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Bolzano's presentation of logic is embedded in the vast Theory of Science (henceforth TS).
The latter dealt principally with history but also included some philosophical material, including a different presentation of logic.
His obsessive concern for the minute detail of axiomatizations was unattractive to many when more streamlined methods were available, and his radical nominalism made the presentation of logic according to his principles a matter of extreme inconvenience.
As Göran Sundholm emphasized in conversation, the "relative completeness" that Heyting mentions, i.e., the richness of the presentation of logic in Principia Mathematica, must have been his reason for not taking Hilbert 1923 as his point of departure.
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Aside from Herbart's textbook presentation of traditional logic (e.g., Section II of LEP), his main logical innovation is his so-called method of relations, devised as an organon for correct metaphysical thinking (SW V: 201, f).
Now consider a one-sided sequent calculus presentation of linear logic where the only introduction rules are right-introduction rules.
For example, just in the chapters devoted to contemporary evolutionary biology, you find a presentation of the logic of speciation, sexual selection, etc. but without real attention to evidence that can be found in nature for those mechanisms or to falsification that potentially excludes other explanations.
See Shapiro's entry on Classical Logic in this encyclopedia for a detailed presentation of first-order logic.
One of the obstacles to penetrate the various presentations of basic logic is the lack of a similar formulation.
This is why the presentation of Bolzano's logic must begin with the examination of several preliminary concepts.
Wittgenstein supplies, in the Tractatus, a vivid presentation of Frege's logic in the form of what has become known as 'truth-tables'.
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CEO of Professional Science Editing for Scientists @ prosciediting.com