How to Formalise Inductive Reasoning
I have been wondering along those same lines. In academia.edu, I have posted Induction and the Uniformity Principle; derivation of the principle from fundamental axioms. Here is the table of contents:
2.0 The inductive process is a syllogism.
3.0 The Uniformity Principle is the major premise of the inductive syllogism
3.2 Objection to the Uniformity Principle; the course of nature changes.
3.3 Objection to the Uniformity Principle; simple enumeration fails.
4.0 The Uniformity Principle is derived from three fundamental axioms.
4.1 The purpose of the axioms
4.2 The axioms of uniformity: Existence, Identity, Continuity
4.3 The axioms are valid and support uniformity.
4.4 The three axioms are necessary and sufficient.
5.0 Uniformity permits the process of falsification that allows the content of the conclusion to be estimated.
I do not know how closely my analysis approximates that of Prof. Toulmin. However, I think that my one-step-at-a-time approach is similar to what he was looking for.
I hope this answer is useful.