Bernard Bolzano: Study of science – Degrees of confidence

Bolzano defines certain or secure judgement as being generated in such a manner that there is no possibility of error. He notes at once that certainty in this sense is a contextual notion, related to a certain thinker who holds a judgement to be true. Thus, Bolzano says, for God, all truths are certain, while for us, only those are certain that we know immediately or that we have derived from immediately known truths without probable inferences.

Bolzano divides erroneous judgements into four groups, on the basis of whether all the premisses of these judgements are true or whether at least one of them is false and whether the judgement in question is a probable conclusion of the premisses or whether it is just imagined to be so. He manages to simplify this classification by noting that in cases of imagined inference, we could take the erroneous rule of probable inference as one premiss, which of course is then a false premiss.

Bolzano divides probable judgements into those where we know their probability and those where we don’t know it. If we know the probability of a judgement, he continues dividing, the judgement might be either held to be true or not (in quantitative terms, this depends on whether the value of probability is greater than ½ or not). If a probable proposition is held to be true, this probability can be either real and objective, if all its premisses are true, or then just assumed and subjective, if some premisses are false. In case of objective probability, Bolzano suggests, the premisses are then partial grounds of the probability of judgement – he emphasises that they are not grounds of the judgement, although they are causes of us having the judgement (that is, what he earlier has called grounds of knowledge).

Bolzano also divides probability into contextual or relative and complete or absolute probability, depending on whether the probability of the judgement is calculated in relation to just a part of judgements held to be true by the thinker or in relation to the whole collection of them. If the absolute probability is so high that it would be ridiculous to even consider its negation to be true, Bolzano calls the judgement believable or morally certain.

Bolzano returns to the notion of confidence in judgement, which has earlier just mentioned in passing, characterising it only as coming in degrees. The question is why we have different degrees of confidence: could it be that before we even assent to a judgement, we ascribe to it a definite degree of probability, on the basis of which we decide how confident we are in its truth? Bolzano admits that this might be the case sometimes, but certainly not with all judgements, because ascribing a probability to a judgment is a new judgement, leading us to an infinite regression, since our degree of confidence in this new judgement would require another judgement on its probability. He notes that especially immediate judgements are not preceded by a judgement of their certainty, since otherwise they wouldn’t be immediate. Furthermore, Bolzano continues, not even all mediate judgements cannot be preceded by a judgement of their probability, because this latter judgement would again be a mediate judgement, leading us to a new infinite regression.

Bolzano suggests as an answer that the probability of judgements, even when it goes unnoticed by us, is connected to a force that the judgement has to affect our confidence in it. Thus, he concludes, our confidence in a judgement is determined, firstly, by the absolute probability of the judgement in relation to all the current judgements in our minds, and secondly, by the degree of confidence these previous judgements have in our minds. Bolzano considers the question, whether there might be other influences, and at once rules out the possibility that forgotten judgements had something to do with our confidence. He does admit that differences in our attention and our wishes on truth or falsity of some matters might affect our confidence, but suggests that this influence happens mediately through judgements they evoke.

Bolzano suggests enumerating our degree of confidence in the immediate judgements and judgements derived from these with no use of probable inferences as 1 – this would be the highest quantity confidence could have. Our confidence in judgements derived with probable inferences would then be less than 1. Bolzano suggests calling such a less than fully certain confidence a moral confidence or conviction, if the respective probability is so high that the judgment is morally certain, while otherwise the judgement would be a mere opinion.

How is the degree of confidence then calculated? If the probability of a proposition is ½, Bolzano thinks, our confidence in it would have to be 0, since then we are not at all confident in the judgement, and indeed, do not even hold it or its negation to be true. Thus, he suggests defining the confidence in a judgement as the difference of the probability of the judgement and the probability of its negation, that is, if the probability of judgement is presented as a fraction m/n, then the confidence would be m/n – (n – m)/n = (2m – n)/n. Bolzano notes that the confidence in a judgement could then get negative values up to –1, but this means just that our confidence in the negation of the judgement has the respective positive value.

Bernard Bolzano: Study of science – Does knowledge have any limits?

Just like the study of propositions was followed by the study of truths, Bolzano moves from the study of judgements to consider their relation to truth. More precisely, his question is how both true and false judgements are generated. Knowing especially the sources of error, Bolzano suggests, helps us to avoid them altogether. On the side of true judgements, he adds, it should be helpful to know if our knowledge has any intrinsic limits.

Before getting to the actual investigation, Bolzano wants to define the notions of knowledge and error and also the related notion of ignorance. He suggests first defining knowledge as truthful judgement, but quickly notes that with this definition, something we weren’t asserting at the moment would not be knowledge. Thus, Bolzano finally lands to the definition that knowledge is a truthful judgement that we have assented to at some point and that we can recollect and would continue to assent to, when explicitly asked.

Bolzano notes that the concept of error refers to judgements. We do say, he admits, that we have an erroneous concept of a thing, but then we just mean that we make erroneous judgements about the thing. Similarly, Bolzano contends, ignorance should refer to judgements, since in case of concepts we can instead speak of their absence. Thus, he defines ignorance as not having a certain piece of knowledge, regardless of whether the respective judgement has never been made or whether it is just not recollected. Error, then, is for Bolzano a false judgement that one has assented to at some point and that they can recollect and would continue to assent to. He also points out that assenting to propositions that imply a specific knowledge or a specific error is still not enough for having this specific knowledge or error without an explicit assent.

Bolzano states that we are clearly ignorant of infinitely many truths, because our mind is too finite to contain all of them. More precisely, he adds, we are ignorant of truths, because we do not have the relevant representations, or in case of mediated judgements, because we haven't been able to do the proper inference, either because the required premisses are never at the same time in our mind or because we lack the capacity for the required inference.

Bolzano starts his investigation of errors by noting that it is not rational to doubt the truth of our immediate judgements: If we doubted one of them, we should doubt all of them, since they are all generated in the same manner, and in fact, we would have to doubt all of judgements, since ultimately they are based on immediate judgements.

