Bolzano’s Concept of Consequence

In the second volume of his Wissenschaftslehre from 1837, the Bohemian philosopher, theologian, and mathematician Bernard Bolzano (1781-1848) introduced his concept of consequence, named derivability (Ableitbarkeit), together with a variety of theorems and further considerations. Derivability is an implication relation between sentences in themselves (Sätze an sich), which are not meant to be linguistic symbols but the contents of declarative sentences as well as of certain mental episodes. When Schmidt utters the sentence ‘Schnee ist weiß’, and Jones judges that snow is white, the sentence in itself expressed by Schmidt is the same as the one to which Jones agrees in thought. This sentence in itself is an abstract entity: in some sense, it exists; but it is unreal insofar as it lacks a position in space and time, does not stand in causal relationships, and is independent of the existence of thinking beings and languages. 3 Sentences in themselves are conceived by Bolzano as the primary bearers of the unrelativized truth-values true or false. This should be understood as meaning, first, other things, such as sentences or judgements, have their truth-values in virtue of the truth-values of the sentences in themselves which are their contents. Second, what is expressed by, e.g., ‘I am hungry’ in each case does not possess relativized truth-values like true/false with respect to person S at time t. It includes elements specifying a particular time and person, which makes it unqualifiedly true or false. Third, there are neither truth-value gaps nor a third truth-value (e.g., indeterminate): every sentence in itself is either true or false. All in all, sentences in themselves are identical with, or at least similar to, Frege’s thoughts. I will frequently use the shorter term ‘propositions’, and I will refer to them by putting sentences in square brackets. [3 is a prime number] is the proposition expressed by ‘3 is a prime number’. A sentence in itself consists of sub-propositional parts which Bolzano calls ideas in themselves (Vorstellungen an sich). [3 is a prime number] can be decomposed, among other things, into an idea of the number 3 and an idea of the property of being a prime number. These ideas in themselves are also neither