Bernard Bolzano: Study of science – Set theory 101

After dealing with some characteristics of individual representations, Bolzano moves next to consider relations of representations toward one another. He begins by insisting that there are no two completely identical representations in themselves. True, Bolzano admits, there can be several completely identical subjective representations, in the sense that different people at different times can think of same content, for instance, with differing degrees of clarity, but this already presupposes that the content in question – the representation in itself – is one and the same. Furthermore, he adds, there may still be similar representations in the sense that these representations share so many commonalities that they can be easily confused.

Although two representations cannot be completely identical, Bolzano continues, they can be identical from a particular viewpoint. For instance, they can have the same constituent parts, but just united in different manners, like permission to not speak and not permission to speak. Then again, Bolzano notes, there can be no two representations with the same constituents combined in the same manner. Of course, he adds, some representations do not share even a single constituent part, while others share some, but not all and might even be combined in a similar manner and could thus be called affine representations. Bolzano warns not to confuse affine and similar representations: for instance, A and not-A are affine, because they share the constituent A, but they certainly won’t be confused, while ethically good and useful for society share no constituents, but are easily confused and thus similar.

In addition to constituent parts, Bolzano states, we can consider the relation of representations according to their extension. We can firstly compare just how many objects there are in the extension: in later set theory this is usually called the cardinality of a set. Bolzano makes the obvious remark that two representations may be equally large or one of them might be larger than the other. His examples are interesting: the representations of human soul and human body are equal in this sense, because although no soul is a body, for every soul there is a body and vice versa. Then again, the representations of a human finger and a human hand are unequal, because for most of the hands there are five fingers. With representations of infinite extensions, Bolzano adds, we can sometimes not say which is the bigger one, like in the case of balls and tetraedras. He also notes that these relations can be extended to collections of representations – actually he says this of the most of the relations later, so I won’t be repeating this comment again.

Extending the viewpoint from the number of objects to objects themselves, Bolzano notes that representations may be either compatible or incompatible, that is, have one or some common objects or not. He notes the obvious truth that if we have several compatible representations, then a smaller selection of these representations will also be compatible, and the not so obvious truth that if two characteristics are compatible, then corresponding concrete representations are also, but not the other way around. Bolzano explains the latter truth with an example: prudence and cautiousness are compatible, because it may be prudent to be cautious, and therefore also prudent and cautious persons are compatible (prudent persons can be cautious), but although pious and learned persons are compatible (that is, a pious person can be learned), piety and learnedness are not (being pious is not being learned). Bolzano also makes a suggestion how to present the relation of compatibility geometrically – an endeavour that had been in the air for a while – and since he does this for the rest of the chapter for other relations, I won’t be noticing it anymore.

Next relation Bolzano mentions is that of one extension or representation comprehending another, where all the objects belonging to the comprehended extension or representation belong also to the comprehending one. Obviously, he points out, the relation of comprehension can go both ways and then we can speak of equivalent representations that have the exact same objects. Bolzano notes that equivalent representations can also share the same constituents, like virtuous who is prudent and prudent who is virtuous. This doesn’t work, he underlines, if both representations are simple (then we would be speaking only of one representation), but it can work if one is simple and the other is composed – just think of A and not-not-A and A that is A. He also makes the remark that representations including equivalent representations in the same manner are not always equivalent, for instance, root of 24  and root of 42.

Obviously the relation of comprehension might also be one-sided, Bolzano points out, and then we speak of subordination. He also notes that a subordinating representation always has more objects than the subordinated one, but a representation with more objects of course does not necessarily subordinate a representation with less objects.

If two representations have common objects, but neither comprehends the other, Bolzano calls them intertwined, chained or disparate. He also points out that although two representations are intertwined with a third one, we cannot say anything about their relation toward one another: if the two representations in question are not chained to one another, Bolzano calls them mediately chained. A series of representations, then, where a representation is chained to the next and the previous in the series, but to no others, Bolzano names a chain, while an otherwise similar arrangement, but such where the first and last member are also chained to one another, he calls a closed chain. Then again, if in a collection of representations any arbitrary pair is chained, Bolzano says them to be intertwined in all sides.

