Bolzano Bernard (1781–1848), Austrian philosopher. He studied philosophy, mathematics, physics, and theology in Prague; received the Ph.D.; was ordained a priest (1805); was appointed to a chair in religion at Charles University in 1806; and, owing to his criticism of the Austrian constitution, was dismissed in 1819. He composed his two main works from 1823 through 1841: the Wissenschaftslehre (4 vols., 1837) and the posthumous Grössenlehre. His ontology and logical semantics influenced Husserl and, indirectly, Lukasiewicz, Tarski, and others of the Warsaw School. His conception of ethics and social philosophy affected both the cultural life of Bohemia and the Austrian system of education.
Bolzano recognized a profound distinction between the actual thoughts and judgments (Urteile) of human beings, their linguistic expressions, and the abstract propositions (Sätze an sich) and their parts which exist independently of those thoughts, judgments, and expressions. A proposition in Bolzano’s sense is a preexistent sequence of ideas-as-such (Vorstellungen an sich). Only propositions containing finite ideas-as-such are accessible to the mind. Real things existing concretely in space and time have subsistence (Dasein) whereas abstract objects such as propositions have only logical existence. Adherences, i.e., forces, applied to certain concrete substances give rise to subjective ideas, thoughts, or judgments. A subjective idea is a part of a judgment that is not itself a judgment. The set of judgments is ordered by a causal relation. Bolzano’s abstract world is constituted of sets, ideas-as-such, certain properties (Beschaffenheiten), and objects constructed from these. Thus, sentence shapes are a kind of ideas-as-such, and certain complexes of ideas-as-such constitute propositions. Ideas-as-such can be generated from expressions of a language by postulates for the relation of being an object of something. Analogously, properties can be generated by postulates for the relation of something being applied to an object. Bolzano’s notion of religion is based on his distinction between propositions and judgments. His Lehrbuch der Religionswissenschaft (4 vols., 1834) distinguishes between religion in the objective and subjective senses. The former is a set of religious propositions, whereas the latter is the set of religious views of a single person. Hence, a subjective religion can contain an objective one. By defining a religious proposition as being moral and imperatives the rules of utilitarianism, Bolzano integrated his notion of religion within his ontology. In the Grössenlehre Bolzano intended to give a detailed, well-founded exposition of contemporary mathematics and also to inaugurate new domains of research. Natural numbers are defined, half a century before Frege, as properties of ‘bijective’ sets (the members of which can be put in one-to-one correspondence), and real numbers are conceived as properties of sets of certain infinite sequences of rational numbers. The analysis of infinite sets brought him to reject the Euclidean doctrine that the whole is always greater than any of its parts and, hence, to the insight that a set is infinite if and only if it is bijective to a proper subset of itself. This anticipates Peirce and Dedekind. Bolzano’s extension of the linear continuum of finite numbers by infinitesimals implies a relatively constructive approach to nonstandard analysis. In the development of standard analysis the most remarkable result of the Grössenlehre is the anticipation of Weirstrass’s discovery that there exist nowhere differentiable continuous functions. The Wissenschaftslehre was intended to lay the logical and epistemological foundations of Bolzano’s mathematics. A theory of science in Bolzano’s sense is a collection of rules for delimiting the set of scientific textbooks. Whether a class of true propositions is a worthwhile object of representation in a scientific textbook is an ethical question decidable on utilitarian principles.
Bolzano proceeded from an expanded and standardized ordinary language through which he could describe propositions and their parts. He defined the semantic notion of truth and introduced the function corresponding to a ‘replacement’ operation on propositions. One of his major achievements was his definition of logical derivability (logische Ableitbarkeit) between sets of propositions: B is logically derivable from A if and only if all elements of the sum of A and B are simultaneously true for some replacement of their non-logical ideas-as-such and if all elements of B are true for any such replacement that makes all elements of A true. In addition to this notion, which is similar to Tarski’s concept of consequence of 1936, Bolzano introduced a notion corresponding to Gentzen’s concept of consequence. A proposition is universally valid (allgemeingültig) if it is derivable from the null class. In his proof theory Bolzano formulated counterparts to Gentzen’s cut rule.
Bolzano introduced a notion of inductive probability as a generalization of derivability in a limited domain. This notion has the formal properties of conditional probability. These features and Bolzano’s characterization of probability density by the technique of variation are reminiscent of Wittgenstein’s inductive logic and Carnap’s theory of regular confirmation functions.
The replacement of conceptual complexes in propositions would, if applied to a formalized language, correspond closely to a substitutionsemantic conception of quantification. His own philosophical language was based on a kind of free logic. In essence, Bolzano characterized a substitution-semantic notion of consequence with a finite number of antecedents. His quantification over individual and general concepts amounts to the introduction of a non-elementary logic of lowest order containing a quantification theory of predicate variables but no set-theoretical principles such as choice axioms. His conception of universal validity and of the semantic superstructure of logic leads to a semantically adequate extension of the predicate-logical version of Lewis’s system S5 of modal logic without paradoxes. It is also possible to simulate Bolzano’s theory of probability in a substitution-semantically constructed theory of probability functions. Hence, by means of an ontologically parsimonious superstructure without possible-worlds metaphysics, Bolzano was able to delimit essentially the realms of classical logical truth and additive probability spaces. In geometry Bolzano created a new foundation from a topological point of view. He defined the notion of an isolated point of a set in a way reminiscent of the notion of a point at which a set is well-dimensional in the sense of Urysohn and Menger. On this basis he introduced his topological notion of a continuum and formulated a recursive definition of the dimensionality of non-empty subsets of the Euclidean 3-space, which is closely related to the inductive dimension concept of Urysohn and Menger. In a remarkable paragraph of an unfinished late manuscript on geometry he stated the celebrated curve theorem of Jordan. See also FREE LOGIC, MODAL LOGIC, PHI- LOSOPHY OF MATHEMATICS , PROBABILITY, SET THEORY, TARSK. J.Be.