Moving on to discuss derived judgements, Bolzano notes that we really cannot doubt the basic rules of inference we use, again, because that would undermine our reliance on all of our derived judgements. What we can doubt and what can cause errors, he suggests, is when our inferences involve probability. In other words, it is not their probability that is to be doubted, but our instinctual habit of accepting highly probable propositions as true, although there is a slight possibility they are not. This means, Bolzano argues, that any entities, like God, that do not use probable inferences, are incapable of error. Furthermore, he adds, erroneous judgements are not completely certain, but they do have a deceptive look of a truth, due to their high probability.

Most judgements based on probability are experiential, Bolzano thinks, but he does not rule out purely conceptual judgements that use probability. Indeed, he states, every time we use recollected premisses in our inferences, we are involved with probabilities, since we cannot be completely certain on what basis we accepted them.

Bolzano notes that the general description of the cause of errors does not yet give us proper tools to avoid errors, since we cannot just reject all probable inferences. Thus, he gives a more detailed explanation on the causes that especially make us think false propositions as probable. The list is not very original, as Bolzano himself admits: ignorance, reliance on incidental connections, unreliable memory, too weak or too lively imagination, inclinations of our own will, ambiguousness of words and influence of other people are all among possibve causes of errors.

The discussion of error has taught us, Bolzano continues, that judgements are necessarily true, if they are immediate or derived from immediate judgements by inferences that do not involve probability. Mediate judgements can be at times or accidentally true, even if derived by probable proofs, he adds, and even if they are derived from false judgements through inferences.

Bolzano reminds us that some of our true judgements are based on knowing the ground of that truth. The question is can we know truths otherwise than from their grounds. Bolzano answers at once that at least immediate judgements are not known from grounds. Then again, he continues, some truths have no ground – these are the so-called fundamental truths. Could all our immediate judgements be fundamental truths? Bolzano denies this, because our immediate perceptual judgements have a ground, since perceptions have a cause. Furthermore, he continues, knowledge based on probability does not presuppose knowledge of the ground, and in fact, when making judgements of experience, we seem to never know the grounds before the consequences. This leaves still the question of conceptual truths, in finding of which we require probability at most for recollecting earlier known truths. Immediate pure conceptual truths could be taken as fundamental truths, Bolzano conjectures, but he is apprehensive about truths derived solely from such fundamental truths: since there usually are many possible ways to prove them, but there should be only one ground for one consequence, we most likely do know some derived conceptual truths on the basis of something else than their ground.

If a judgement is not immediately made, Bolzano reminds us, it is derived from other judgements, which are then the cause of the first judgement. If the first proposition is true, the collection of the judgements from which it is derived could be called the ground of this knowledge – actually, Bolzano thinks, it should be more proper to call them the cause of knowledge, but this is not a commonly used term. Grounds of knowledge can, of course, contain false judgements and they need not be grounds in the proper sense of the word, thus, Bolzano divides them into objective grounds of knowledge, which are also proper grounds, and subjective grounds of knowledge, which are not.

As Bolzano has stated a number of times, we can know only a finite amount of truths at a time, but this does not yet imply that there would be a definite limit on what we can know. Considering what finding limits of our knowledge would mean, he first suggests simply that it would imply determining what we can know what we cannot. Then again, Bolzano reminds the reader, we can know truths accidentally even through errors. To make the idea of the limits of knowledge more stable, he then suggests including within them only truths that we can reach in a regular fashion.

Bolzano continues by stating that the notion of the limits of knowledge is still ambiguous: we might speak of limits for a certain purpose, my individual limits or limits of all humanity or even limits for all conscious beings, limits for the time being and limits for the end of times. He suggests that the aim of determining these limits in logic should be to show where we need not put our efforts in searching for the truth. Thus, it would make sense to speak either of our own or of the limits of humanity. It would also not be useful to know only the current limits of our knowledge, Bolzano suggests, since those can change in the future.

Going to the concrete question of determining these limits, Bolzano notes that this cannot be done by simply stating the truths we cannot know, since then we would in fact know them. He suggests that we could instead mention generally the objects, of which we can know nothing, but notes at once that we can know something of every object, for instance, characteristic it shares with all objects in general: and indeed, even the judgement “we can know nothing of this object” would also be a truth we know of this object.

Moving further with the idea, Bolzano suggests that maybe we could point out objects, of which we cannot know anything that is proper to these objects. His answer is that since we have to be able to have a representation of the object even to speak about it, we do know something that is proper to it (this representation). Another possibility is that the limits of knowledge would limit knowing any such proper characteristic of the objects in question that are not included in their representation. Bolzano thinks proving such unknowability would have to be based on the proper characteristic of the object included in its representation, but admits knowing no such proof. In addition, he sees a possible contradiction lurking here: if we could probe unknowability in this sense, it would actually be a characteristic proper to the object, but not included in its representation.

The only possibility Bolzano can think of anymore is that we might point out propositions of which we could never know whether they are true or false. The problem is that he sees no way to prove propositions to be unknowable: although we haven’t decided their truth now, we could find means to decide the issue in the future. Indeed, Bolzano states, it has sometimes taken hundreds of years to find out proofs for conceptual truths, and in case of judgements with experiential elements, further experiences in the future might decide the issue about their truth,

Bolzano again admits that his conclusions on the limits of knowledge are not revolutionary and points out that most non-sceptical philosophers would agree with him. The clear exceptions are Kant and his followers, who thought they could define strict limits to human knowledge. Bolzano has already rejected some key elements of Kant’s philosophy, which underlie also Kant’s notion of the limits of knowledge. Still, Bolzano criticizes the idea yet again from another angle.

Bolzano finds even Kant’s division of proper sciences into just four parts (logic, arithmetic, geometry and pure physics) faulty, because it lacks all practical sciences and aesthetics. He also considers it absurd that Kant insisted logic would consist of mere analytical judgements. Indeed, Bolzano lists many synthetic judgements of logic, such that there are representations and that every proposition can be divided into at least three parts. Even rules of inference are synthetic judgements, he thinks. Then again, as we have already seen, Bolzano considered arithmetics to be more analytical than logic and definitely not dependent on any intuition of time. What Bolzano finds particularly faulty in Kant’s notion of the four sciences is that they contained only universally recognised truths, because he thinks their great certainty rises only from the ease of empirically testing their most important theorems, possibility to infer the rest of their theorems and lack of human passion concerning them.