Bolzano points out that there clearly cannot be any representation with such a number of objects that it exceeds every other representation, because there are always infinitely many representations with the same number of objects. Still, he thinks that we can find a representation that has so many objects that none can have more. Such a representation would also have no representations subordinating it – it would be unconditionally highest in a sense – and indeed, a concept of something or object in general is a clear example of such a representation. There are others, Bolzano adds, including not-nothing and something that is self-identical, but they all share the same extension.Similarly, he continues, there are infinitely many representations that have the lowest number of objects and that subordinate no other representations, namely, all singular representations – indeed, there are infinitely many extensions such a representation can have, because there are, according to Bolzano, infinitely many objects. He also answers positively the question, whether there are general representations that subordinate no other general representations, because we have representations with only two objects, such as sons of Isaac, numbers between 3 and 6 and roots of x2 – 1 = 0.

Some representations can be put in order according to the number of objects they have, Bolzano continues, and in some cases we can find between two such representations a third that lies in the middle of them. Then again, he adds, two such ordered representations can also follow one another immediately in the sense that they have no representation between them (just think of representations with two and three objects). Similar relations can be defined also with the relation of subordination, Bolzano points out and moves on to ponder the interesting case of representations with infinite extensions. There are, he says, representations with infinite extensions and in such a relation of subordination that there is nothing in between them – his example is a representation of created substance and a representation of substance in general, both of which have infinite extension, but where the latter contains one substance more than the first one (God). Then again, Bolzano adds, there are also such representations in a relation of subordination with infinite extensions that they have infinitely many different representations between them: his examples are the representation of a right angle and a representation of an angle in general.

More difficult conundrums are whether all subordinated, but not subordinating general representations have exactly two objects and whether extensions of two representations following each other immediately always differ by just one object. Bolzano notes that the answer to both questions is affirmative, if there is a common concept for every arbitrary set of objects. Of course, he points out, every set of objects is comprehended at least by the representation of an object in general, but it is not as obvious whether there’s such a representation that comprehends only this set. Bolzano argues that the answer is affirmative if we can compose the representation from parts representing the individual objects: each individual must have their own representation, because it a subject of many truths, such as that it is individual, and then the distributive representation of the collection of these individual representations does the trick (of course, Bolzano adds, this objective representation might be infinitely long and thus not subjectively in our reach). The question seems more doubtful, he continues, if the desired representation should not contain individual representations nor intuitions, but be a pure concept: is there a common concept for just triangle, proposition and virtue? Of course, we have just discussed such a representation, so we can have a representation of such a concept, which could then be called a symbolic general concept, as opposed to actual general concept. Even if it would be impossible to find an actual general concept comprehending only a certain set of objects, Bolzano concludes, we might be able to find some concept lower than object in general that comprehends them and there might be lowest of them in the sense that there is no lower.

Representations with a finite extension, Bolzano notes, can be measured by singular representations. Then again, he adds, all representations with infinite extensions cannot be measured with any finite set of measures. As an example Bolzano considers sets of numbers with forms n, n2, n4, n8, n16…, that is, natural numbers, squares of natural numbers etc. All of these sets are clearly infinite, and the smaller the exponent, the more there are objects in the set. Bolzano argues that the relation of the extensions in these series must be infinite – suppose N is “the number of all natural numbers”, then nm = N/m, and with m < s, nm : ns = N/m : N/s = 1 : N/(s – m), which exceeds any finite relation.

If a number of representations is incompatible, Bolzano notes, this means only that there isn’t a object comprehended in all representations, but there may be an object common to some of these representations. Thus, we can also define an all-sided incompatibility or exclusion, where a number of representations and also any pair of them is incompatible. With two representations, Bolzano points out, incompatibility is equivalent to exclusion. Exclusion does not mean, he continues, that an object not part of one representation would be part of the other, but this is possible – then we are speaking of contradictory representations. Contrary representations, on the other hand, Bolzano defines as such that exclude one another, but are not contradictories. He also remarks that every representation A has infinitely many contradictories, but these are all equivalent to not-A, while there are infinitely many non-equivalent contraries for any representation.

