The Zahl-Anzahl Distinction in Gottlob Frege: Arithmetic of Natural Numbers with Anzahl as a Primitive Term

The starting point is Peano’s expression of the axiomatics of natural numbers in the framework of Leśniewski’s elementary ontology. The author enriches elementary ontology with the so-called Frege’s predication scheme and goes on to propose the formulations of this axiomatic, in which the original natural number (N) term is replaced by the term Anzahl (A). The functor of the successor (S) is defined in it.

Relation of Leśniewski's mereology to Boolean algebra

It has been stated by Tarski [5] and “proved” by Grzegorczyk [3] that: (A) The models of mereology and the models of complete Boolean algebra with zero deleted are identical.Proved has been put in quotes, not because Grzegorczyk's proof is faulty but because the system he describes as mereology is in fact not Leśniewski's mereology.Leśniewski's first attempt at describing the collective class, i.e. mereology, was done in ordinary language with no rigorous logical foundation. In describing the collective class, he needed to use the notion of distributive class. So as to clearly distinguish and expose the interplay between the two notions of class, he introduced his calculus of name (name being the distributive notion), which is also called ontology, since he used the primitive term “is.” At this stage, mereology included ontology. Then, in order to have a logically rigorous system, he developed as a basis, a propositional calculus with quantifiers and semantical categories (types), called protothetic. At this final stage, what is properly called mereology includes both protothetic and ontology.What Grzegorczyk describes as mereology is even weaker than Leśniewski's initial version. To quote from [3]: “In order to emphasize these formal relations let us consider the systems of axioms of mereology for another of its primitive terms, namely for the term “ingr” defined as follows:A ingr B.≡.A is a part B ∨ A is identical to B.The proposition “A ingr B” can be read “A is contained in B” or after Leśniewski, “A is ingredient of B”.”

Computational logic

We make certain notational conventions. These are referred to as CL (Computational Logic). The relation of interchangeability is introduced as the basic connection between logical formulas. This approach lends perspicuity to the results of sentential calculus. With its technical devices CL is able to rephrase logical theorems in rather succinct manner. Our exposition tries to steer a middle course between informality and strict rigor.It is felt that the method of CL offers advantages for the teaching of logic. Proofs are algorithmic and resemble those of elementary algebra. All inferences are reversible and practically non-tentative.Sign is a primitive term of CL. Its denning property is a capacity for entering into binary combination with other signs, or with itself, according to this convention:x and y are signs if, and only if, (x y) is a sign. Bracketing is looked upon as an operation on signs in terms of which other operations are definable. The practice of some authors in classifying brackets under the heading of symbols, seems to us questionable; for unlike symbols or signs brackets are never used to denote anything. Brackets enter into the composition of signs, not to denote a grouping, but rather to exhibit it, in the manner of a diagram.An unending list of letters, with or without subscripts,serve to denote arbitrary signs. An arbitrary sign may or may not have other signs as parts. The numeral “2” is used as a constant. Bracketing abbreviation is as follows:and so on. Outermost brackets will ordinarily be omitted. This kind of bracketing may be termed left-associative. For convenience, these bracketing conventions are crystallized into a rule.