Moving on to relations of propositions, Bolzano begins by noting, just like with representations, that there are no two completely equal propositions in themselves. Again like with representations, he continues, propositions in themselves can still be similar in the sense they have so many commonalities that they can be easily confused with one another. Bolzano points out that propositions with similar representations are not necessarily similar and that similar propositions need not have very similar representations. For instance, “some As are B” is similar to “representation of an A that is also B has objectivity”, although the representations in them are not, and “Sun lights the Earth” and “Earth lights the Sun” have similar representations, but are not similar propositions.
Just like with representations, Bolzano states, we can compare the content of propositions, that is, their constituents. He points out that all propositions share at least the copula “have”. Thus, to make the definition sensible, Bolzano takes affinity of propositions to mean that they share at least one constituent beyond “have”. Clearly, he points out, such affinity can have degrees, and propositions with affine representations are affine, and affine propositions have some affine or same representations.
Bolzano goes on to define some types of affine propositions. Some of them, he says, can have the same subject or predicate or a contradictory subject or predicate. Bolzano particularly defines converted propositions as such that are otherwise completely same, but two representations have switched their places. Such switches can occur in different places of the propositions, but Bolzano defines as perfectly converted the propositions “what has a has b” and “what has b has a”; technically “a has b” and “b has a” are even more converted, he admits but it is so rare to find two characteristics that characterise one another. Bolzano defines as conceptually converted two propositions “representation of A that has the characteristic b has objectivity” and “representation of B that has the characteristic a has objectivity” – in other words, this is the familiar relation of “some As are Bs” and “some Bs are As”. He also mentions the traditional notion contraposition as the relation between contraposed “what has a has b” and contrapositing “what has not-b has not-a”.
Bolzano notes that the extensions of the propositions bring forth new relations. Still, he passes these quite quickly, since the extensions of the propositions were defined by the extension of the subjects of the propositions so that the relations really introduce nothing radically new. Bolzano also adds that similar relations could be defined in terms of extensions of predicates.
What Bolzano considers the most important relations of propositions come up with the use of variables, introduced with the notion of validity of propositions. What the notion of variables allows Bolzano to do is to introduce in modern logic very familiar analogies between propositional logic and algebra of sets (in Bolzano’s logic, relations of extensions of representations). For instance, he defines compatible propositions as such that, when taking the same representations in them as variable, some choice of these variables makes all the propositions true – similarly, incompatible propositions are then such that taking the same representations as variable, no choice of these variables makes all propositions true. Clearly the notions of compatibility and incompatibility of propositions are analogical to the compatibility and incompatibility of representations, and indeed, one could be defined in terms of the other – propositions are compatible, if the collections of representations making them true are compatible, and representations are compatible, if propositions saying that an object represented by one of these representations are compatible.
Yet, there is a catch in Bolzano’s definitions: the relations of compatibility and incompatibility depend on the choice of variables, so that propositions that are compatible with one choice could be incompatible with another choice. In practice, this dependence on the chosen variables does not usually affect that much of Bolzano’s discussion, just as long as the chosen variables are always the same.
Bolzano goes on to describe simple features of these new concepts, such as that if a proposition is compatible with another and this with a third, the first and the third might not be compatible (compatibility is not a transitive relation, we would say nowadays). What makes his discussion a bit more complex is that instead of propositions he is usually discussing collections of propositions, and as he notes a few times, what holds for individual propositions might not hold for collections of propositions (this is more evident with notions a bit more complex than just compatibility and incompatibility).
Bolzano shows special interest in propositions of the form “X has a” or “A has x”, where either the subject X or the predicate x is the only variable. Thus, he notes, propositions with predicates as variables – say, “A has x” and “B has y” – are compatible, just as long as their subjects refer to some objects: we just have to pick suitable predicates that fit with the kinds of objects. Similarly, propositions with the subjects as variables – “X has a” and “Y has b” – are compatible, just as long as their predicates are characteristics of something: we just have to pick suitable subjects that have these characteristics. Then again, Bolzano points out, the case is somewhat more complex, if the propositions had the same subject that was then taken as variable – for instance, with “X has a” and “X has b” – since such propositions are compatible if and only if the predicate representations are compatible or are characteristics of some shared objects.
So, what are the other similar notions discussed by Bolzano. Well, he defines derivability of a collection of propositions, in analogy with comprehension of representations, as a relation where the derivable collection of conclusions is always true, when the collection of compatible propositions or premisses from which it is derived is also. Here we find an example of the complexity involved with Bolzano speaking of collections of propositions, instead of just individual propositions. Thus, an individual proposition can never be derivable from both an individual proposition and its negation, Bolzano points out, unless it is formally true, that is, always true with this particular choice of variables. Then again, if a proposition is derivable from a collection of several propositions, it could be derivable, even if some or even all propositions would be replaced by their negations: for instance, conclusion “B and C are not equivalent” is derivable from premisses “every A is B” and “it is false that every A is C”, but also from their negations.
