Bolzano rounds up his discussion of propositions with the favourite pastime of the later logicist school: that of revealing the hidden logical structure of ordinary linguistic expressions. Remember that Bolzano's ultimate goal is to show that all propositions bend to the universal form of “A has b”. Thus, he uses the rather convoluted terminology that a representation has objectivity, when he wants to say simply that a representation refers to an object. In line with this, Bolzano suggests that “nothing is B” or means actually that the representation of something, which has the characteristic b, has no objectivity, rendering it unnecessary to take the word “nothing” as the subject of this proposition. Similarly, he thinks that “an A is B” means actually that the representation of an A, which has b, has objectivity.
Bolzano moves on to impersonal sentences, using in German the “Es gibt” -structure, translatable often with the words “there is”, which is again another way to say that a representation refers to an object. This is true, he insists, even of weather reports, which form one subspecies of “Es gibt” -sentences (for instance, “it snows” should mean that a representation of snowfall happening now refers to an object). Other expressions with the “Es gibt” -structure have different readings for Bolzano: “they say…” means that there are people who say something, while “it is true that…” means that a certain proposition is true.
Bolzano points out that “some A are B” in its usual sense means that characteristic b belongs to several, but not all and not even to remarkably many A. On the other hand, he states, “many A are B” means that the characteristic b belongs not to all, but to a great number of A. Indeed, Bolzano continues, the word “many” is relative and could mean many things: an absolutely great number of things, like 3 billion, a great number in relation to something else, like the number of stars we can see with bare eyes, in comparison with the number of all of them, and a great relation of numbers in comparison with something else, for instance, when many of our family are sick, although it covers only a tenth, in proportion to other families this is much.
Bolzano goes through some relatively simple cases. “n As are Bs” means either that there are at least n As that are Bs or that exactly n As are Bs, both easily turnable to his favoured form. Comparative proposition “A has characteristic b more than C has” speaks actually of the degree or magnitude of the respective characteristic of the objects, while exclusive proposition “only A alone is B” means that the characteristic b belongs exclusively to A and implicitly to some A (whether it belongs to all of them depends on the context). On the other hand, Bolzano notes, “A is only B alone” does not mean that it would have no other characteristics, since everything has infinitely many characteristics, but that it has of some relevant characteristics only that of b (for instance, “Caius is only good to invent rhymes” suggests that he has no other characteristics of a good poet).
What Bolzano calls a restrictive proposition – “A, as C, is B” – can have many different meanings. Firstly, “Earth, as consisting of many substances, is destructible” suggests that consisting of many substances is a ground for Earth being destructible. Then again, “Caius, as a musician, is wonderful” does not mean that Caius himself would be wonderful, but his musicianship is. Finally “Titus, as a judge, should accept no gifts” does not mean that Titus should accept no gifts, because he is judge, nor that his judgeship should accept no gifts, but points to a special relation between the three concepts – Bolzano does not explicate this relation more precisely, but it is something to the effect that it is a characteristic of his position of a judge that while doing his duty in that position, he should not accept any gifts, which would be considered bribes. Bolzano also notes that a proposition “A as A is B” is equally ambiguous, but usually means that the characteristic b, ascribed to A, is derivable already from concept A.
We have already seen Bolzano deal with the phrase “A is, because B is” which means that the B is a ground or at least a partial ground of A. He adds that often this form is used merely to indicate that A is derivable from B. Bolzano points out that if-then -structure is also often used to describe the relation of derivability, but it is in some cases applied elsewhere. For example, “if digits in a number are arbitrarily switched and the result is subtracted from the original, then the result is always divisible with 9” indicates no derivability, he thinks, because there are no obvious variable representations, and instead it describes certain relations between numbers (that the number differing from a given number only thus that its digits are in a different order, is to this one in such a relation that their difference is always divisible by 9). Similarly “if Caius is silent in this situation, he is unthankful” does not indicate derivability, but describes a general proposition that anyone who is silent in the situation of Caius is unthankful, while “if Caius is dead, then Sempronius is a beggar” deals with relations between Caius and Sempronius, which make the latter necessarily a beggar with the death of the former.
