Two Formal Interpretations of Bolzano’s Theory of Substances and Adherences

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. (Bolzano’s theory of ideas is often suspected of antinomial consequences.) 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.

if any abstract individual belongs to the extension of any property, then all abstracts belong to it. This observation alone gives enough motivation to investigate another possible semantical framework for our theory. To this end, we turn to the semantics elaborated by Zalta (1997Zalta ( , 1999, which was inspired by Aczel. We adopt a version of Aczel's structures and show models with a non-empty set of all adherences. We argue that the question of 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.

Philosophical Background and Formal Framework
In this section we briefly present the main ideas formulated by Świętorzecka (2017), but adding a number of changes to them. We also revise some translations of the original texts and refine their interpretation in light of additional sources. Most of these modifications are inspired by suggestions from Professor Wolfgang Künne.

Bolzano on Substance-Adherence, and the Collection of All Adherences
In Bolzano's philosophical system, the concepts of substance and adherence function as metaphysical categories (WL, I.560). 1 The first extensive passage in which this pair occurs comes from Athanasia-the work in which Bolzano tries to defend the hypothesis of the immortality of the human soul. There, Bolzano sketched an argument for the existence of substances based on the concept of adherence. 2 This argument is reconstructed in our formalization: [1] If there is one actual [Wirkliches] thing, it is evident that there must be one or more than one substances. [2] For an actual thing that is not a substance must be an adherence, hence a property [Beschaffenheit], and so it presupposes another actual thing in which it occurs [befindet sich an]. [3] This other actual thing might in turn be an adherence or property because properties, too, have their own properties, and if the former are actual things then so are the latter.
[4] But, if we think of all actual things that are merely adherences as united in one collection [Inbegriff], it soon becomes clear that this [collection], too, cannot exist [bestehen] by itself, but it requires one or several other actual things in which it occurs. [5] Now this one or these several actual things are not again adherences, for if they were they would belong to this collection; so they are 1 References are to volume number and page number of the original German edition of WL (Bolzano 1837a). Readers may find detailed references to Bolzano's texts and an extensive study of these concepts especially by Künne (2015) and Schnieder (2002). We refer only to the passages that are relevant to our later approach introduced in Sect. 2. 2 Bolzano distinguished between simple substances and complex substances (aggregates of the former), and argued for the existence of simple substances from the existence of such aggregates (cf. Bolzano 1851, §50-65). substances.
[6] Therefore, it is certain that there are substances as it is certain that there is at least one actual thing.
(original text: Bolzano, ed. 1838, 39, we take author's translation, cf. Świętorzecka 2017) The adjective 'wirklich' is translated as 'actual' (from 'wirken'-'to act'). Throughout the whole text we adopt Künne's translation for most key Bolzanian terms (Künne 2008). The above argument directly applies to souls only if Bolzano understands 'substance' as 'simple substance'. Bolzanian substances, if there are any, are actual things, that is, they are elements of the causal order. They are bearers of adherences, which also are actual things. Such substances are simple, they are not perceivable, and exist omnitemporally (human souls are supposed to be a subclass of them). In many respects Bolzanian simple substances are like Leibnizian monads, but interact with each other. 3 Substances and adherences constitute the realm of all actual things. Bolzano's general notion of a substance and his distinction between substances and adherences is clearly inspired by Aristotle's dichotomy of substances and accidents. Aristotelian substances possess their accidents as attributes (cf. Aristotle 1928 Met 1028a-29; Cat 15b; Met 1022b). The same applies to Bolzano's substances and adherences (WL, I.379), although for Bolzano the relation of possessing may also occur between adherences, when one adherence is the subject of another (cf. [3] in the block quotation above). In order to grasp Bolzano's concept of adherence, let us focus on a few examples from Athanasia. There, he claims that the color, smell, and weight of a given body are adherences (1838, §1). An adherence is also this redness of a given flower ( §1, 24). Unusual power may also be an adherence of memory, which in turn is an adherence of any given person ( §1, 23). 4 Possession of an attribute by a subject is expressible in Bolzano's system by a sentence in its canonical form: A has b, where "b represents the abstractum that belongs to the concretum B" as used in the traditional construction: A is B (WL, II.9). Bolzano's canonical sentences consist of two terms connected by the copula has. 5 Sentences express (drücken aus) the so-called propositions in themselves (Sätze an sich), which are built from ideas in themselves represented by the terms A, b and the copula has. Propositions and ideas in themselves form the Bolzanian world of non-actual objects.
The canonical form A has b may be substituted with a certain restriction: 3 Bolzano's diary notes entitled Verschiedenheit zwischen Leibnitzens und meinem Ansichten (Bolzano 1837b) are very informative for this comparison (they were overlooked by Świętorzecka (2017)). 4 Examples such as these seem to show that possessing is neither reflexive, nor symmetric, or transitive. These features are not captured by our formalization. We shall also put aside the fact that Bolzano distinguishes between properties of actual objects that are identity-dependent on their bearers and properties of actual objects that are shared by several objects. The former are particulars, that latter universals. If the opposition between substance and adherence is to be exactly grasped and preserved then one has to understand by substance only that actual thing (an entity) that can only be thought of as a subject, never as a predicate. (Bolzano 1833, 144) Thus, the variable b in the scheme A has b cannot represent any substance. The variable A, however, may be substituted by terms designating (bezeichnen) substances or adherences. According to Bolzano, the term b in A has b designates an adherence if, and only if A designates an actual object, and A has b expresses a truth. If these conditions are fulfilled, we can call b an 'adherential term' and the idea expressed by it an 'adherential idea'. Thus understood, an adherential idea is never empty or, as Bolzano would put it, objectless (gegenstandlos).
The one-to-one relationship between adherential terms and adherential ideas will be used in our rather simple interpretation of Bolzanian Inbegriff of all adherences. Let us focus one of the many meanings of the term 'Inbegriff', explained in WL (I.393) as a collection of individually listed objects: we are supposed to indicate the cause of a certain result M, and it turns out that several objects A, B, C, D, ... contributed to its production, 6 we must, in order to think of the complete cause of M, think of A, B, C, D, ... together, that is, think of the collection of A, B, C, D, ... And it is obvious, that [...] we do not think about the order and sequence in which they contributed to bringing about M.
(WL, I.394) Surely, adherences are not the cause of substances. However, all of them taken together need (require, bedürfen) existence of substances. We consider the components of such a whole in an intensional way. The very concept of Inbegriff of all adherences will get both extensional and intensional interpretation. We take as the extensional counterpart of this concept, a set of all adherential ideas. From the other hand, it will be also represented by certain intensional object attributed to every adherential idea on the basis of inherence relation. What we should say is, that the proposed extensional interpretation may be considered as not adequate in case of Bolzanian Inbegriff in general. 7 However, our approach is just not general but only dedicated to adherential Inbegriff. It enables us to trace the original structure of the analyzed argument and for this reason, we think that our solution has explanatory power.

