Moving on to more particular rules of heuristics, Bolzano states that the essential question is what problems it is meaningful to solve. This meaningfulness, he adds, consists of two elements, firstly, whether it is probable that we can solve them, and secondly, whether the solutions would be useful. Concerning the latter, Bolzano especially highlights questions concerning our duties.
In order to choose the most meaningful problems, Bolzano contends, we should have some understanding of what kinds of problems we could have. Thus, the rest of this part of his logic is concentrated on presenting a wide array of types of problems that could come up in many disciplines and strategies for solving them. Especially the first half of this discussion contains problems that are properly logical, and a number of them seem rather artificial and make the reader wonder if they would really come up so often in particular sciences.
The first problem is the analysis of representations: are they simple or complex, and if they are complex, from which representations and in what manner are they put together? Bolzano notes that answering these questions is very difficult, because most of our representations are not clear and therefore their potential parts are not intuited. Thus, he says, we commonly make the mistake and state that a complex representation is simple, just because it is so familiar to us from childhood. Then again, Bolzano adds, it is also very common to take simple representations as complex, especially when they are subordinated to other representations that we therefore consider their parts (Bolzano's favourite example is trying to define actuality in terms of possibility).
Besides analysis, a fundamental problem is to check whether a representation corresponds to any actual or non-actual objects. This task, he adds, should usually be preceded by analysis, because simple representations are always real in the sense that they are not contradictory and thus possibly refer to an object. On the other hand, Bolzano states, from a complex representation it is often simple to check whether they have any obvious contradictions and are thus imaginary.
Even if we have established a representation to be real, Bolzano explains, it might still not actually refer to an object. Thus, he notes, we should proceed to check whether the representation presupposes an existing object or not. If it would refer to a non-existing object, like a number, Bolzano thinks, the consistency of the representation is enough to establish that it has an object. On the other hand, if it would refer to an actual object, the existence of such an object has to be established from conceptual truths or experience.
A somewhat more difficult problem is to check whether a representation refers to a singular object or several. Bolzano notes that if we can divide the representation into two representations contradicting one another and referring to objects (like triangles can be divided into those having a straight angle and those not), the answer is obvious. If we cannot find such a division, he adds, we can only with more or less confidence determine that it is singular. Sometimes, Bolzano points out, the very form reveals it (e.g., when we speak of a sum of all B), and even if it does not have such a form, but its objects do not exist, the determinations contained in the representation might be enough to determine the unique object (e.g. even numbers between 6 and 8). Bolzano thinks it a more difficult problem, if we have a pure concept, the objects of which should be actual: if the uniqueness is not determined by form, Bolzano is certain only of the notion of God as referring to a single object; of the other concepts of this type, if experience has shown that they refer to at least one object, he thinks it probable that they refer to more than one.
Because science should be as exact as possible, Bolzano considers it an important problem, whether a representation could be simplified by removing unnecessary determinations. In order to do this, he explains, we need to first analyse the parts of the representation and check if it has objects or not: if it has objects, we should try dropping its determinations one by one and seeing if it still refers to all the same objects; if it does not have objects, we should vary its constituents and thus turn it into a representation with objects and then do the dropping of determinations.
Moving on to cases with several representations, the obvious problem is what relations their extensions have. Bolzano goes into great detail to discuss these relations, but ultimately they all boil down to the question whether some combination of these representations or their negations refer to objects: for instance, representations A and B are compatible, if representation “something that is both A and B” has objects.
A somewhat different problem is that of finding similarities between objects. Bolzano notes that in order that we may give an answer other than trivial one that they are all things, we must have some representations of them. The problem then becomes analysing the representations and finding common characteristics shared by all of them. Bolzano actually gives not that much instructions on how to do this, but suggests that we should practice this kind of problem by thinking and comparing different kinds of things.
A natural contrast to the previous problem is, of course, that of finding differences between objects, in other words, characteristics that belong to one object, but not to another. Again, Bolzano says, we need some representations for the objects, but this time such that they are mutually exclusive, so that the representation referring to one object does not refer to the other object. Again, Bolzano has not so much instructions for this problem, but he suggests analysing the representations, in the hope of finding some characteristic belonging to one representation and not to the other. He also points out that although a single characteristic might not be the point of difference, a number of them put together could be it.