Kant especially thought we could not say anything about supersensuous objects, because we can make synthetic judgements only of what we perceive immediately, since space and time are a necessary condition on what we can know. Bolzano thinks it not at all clear what we should include among such objects. Usually, he says, by sensuous objects we mean such that can generate perceptions by affecting our organs, that is, bodies, which would mean that simple substances, of which bodies consist, are not sensuous – still, they exist in time and space. Similarly, Bolzano continues, the soul should be supersensuous (since we do not perceive it as such), but it also exists in time (because it changes) and in space (because it interacts with things in space). In fact, he concludes, only God is not in space and time, because God does not change, although in a sense God is in all space as time as their creator.

Bolzano also thinks that there are many true metaphysical judgements about these supersensuous objects that we know – his examples include “God exists”, “no simple substance can be destroyed”, and “representations can occur only in simple substances”. He admits that we have not found the proper grounds for these judgements, but they still are known to be true judgements, and indeed, we still lack proper foundations in most of our sciences, since we haven’t been able to properly define even such crucial notions as space in geometry.

Bolzano goes into great detail in pointing out missteps in Kant’s proofs of antinomies and we need not follow him there – the faults of these proofs have been many times attested by several philosophers. Instead, we should point out that Bolzano finds Kant’s notion of practical proofs of God, freedom and immortality preposterous: surely we cannot first suppose that we ought to strive for some end (in Kant’s case, the greatest good) and then just say that the preconditions of this end must be possible, but instead, we should first ascertain that achieving the very end is possible and only then conclude that it is something we ought to strive for.

Bernard Bolzano: Study of science – What are judgements made of?

From subjective representations, Bolzano moves to judgements. He underlines that judgements are not representations – if we just represent a proposition, we do not judge or assert the matter to be so. Still, he admits, judgements do share some characteristics with subjective representations. Both have a stuff: with subjective representations it is an objective representation, with judgements it is a proposition. Furthermore, just like subjective representations, judgements are actual, but only as adhering to a soul.

Propositions are constituted of representations, Bolzano reminds us, and so should judgements consist of subjective representations. He states that parts of a judgement coexist or at least the earlier parts have not completely vanished when the later parts occur in our mind. Then again, Bolzano adds, just like coexistence of representations does not mean that a combined representation exists, the coexistence of the parts of judgement does not mean that the judgement exists. He also suggests that the formation of judgements is not completely voluntary, although we can affect it by choosing what representations we attend to.

Just like with representations, Bolzano finds it convenient to speak of one judgement, when the same person judges the same objective proposition uninterruptedly. This means that judgements are different, if they have different propositions as stuff, belong to different thinkers or are discontinuous in time. Bolzano also suggests that similarity of judgements presupposes at least that they have the same proposition as their stuff.

Bolzano thinks that many other features and types of representations apply also to judgements. Examples of these include clarity, distinctness and liveliness (then again, he notes that confidence or certainty is a feature unique to judgements). Other classifications, such as the distinction between analytical and synthetical, judgements derive from propositions.

Just like representations, Bolzano points out, our judgements come and go: indeed, it is very common that our opinions change over time. Judgements also leave traces on our soul, which makes it possible to recollect our judgements. Of course, Bolzano adds, we may deny a judgement we recollect asserting in the past or our confidence about it might have changed.

Bolzano moves on to the question how our judgements are generated. Some of them, he points out, have been caused or mediated by other judgements through the use of inference. Bolzano recognises three types of mediation: firstly, that based on objective consequence relation, secondly, that based on other relations of derivability, and finally, that based on probability. He finds the first two unproblematic: objective grounds can also cause judgements in us, while knowing relations of derivability is just something where we resemble God, although in a restricted manner. Probability relations, on the other hand, are something God in their omniscience cannot be dealing with. Indeed, Bolzano suggests, using probabilities in inferences arises from the limitedness of our powers of cognition – great enough of probability makes us confident of a judgement, and after a time, we forget that it is just probable and remember only the judgement.

Mediated judgements must ultimately be based on some unmediated judgements, Bolzano argues. These include, he says, judgements like “I have the appearance A”, where A could be, for instance, a representation, a judgement, a sensation or a decision, and also judgements of the type “this, what I intuit here and now, is an A”, where the A could be something like red, fragrant or sour (he does note that sometimes the latter kind of propositions are mediated, for instance, if we know that something that we see is of a kind that is salty, we can deduce that this what I see is salty). Bolzano calls immediate judgements of both kinds perception judgements, while all mediated judgements referring to intuitions he names experiential judgements.

All mediated judgements cannot arise from mere perceptual judgements, Bolzano contends, for instance, because we need something more to show that our intuitions presuppose some external thing that has generated them. Thus, he concludes, we need pure conceptual judgements that describe connections between pure concepts. In this way, Bolzano goes against any mere empirical explanation of the origin of our judgements. This does not mean, he adds, that just insisting on two sources of judgements – intuitions and concepts – is enough.

Bolzano goes into great detail in studying especially the views of Kant and some of the more prominent Kantians, but finds their explanations unsatisfying. While Kant found the explanation of analytical judgements simple, Bolzano thinks these judgements contain unexpected difficulties. Analytical judgement, as defined by Kant, should be such that the subject contains the predicate – so basically, Bolzano notes, of the form “something that is both A and B is A”. Although seemingly self-evident, he points out, they presuppose that there is some A that is also a B (taking the often used example, “all bachelors are unmarried” presupposes that there are bachelors, that is, unmarried men).

Bolzano’s take on Kant’s analytic judgements might seem a bit pedantic (and dependent on his reading of universal propositions as making an existential commitment), but his take on what Kant calls a posteriori judgements is more interesting. Kant had suggested, Bolzano says, that such judgements are based on intuitions. Bolzano thinks this is an inadequate explanation, since singular intuition cannot justify universal judgements like “all bodies have weight”. Indeed, he goes further and insists that even a singular judgement like “this body has weight” is not explained by mere intuitions. Either the notion of the weight is simply included in our intuition of this body, and then we would be discussing simply an analytic judgement, or more likely, this singular judgement says something like “this body that is the cause of other intuitions (e.g. colour, smell etc.) is also a cause of our intuition of weight”, which is something not justified by mere intuitions.