Zalta's Theory
The framework we will consider is a fragment of a rich formal ontology originally proposed by Zalta (1983). The first attractive feature of this formalism is the structure of its language. Zalta uses predicate terms that represent the so-called gerundive versions of certain verbs, predicate adjectives, and predicate nouns (Zalta 1983, 37-38). The predication analysed by Bolzano assumes the same grammar when he tries to bring all simple sentences into the canonical form A has b. Secondly, Zalta's language enables one to distinguish between extensional and intensional contexts.
To this end two different concatenations of individual and predicate terms are used. Formulas x read as follows: (individual) x exemplifies (the property) and mean that some individual object belongs to the extension of a certain property. Formulas x -(individual) x (represents) encodes (the property) are used to describe the fact that some individual belongs to the intension of a certain property. Bolzano was obviously aware of the extension-intension distinction. In his account, ideas in themselves that are not objectless may have the same extension (Umfang) even though they are different ideas. Also, an idea in itself is an intensional object that can be both the Stoff (matter, content) of thinking and the Sinn (sense) of a linguistic expression used to express the act of thinking (Künne 1997). Zalta's theory deals specifically with Meinongian abstract objects, which have the status of individuals. We claim that simple Bolzanian ideas have exactly the same status, although their number and variety are much more modest than Meinongian objects. This difference is kept in our proposed formalization of Bolzano's concepts. We contend that the specific axiom schema assumed by Zalta to introduce 'the paradise' of Meinong's abstracts, is not used in its full range. We start our brief presentation with a description of the language employed. We use x as a symbol for individual variables: x, y, z, … ; F for predicate variables: F, G, … ; the primitive constant E! reads as follows: exists (concretely); the binary constant = E -is (concretely) identical with; the logical symbols: ¬, →, ∀, and parentheses.
The predicate terms and formulas are the following: Expressions having the form x are called encoding formulas. Propositional formulas are expressions that do not contain encoding subformulas nor quantifiers for predicate variables. Let us use , as representing individual or predicate variables. Symbol t represents individual or predicate terms.
The assumed logic is characterized by classical propositional tautologies and all instantiations of schemata: The theory of abstract objects is characterized by adding the following proper axioms and definitions: Axiom A3 is assumed by Zalta to express the so-called Meinongian principle of the independence of being from being-so. 8 Abstract objects, like all objects, existso, even when they represent empty properties (like the property of being a round square). This perspective departs from Bolzano's strategy: his ideas mustn't have empty extensions, otherwise they are objectless. In the context of the above formalism, 'a surplus' of abstracts is irrelevant in our framework. We do not assume that all abstract individuals acknowledged by Zalta's theory are to be considered as Bolzanian ideas. Furthermore, we claim that abstracts, considered as ideas of adherences, are not objectless: adherences are always attributes of something.
We introduce the following constant for abstract object that does not encode any property 9