A problem combined of the two previous is that of finding a characteristic shared by all given objects, but differentiating it from another group of objects or even from all other objects. As we might guess from the two previous problems, Bolzano notes that to even begin solving the problem, we need two sets of representations: in other words, to find out similarities within one group of objects, we need representations for each of the objects, and to differentiate this group from another group, we need another set of representations, the two sets being mutually exclusive. In fact, Bolzano points out that we can even weaken these conditions and assume only that we have one set of representations exclusive to one group objects and another set covering both groups, since representations required for the second group can be defined in terms of the common representations and of the negations of the representations in the first group.
Bolzano moves on tackle a group of problems that involve finding a representation with a certain relation to a given group of objects, whether it is a case of this representation covering at least these objects, covering these objects and also more objects, covering only a part of these objects, covering exactly these objects, covering less than the whole group, but more than a given part of it or even covering a part of these objects together with some other objects. Especially the more complex problems feel very contrived, although they do include the problem of dividing a group of objects into a number of mutually exclusive representations. As with the problems of finding similarities and differences, Bolzano states, we require some representations that cover these objects (and in some cases, excluding all other objects), and indeed, pretty much all of them hinge on finding some similarities between these given representations.
Bolzano moves on to suggest problems involving judgements or propositions. Just like with representations, the fundamental problem is to analyse the judgements into their constituents. From these constituents, he continues, we can determine most of the internal characteristics of a judgement. The only exception, Bolzano points out, is the question whether the judgement is analytical or synthetical, which we must test by varying the parts of the judgement and seeing whether it remains true or false.
Just like with representations, Bolzano says, one problem concerns the relation of different propositions. The peculiarity of his logic is that many of these relations are defined in terms of variable constituents of propositions. Thus. to see whether two propositions are compatible, we need to see whether we can choose such values of variables that make both of them true. In some cases, Bolzano points out, such tests would have to be done infinitely many times, for instance, in case of derivability, which is defined as the derivable proposition being true with all values that make its premisses true. Bolzano notes that establishing such derivability must then be done by formal considerations used in the doctrine of inferences or by means of concrete sciences.
An obvious problem involving propositions is to test whether they are true. Bolzano notes that a necessary preliminary for this task is to decide whether the proposition in question is purely conceptual or empirical. In the former case, he thinks, a true proposition would be either self-evident or based on some self-evident truths. In the latter case, Bolzano continues, only the falsity of the proposition can be established conceptually, that is, by showing that it contradicts some conceptual truth. If we want to confidently assert the truth of an empirical proposition, he notes, we must first show that its subject term refers to some object and then establish that there is no perception that contradicts the proposition and that many of our perceptions agree with it in the sense of being derivable from it. All of these questions, Bolzano states, can usually be answered only with probable answers, while in some cases we may even know that the question cannot be settled through mere empirical means.
Bolzano defines proof, firstly, as anything used for directing the attention of a thinker in order to generate in their mind a judgement that they did not assent to before or at least not with the same degree of confidence. In this sense, proofs can be even sensuous objects and changes, but Bolzano gives a more restricted definition of proofs, covering only sets of judgements or propositions that mediate our judgements. An obvious problem is to test how convincing a proof for a proposition is, especially from the standpoint of the thinker searching for truth (Bolzano saves the question of how convincing a proof is from the standpoint of a learner to another place). This problem boils down to two questions: can we assume all our premisses (even the implicit ones) with sufficient degree of confidence and is the supposed conclusion derivable from these propositions or from a part of them or do they at least provide for the conclusion a degree of probability over doubtfulness? Bolzano concentrates his attention especially on the second question, pointing out signs how to recognise traditional fallacies in proofs.
