The work opens with preliminary explanations of ideas and notations, establishing variables, propositional functions, and the fundamental functions of propositions. The authors define contradiction, logical sum, logical product, implication, and equivalence, while introducing the assertion-sign and the use of dots for bracketing. Definitions are shown to be theoretically superfluous yet practically indispensable for analysis.
The text then addresses the theory of logical types, setting forth the vicious-circle principle and the nature of propositional functions. The hierarchy of functions and propositions is constructed to avoid paradoxes like those of Burali-Forti and Epimenides, and the axiom of reducibility is introduced to legitimate ordinary mathematical reasoning without assuming the independent existence of classes.
Incomplete symbols are examined extensively through the theories of descriptions and classes, demonstrating how grammatical subjects can disappear through logical analysis. The distinction between extensional and intensional functions is established, along with the precise requisites that any theory of classes must satisfy.
The volume proceeds to develop mathematical logic proper, beginning with the theory of deduction and the immediate consequences of primitive propositions. Formal rules, equivalence, and the logical product of two propositions are established through systematic deduction.
The theory of apparent variables is systematically extended from lower to higher types of propositions. Formal implications are analyzed, and identity is formally defined through the axiom of reducibility to ensure that identical objects share all properties.
The volume concludes by laying the groundwork for the structural hierarchy of functions and propositions, providing the precise logical machinery required for all subsequent mathematical derivations across the later parts of the work.