Theory
AdBℑ [ ] In this section we form a theory motivated by Bolzano's concepts of substance and adherence.
We use [X] to name a theory obtained from the theory by adding a set X of formulas or a single formula X as axioms to it. In the lower index we note the new primitive constants added to the language.
Formulas x are now to be read as x has (the property) . The language is enriched by adding two constants Ad and B, used in contexts Ad , B to be read as: (property) is an adherence; (property) is conditioned (by something actual).
Firstly, we formulate a thesis about the Bolzanian translation of sentences having the general form A is B into sentences having the general form A has b.
We define the concept of inherence: The theory of abstracts objects expressed in the above language will be called AdB .
To give an explanation of Inbegriff we use the predicate term ℑ introduced following Comp . However, our proposal breaks the restriction added to Comp . For this reason all ℑ substitutions of logical tautologies, A3 and A4 must be weakened by the condition about the existence and uniqueness of a property named just by ℑ : Now we list the following axioms motivated by the previously quoted passage from Athanasia and some of Bolzano's remarks on substances and adherences: 10 Let us call the set of the above axioms . Theory AdBℑ [ ] is a proposed formalization of the considered part of Bolzano's lecture on substances and adherences.
Our AB4 differs from its previous version of Świętorzecka (2017). 11 It is a simpler and more intuitive formulation of the assumption that justifies [5]. Let us also note that AB4 follows from AdF ∧ Fx ∧ ∀G(AdG → ¬xconG) → Sx which could be called the principle of the substantial foundation of adherences.
We now modify the proof of the main thesis, which is a counterpart of the theorem justified by Bolzano in Athanasia, weakened by condition IN: 10 We refer to numbers introduced in the quoted Bolzano's argumentation to note the inspiration for our axioms. 11 Originally, A4. IN → (ℑ x ∧ ¬xconS ∧ ¬ℑ y ∧ y x → ∃z(y z ∧ zconS)) was called the principle of substantial foundation of ℑ .