Bolzano takes representations to be coordinated under some different representation X, if they exclude one another and are subordinated under X. Because all representations that do not include all objects are subordinated in some representation, all representations excluding one another are in some sense coordinated. If the coordinated representations also exhaust the extension of X in the sense that there is no object of X that would not be an object of some of the coordinated representations, Bolzano calls them complementing or integrating parts of the extension of X. He says that representations complement one another unconditionally, if they exhaust the extension of the representation of something in general, and points out that contradictory representations complement one another unconditionally.

Sometimes with coordinated objects, Bolzano notes, we can order the representations in the sense that some share characteristics more than others, like a youth is closer to an adult than a child is. This sense of ordering differs from that made with the number of objects under representation and especially never coincides with the ordering by subordination (excluding representations cannot subordinate one another). Bolzano notes that there are representations that have infinitely many representations between them (take the representations of two magnitudes of angles), but also representations with no representations between them (e.g. straight and crooked angles).

Bolzano proves a number of rather simple theorems, which would nowadays be probably simple exercises for students of set theory. Just to give an example, here’s a few of them: if two representations are compatible, so are all representations subordinating them; if representations A and B are chained, then X that is subordinated to A does not subordinate B, but is either subordinated to B or chained with it or excludes it; two merely contrary representations can be excluded by some representation; if representations A and not-B or B and not-A are compatible, A and B are either chained or exclude one another.

Bolzano goes on to define kind (or as he points out, species, genus or class, depending on the context) as a collection or sum of all objects represented by a general representation. He notes that we can define the same relations to kinds as we have done with representations in general, but there’s a clear difference – kind is always a singular representation (it represents a single sum of objects).

Bolzano has an interesting and complicated take on the notion of opposition, which he takes to be primarily a representation between objects, although it can be extended to representations exclusive only to these objects. What he means by opposition is a relation, where it is possible to compose from a representation applicable exclusively to one object through mere addition of some pure concepts a representation applicable exclusively to the other object and of such a characteristic that as soon as we change the representation applicable exclusively to the first object with a representation applicable exclusively to the second object, the new representation applies exclusively to the first object. This definition is a mouthful, and Bolzano tries to explain it through an example, but manages to pick so complicated one that it almost requires explanation itself: essentially, he takes lines OR and OS moving to opposite directions from the same point O and notes that a sum of OR and OS (and indeed, a sum of any OM and ON, where M lies in OR and N lies in OS) equals the line RS (or MN), which is not true if we take any other line OP to some other direction (essentially because R, S and O lie on the same line, but O, P and S or R form a triangle), and then tries to state all of this in the terms of his definition, with not that much clarity gained. What is important is that opposed representations are singular representations and that they exclude one another and are contraries, but only a special kind of them.

Bolzano concludes the chapter by noting that all the definitions have assumed that the representations in question have objects. Then again, he points out, we seem to be able to say that, for instance, “mountain that is golden” and “gold that is shaped into a mountain” are equivalent representations, that the representations of five-sided and seven-sided regular polyhedra exclude one another and that the representation of humans with nothing good in them is subordinated to the representation of entities with nothing good in them. Bolzano suggests extending the definition in the same way as he did when speaking of superfluous parts of representations, by considering what happens when we change some constituents to make the representations non-empty: for example, by changing the part “mountain” in the first pair to something else (say, a ring), we can easily see that the resulting representations are equivalent, making the original ones also.

Bernard Bolzano: Study of science – Collections of representations

Bolzano's journey through characteristics of representations has reached the concept of characteristic itself. He really has no definition of the concept, except that a characteristic is something that an object has, no matter how temporary it is. This having, Bolzano thinks, has then no better explanation than being a relation of a thing to its characteristic (and not, say, possessing another thing, like when we say that I have money). It is then completely arbitrary, which of the two – characteristic or having – is taken as primary notion and which as defined, although Bolzano prefers having as the primary one. He also notes that spatial and temporal determinations are to not to be taken as characteristics, since in propositions they more conveniently characterise the subject position, while characteristics always fall to the predicate position.

