Indeed, at the time of their correspondence, Frege's work was considerably more "formal" than Hilbert's, since Hilbert at this time was not using an explicit syntactically-defined system of deduction.
Thus the Stoics concerned themselves with several issues we would place under the heading of predicate logic; but their main achievement was the development of a propositional logic, i.e. of a system of deduction in which the smallest substantial unanalyzed expressions are propositions, or rather, assertibles.
Chrysippus developed a syllogistic or system of deduction in which he made use of five types of basic arguments or argument forms called indemonstrable syllogisms, which played the role of axioms, and four inference rules, called themata by means of which complex syllogisms could be reduced to these axioms.
Designed as "revenue-neutral," the idea was to reduce tax rates while eliminating almost all of what had become a complex system of itemized deductions, including all deductions for state and local taxes.
Long before the debate with Hilbert, Frege already held that logical rigor requires the use of formal systems of deduction, "formal" in the sense that all thoughts are expressed via precisely-determined sentences, and that all inference-rules and axioms are presented syntactically.
The reason is that most income must be reported to the I.R.S. by those who pay it, but there is no corresponding system of reporting deductions.
In Japan, and many European countries, private health insurers — all of them nonprofit — finance visits to private doctors and private hospitals through a system of payroll deductions.
Gentzen developed and studied the system of natural deduction during 1932, and had arrived by September 1932 at a calculus of natural deduction (ND for short) that is standard today.
The system of natural deduction lay mostly dormant for some thirty years, until the thesis of Dag Prawitz of 1965, Natural Deduction: A Proof-Theoretical Study.
Prawitz gave first a normalization theorem and subformula property for a system of natural deduction for classical logic.