More About AdBℑ [ ] : Towards a Russellian Antinomy
Our theory states that: On the basis of AdBℑ [ ] we may put a question related to the conjecture that the Bolzanian theory of ideas is inconsistent: What about the self-referentiality of adherences and the very concept of Inbegriff? The problem of whether some of Bolzano's considerations on ideas in themselves drift towards contradictions in the style of Russell's antinomy, was raised by Berg (1987) and Morscher (2007). 12 The following passage recorded by Berg demonstrates that Bolzano was fully aware of the danger of an uncontrolled use of the concept of set: Problems involved in the first concepts of the Theory of Quantities. The total of all A, where A denotes an idea of limited scope, e.g. Man-is considered rather frequently. But as soon as we substitute for A the widest of all ideas, namely, the idea of Anything, then the problem arises that the idea: the total of all thing-or the total of all or the absolute All actually should comprehend even itself, since this All is something also; which is absurd, however (citation following Berg 1987, from Misscellanea mathematica, manuscript Dep. of the Austrian National Library, Series nova 3455, 1968) Berg claims that these explanations should be linked with multitudes which are sets in sense of Mengen and not with Inbegriffen of objects that stands under (fallen unter) ideas in themselves. Actually, Bolzano considered the idea of an object in general (and the idea of something) that stands under itself (WL, I.459, WL I 496). Had he also considered the idea of all non-self-referential ideas (which do not stand under themselves), then perhaps he would have become aware of the dangers of antinomial Russellian's reasoning. Indeed, if we assume some kind of comprehension schema for multitudes of objects standing under ideas (cf. WL, I.468-473), then we have the following problem: if, on the one hand, the idea of all ideas which do not stand under themselves is self-referential (means: stands under itself), then it is non-self-referential; on the other hand: if it is non-self-referential (i.e. it does not stand under itself), then it stands under the idea of all selfreferential ideas, hence it is self-referential. We leave the open question whether Bolzano's system was explicit enough to allow the reconstruction of such antinomy [and so to justify the formal approaches of Berg (1987) and Morscher (2007)].
In frame of our proposal, it is worth noticing that already allows to speak in a consistent way about the (non)self-referentiality of some abstract individuals in the sense of the predicate .
Let us introduce the following definition: An attempt to test the resistance of ℑ to Russell's antinomy, would not succeed in a way similar to that of Clark, 13 although the consequent of our AB0 breaks the restriction put on Comp . Let us assume IN and the definition ℑ − x = df ¬ℑ x . We call − the characteristic coder of ℑ − . From ℑ − we get only Adℑ − and ¬ℑ − with − conℑ − imply only ¬Adℑ − .
The connection between the self-referentiality of the characteristic coder of ℑ and the fact that ℑ is an adherence is described by the following theorem of Let us take the following definition of (weak) self-reference of properties: The self-referentiality of properties consists in the fact that some individuals representing them belong also to their extensions. We can easily notice that We cannot transfer the above connection to intensional subsumption. Now, the idea of taking adherences to be non-self-referential properties in the sense of ¬Ref leads to Russell's antinomy on the ground of AdBℑ [ ].
13 The purpose of Clark's argument is to demonstrate that we fall into Russell's antinomy when Comp is used without the restriction to propositional formulas. Let as assume that we accept the definition R = df x.∀F(xF → Fx) . The complement of R is described in the following way: . We take such that: conR − . The question now is whether R − or ¬R − ? We assume that R − . We get ∃F( F ∧ ¬F ) . We know that R − ∧ ¬R − . So: ¬R − . Now we assume that ¬R − . Then R , and from this ∀F( F → F ) . By logic we get: R − → R − , so: R − . This chain of reasoning refers to explanations given in Rapaport (1978).
We define an empty property as follows: Extensional identity of properties is understood as usual: The following equivalence enables to reach contradiction in Russel's style: One thing to notice is that in our formalization we can consistently speak of the self-referentiality of the characteristic coder of ℑ (if IN is fulfilled). It will be shown, that in the case of Aczel's semantics, there are models for AdBℑ [ , IN] that allow its self-referentiality.

Two Interpretations of AdBℑ [ ]
We consider two semantic frameworks originally designed for different versions of Zalta's formalism. The first one was suggested to Zalta by D. Scott. Classical firstorder fragment of the theory of abstracts objects is sound in this semantics (Zalta 1983). Our AdBℑ is a system of this kind. The second approach follows ideas that Zalta owes to P. Aczel. This interpretation was developed for certain extension of Zalta's system (Zalta 1997(Zalta , 1999. Here we will simplify Aczel's semantics and adopt it to our theory as more 'attractive' than the first one. Both interpretations taken in account, are based on the same idea of expressing extensional and intensional nature of predicates by the simultaneous use of two functions. Extension function assigns to every property corresponding to some predicate, a set of individuals that posses this property. Intension function assigns to every property a set of individuals which represent it. In the universe of individuals we meet concrete (ordinary) and abstract individuals. Objects of the first kind do not belong to a value of any intension function but individuals of both kinds can belong to extensions of properties. The way of treating abstracts as elements of extensions is different in the considered approaches. In Aczel's semantics there is used a technical trick consisting of an imitation of abstracts by their proxies. Thanks to this we can consider models including self-referential properties in the sense of D11 and such that do not have all abstracts in their extensions.
Scott and Aczel structures presented here are adequate to the fragment of Zalta's theory used in our formalization. However, they verify different sets of formulas and so they can behave in different ways when they are used for interpretation of some AdBℑ extensions. Indeed, as we will see, Scott's semantics does not allow to consider structures with non-empty set of adherences. This problem is not present in Aczel structures.