Bolzano divides characteristics into internal and external or properties and relations. Of these, he insists, relation between things is actually a characteristic of the collection or the whole composed of the related things, for instance, the relation of three points forming an equilateral triangle is the characteristic of the collection of the three points, not of the points themselves, although we can say that the points have the (external) characteristic of being a part in a collection with such characteristic. Bolzano adds to his definition of relation that both the parts and the characteristic must be variable, because otherwise we would have to say that the primeness of number 13 is a relation, because it refers to the whole system of numbers.

Having defined relations and external characteristics, Bolzano can easily explain internal characteristics or properties as characteristics that are not external. From these definitions it easily follows that a relation between things is a property of their collection. Furthermore, Bolzano adds, an internal characteristic of even a simple thing can be seen as a relation, that is, between the thing and its characteristic (the having described earlier). He then briefly defines similarity or equality as a reciprocal relation, where objects have a same part in a characteristic belonging to the collection they form. A dissimilar or unilateral relation is then a relation that is not reciprocal.

Bolzano notes that with complex objects we can distinguish between their matter – the parts from which they are constituted – and their form – the manner in which the parts are combined. He notes that similar definitions can then be applied to representations of things, so that representation of their matter means representations showing constituent characteristics of the things and representation of their form means type of combination for these parts. Bolzano notes that the distinction between matter and form can be related to the distinction of internal and external characteristics. Matter of an object, he says, can always be determined as consisting just of internal characteristics, when the parts are thought as simple representations, but they can also be seen as external characteristics, if described by representations referring to other things: for instance, “not-human” consists of “not” and “human”, but also of “not” and “the most perfect type of living beings on Earth”. Then again, Bolzano thinks, some types of combinations in form can be represented only through relations.

We have already mentioned a couple of times the notion of a collection (Inbegriff). By this, Bolzano means simply a representation composed of other representations A, B, C…, where the order of these collected representations or parts of the collection is not yet considered: it might be relevant or irrelevant, depending on the type of collection. He also notes that we can either discuss this collection collectively – the whole formed of these parts – or distributively – each part of this collection (for instance, all players are a team, but each player is not). Bolzano also points out that sometimes we leave implicit whether the collection includes parts beyond those explicitly mentioned.

We just mentioned that the order of the parts might be irrelevant to the collection. In case it is explicitly so, Bolzano speaks of sets (Menge). Now, both in collections in general and in sets, the parts of parts are not usually parts of the collection or set – a set of people does not include their hands. Then again, Bolzano notes, there is an important type of sets – he calls them sums – where parts of parts are parts: lines are of this kind.

If the order of the parts does not matter in a set, a series is a collection where it does. Bolzano defines series through a law or rule, so that for any part or member M of the series can be found another member N, where either N is determined by the law from M or conversely. The N and M are then said to follow immediately one another, the determined member being later and the determining member earlier. Bolzano also defines internal members of a series as such that have both earlier and later members, while for external or limit members, either of them cannot be found. Limit members include the first or starting member that has no earlier member and the final or end member that has no later member.

Another important concept Bolzano mentions is that of unity or unit (Einheit can be translated in both ways), which is simply something that has a certain characteristic A (in the concrete sense) or then a property that makes something a unit of type A (in the abstract sense). Plurality in the concrete sense is then, for him, a collection of concrete units of some type A, while plurality in the abstract sense is the property, by which something is a concrete plurality. Bolzano notes that we could go on defining twos and threes and so forth by determining that the plurality is to have a unit and then a unit etc., but turns then to define a concrete whole or all as a collection that has each object belonging to some representation A and nothing else; similarly allness in abstract sense is a property that makes a concrete whole into such. He also points out that when the expression “all A” is used to mean A in general, it is used distributively, but here it is used collectively.