What Kant finds truly difficult, that is, the so-called synthetic a priori judgements, but insofar as they coincide with what Bolzano himself calls pure conceptual judgements, he thinks they are rather unproblematic – we shall see a bit later how Bolzano accounts for them. On the contrary, Bolzano finds Kant’s explanation of them wanting. Kant’s solution is to rely on the so-called pure intuitions of space and time, but if they are supposed to work like the regular intuitions with a posteriori judgements, Bolzano argues, they do not work at all.

Bolzano goes further to analyse the Kantian notion of pure intuition, which should differ from empirical intuitions by not being generated by actual objects, but by imagination. Bolzano wonders why this would make any difference, since imagined objects are just less lively than intuitions of actual objects. Of course, Kant had suggested that it was not regular imagination that he was discussing, but pure imagination – Bolzano finds this a meaningless wordplay. Kant did suggest as a difference that empirical imagination generated images and pure imagination only pure schemes, that is, rules for finding images for a concept. Bolzano finds this even more perplexing: why call such a scheme an intuition? Indeed, he says, Kant’s scheme seems to coincide with what was traditionally called a genetic definition, which is essentially something conceptual.

Looking at concrete examples of synthetic a priori judgements, as discussed by Kant, Bolzano comes to the conclusion that Kantian notion of pure intuition shows infatuation with the idea that geometric proofs are based on constructions of figures in our imagination. Bolzano himself suggests that using such constructions as substantial steps in geometric proofs just shows the still inadequate condition of this science. Indeed, while they make proofs more lively in simple cases, with more complex figures they soon become too imprecise.

Bolzano finds even more to complain in Kant’s extension of the need of construction to arithmetics. Looking at the famous Kantian example, 7 + 5 = 12, Bolzano contends that a true notion of arithmetic sum would imply its associativity, that is, the fact that (a + b) + c = a + (b + c). If numbers are then defined by their relation to the predecessor (e.g. 7 = 6 + 1), proving the required judgement becomes just a simple task of using these definitions and associativity. True, Kant insists that such a process would require using time, but Bolzano finds this objection preposterous: equally well one can say that any sorites, that is, combination of several steps of syllogisms, takes up time, but Kant is still not saying that they require construction in time.

Returning to Bolzano’s own explanation of the origin of judgements, the immediate judgements he finds easy to account for. The first type of perceptual judgement, “I have appearance X”, just requires that we can have clear representations of our mental phenomena. With the second type, “what I intuit here is A”, the only problem Bolzano finds is that A can be a very complex representation, but he suggests that even such complex representation could be gradually built over time, when we attend to changes in our soul. The immediate conceptual judgements Bolzano finds even more unproblematic: we just require an ability to generate the pure concepts in question, perhaps occasioned, but not justified by other judgements, while combining these pure concepts into judgements is based on nothing more than immediately observing their relations.

The only real problem, Bolzano argues, is that many of our judgements have been generated at such an early time in our life that we are not anymore aware of the inferences we have required in generating them. Especially our judgements of temporal relations are problematic, he emphasises and adds that Kant never really addressed this question, at least when it comes to temporal relations of our representations, just assuming that we have an immediate awareness of some of our representations being earlier than others. Bolzano contends that this is something we cannot intuit immediately, but then again, we cannot derive it from observing external changes, since seeing something as change already requires an understanding of temporal relations.

Bolzano suggests as an ultimate basis of temporal judgements that we must be able to perceive that our mental phenomena occur at this moment, now. Furthermore, he continues, we can also observe that some of our representations conflict with one another in the sense that they cannot occur at the same time (for instance, judgement that A is true and judgement that A is false, or wish and sensation of its fulfilment). Sometimes, Bolzano argues, we are aware of an appearance as occurring now and a trace or a memory of a conflicting appearance. Thus, he suggests, we get our first notion of some of our appearances occurring at different times: we remember something that occurred, but is not occurring now. Series of representations can form even our first, although rather crude way of measuring time (think of a person counting in their head “one, two, …”).

According to Bolzano, we can distinguish between external and internal intuitions, and of the former we know that they have been brought about by external objects with an ability to generate intuitions in us (apparently this is based on a conceptual judgement on the nature of external intuitions). If we have similar intuitions, he continues, we can draw the probable conclusion that they have been generated by the same thing: this is not a certain conclusion, because sometimes different things provide similar intuitions. Furthermore if a group of intuitions always occurs at the same time, we can draw the probable conclusion that they have been caused by one thing with different powers. We still have to explain why this thing does not cause these intuitions always, and in some cases, the examples of which we do not need to deal here, we can infer that it has been a question of our spatial distance from the thing being different.

Bolzano goes into more details in his explanation of these processes. The important point here is that these inferences, as they occur in an animal or in a small child, are more like associations that slowly build up a set of judgements. Of course, they are not clear judgements, that is, judgements that the animal or the child would be aware of. In case of a child, Bolzano states, when they reach a suitable age, the resulting judgements start to gain clarity.

Bernard Bolzano: Study of science – Thinking representations (again?)

After a rather modern take on logic in the so-called elementary study, Bolzano returns to the idea that logic deals with human cognitive capacities. His justification is that such epistemological considerations are necessary for evaluating the suggested rules for finding truths and for organising them into individual sciences. Bolzano states that studying cognitions means essentially studying judgements and their constituents or representations. Furthermore, he adds, since cognitions form a specific subset of judgements, because all cognitions are true, logic should also study the relation of judgements to truth. Finally, Bolzano suggests investigating the degrees of reliability of judgements.

Starting with representations, we have to emphasise that it is not what Bolzano called representations in themselves that we are dealing with here, but subjective representations, which are more like appearances of representations in themselves in some mind. He also emphasises that in difference from a common way of speaking, he does not refer by representations only to representations of absent things, but to any constituents of our judgements that are not full judgements.

