“The first basic notion of Bolzano’s ontological system is the part relation. Its domain, i.e., the set of all objects bearing it to something, embraces concrete substances, abstract objects, and collections. The converse domain of the part relation, i.e., the set of all objects to which it is borne, contains collections only.
Some collections are concrete entities existing in space and time, the rest are abstract sums or other sets. Concrete sums are composed of substances and adherences, i.e., forces. Forces applied to certain substances give rise to subjective ideas or judgements. Further results of such applications are the concrete sentence occurrences. A subjective idea is a part of a judgement which is not itself a judgement. The set of judgements is ordered by a special causal relation.
Bolzano’s abstract world is constituted of sets, abstract sums, certain attributes (i.e., properties or relations), ideas-as-such, and objects constructed on the basis of these entities. Thus, sentence shapes are a kind of properties, and certain complexes of ideas-as-such constitute propositions. The notion of an idea-as-such can be constructed from expressions of a language by means of axioms for the relation of being an object of something. Analogously, properties can be generated by axioms for the relation of something being applied to an object. The converse of this relation, i.e., the relation of an entity having a property, and the relation of being an object of an idea-as-such are fundamental ontological constants of Bolzano’s.
Natural numbers are defined as properties of bijective sets, and real numbers are essentially conceived of as properties of sets of certain infinite sequences of rational numbers. The analysis of infinite sets leads to a generalization of the part relation by scrapping the doctrine that the whole is always greater than any of its parts. The extension of the linear continuum of finite numbers by infinitesimals within the coarsest free algebraic filter settles definite limits to Bolzano’s approach to analysis.
A part relation in a narrower sense, viz., the relation of being a subsequence of a sequence of abstract objects, holds among ideas-as-such and propositions. Furthermore, the relation of derivability holds among propositions, and true propositions are ordered by the relation of entailment.
Among the relations holding between the constituents of the concrete world and the abstract world there are the relations of a substance having a property or being an object of an idea-as-such. Moreover, the relations of an idea-as-such or proposition being the subject matter of a subjective idea or a judgement, respectively, establish ontologically important connections between the abstract world and the concrete world.
The main features of Bolzano’s ontology may be schematized as follows:
The question whether a rational reconstruction of Bolzano’s ontology is possible will be sustained like a pedal point throughout the present study. In many respects, indeed, his ontological system is a model of thrift, comprehensiveness, and deductive cogency. He shows us how to grasp a self-contained, abstract “third” world (in Popper’s sense) embracing the realms of classical logical truth and additive probability spaces without indulging in possible worlds, states of affairs, facts, and all that. Admittedly, from a modern point of view certain aspects of his ontology may look like Dr. Johnson’s dog walking on its hind legs: it is not always done quite well, but you are surprised to find it done at all. To rational bipeds of our time it should be more instructive, though, to watch this performance rather than amazing at metaphysical cephalopods wallowing in clouds of ontological splendors, or gazing at recondite cogitators crawling on all fours through a self-induced verbal fog.” (pp. 31-32)
“Ontology without possible worlds.
A minimal requirement for pursuing philosophy of science and mathematics is the access to sentence (or formula) shapes, an adequate truth definition, substitution, and some set-theoretic principles. The first three notions allow a semantic demarcation of the
realms of classical logical truth and additive probability spates. Apart from syntactic identity, the strongest semantic principle of individuation for sentence shapes is logical equivalence. If one should insist on abstract objects with stronger semantic identity conditions, as Bolzano did for reasons of philosophical foundations, then non-linguistic propositions may be tendered.
Bolzano proceeded from an expanded and standardized ordinary language by means of which he could describe the universe of propositions and their parts. We have seen that this exposition can be organized into explicit postulate systems. The existence of propositions and their parts being thus guaranteed, Bolzano defined the semantic notion of truth and introduced the function corresponding to a “replacement” operation on propositions. He could also easily have rendered an exact definition of the notion of a sentence shape. The replacement of conceptual complexes in propositions enabled him to develop the essential parts of classical logic and probability theory without resorting to ontologically lavish constructions.
Bolzano’s notion of proposition offers an interesting alternative to the corresponding concepts developed in modern possible-world semantics. (For a lucid survey, see Edgar Morscher, Propositions and all that: Ontological and epistemological reflections, in: L. M. de Rijk (ed.), Logos and Pragma. Essays on the Philosophy of Language in Honour of Professor Gabriel Nuchelmans, Nijmegen, 1987, pp. 241-257) According to a representative theory of this kind, a proposition is a function sending possible worlds onto truth-values. A possible world is a maximally consistent set of states of affairs. A state of affairs is somehow conceived of as being built up from members of the domain of individuals and their attributes. Moreover, a fact is a real state of affairs. Thus, a concrete object and its attributes can be parts of a state of affairs. For example, the concrete individual Kurt Waldheim and the property of being the 42nd president of the United States of America would, according to this view, be parts of the state of affairs that Kurt Waldheim is the 42nd president of the U.S.
The main flaws of this approach to the ontology of propositions are, first, that propositions expressed by logically equivalent sentences conflate and, second, that a concrete object can never be part 01 a state of Alai’s which is nut a fact. lot example, the real Kurt Waldheim can never be part of the state of affairs of someone being the 42nd president of the U.S.
The latter obstacle can be removed by representing concrete things by bundles of world-lines, i.e., by sets of sets of world-points. The real Kurt Waldheim, e.g., is thereby represented by a bundle of world-lines which will never enter into a state of affairs containing the property of being the 42nd president of the U.S. The fictitious Kurt Waldheim figuring in such a state of affairs branches off from the bundle representing the real Kurt Waldheim at a certain space-time point in the world of 1993. In view of the highly abstract character of this approach, an alternative remedy might be to leave states of affairs unanalyzed and take them as primitive entities. From the ontological point of view, however, we could then as well get on directly with propositions.
An attempt to evade the former difficulty of propositions conflating under logical equivalence of the corresponding sentences by proffering new categories of intensional objects will be a great expense to unyielding ontologians. One device may be to take the functions sending the possible worlds onto truth-values in intension. Hence, a practicable theory of propositions based on a possible-world semantics would have to postulate the existence of sets of sets of world-points, and moreover of properties, relations, and function concepts. An attempted entity-saving measure of introducing the attributes by functions in extension from possible worlds onto sets of individuals or sets of n-tuples of individuals would be redundant, however, since attributes are parts of the constituents of possible worlds.
An ontology based on Bolzano’s system of propositions would only have to postulate the existence of one category of intensional objects, namely ideas-as-such, and could otherwise employ purely extensional set-theoretic and algebraic methods. A possible objection to Bolzano’s ontology might be raised on account of the fact that it cannot yield the semantics of epistemic and other non-classical logics. In these regions outside the analysis of the foundations of science and mathematics, it may be argued, real philosophy begins with the search for new semantic superstructures while the metaphysical dusk of possible worlds approaches.”
(From: Jan Berg, Ontology without Ultrafilters and Possible Worlds. An Examination of Bolzano’s Ontology, Sankt Augustin: Academia Verlag 1992