Bolzano’s account of the representations of set and series is quite close to later logicist ideas of defining basic mathematical concepts, so it is no surprise that he next tackles magnitudes. Types of magnitudes, he says, are characterised by the property that no matter which two of them are taken, they are either equal or have the relation of one of them being greater, that is, a whole with a part equal to the other. With this definition, Bolzano notes, pluralities, wholes and units can be seen as magnitudes.

Bolzano has something to say even about the notions of finite and infinite. A plurality of finite magnitude, he defines, is any plurality of type A that appears as a member in a series, where two of As is the first member and next member is reached from the previous by adding a new A. Plurality of infinite magnitude, on the other hand, is a plurality of type A, where each finite plurality of A appears only as a part. Furthermore, Bolzano defines number as a member of a series, the first member of which is a unit of any type A and each next member is a sum of previous with a new unit. Every finite plurality, he points out, is a number, while infinite pluralities are innumerable.

From these novel considerations, Bolzano moves on to more traditional logical notions, first of which is what he calls an exceptive representation, that is, a collection from which certain objects are excluded, either by individually naming the excluded objects or through general characteristic, for instance, by discussing a collection of all A that do not have the characteristic b, although Bolzano is not certain whether to really call this latter an exceptive representation. The second traditional notion Bolzano discusses is the concept of negation or “no”, which he considers undefinable. He defines all representations having “no” as constituent to be negative, although since two negatives cancel one another, he delineates a stricter sense of negative representations, which do not have an even series of negations – other representations are then affirmative.

Bolzano divides quite traditionally negative representations into two kinds. First of these he calls purely or completely negative representations. These deny a certain representation A without requiring any other representation in its stead – not even the very indeterminate representation of something. Such purely negative representations include, Bolzano notes, at least the concept of nothing. The other kind is, then, those of partially negative representations. In these, Bolzano explains, negation is only one of its constituents, like in the representation of A that is not B.

Bolzano closes off this section with the notion of symbolic representation. Quite appropriately, this notion is defined by the form “representation that has a characteristic b”. Thus, all the various concepts described in this section are of such a kind. Furthermore, all the concepts described are what Bolzano calls real or objective symbolic representations, since there have been representations having the described characteristics. He notes that we could also delineate the notion of mere symbolic representation, which are completely imaginary in the sense of having no objects: a notion of a representation that is also a judgement would be of such kind.

Bernard Bolzano: Study of science – Intuitions and concepts

Still going on with the study of the characteristics of representations, Bolzano notes that most of the representations relate to an object and underlines that this relating is different from a proposition, where a representation or a characteristic is related to an object – this is asserting something of the object. Representations with objects or objective representations, he continues, have an extension or a sphere of objects to which they relate to, and these objects and any incomplete collection of them is said to belong to this extension. Bolzano notes that the set of these objects has a magnitude, which can be infinite. The existence and magnitude of the extension can depend on contingent matters, like the representation of the sons of Chingish Khan, but Bolzano thinks it is permanent – even a representation like “all the people living now”, since the phrase “now” fixes this particular representation to this particular point in time. Some representations, he continues , have no objects, such as nothing, round square of golden mountain. Some representations, which Bolzano calls singular representations, relate only to a single object (for instance, universe), while others relate to a finite number of objects (e.g. all the propositions in Euclid’s book of geometry).

Bolzano points out that a representation can have more parts than required for referring to its objects, for instance, a representation of a triangle that is both equiangular and equilateral. An interesting case he considers are the representations of the form “this A”: the A is superfluous, because “this” already refers to the object in question perfectly and the determining A is merely used to simplify the identification of “this” for humans. Although the notion of superfluous parts has been first defined only for representations with objects, Bolzano finds a clever way to extend it to representations with no objects, like round and angular quadrangle: we can say that some parts in such a representation are superfluous if always when we change some other parts of it and the result refers to an object, the parts in question are superfluous in the original sense.