After these rather logical questions, Bolzano moves on to deal with more substantial problems involving grounds of propositions. In fact, these are so fundamental questions that they deserve a post of their own.
sunnuntai 8. marraskuuta 2026
keskiviikko 28. lokakuuta 2026
Bernard Bolzano: Study of science – Heuristics
The ultimate aim of Bolzano’s Wissenschaftslehre or logic is the division of the whole field of human knowledge in disciplines, thus, he thinks it important to investigate how truths are discovered. Then again, he adds, logic should concern only with such heuristic rules that relate to thinking and its general applicability, because these are often of use in making scholarly treatises, while rules about, for instance, the proper use of senses fall out of its scope. Bolzano divides these rules into general rules applicable to all science and particular rules applicable to many, but not all kinds of investigation: we shall follow him in this division and leave the particular rules to the next post.
Before going to the general rules, Bolzano investigates the notion of thinking or meditation (Nachdenken in German). Commonly, he says, thinking means any series of representations and judgements, but Bolzano does not find this a good definition, since representations and judgements are constantly going on in our minds, even when we are not actively meditating anything. Thus, Bolzano excludes from the realm of meditation any stream of representations and judgements that are generated with no help of our will, like impressions caused by external objects. Even when our will has an influence, it is not meditation, he adds, if the aim is something else than just generating a certain species of judgements, for instance, when we seek satisfying representations (even dreamers do such things).
Meditation requires, Bolzano explains, that we awaken in ourselves representations and judgements with a definite and conscious purpose that we finally arrive at a judgement which has a certain characteristic: for instance, when someone tries to convince themselves that a judgement is true or when someone calculates something. Furthermore, he adds, if we are speaking of a pure mediation, the means by which our will generates the representations and judgements must be of a certain kind, so that our thinking is not influenced by the perceptions of external objects. Still, Bolzano contends, even pure meditation cannot be completely removed from all physical interaction, because the effect of the will on our representations is always mediated by our body. Then again, he notes, pure meditation cannot use the body as means, since bodily observations are not meditation.
Bolzano finally comes to the conclusion that pure meditation can be conjoined with unconscious influence on the body and even on other bodies: think of a person in deep concentration, who is mumbling or tapping their foot unconsciously. Thus, he explains, in meditation we generate, with nothing more but this possible unconscious interaction with the external objects, representations and judgements that we require to arrive at a judgement we search for.
Logic, Bolzano explains, is especially interested in meditation that is primed for searching for truth: in other words, meditation in logic tries to find an answer to some question. He divides truth searching meditation into recalling, where the truth is already familiar to us, but not just present in our mind, and discovery, where the truth is new and unknown (just to us or even to all humans). discovery; it is indifferent whether the truth is unfamiliar just to us or also to all humans). Furthermore, he defines speculation as meditation, where the truth that is searched and the propositions that we observe for the sake of finding the truth are conceptual propositions.
The actual rules Bolzano suggests are, as he himself admits, nothing original. Thus, he says, we should choose what kind of truths we want to search, since we can know only a finite number of truths. Furthermore, Bolzano notes, some questions cannot even be answered, and if it can be done easily, we should first check this possibility. Indeed, he adds, often we must solve a set of preliminary questions, before we can answer the primary question.
As for the actual methods of discovery, Bolzano mentions first the direct method, where we move forward from truths we already know through inferences and – hopefully – arrive at the answer we are searching for. The other possibility is an indirect method, where we construct a proposition that might be the truth that we are looking for – a hypothesis – and test whether it really is true. Indirect method, Bolzano explains, has many subspecies, such as apagogical method, where we start from a negation of the assumed truth and try to derive absurdities from it, analytical method, where we start from a hypothesis and test its possible grounds, and induction, where we test a sufficient number of instances of the hypothesis and determine at least that it is probable. He also notes that often only one method is not enough, but e.g. direct method must be used in conjunction with some form of indirect method.
This would exhaust the scope of pure meditation, but Bolzano still includes some rules that are commonly useful in investigating problems. Thus, he notes the usefulness of experiences, both those that we hear from reliable witnesses and those we have made ourselves. Furthermore, Bolzano reminds the reader of the importance of testing occasionally even judgements we have earlier deemed to be true and of analysing their reliability, although he notes that we cannot really do the so-called Cartesian doubt, since testing always requires assuming some judgements as criteria. Bolzano also emphasises the role of attention in guiding our thinking, of the capacity to use signs to give our thoughts more stability, of mental images in making our thoughts more vivid and of logical analysis to help us make inferences from our judgements.
