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Relation of Leśniewski's mereology to Boolean algebra

Published online by Cambridge University Press:  12 March 2014

Robert E. Clay
Robert E. Clay*
Affiliation:
University of Cape Coast, Cape Coast, Ghana University of Notre Dame, Notre Dame, Indiana 46556
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Extract

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”.”

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 39 , Issue 4 , December 1974 , pp. 638 - 648
Copyright
Copyright © Association for Symbolic Logic 1974

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References

REFERENCES

[1] Clay, R. E., Sole axioms for partially ordered sets, Logique et Analyse, vol. 48 (1969), pp. 361–375.Google Scholar
[2] Clay, R. E., The relation of weakly discrete to set and equinumerosity in mereology, Notre Dame Journal of Formal Logic, vol. 6 (1965), pp. 325–340.Google Scholar
[3] Grzegorcyk, A., The systems of Leśniewski in relation to contemporary logical research, Studia Logica, vol. 3 (1955), pp. 77–97.Google Scholar
[4] Słupecki, J., Towards a generalized mereology of Leśniewski, Studia Logica, vol. 8 (1958), pp. 131–154.CrossRefGoogle Scholar
[5] Tarski, A., On the foundation of Boolean algebra, Logic, semantics, metamathematics, Clarendon Press, Oxford, 1956, pp. 320–341.Google Scholar