Bolzano begins familiarly with characteristics common to all subjective representations, first of which is that all of them are connected to some representation in itself. He sets out to prove this statement by asking the reader to imagine someone thinking of a representation A. Then, Bolzano states, “this person has representation” is a true proposition, even if no one is thinking of this proposition. This proposition, he concludes, has as one of its constituent the representation in itself of A, corresponding to the subjective representation A. Bolzano suggests calling this representation in itself the stuff (Stoff) of the subjective representation, distinguishing it clearly from its matter – the further subjective representation, of which it is constituted – and its object, which all subjective representations might not even have.

Bolzano emphasises that all subjective representations exist. He admits that we do say that something is just a representation, but then we merely mean that it does not refer to an existing object. Although existing, Bolzano explains, subjective representation is not an independent substance, but a characteristic of some mind: not an external characteristic or relation, but an internal characteristic or property.

Bolzano moves on to consider the question when subjective representations are same or similar. He suggests that subjective representations are numerically different at least when the corresponding representations in themselves are different or when they are thought by different persons. Bolzano notes that God is always thinking of all of the representations, so that there are as many divine subjective representations as there are representations in themselves. With humans, on the other hand, the subjective representations are constantly changing, but Bolzano thinks it convenient to think of them as holding their identity, as long as there is temporal continuity, so that any discontinuity in time would make the representations different.

Similarity of subjective representations, Bolzano goes on, requires at least the same representation in itself as stuff. Clearly, he adds, we can say that two subjective representations at different times are similar, but can we have similar representations at one and same time? Bolzano notes that we can have many different representations at the same time, for example, when we are thinking of judgements. Thus, he continues, we might think that, for instance, in a judgement like “a triangle with all angles similar has all sides similar” has two similar representations of similarity. Still, Bolzano finds it more plausible that it is just the same representation of similarity that is mentioned at two spots of this judgement, thus preferring the notion that we can speak of similarity of representations only in case of representations had at different times.

As actual entities, Bolzano points out, subjective representations have different effects. Such different effects may result from the corresponding representations in themselves – for instance, thinking snakes feels different from thinking flowers – or from the relation that the representation in itself has to other representations in themselves – thinking snakes is a different experience, depending on whether we are considering venomous or non-venomous snakes. Bolzano also points out that much depends also on an internal characteristic of the subjective representation he calls its strength or liveliness: stronger representations move us more.

Evidently all divisions of representations in themselves apply also to subjective representations, but Bolzano considered only few more prominent cases. Thus, he notes that we clearly have complex representations, which means that we must also have simple representations. Bolzano thinks it plausible that we never have just a single simple representation, because we are also aware of, for instance, words describing such a representation. Then again, he carefully distinguishes thinking of a mere collection of simple representations from thinking of a complex representation, since the latter still requires combining the constituents into a unity.

Important subsets of subjective representations, Bolzano states, are that of subjective intuitions – these are simple and refer only to one object – and that of subjective pure concepts – these are not intuitions and do not also have intuitions as constituents. Bolzano argues that we have intuitions, since we have representations that we regard as effects of a single actual object represented by this representation (say, the representation of “this smell”). He also thinks that we have pure concepts, using the notion “and” as an example. Bolzano underlines as an important type of intuition those that we have of our own representations.

A very common way to divide subjective representations, Bolzano points out, has been that between clear and obscure representations, but he finds most explanations of this difference lacking. The best explanation suggested, Bolzano thinks, has been to define clear representations through our being conscious of them, but what does this consciousness mean? Clearly, if we judge that we have a representation, this is a clear representation we are conscious of, but Bolzano deems the existence of such a judgement to be a too strong demand of clarity: picture two otherwise similar representations, only one of which we judge to be our representation – wouldn’t both still be clear? Of course, at least an ability to make such a judgement characterises our consciousness of representation, but Bolzano finds an ability to do something a rather weak basis for a definition.

Bolzano continues by noting that in order to judge that we have a representation, we need a representation of this representation. Then again, the converse is not true or we can have a representation of a representation without judging that we have this representation. Thus, Bolzano suggests as the first definition of clarity of a representation that we have a representation referring exclusively to this clear representation. He notes that this definition still leaves undetermined whether the representing representation is an intuition or a complex representation. Considering the complex representation of a representation we have a month ago, he states that it is obvious that only intuitions can be representations representing a clear representation. Bolzano also points out that mere ability to intuit a representation is not a sufficient mark for its clarity, since it is the actual intuition that gives our representations clarity, for instance, when we actually intuit having pain, instead of just being in pain, but not intuiting it.

Armed with this definition of clear representations, Bolzano thinks it obvious that we do intuit some of our representations. He is also adamant that some of our representations are obscure or such that we do not intuit them. Otherwise, Bolzano argues, we would have infinitely many representations, since we would intuit a representation, then intuit this intuition and so forth.

Another common division of representations is that between distinct and confused representations, and Bolzano follows the usual definitions quite faithfully. Thus, he takes distinct representations to be such where we know its possible constituent representations and how they are combined. A clear difference in Bolzano’s definition from those of some of his predecessors is that he can accept also simple representations as distinct, as long as we just know that it does not have any constituents. He also notes that distinctness has degrees and defines a complete distinct representation as such where we know how it is formed of its ultimate, simple constituents.

Bolzano has already stated that God’s subjective representations are eternal, but is it so with humans? Looking specifically at intuitions of external things, Bolzano notes that they are caused by external things affecting us, thus, making at least their come and go with this interaction. Then again, other representations are more dependent on our own activities. For instance, intuitions of our own representations depend on our attention toward our own mind, while simple concepts, Bolzano explains, are generated together with activities by which form representations from other representations (e.g. the concept of “not”) or judgements from representations (e.g. the concept of “having”). Thus, he concludes, these representations come and go together with such activities.

Although representations come and go, Bolzano suggests that they do leave some traces in our soul. Such traces, he explains, are the main reason why we can later draw from our memory representations that we know resemble those that we have had earlier. This drawing of memories or recollection follows the so-called laws of association, which Bolzano reduces to the single rule that when we have regularly had two representations of a certain kind occur at the same time in our mind, the existence of one type of representation can awaken also a representation of the other type. The possibility to recollect has degrees, he notes and surmises it has something to do with the strength of the representations in question.