Bolzano notes that complex representations can be contradictory in the sense that they are of the form “A that is also P”, where A implies M and P implies not-M (e.g. triangle with four sides). Such contradictory representations, he continues, have sometimes been called empty, impossible or imaginary, but these names have some room for confusion. Firstly, Bolzano explains, emptiness in this case does not refer to the content of the representation, but to its empty extension – it has no objects. Then again, he notes that there are in this sense empty representations that are not contradictory (just think of a golden mountain).

Secondly, representations as such cannot be necessary, actual or possible, since they do not exist, and not even impossible in the sense that they would otherwise be able to exist, but something hinders their existence. Such modal terms, Bolzano thinks, concern our capacity to find objects for the representations, but again, some non-contradictory representations are impossible in this sense.

Finally, he concludes, representations in themselves are never just imaginations, but we can say that contradictory representations are imaginary, because their objects can never be found outside our thoughts in existence. All other representations can then be called real, but this does not mean they have objects, because some non-contradictory representations do not have them. In any case, only complex representations can be imaginary, which implies that simple representations are always real. Furthermore, Bolzano states, imaginary representations can be parts of real representations, like in the case of a representation of a mathematician who first thought of the square root of –1.

Bolzano presents the intriguing question whether simple representations can ever be singular. At first sight, it seems apparent that limiting a representation down to one individual requires always adding new determinations. Yet, Bolzano answers, we do have simple and singular subjective representations, when we attend to immediate sensations produced by external objects for instance, for instance, when seeing a rose, we say “this is red”, “this” is a simple, singular representation. Corresponding to these subjective representations, there must be corresponding simple and individual representations in themselves, which Bolzano calls intuitions.

Another important subtype of representations, beside intuitions, Bolzano thinks, is formed by concepts. Concepts, he defines, are representations that are not intuitions and have no intuition as their part: examples include something, which refers to an infinity of objects and has no intuition as its part, because it has no parts, and representation of God, which does refer to one object, but none of its parts is an intuition. Clearly we need a third category of mixed representation, which Bolzano explains as complex representations with intuitions as their parts, for instance, rose that spreads this smell. Depending on the prominent part of a mixed representation, it can be either a mixed intuition (e.g. this that is a colour) or a mixed concept (truths contained in this book).

Examples of intuitions we know of, Bolzano notes, belong to the realm of actuality as certain changes in our soul. He adds that these are probably the only intuitions we humans can know, since the only simple representations of an individual object we are aware of seem to be those caused by something actual. We cannot intuit all actual objects, Bolzano states, but for each actual or non-actual object that we have an exclusive representation of we can form a representation with intuition as a part, by relating the original intuition to other things that we can intuit (e.g. seed of this flower). According to Bolzano, with few exceptions, such as God and the universe, we cannot construct from pure concepts a representation that would pick out exclusively one object. Even if we managed to pick a set of characteristics that seem to individuate a finite substance, in the infinity of things there might be lurking another one with the very same set.

Bolzano points out that we can never have the same intuition twice, because the object of an intuition is a certain change outside or inside us causing this intuition and this exact thing cannot occur twice, although there can be representations similar to it. In the same manner, no one can have the same intuitions as I do and intuitions cannot be transmitted, unlike pure concepts. True, Bolzano admits, we do say that we transmit intuitions, but actually we just tell others of their characteristics, for example, by indicating that they are caused by a certain external object.

If interactions with a certain object are frequent, we may even give it a name, which is then a mixed representation (“object that has been the cause of such and such intuitions”). This is true, Bolzano notes, even of names of long gone objects (“person who lived in ancient Athens and was called Socrates”). In any case, he underlines, names are reserved only for some objects, and usually we just use words like “now”, “then”, “here”, “this” etc. and sometimes add some species for explanation (e.g. “this rose”). The last, which is a designation of a mixed representation, Bolzano muses, could be used in scientific treatises as indicating pure intuitions that otherwise are difficult to express.