Before going to the general rules, Bolzano investigates the notion of thinking or meditation (Nachdenken in German). Commonly, he says, thinking means any series of representations and judgements, but Bolzano does not find this a good definition, since representations and judgements are constantly going on in our minds, even when we are not actively meditating anything. Thus, Bolzano excludes from the realm of meditation any stream of representations and judgements that are generated with no help of our will, like impressions caused by external objects. Even when our will has an influence, it is not meditation, he adds, if the aim is something else than just generating a certain species of judgements, for instance, when we seek satisfying representations (even dreamers do such things).
Meditation requires, Bolzano explains, that we awaken in ourselves representations and judgements with a definite and conscious purpose that we finally arrive at a judgement which has a certain characteristic: for instance, when someone tries to convince themselves that a judgement is true or when someone calculates something. Furthermore, he adds, if we are speaking of a pure mediation, the means by which our will generates the representations and judgements must be of a certain kind, so that our thinking is not influenced by the perceptions of external objects. Still, Bolzano contends, even pure meditation cannot be completely removed from all physical interaction, because the effect of the will on our representations is always mediated by our body. Then again, he notes, pure meditation cannot use the body as means, since bodily observations are not meditation.
Bolzano finally comes to the conclusion that pure meditation can be conjoined with unconscious influence on the body and even on other bodies: think of a person in deep concentration, who is mumbling or tapping their foot unconsciously. Thus, he explains, in meditation we generate, with nothing more but this possible unconscious interaction with the external objects, representations and judgements that we require to arrive at a judgement we search for.
Logic, Bolzano explains, is especially interested in meditation that is primed for searching for truth: in other words, meditation in logic tries to find an answer to some question. He divides truth searching meditation into recalling, where the truth is already familiar to us, but not just present in our mind, and discovery, where the truth is new and unknown (just to us or even to all humans). discovery; it is indifferent whether the truth is unfamiliar just to us or also to all humans). Furthermore, he defines speculation as meditation, where the truth that is searched and the propositions that we observe for the sake of finding the truth are conceptual propositions.
The actual rules Bolzano suggests are, as he himself admits, nothing original. Thus, he says, we should choose what kind of truths we want to search, since we can know only a finite number of truths. Furthermore, Bolzano notes, some questions cannot even be answered, and if it can be done easily, we should first check this possibility. Indeed, he adds, often we must solve a set of preliminary questions, before we can answer the primary question.
As for the actual methods of discovery, Bolzano mentions first the direct method, where we move forward from truths we already know through inferences and – hopefully – arrive at the answer we are searching for. The other possibility is an indirect method, where we construct a proposition that might be the truth that we are looking for – a hypothesis – and test whether it really is true. Indirect method, Bolzano explains, has many subspecies, such as apagogical method, where we start from a negation of the assumed truth and try to derive absurdities from it, analytical method, where we start from a hypothesis and test its possible grounds, and induction, where we test a sufficient number of instances of the hypothesis and determine at least that it is probable. He also notes that often only one method is not enough, but e.g. direct method must be used in conjunction with some form of indirect method.
This would exhaust the scope of pure meditation, but Bolzano still includes some rules that are commonly useful in investigating problems. Thus, he notes the usefulness of experiences, both those that we hear from reliable witnesses and those we have made ourselves. Furthermore, Bolzano reminds the reader of the importance of testing occasionally even judgements we have earlier deemed to be true and of analysing their reliability, although he notes that we cannot really do the so-called Cartesian doubt, since testing always requires assuming some judgements as criteria. Bolzano also emphasises the role of attention in guiding our thinking, of the capacity to use signs to give our thoughts more stability, of mental images in making our thoughts more vivid and of logical analysis to help us make inferences from our judgements.
sunnuntai 18. lokakuuta 2026
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.
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.
perjantai 16. lokakuuta 2026
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.
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.
lauantai 10. lokakuuta 2026
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.
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.
torstai 1. lokakuuta 2026
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.
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.
sunnuntai 23. elokuuta 2026
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.
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.
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