Introduction
The algebra of logic, founded by George Boole and further developed by Ernst Schröder, establishes fundamental laws to express reasoning principles While it can be viewed as a mathematical algebra based on arbitrary principles, its philosophical implications regarding the mind's operations remain a separate discussion The formal value of this calculus holds significance for mathematicians, independent of its interpretation or application to logical issues Thus, our focus will be on the algebraic aspects rather than its logical implications.
The Two Interpretations of the Logical Calculus
The algebra in question can be interpreted in two distinct ways, similar to logic, depending on whether letters represent concepts or propositions By aligning with Boole and Schröder, we can unify these interpretations by viewing concepts and propositions as corresponding to classes; a concept defines a class of objects (its extension), while a proposition defines the class of instances or moments in which it holds true (also its extension) This leads to the conclusion that the calculus of concepts and propositions can be merged into a single calculus of classes, or the theory of whole and part, as termed by Leibniz However, notable differences between the calculus of concepts and propositions remain, preventing their complete formal identification and reduction to a singular calculus of classes.
In essence, there are three distinct calculi or interpretations of the same calculus, but it's crucial to understand that the logical value and deductive sequence of the formulas remain unaffected by these interpretations To facilitate this necessary abstraction, we will use the symbols C I (conceptual interpretation) and P I (propositional interpretation) before the relevant phrases These interpretations are intended solely to enhance clarity and comprehension of the formulas, without serving as justifications for them, and can be omitted without compromising the logical integrity of the system.
To maintain clarity and avoid bias towards any interpretation, we define letters as representing terms, which can refer to either concepts or propositions depending on the context We specifically use the term "term" in its logical sense Additionally, when referring to the components of a sum, we will use the term "summand" to prevent any confusion between its logical and mathematical meanings.
A term may therefore be either a factor or a summand.
Relation of Inclusion
The algebra of logic, like other deductive theories, can be based on different foundational principles For our discussion, we will select the system that closely aligns with Schröder's exposition and contemporary logical interpretations.
The core concept of this calculus is the binary relation known as inclusion (for classes), subsumption (for concepts), or implication (for propositions) We will refer to it as inclusion, as it reflects both logical interpretations, represented by the symbol