From general characteristics of all or most propositions, Bolzano moves to characteristics that could be used to classify different propositions. He begins by noting that no propositions are truly simple, because they all have subject, copula and predicate. Still, he points out, we can differentiate between simple propositions, where both the subject and the predicate are simple (copula is, of course, always simple “have”), and complex propositions, where at least one of the subject and the predicate is not simple.
Another of Bolzano's divisions concerns the question, whether some of the representations in a proposition are intuitions. Thus, he calls a proposition with nothing but pure concepts a conceptual proposition (conceptual truth, if it is a true proposition) and a proposition with at least one intuition an empirical proposition.
Bolzano notes that in his suggested form of propositions, the predicate is always abstract. Then again, depending on whether the subject is abstract or concrete, we could speak of abstract and concrete propositions. Bolzano points out that some representations are not abstract nor concrete (e.g. pure intuitions), so if such happens to be the subject of a proposition, this proposition is then not abstract nor concrete.
Bolzano lists types of propositions involving notions of collections, which due to their simplicity we need not go through in detail. Thus, he mentions e.g. collective and distributive propositions, with collective and distributive representations as their subjects (“ A, B and C together…” and “each of A, B and C…” being exemplary forms) which differ from propositions with collective and distributive predicates (both have the form “A has b and c”, but in one the predicate is understood collectively, in the other distributively). We might still mention Bolzano’s assertion of equality (“characteristic m belongs to objects A, B, C… together”), assertion of difference (“relation of A, B, C… to F, G, H… is such that the first have a characteristic m, the latter don't”) and determination (“characteristic m belongs exclusively to A, B, C…).
Moving on to propositions with negative representations, Bolzano thinks that since completely negative representations represent no object and thus no characteristics, they cannot appear as subject or predicate in any propositions. On the other hand, he continues, “something that is no A” can, so that we could at least have propositions with negative subjects.
Bolzano notes that often negation is said to be attached to copula, but he thinks it is actually the predicate that is actually then negative. True, he admits, language does often attach negation to the verb, but so it does with probability or necessity, which, according to him, do not characterise the copula, but the whole proposition. Thus, Bolzano concludes, negation seemingly attached to copula is actually attached to the whole proposition and indicates that the proposition is false. He justifies this statement with the example of a proposition with many objects in the subject: “all A do not have b” does not mean that for each A, they would not have b, but only that it is not true that all A have b. Even if the subject is singular object, we can say “A has no b”, which indicates a lack of characteristic b, that is, a characteristic of not-b, which Bolzano then takes as the defining moment of negative propositions, in difference from affirmative propositions.
Bolzano notes an interesting connection between negative and collective propositions. While propositions with collective and distributive predicates are equivalent when affirmative, they are not so when negative: “A has not the collection of characteristics b, c …” means not the same as “A has not any of characteristics b, c…”, because in the former case it still can have some of the characteristics in the collection. Generally all types of propositions have their affirmative and negative version, and if there’s nothing remarkable about their difference, I shall not mention them, even if Bolzano does.
From the standpoint of logic itself, Bolzano notes, it is remarkable that some propositions handle either representations or other propositions. If a proposition deals with a representation, he points out, its subject must then be a representation of a representation. A simple example is a proposition stating that a representation has – or has not – objects. Indeed, all the notions dealt in Bolzano's discussion of representations have their own corresponding propositions – for instance, we might assert a representation to be general or to be comprehend by another representation – and I will also skip this part of Bolzano's account as rather simple repetition.
Moving on to propositions about propositions, Bolzano notes that at this point he can deal only with propositions about characteristics of propositions, since their relations haven't been considered yet. In fact, he at this point mentions only the affirmation of A, saying that a proposition A has truth, and the corresponding negation of A, which says that proposition A has no truth.
Previous types of propositions appear in all sciences, Bolzano continues, but there are some that are useful, although they appear only in some sciences. His first example is the existential proposition that asserts or denies the existence of something.
An important type of propositions for Bolzano is formed by those describing mental phenomena, that is, effects caused by the soul, either within the soul itself or outside it. Two of such phenomena are already familiar to us, namely, subjective representations and. judgements. In addition to these, Bolzano mentions sensations of convenience or inconvenience, accompanying representations, wishes or desires, caused by judgements that certain objects would cause a sensation of convenience, volitions, which differ from desires by concerning what we should do, no matter if it is inconvenient, and actions, which are changes caused by volition on our soul or on certain other substances, primarily our organs and through them surrounding objects. All of these phenomena, Bolzano suggests, come with their own propositions, which may or may not assert the person e.g. sensing something.
