Abstract

Our research concerns a formal representation of Bolzano’s original concepts of Substanz and Adhärenz. The formalized intensional theory enables to articulate a question about the consistency of a part of Bolzano’s metaphysics and to suggest an answer to it in terms of contemporary model theory. The formalism is built as an extension of Zalta’s theory of abstract objects, describing two types of predication, viz. attribution and representation. Bolzano was aware about this distinction. We focus on the consistency of this formalism and the description of its semantics. Firstly, we explore the possibility to reconstruct a Russellian antinomy based on the concept of the Bolzano’s Inbegriff of all adherences. Our aim is to show limitations of his theory that prevent a contradiction when the Inbegriff consists of non-self-referential adherences. Next, we discuss two competing semantics for the proposed theory: Scott’s and Aczel’s semantics. The first one yields a problematic result, that there are no models for the considered theory, containing a non-empty collection of all adherences. This is due to the fact that Scott’s structures verify the formula on reloading abstracts in extensional contexts. We show that Aczel’s semantics does not contain this difficulty. There are described Aczel’s models with a non-empty set of all adherences. The self-referentiality of such a collection becomes irrelevant here. Finally, we show that there are Aczel’s structures verifying the formula on reloading abstracts and we exclude them from the class of models intended for our theory.