A special case of recollection Bolzano considers is such where one representation is supposed to refer to an object and we then generate another representation or image of this object (Bolzano also calls this image scheme, which is a term reminiscent of Kant’s philosophy). The image is often more determined than the original representation – for instance, when we imagine a triangle with a definite shape, when thinking of triangles in general – and we might have such an image even of things we cannot actually perceive (e.g. God) or even of completely imaginary things (say, when we think what a body limited by ten regular figures would look like).

Bolzano goes into great detail to discuss the use of representations as signs for other representations, which obviously leads to the formation of language, but he mostly suggests some simple definitions, so we can skip this part of his account. A more interesting digression is his suggestion that especially in case of complex representations formed in early childhood, we are not aware of their constituents. This is especially true, Bolzano thinks, with generic representations related to intuitions, like “red”, “blue”, “sweet”, “bitter” etc. These must be complex, he insists, because they are not intuitions, which always refer to just a single object (in other words, we have intuition of this particular red we sense now, but not of red in general), and also not simple concepts, since they do not have as clear limits (for instance, the line between red and orange is fuzzy). Indeed, Bolzano goes even so far as to deny that they can have intuitions as their constituents, since they do not have any reference to a specific moment in time (as my representation of e.g. my parents do). Instead, he suggests, these generic representations must be concepts of laws describing how certain types of intuitions can occur in our soul through interaction with other objects.

Bolzano ends his discussion of subjective representations with the rather obvious point that there are representations that we do not at the moment have and even never have had. Indeed, since intuitions are dependent on interaction with external objects, there are many objects we have never had interaction with and thus have never intuited.

Bernard Bolzano: Study of science – Inferences

No book of logic is not complete without some mention of inferences, and Bolzano’s is no exception. At worst, such an account devolves into a tedious listing of disparate specimens of inferences the author has deemed important. Bolzano is in danger of doing just that, but he does admit that such a listing can never be complete and tries at least to provide an inkling of systematicity to the proceedings.

Just to see more closely, what the method of Bolzano is in going through inferences, I shall give more details of the beginning of Bolzano’s account. He starts with what he calls the simplest kind of proposition: “A has b”. Note that by a lower case letter Bolzano always refers to a characteristic, while a corresponding upper case letter refers to all the objects that have such a characteristic. Thus, Bolzano’s simplest proposition means actually “all A have b”, and indeed, “all A are B” – the traditional form of universal proposition. Just like traditionally, the subject term has some objects to refer to, which Bolzano indeed takes as the first conclusion that can be drawn from this simple type of proposition.

After going through some equally simple conclusions that can be drawn from a single proposition of the simplest type, Bolzano moves on to discuss combinations of this type of propositions. To facilitate the discussion, he moves to an equivalent form “what has a, has b”. Bolzano points out that if we start with two propositions, they could share both representations a and b, only one of them or none.

Starting with the first case, Bolzano points out that the propositions could literally taken be only “what has a, has b” and “what has b, has a”, but he suggests adding the possibility of negating either a or b, leading to eight possible propositions:

1) what has a, has b,

2) what has b, has a,

3) what has a, has not b,

4) what has b, has not a,

5) what has not a, has b,

6) what has not b, has a,

7) what has not a, has not b and

8) what has not b, has not a.

The first proposition could always be taken as 1), because the important thing is the relation of the representations in the two propositions, and if needed, we can always say that not-not-a is equivalent to a. This leaves us seven other options for the second proposition, and even from these, we can take the options 3), 4) and 6), which cannot be connected with 1). Thus, we can infer from 1) and 2) that A and B are equivalent, from 1) and 5) that everything has b, and from 1) and 8) that B is not subordinate to A, but is also not equivalent to the representation of something in general (1) and 7) lead to various conclusions, which I’ll leave unmentioned for simplicity’s sake).

Bolzano goes on for quite a while with such a level of detail, although he eventually moves into a more summarised presentation. We do not need to follow him that far, but we can just note that next he goes on, as he promised, to discuss combination of propositions like “what has a, has b” and “what has a, has c”, sharing only one representation, which are, in effect, Aristotelian syllogisms using only universal premisses. He also considers the case where the propositions do not share any representations – “what has a, has b” and “what has c, has d” – the conclusion of which is that the whole combined of objects of A and B is a part of a whole combined of objects in B and D. Bolzano completes the study of inferences using only the simplest type of proposition with a consideration of cases with more than two premisses.

The next move Bolzano takes is to consider negations of type “it is false that A has b” (not to be confused with “A has not-b”, which would still imply that A refers to some objects, which the negation leaves open). Here we see the pattern Bolzano is making: he starts with inferences from a single negation, then proceeds to discuss inferences using both negations and affirmations of the first type and finally moves on to the case of several negations.

Bolzano follows the same systematic approach with propositions stating that a representation refers (or does not refer) to objects: he looks at individual propositions, then its combinations with propositions of the same or one of the previous types. The only interesting new feature is that in addition to propositions “representation A refers (or does not refer) to objects” he also considers propositions where the A has been analysed further, for instance, “representation of something that has characteristics a and b refers to an object” (these form the traditional particular premisses in Aristotelian syllogisms).

The next couple of types of propositions include such that indicate that a representation refers to one, several of a certain number of objects. What is interesting in these cases is that they imply further propositions about the sums of these sets of objects, at least if we know that they do not share any objects.

Bolzano moves on to discuss propositions about relations of representations, such as “representations A and C are compatible”. He notes that when we speak of mere individual representations, many of these propositions are actually equivalent to propositions considered earlier (thus, the just mentioned example says that some A are C), but his habit of speaking about collections of representations makes the discussion more complex. Thus, a proposition about comprehension between collections of representations (“every object that falls under one of the representations A, B, C… falls also under one of representations M, N, O…”) leads to a complex disjunction, where each individual object belonging to some representation of the first collection belongs to some representation of the second collection.

Bolzano has thus far discussed propositions about representations, but moves next to propositions about propositions, for example, the relations of derivability, equivalence and contradiction. What makes his account hard to follow is his use of variable representations to indicate that it is not individual propositions, but propositions of certain form he is discussing here. This feels an unnecessary complication at this point, when he is, for the most part, speaking only of inferences involving relations of propositions in general (e.g. variations of modus ponens and tollens), without indicating what the propositions are like (the only exception consists of such simple cases as premiss “if A is B, then A is D” leading to conclusion “every B is D”).

