Bolzano has already mentioned the possibility of propositions about propositions, but he then left a fuller account of that notion to a time after the discussion of relations between propositions, since many propositions about propositions deal with just such relations. As he is now finally in a position to deal with the topic, Bolzano is especially interested in showing how to present such propositions in his official form of all propositions: “subject has a predicate”. Thus, for instance, the notion of compatibility of propositions would be fully expressed as “representation of a collection of certain representations, which put in place of i, j… in the propositions A, B, C… makes them all true, refers to some object” (the notion of incompatibility is, of course, expressed by a negation of this proposition). As he is well aware, we express such complex propositions in a much simpler manner and especially the notion of variable representations in the propositions is usually left implicit (for instance, we just say that the propositions can be true at the same time).
Bolzano goes quickly through all the various forms of compatibility and incompatibility, but we really need to look at just the notin of derivability, as it is the only one where he feels the need to give the full official form. It is really no wonder he refrains from doing it in other cases, since even this one is a mouthful: “representation of certain parts in A, B, C… M, N, O…, which are such parts that every arbitrary collection of representations, which put in their place makes the A, B, C… true, makes also the M, N, O… true, refers to objects”.
From the species of compatibility and incompatibility of propositions, Bolzano moves on to discuss the relation of complementarity, which is obviously important to disjunctions. We may quickly go through Bolzano's official structure for the basic notions in this regard, that is, the material complementarity – “representation of a true proposition among M, N, O… refers to some object” – and formal complementarity – “representation of a true proposition among a collection of propositions, whatever else is put in place of representations i, j…, following rule R, refers to some object” (it is not clear what this rule R refers to). Bolzano is as quick in presenting his official structure of propositions on probability – “relation of the set of all such representations that make the propositions A, B, C… true to the set of those that in addition to A, B, C… make also M true is m” – although he does add that we commonly say something to be probable, when this m is more than ½.
Bolzano's official structures for material and formal consequence are by now probably not surprising, former being “relation of truths M, N, O… to truths A, B, C… has the characteristic of a relation of consequence to its ground”, and the latter “the relation of propositions M, N, O… to the propositions A, B, C… has the characteristic that each collection of representations that put in the place of i, j… makes the A, B, C… true, makes also the M, N, O… into such truths that are the consequence of the former”. What is more interesting is his remark that we often speak of grounds and consequences, when we should actually speak of partial grounds and consequences. Furthermore, Bolzano defines cause as an existing object, the existence of which is a partial ground of some truth, and effect as an existing object, the existence of which is one consequence of an existence of a cause. He also points out that the existence of a cause implies the existence of an effect, but not necessarily the other way around, since an effect could have many possible causes. Thus, he defines condition as a full or partial cause, the existence of which is derivable from the existence of the effect.
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