“A determines B” (for instance, “two sides of a triangle and the enclosed angle determine the whole triangle”) means, Bolzano thinks, that certain propositions expressing characteristics of B are derivable from certain propositions expressing characteristics of A in regard to representations that concern characteristics of A. “A determines B wholly or completely” means then that all propositions concerning characteristics of B are derivable from certain propositions concerning characteristics of A (for example, centre and radius determine circle completely, because all characteristics of the circle can be derived from their characteristics).
“Either-or” -structure, Bolzano says, means in its strongest sense that only one of the options presented is true, but it can also mean that at least one and possibly several of the alternatives are true. Similarly, he adds, it can refer to options present in the sentence or use some variables. Furthermore, Bolzano points out, “all A are either B or C” does not mean that either all A are B or that all A are C, but that each individual A is always either B or C.
Bolzano goes into great detail with the so-called modal terms or necessity, impossibility, possibility and contingency. In their strongest sense, he thinks, these concepts refer to actual existence of objects – “must” means “must exist” and “can” means “can exist”. More precisely, Bolzano explains, if something (say, God) is necessary = there is a conceptual truth of the form “A exists”, where A comprehends the object in question. Something is then impossible if its existence is a conceptual falsity. Possible thing is then defined as not impossible and contingent thing as existing, but not necessarily. Bolzano's definitions imply that any object that can be represented with a pure concept referring only to it is either necessary or impossible, thus, contingent is something that we can represent only through mixed representations or intuitions.
This strong sense of the modalities can be relativised, so that e.g. something is necessary relative to another thing (Bolzano speaks also of external necessity) if neither is by itself necessary, but the existence of one is derivable from the existence of the other (like punishment is necessary relative to a sin). Furthermore, Bolzano adds, in an improper sense these concepts can be extended to truths in themselves. Thus, b belongs necessarily to A, if “A has b” is a conceptual truth; b is impossible for A = “A has no b” is a conceptual truth.
Bolzano points out a final way to use especially the concept of possibility. In this sense, something is possible or something could be, if we do not know it to be impossible, that is, if we know of no conceptual truth that would express the opposite. Obviously this does not mean that it might not really be impossible.
If a proposition speaks of an actual object and the characteristics ascribed to it in the proposition do not always belong to it, Bolzano thinks, the proposition must determine a time, when the characteristics belong to the object. Thus, we have verb tenses for present, past and future times, and sometimes the determination of time is even the object dealt in a proposition (Bolzano’s example is “cherries bloom earlier than grapes”, which should mean that the relation of the season of cherries blooming to the season of the grapes blooming is the relation of an earlier to a later season).
Bolzano is especially interested in propositions about the beginning, the enduring and the end of a state. A state endures through some period of time, he says, when the proposition that the state is actual at the moment x remains true, whatever moment included in the period of time is put in the place of x. A state enduring now means then that it endures through a period of time including the present. Bolzano finally defines a beginning or an end of a state at a certain moment in such a way that the proposition that the state is actual at the moment x is true, as long as x is replaced with a moment that is a bit later than the moment of beginning or a bit earlier than the moment of ending, but false, if the x is replaced with a moment earlier than the moment of beginning or later than the moment of ending.
Bolzano also emphasises the importance or propositions speaking of becoming. Such becoming, he says, involves actual existence in a future time, but mere existence in future time is not always becoming. The difference lies not just in how far in the future the event is, Bolzano thinks, because we do speak of events centuries in the future as already becoming. Instead, he suggests, becoming requires change, and the future event becoming must be caused by this change. Then again, Bolzano continues, change requires something that is changed: an internal characteristic of the changing thing does not remain the same even for the smallest time, when the change is going on. Given this definition of change, becoming of M can then be defined as an on-going change causing M to exist in the future. If we say that M becomes from A, this means that the object A is undergoing the change, which causes the future existence of M. Finally, Bolzano concludes, if M becomes through or because of P, the object P is a total or partial cause for the change that causes the future existence of M.
Bolzano concludes the section on propositions by suggesting a few examples, where one and the same sentence actually combines several propositions. Thus, differentiating proposition “not X, but Y does have b” is a combination of proposition “X does not have b” and “Y has b”, compared to one another, while extensive expression “all A, including those that are X, have b” says that all A have b but that the listeners probably won't believe this and think that X are an exception.