Bolzano also defines the stronger relation of exact or adequate derivability, where none of the premisses or their constituents could be dropped without removing the relation of derivability. In this case, he points out, neither the conclusion nor any of the premisses can be formally true and no premiss is derivable from other premisses. While derivability is a transitive relation, exact derivability is not, although it has particular instances of transitivity. Thus, Bolzano can define composed derivability as a relation where the premisses of an exactly derivable conclusion are again exactly derivable from further premisses and the final conclusion is then also exactly derivable from these more ultimate premisses. Then again, he thinks, there are also cases of simple or non-composed derivability such as a proposition “all A are C” being derivable from “all A are B” and “all B are C”.
Further relations of this kind are equivalence (collections of propositions being derivable from one another), subordination (one-sided derivability), independence (collections of propositions being compatible, but not derivable from each other), exclusion (when all propositions of a collection are true, propositions of another collection are false), mutual exclusion (two collections excluding one another), contradiction (if all propositions in one mutually exclusive collection are false, all propositions in the other are true) and contrariety (mutually exclusive collections are not contradictories). None of these have anything surprising in them, so we can skip Bolzano’s discussion of them.
Bolzano moves on to discuss relations concerning not just what happens if propositions or their collections are true, but determining what propositions in a collection are true. Here, it can be a case of speaking about the truth and falsity of the propositions as such or as case of speaking about the truth and falsity of the propositions with some representations taken as variable – in the former case, the concepts are defined, Bolzano says, materially, and in the latter, formally, although most of the concepts make best sense in their formal meaning. The simplest example is when all propositions of a collection are true or all are false, but more interesting examples are discovered, when some, but not all propositions in a collection are true – Bolzano calls this a case of complementary propositions. Complementation, he points out, can be one-membered complementation or disjunction, so that only one proposition in a collection is true, and disjunction can be exact, when no proposition could be removed from the collection, without cancelling the feature that it has always at least one true proposition. Complementation might also be multimembered, if it has more than one true proposition, and in this case, it might even always have some exact number of true propositions. Finally, even if a collection of complementary propositions does not have any exact number of true propositions, its propositions might still be compatible, that is, all its propositions could be true in some cases. At the end of all this, Bolzano notes that all thus defined notions could also be conditional, that is, we could restrict our attention to such representations that make certain presuppositions also true.
Bolzano extends this notion of presuppositions to his previously defined idea of validity of propositions, calling it comparative validity and identifying it with the notion of probability. In principle, he suggests that we find out representations making the presuppositions true and then from these the particular representations making a proposition we are interested in true – the probability of this proposition is the number of the latter collection of representations divided with the number of the former collection. In practice, Bolzano admits, we have to make some limiting assumptions about the number of the cases, because the number of possible representations that could be used is infinite, due to all equivalent representations: basically, he restricts his whole discussion to rather simple cases of pulling balls out of a box. As always, Bolzano is interested to extend his discussion from individual proposition to collections of propositions, which he manages to do by proving that with suitable choice of variables and presuppositions, the probability of a collection of propositions is the product of the probabilities of the individual propositions
Bolzano discusses the relation of a proposition that is ground to a proposition that is its consequence. He notes that this relation does bear some affinity with the earlier notion of derivability, but is stronger: for instance, the proposition of weather getting warmer is derivable from the proposition of thermometer rising and vice versa, but only the former is the ground for the latter, not the other way around. Bolzano points out that the consequence relation can also be connected with the notion of probability. Thus, the presupposition of summer coming increases internally the probability of a proposition that thermometer rises, because the coming of summer is, together with other propositions, a ground for thermometer rising, while a weather report increases its probability only externally, because this report is no part of a ground of thermometer rising.
The final relation Bolzano considers is that of an answer to a question – or actually we have here many relations to consider. We may speak of an answer fitting a question, which is simply the truth desired in the question, if it is not absurd and has one. Furthermore, Bolzano notes, we can also define an answer given to a question as any proposition that someone suggests as the required truth. Such an answer can be correct or fit the question or incorrect, he adds, and while a correct answer must be a true proposition, incorrect can be true or false. A determined question, Bolzano points out, can have only one answer, or at least just many equivalent answers, while there are innumerably many incorrect answers, even for absurd questions. He also defines solution as an answer to a problem in the restricted sense: such a solution, at least if it is a correct one, will consist of rules, that is, of propositions expressing how a certain undertaking for a determined goal is to be effected.
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