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.