The final kind of inferences Bolzano considers involve probability, where the conclusion is not certainly true, but only of certain probability. Bolzano does not really dive deep into such probable inferences, but after some simple inferences (e.g. if probability of proposition is m, probability of its negation is 1 – m), he merely notes induction and analogy as important subtypes or probable inferences.

Bernard Bolzano: Study of science – Truths and consequences

 After propositions, Bolzano moves to discuss truths in themselves or objective truths that may or may not be known by anyone. He considers this a field not yet investigated by anyone and finds even his own results very preliminary.

Like with propositions, Bolzano begins with characteristics common to all truths, the first one being the familiar one that truths do not actually exist, making it thus incorrect to say that some truths exist eternally and some temporarily. He notes also that all truths deal with objects, so that their subject is a representation that refers to at least one object. Bolzano admits that some truths appear to have propositions as their subject, but thinks that the subject in these cases is actually a representation of a proposition. Similarly, he continues, truths with imaginary objects as their subject (e.g. round square does not exist) usually have a representation of an imaginary object as the real subject (in the example, the truth  is actually “the representation of a round square refers to no objects”). Sometimes, Bolzano adds, such truths are actually hypotheticals: “golden mountain is barren” means actually “if a mountain is golden, it would be barren”.

If the subject of a truth must be a representation referring to some object, Bolzano says, its predicate must be a representation referring to such a characteristic that every object referred to by the subject has. Furthermore, he continues, all truths must be compatible with one another (they are true at the same time), and every truth is derivable from infinitely many other truths, while infinitely many truths are derivable from every truth. Finally, every truth and every collection of truths is a part of another truth (at least the truth saying that all these propositions are true).

Just like with propositions, Bolzano goes on to describe different types of propositions. Actually he mentions only one possible division of truths, namely, that of analytical and synthetical truths. Analytical truth, Bolzano defines, has as its constituent at least one representation that can be switched with any other representation so that the proposition remains a truth, just as long as the proposition (that it, its subject) refers to an  object; any other truth is synthetical truth.

Evidently, Bolzano thinks, there are analytical truths (his example is “every A, which has b, has b”). Then again, he thinks, synthetical truths are a bit more difficult to find. Bolzano notes that there are characteristics that belong to all objects referred to by a representation without being constituents of that representation, but this is not enough to establish the existence of synthetical truths. For instance, the characteristic of the sum of the angles of a triangle equalling two right angles is something featured in all triangles, although this characteristic is not a constituent of the representation of a triangle. Still, Bolzano surmises, the truth “this triangle has angles equal to two right angles” is still analytical by his definition, because in switching the representation “this” with another representation that makes the subject objective (in other words, by changing the object referred to by the word “this” to some other triangle), the proposition remains always true. 

Bolzano notes that the problem just described can be circumvented, if we just pick a subject that is either simple or then of the type “something that has x”. The predicate of the synthetical truth we are searching for, he adds, should not be such that holds of all things when some of its constituents is changed (any characteristic of the type “being x or not-x” is of this kind. Bolzano suggests using either simple predicates or combinations of simple predicates of the kind “x and y and z…”. Finally, the constructed proposition must also be true – Bolzano’s own example is “A is actual” where A is meant to be some intuition. 

Bolzano thinks that there are clearly also negative synthetical truths (his example is “A has no b”, where A is intuition and b is a characteristic that A does not have). Furthermore, he notes, “representation of something, which has characteristics a, b and c, refers to an object” is always a synthetical truth, because if the characteristics are switched to such that are incompatible with one another, the result is not true. Finally, Bolzano points out that “representation of something, which has characteristics a and b, is imaginary” is synthetical, if it is true, because then a and b are incompatible, and when switched with compatible representations, make the resulting proposition untrue.

Bolzano considers it evident that there are both analytical and synthetical truths that are either intuitive propositions or conceptual propositions. Thus, “each triangle is a figure” is a conceptual analytical truth and “this triangle is a figure” is intuitive analytical truth. Similarly, the aforementioned “A is actual” is an intuitive synthetical truth, while Bolzano’s other examples of synthetical truths are conceptual.

The rest of Bolzano’s investigation of truths discusses what he considers the most remarkable relation between them, that is, the already mentioned relation of ground to its consequence relation. He again notes the example of a thermometer rising, which is at most a ground for us knowing that the summer is coming, but not an objective ground for the truth of the summer coming, where the consequence relation would hold independently of our thoughts. This example shows also that the consequence relation is not the same as the relation of derivability: “summer is coming” is derivable from “thermometer is rising”, but not its consequence.

Is consequence still always derivable from its ground? Bolzano thinks this is not so, because of the following example. All practical truths, he says, must be founded in a truth of the form “we should do A”, which with some propositions of the form “doing A requires doing X” is the ground of all practical truths. Now, Bolzano suggests, this highest law of ethics appears to be founded in the truth that A is something that we can do, but there seems to be no rule of inference that would account for this unique consequence relation, since they can prove obligations only from obligations.

As we already saw with the example of the ground of practical truths, the complete ground of a truth is usually a collection of truths, while the individual truths in that collection are then just partial grounds. Bolzano ponders the question whether the rules of inference used for deriving the consequence from this collection of truths should be regarded as part of the ground. He comes to the same conclusion as Lewis Carroll that this would lead to an infinite series of ever new rules of inference.

Bolzano then considers the question whether the consequence relation is the same as causality. No literally, he answers, because causes and effects are actual objects, while grounds and consequences are truths, which are not actual. This leaves still the possibility that causes and effects could still be constituents of grounds and consequences. Indeed, some consequence relations involve causes and effects (e.g. “God is” could be a partial ground of “world is”). Still, Bolzano thinks, not all consequence relations involve existing objects, such as consequence relations involved with mathematical truths. Thus, he concludes, concepts of cause and effect must be defined by concepts of ground and consequence, as he has already suggested, but the latter cannot be defined by the former.

Bolzano does not find ground and consequence to be simple concepts, because they are concrete and least have “something” as their constituent (that is, ground is something that “grounds” something else and consequence is something “grounded” by something else). He has no idea whether this abstract notion of “grounding” has a further definition. All the definitions suggested by earlier philosophers, Bolzano says, are either circular or then faulty – for instance, grounding cannot be defined as determining, because an equation determines a magnitude, but is not its ground.

