answer:
The rejection of axioms in formal systems can lead to inconsistencies and a reduced ability to generate reliable outcomes, as seen with the development of non-Euclidean geometries or non-classical logics. However, such rejections can also lead to new insights and frameworks that, while internally consistent, must also be judged on their applicability and alignment with real-world problems to determine their usefulness.
Step-by-step explanation:
Rejecting axioms in formal systems impacts their consistency and usefulness by potentially creating contradictions and lessening the system's ability to produce coherent and reliable outcomes. Axioms function as foundational truths in logical frameworks, and when they are altered or dismissed, the entire structure may be compromised. Philosophical and mathematical systems depend heavily on axiom-based logic to maintain internal consistency and to derive further truths.
An example of this is the rejection of Euclid's parallel postulate in geometry, leading to the development of non-Euclidean geometries. Initially, such rejections seemed to yield systems with contradictory or illogical geometrical interpretations. However, they ultimately proved to be consistent within their own frameworks and provided valuable insights into the nature of space and the foundations of mathematics. Similarly, in philosophy, the questioning of axioms can lead to periods of upheaval, as was seen in the transition from classical logic to non-classical logics such as fuzzy logic, which relaxed the principle of bivalence and permitted a range of truth values beyond the binary true and false.
The usefulness of a logical system, while influenced by its axioms, is not solely determined by them. The system must also align with our intuitions and be applicable to the kinds of problems we wish to solve. Hence, some logical systems may be perfectly consistent internally but not particularly useful because they don't adequately reflect the complexity of the world or do not allow us to interact with empirical evidence in a meaningful way.