An important subtype of propositions concerning mental phenomena is formed by propositions with the concept described by the German word Sollen (what ought to or should be done). Bolzano notes that the concept of Sollen properly applies only to actions or actually to decisions of will: each decision has a characteristic it should have. Thus, he defines an ethically good decision as such that is as it should be, whether it is a duty or just commendable decision, and the corresponding proposition he then calls an ethical proposition. On the other hand, Bolzano notes, propositions like “it should rain” use the concept improperly, expressing merely the uncertainty of what is asserted. A related notion is that what we may or are allowed to do, which can be defined as not something we should not do, and the corresponding proposition Bolzano calls an assertion of permission.
Another important subtype of propositions involving mental phenomena Bolzano considers are propositions about desires or problems (something is desired to be done). He is especially interested in problems that assert a wish for truth with certain characteristics, that is, in questions. More precisely, Bolzano explains, questions do not ask for truths in themselves (as he has said many times, these do not exist), but their appearance in mind as a thought or their linguistic expressions. He lists some subspecies of questions, such as questions for truth or falsity of a given proposition (e.g. is it true that God exists), questions about predicates for given subjects (e.g. what are the characteristics of triangles) and questions about subject for given predicates (e.g. who is the tallest person on Earth). An important subtype for sciences is formed of practical or technical questions, which Bolzano also calls problems in a stricter sense and which require truths describing how to reach a certain goal. Some questions, he notes, are determined in the sense that they correspond to only one or several equivalent truths, some are undetermined or correspond to many non-equivalent truths, while others are impossible or imaginary in the sense that they demand characteristics that are not to be found in any truth.
Bolzano goes on to mention some quite basic divisions of propositions: they can have a subject that represents no object or a subject that represents at least one object, and if latter, they can have only one object or many and maybe even infinite objects. He also points out the division that has already been mentioned many times, at least covertly, namely, that of true and false propositions.
Furthermore, Bolzano again reminds the reader that truth or falsity is an unchanging feature of a proposition, and if we appear to speak of propositions changing truth value, we are actually speaking of linguistic expressions that can change the proposition they signify, if they include words like now or this. He develops this idea by suggesting that we could think of some parts of propositions as variables, where the change of the variable part could generate propositions with different features – proposition referring to no object could become a proposition referring to some object or a true proposition could become false. If we determined the possible results of the variable, Bolzano notes, we could then measure how many of these variations are true and how many false. Thus, he defines validity of a proposition as a relation of the number of its true variations to the number of all variations: of course, this proportion is dependent on what parts are chosen as being variable. Bolzabo then defines a universally valid or formally true proposition as having the validity of 1, while a universally invalid or formally false proposition has then the validity of 0.
Related to this notion of validity, Bolzano points out that no proposition is formally true or false, if all its representations are taken as variables. Still, he thinks, it is of interest if there is at least one representation, such that taking it as variable, the proposition is formally true or false (for instance, “a human that is evil deserves no praise”, when the representation “human” is taken as variable). Bolzano decides to call such propositions analytical, while a synthetical proposition is then such that it has no representation that could be taken as variable so that the proposition would then be formally true or false. He then goes on to list some examples of very general analytical truths: identical or tautological proposition “A is A” – or as Bolzano prefers to say, “A has a” – “A that is B is A”, “A that is B is B” and “every object is either B or not-B”. He points out that all the examples he just listed were such that they are formally true, if all but logical parts (whatever that means – Bolzano admits that the notion is hazy) are taken as variables and decides to call them logically analytical propositions.
Another feature related to the notion of variables in propositions is what Bolzano calls a conversion of a proposition, where two representations within the same proposition change their place. If such replacing does not change the truth value of the proposition (for instance, if it is true both that Titus loves Cajus and that Cajus loves Titus), he calls the proposition convertible or reciprocable proposition. Bolzano then defines analytically reciprocable proposition as such where the changed representations could be anything without changing truth or falsehood, like in the proposition “A that is B is A”.
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