Scott's Semantics for
AdBℑ [ ] Following Zalta (1983), we formulate a definition of a Scott structure: For elements of Prop ⟨+, X⟩ , ⟨−, X⟩ we take notation +X , −X and we represent them using variables p, p � , … ( p = +X is an abbreviation for: p = +X or p = −X). Elements of 2 Prop are called abstract individuals s and the sum:  ∪ 2 Prop is the set of all individuals-objects s . We use o, o ′ to represent elements of  ∪ 2 Prop .
(iii) ext ∶ Prop → 2 ∪2 Prop is the function of extension s of properties, which fullfills the following condition: o ∈ ext(p) ⟺ ∃ X∈2  (p = +X and o ∈ X) or ∃ X∈2  (o ∈ 2 Prop and p = +X) (iv) int ∶ Prop → 2 ∪2 Prop is the function of intension s of properties, which fullfills the following condition: We connect Scott structures with language.
DF2 For any given  =< , Prop, ext, int > , valuation v is a function such that: If is derivable, then is valid in every  (based on Zalta 1983, 161-164).
Let us mention that Scott's semantics is extensional in this sense that the formula: is valid in every  (cf. Zalta 1983, 160-161). The implication converse to follows directly from D2, A4, D13. Now we consider an interpretation of the AdB language and take a 6-tuple   = ⟨, Prop, ext, int, Ad * , B * ⟩ such that S = ⟨, Prop, ext, int⟩ is a Scott structure and Ad * , B * ⊆ Prop . We assume also that: We extend DF2 by the following conditions: All formulas which are valid in  are also valid in any   .

All
AdBℑ derivable formulas are valid in every   .
Let us take the structure       with    = {s} , Prop Prop Prop = {+{s}, −{s}, +�, −�} , functions ext and int are determined as in DF1, and Ad * Ad * Ad * = �. We can observe that: The above theorem is actually mentioned by Świętorzecka (2017) and it shows the consistency of AdBℑ [ ] . However, it refers to rather 'trivial' structures in which there are no adherences (sic!). The search of more interesting Scott structures is rather doomed to failure in view of the following: Let us remind the condition ext: o ∈ ext(p) ⟺ ∃ X∈2  (p = +X and o ∈ X) or ∃ X∈2  (o ∈ 2 Prop and p = +X) . We take any p * and some o * such that they validate ∃x(A!x ∧ Fx) . This means that o * ∈ 2 Prop and o * ∈ ext(p * ) . If o * ∈ 2 Prop then o * ∉  and so we know that ∃ X∈2  (p * = +X) . But from the condition for ext by classical logic we have: Theorem Th2 expresses the idea of reloading abstracts in extensional contexts. Now we can observe that Th3 IN is not valid in any   structure with Ad * ≠ �.
In this way we know that: This is the reason why we look for another semantics.

Aczel's Semantics and Its Application to AdBℑ [ ]
In view of Th2 it is obvious that adding to the theory interpreted in Scott structures an assumption about existence of any property which has in its extension some but not all abstract objects results in contradiction. Aczel's semantics is not so 'bearish' and it falsifies the formula from Th2. In this frame we can speak about self-referential properties which do not contain in their extensions all abstracts. This effect is due to adding to the set of ordinary objects, mentioned proxies of abstracts that are 'invisible' on the level of an object language. Another intensional profit coming from Aczel's semantics is the possibility to distinguish between identity of extensions and identity of intensions of properties. The formula is not valid here. This effect is irrelevant to our particular approach although one could think about this advantage in connection with Bolzano's views about the difference between extensions and intensions of ideas in themselves (WL, I.428-I.459).
Let us define Aczel structure in the following way: DF4 An Aczel structure is a 7-tuple  = ⟨, , , , � �, , ⟩ , such that: (i)  is, as in DF1, the set of concrete individuals, possibly empty 15 (ii) is a nonempty set of objects thought as proxies of abstract objects a introduced in (iv); ≠ ∅ and ∩  = � (iii) is a set of proper ties a : , � , … , such that ∈ , card( ) ≥ card(2 ∪ ) (iv) Elements of 2 are abstracts a (imitated by individuals from ) Elements of  ∪ are called ordinary* objects. The sum of ordinary* and abstract objects a :  ∪ ∪ 2 is the domain of all objects a (represented by variables o, o ′ ). ( ∶ → 2 ∪ is the function of extension a which assigns to every property a its extension and ( ) =  (vii) ∶ → 2 ∪2 is the function of intension a : o ∈ ( ) ⟺ o ∈ 2 and ∈ o.