Since we cannot define the consequence relation precisely, Bolzano suggests describing its features as much as possible, in order to distinguish it from other similar relations. Thus, he says, derivability holds between propositions, whether true or false, while consequence holds only between truths. Sometimes we do say that a consequence relation holds between falsities, Bolzano admits, but then we are actually speaking of derivability or then we consider cases where switching some representations in these false propositions and thus making them truths, creates also a consequence relation between them (Bolzano’s example is when we say that “monkey can speak” is a consequence of “monkey has distinct representations”).

Bolzano notes some other similar confusions. Sometimes we confuse causality with consequence and say e.g. that God is the ground of the existence of the world. Sometimes we speak as if mere representations could be grounds, for instance, when the ground of our fear of death should be our representation of it – actually, it is then not representations in themselves, but our thoughts of them we are speaking and they are not grounds, but again, causes. Bolzano also notes a curious case, where the ground of a proposition being true or false are said to be its constituent representations. He thinks that in this case, the ground lies actually in the truths involving the constituents of the proposition.

Comparing further derivability and consequence relation, Bolzano points out that a proposition can be derivable from itself, but nothing is a ground of itself. Then again, derivability can hold between collections of propositions, and Bolzano thinks that the same is true of the consequence relation. We have already noted that while a ground can be a single truth (thus, A is a ground for “A is true”), it is often a collection of propositions and the individual propositions in that collection are just partial grounds. Similarly, individual propositions based on some ground are just partial consequences, and the collection of all these partial consequences is the complete consequence of the ground.

The notion of total consequence reveals yet another difference from the notion of derivability. While we can derive several propositions from one proposition, the total consequence – consequence in the proper sense of the word, Bolzano thinks – is always unique, as defined by all the partial consequences. It seems that a consequence can then have many grounds, but these are all just partial grounds, Bolzano argues, but the total ground is always unique.

Derivability is a transitive relation (that is, if A is derivable from B and B from C, A is derivable). Due to the just mentioned uniqueness, Bolzano points out, the consequence relation cannot be transitive. Similarly, while A can be derivable from B and B from A, the same sort of reciprocity does not hold for grounds and consequences: we do speak of e.g. reciprocal causality, Bolzano admits, but then it is actually a case of the interacting things having forces that cause some changes in the other thing – not both things literally causing one another. Then again, just like what is a derived conclusion can in another context be a premiss, so a consequence of something can be a ground of something else.

Derivability always survives summing up collections of propositions (e.g. if A and B are derivable from C and D and E and F are derivable from G and H, then A, B, E and F are derivable from C, D, G and H). With consequence relation this is not so easy, Bolzano states, since one collection might contain consequences of the other collection, which would cause difficulties, if the collections contained same truths (e.g. if A and B are consequence of C and D and C and D are consequence of E and F, the collection of A, B, C and D is not the consequence of C, D, E and F). Even if we avoided the cases where the collections contained same truths, the sum might still not work, Bolzano thinks, because the consequence relation requires a more intrinsic connection between the truths (e.g. the collection of “God is almighty” and “isosceles triangle has two equal angles” is not a ground of the collection of “the actual world is the best possible” and “the equilateral triangle has all angles equal”).

Collections of premisses and collections of conclusions appear to have no hierarchical order, and same seems to be true of the collection of partial grounds and the collection of partial consequences, Bolzano thinks: although there may be a preferred order how to present grounds in a scientific treatise, this is just a subjective preference. A more difficult question, Bolzano thinks, is whether just like with premisses, there could be partial grounds, one of which is a consequence of the other. Surprisingly, he suggests that consequence relation does resemble derivability in this respect: “every truth is something immaterial” is ground for the truth “the proposition that every truth is immaterial is itself a truth” and both of them are partial grounds for the truth “the proposition that every truth is immaterial is itself immaterial”.

Every proposition is both a premiss and a conclusion, but is every truth a ground and a consequence? According to Bolzano, every truth is a ground for further truths (at least for the truth that the truth is true). Then again, he thinks it probable that some truths are not consequences of anything (his example is the truth that something exists). Bolzano calls such truths fundamental. He suggests that there must be several fundamental truths, because it seems impossible that all others could be consequences of just one truth.

Bolzano considers what will happen if we ask for all partial grounds of a truth and then for partial grounds of these new truths and so on, creating a tree-like structure. If we reach fundamental truths, he notes, this process ends. Still, Bolzano suggests, there may be truths, where this search for further and further partial grounds continues indefinitely: his example is that of variable states of created substances, which seem to depend on an infinite series of causes.

Bolzano also coins the term auxiliary truth to refer to any truth in the just described tree-like structure, created from a given truth. Unlike the notion of ground, the notion of auxiliary truth is transitive, but like with ground, no truth can be auxiliary truth of itself. Then again, Bolzano notes, the same truth can be auxiliary for another truth in many different places of the same structure: just look at mathematical proofs, where we have to use some truth many times.

Bolzano notes that truths involving only pure concepts are grounded in other truths involving only pure concepts, although they might be known through experiences. Furthermore, pure conceptual truth that is a ground of another pure conceptual truth can never be more complex than the consequence (Bolzano thinks that this is not true of intuitive truths, since they are often grounded in general truths that are more complex). Since Bolzano has assumed that there is only a finite number of simple concepts, he can also assume that there is only a limited number of fundamental conceptual truths. Thus, at least with conceptual truths, the consequence relation is ordered so that the greatest number of consequences are based on the smallest number of fundamental truths.

Bolzano closes the discussion by noting that sometimes grounds are derivable from consequences and sometimes not: as an example of former we have “every A is B” and “every A is C”, which together form the ground of “every A is B and C” and are derivable from it, while as an example of latter we have “what has a, has b” and “what has b, has c”, which form the ground of “what has a, has c”, but are not derivable from it. Just like with causes, Bolzano defines condition in a wider sense as an auxiliary truth of some truth that is derivable from this truth.

Bernard Bolzano: Study of science – Analysing propositions

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.