Final answer:
The axiomatization for formulas in propositional logic using '=>' and 'not' can be sound and complete with a set of axiom schemas and inference rules. These include axioms and modus ponens, although the applicability of mathematical axioms is limited to their domain and should not be generalized.
Step-by-step explanation:
The question pertains to the axiomatization for formulas in propositional logic, specifically involving the use of implication (=>) and negation (not). An axiomatization is said to be sound if all theorems that can be derived from the axioms are true. A complete axiomatization means that all true formulas can be derived from the axioms.
In propositional logic, a common sound and complete axiomatization can be achieved by using a few key axiom schemas and inference rules. One such system uses modus ponens, which allows one to infer 'Q' from 'P => Q' and 'P', and includes axioms such as:
Axiom K: P => (Q => P)
Axiom S: (P => (Q => R)) => ((P => Q) => (P => R))
Axiom N: (~P => ~Q) => (Q => P)
With these axioms, plus the rules of substitution and modus ponens, one can derive the theorems of propositional logic that involve '=>' and 'not'.
It is crucial to note, however, that mathematical truths and axioms do not necessarily generalize to other domains, such as morals or chemistry, as highlighted in the provided text. While mathematical axioms are precise within their scope, their application outside of mathematics can be misleading or incorrect.