Final answer:
The question addresses the relationship between logical division in substructural logic and mathematical division, pondering on undefined and indeterminate results in expressions such as a\0. This analogy is informed by Frege's symbolic logic and philosophies like structuralism but maintains distinctness due to the unique properties of logical and mathematical systems.
Step-by-step explanation:
In the context of substructural logic and the discussion on the two flavors of negation, the analogy is drawn between logical division and mathematical division, particularly regarding expressions like a\0. Such analogies delve into understanding the undefined nature, akin to 1/0 in mathematics, which represents an undefined result since division by zero is not permissible in the mathematical framework. In contrast, 0/0 represents an indeterminate form, implicating multiple potential values. Philosophical logic, building upon Frege's work, strives to eliminate the ambiguity of natural language by using formal symbolism. However, it's crucial to differentiate from mathematical division, where undefined and indeterminate have very particular meanings that do not transfer simplistically to the domain of logic.
The exploration of the concepts of negation and the philosophies surrounding them, such as structuralism and post-structuralism, highlight the complexities and norms of logical reasoning, which often mimics mathematical reasoning in its precision and adherence to established norms. However, they do not always completely align, as logicians must consider more than just the formal structures and absolute truths often associated with mathematical constructs.