“The Wissenschaftslehre (1837) by Bernard Bolzano (1781-1848) is one of the masterpieces in the history of logic. In this encyclopedic work Bolzano intended to construct a new and philosophically satisfactory foundation of mathematics. The search for such a foundation brought forth valuable by-products in logical semantics and axiomatics. For example, Bolzano introduced the notion of abstract, non-linguistic proposition and described its relations to other relevant notions such as sentence, truth, existence and analyticity. Furthermore, he studied relations among propositions and defined highly interesting notions of validity, consistency, derivability and probability, based on the idea of “replacing” certain components in propositions. In set theory, he stated the equivalence of reflexivity and infiniteness of sets and considered isomorphism as a sufficient condition for the identity of powers of infinite sets. He conceived of a natural number as a property characterizing sets of objects, even though he did not base his development of arithmetic on this notion, and analyzed sentences about specific numbers in a way reminiscent of Frege and Russell. In a posthumous manuscript from the 1830’s (recently published) he developed a theory of real numbers, which differs from those of Dedekind, Weierstrass, Méray and Cantor. Bolzano’s real numbers may be identified with certain sequences of rational numbers.
Logic in Bolzano’s sense is a theory of science, a kind of metatheory, the objects of which are the several sciences and their linguistic representations. This theory is set forth in Bolzano’s monumental four-volumes work Wissenschaftslehre (hereafter referred to as WL). Bolzano’s very broad conception of logic with its strong emphasis on methodological aspects no doubt accounts for the type of logical results which he arrived at. The details of his theory of science proper are given in the fourth volume of the WL and belong to the least interesting aspects of his logic. On the other hand, Bolzano’s search for a solid foundation for his theory of science left very worthwhile by-products in logical semantics and axiomatics. His theory of propositions in the starting-point of these results.
Bolzano became more and more aware of the profound distinction between the actual thoughts of human beings and their linguistic expressions on the one hand, and the abstract propositions and their components which exist independently of these thoughts and expressions on the other hand. Furthermore, he imagined a certain fixed deductive order among all true propositions. This idea was intimately associated with his vision of a realm of abstract components of propositions constituting their logically simple parts.
For the following presentation of Bolzano’s theory of propositions I have to define some terms. A concrete sentence occurrence is a sequence of particles existing in space and time, arranged according to the syntactic rules of a grammar, and contrasting with its surroundings. A simple sentence shape, on the other hand, is a class of similar concrete occurrences of simple sentences. A compound sentence shape is built up recursively from simple sentence shapes by means of syntactic operations. Not every compound sentence shape has a corresponding concrete sentence occurrence. Two compound sentence shapes may be considered identical if they are built up from identical simple sentence shapes in the same way. Two simple sentence shapes are identical if they contain the same sentence occurrences.
Now consider the compound sentence containing the following concrete sentence occurrence: ‘a simple sentence shape is a class of similar sentence occurrences or it is not the case that a simple sentence shape is a class of similar sentence occurrences’. In another sense one could say that this sentence shape, which is an abstract logical object outside of space and time, contains two sentence occurrences, i.e., two abstract “occurrences” of the simple sentence shape containing the following concrete inscription: ‘a simple sentence shape is a class of similar sentence occurrences’. In the following, I will use the expression ‘sentence occurrence’ exclusively in the first, concrete sense.
Bolzano’s notion of abstract non-linguistic proposition (Sätz an sich) is a keystone in his philosophy and can be traced in his writings back to the beginnings of the second decade of the 19th century. I shall try to characterize Bolzano’s conception of propositions by means of certain explicit assumptions. These assumptions also give information about the relation between propositions and other logically interesting objects.
In his logic Bolzano utilizes a concept which is an exact counterpart of the modern logical notion of existential quantification. Therefore, he could have stated that (1) There exist entities, called ‘propositions’, which fulfill the following necessary conditions (2) through (15). (Cf. WL 30 ff.)
Thus, propositions possess the kind of logical existence developed in modern quantification theory. However, (2) A proposition does not exist concretely in space and time (WL 19).
According to Bolzano, both linguistic and mental entities such as thoughts and judgments are concrete (WL, 34, 291). Hence, propositions could not be identified as concrete linguistic or mental occurrences. Furthermore,(3) Propositions exist independently of all kinds of mental entities (WL 19).
Therefore the identification between propositions and mental dispositions sometimes made in medieval nominalism cannot be applied to propositions in Bolzano’s sense.
A proposition in Bolzano’s sense is a structure of ideas-as-such. Hence, an idea-as-such (Vorstellung an sich) is a part of a proposition which is not itself a proposition (WL 48). But to he able to generate propositions we have to characterize ideas-as-such independently of propositions. This is in fact implicit in Bolzano. He worked extensively with the relation of being an object of an idea as-such, which corresponds in modern logic to the relation of being an element of the extension of a concept. In terms of this relation, taken as a primitive by Bolzano, certain postulates may be extracted from his writings which concern the existence and general properties of ideas-as-such.
Independently of human minds and of linguistic expressions there exists a collection of absolutely simple ideas-as-such. As examples Bolzano mentions the logical constants expressed by the words ‘not’, ‘and’, ‘some’, ‘to have’, ‘to be’, ‘ought’ (WL, 78); but he admits being unable to offer a more comprehensive list. He seems to mean that each complex idea A can be analyzed into a sequence S(A) of simple ideas which would probably include certain logical constants.
I shall call this sequence S(A) the ‘primitive form’ of A. The manner in which a complex idea is built up from simple ones may be expressed by a chain of definitions. So it appears that some complex ideas behave somewhat like the open formulas of a logical calculus. Bolzano assumes that two ideas are strictly identical if and only if they have the same primitive form (WL 92, 119, 557).
From: Jan Berg, Bolzano’s Contribution to Logic and Philosophy of Mathematics, in: R. O. Gandy, J. M. Hyland, Logic Colloquium ‘76, Amsterdam: North Holland, pp. 147-150.