Bolzano notes that every language has words that are used sometimes for pure concepts, sometimes for mixed representations, especially when speaking of natural objects. He gives as an example the concept of a human being. By human, Bolzano says, we can mean any reasoning creature (pure concept), whether they live on the Moon or Earth, but often we restrict the concept of humans to reasoning creatures on Earth (mixed representation, due to the reference on Earth). We can also find cases with a converse relation, Bolzano thinks and explains that by gold we have originally meant substance connected to certain intuitions, but we can purify this into a pure concept by describing it as substance causing certain changes in humans etc.

Bolzano considers the question whether the names he has picked for intuitions and concepts are suitable. His answer is that intuitions usually mean singular subjective representations, and that he has just extended the concept to objective representations. Then again, Bolzano notes that the term “concept” has often referred to objective representations, except that concepts have always been taken as either consisting of multiple parts or as having an extension with multiple objects. All in all, Bolzano is not very satisfied with the way intuitions and concepts have been treated by his predecessors, who generally recognised them as subjective representations.

All intuitions are simple and singular, Bolzano notes, but there’s more variety with concepts. Evidently there are complex concepts, but there must be also simple concepts, because all complex representations need at least the help of the simple representation “and”, which isn’t an intuition (other similar notions include such terms as have, should etc.) Some concepts have an infinite extension (e.g. created substance), some a finite extension (e.g. one of the cardinal virtues), some only a single object (e.g. community of all morally good entities) and some none (e.g. means for undoing past).

Bolzano considers also a question, important from the time of Kant, whether time and space are intuitions or concepts. Bolzano notes that there are actually many temporal and spatial representations, and for some the question has an obvious answer: general notions like moment in general and point in general cannot be intuitions, because they have many objects – thus, they must be concepts.

This still leaves unanswered the case of temporal and spatial representations with only one object, such as the whole infinite time or space and this particular point in time or space. Bolzano argues that none of these can be an intuition, because they do not have any existing objects: space and time have no effect on anything, and if they seem to, like when we say that time heals wounds, we actually mean that some object has this effect. Indeed, he concludes, they must be pure intuitions, since none of their parts is an intuition.

Although not really a task of logic, Bolzano quickly defines what space and time are. All existing things, except perhaps God, he says, are at a certain time. Thus, we could define time as the determination in something actual that must appear as a condition, in order that we can truly ascribe to it any characteristics. A similar definition can be given of space or a sum of all places, Bolzano suggests. All actual objects act and are acted upon, and the characteristics of these interactions depend on the forces of the objects and their places. Bolzano can then define the place of an actual thing as a determination that must be added to the forces of the thing, in order to comprehend the changes that they generate in one another.

Bernard Bolzano: Study of science – Parts of representations

Bolzano begins his study of the characteristics of representations with a preliminary plan: he shall begin with characteristics common to all representations, then move to remarkable types of representations, first to general types and then to types specified by parts of representations. Starting with the characteristics common to all representations, he notes a couple of features that have surfaced in his discussions earlier. Firstly, Bolzano notes, representations in themselves do not actually exist: thoughts of representations do, but representations in themselves are not thoughts, but their content. Secondly, no representation in itself is either true or false. Bolzano admits that sometimes representations are called true or false, but thinks these cases are clearly derivative. For instance, sometimes we say that a representation is true, if we apply it to some object and want to indicate that this object fits with that representation, which is a proposition, the truth of which is actually in question. Furthermore, sometimes we call a representation true, if we want to suggest that there are in general objects that fit representation, and similarly another representation – say, a square circle – false, if no object fits that representation: Bolzano underlines that this is not the original sense of true and false.

Bolzano points out that many representations are constituted from parts that are also representations, just like our thoughts of these representations consist of parts. He makes the observation that in this case the representation is not a mere sum of its parts, because the way these parts are connected also affects it: for instance, a poor brother of a rich father is different from a poor father of a rich brother. Although Bolzano’s discussion about the parts of representations seems to hinge on what words are used to describe the representations, he notes that sometimes the verbal expression of a representation is deceptive. For example, we have a habit of making implicit restrictions of the words we use, for instance, when we speak of animals and mean specifically animals. For this reason we have often a need to use a term like “animal in general”, which does not mean that something would be added to the representation of animal, but only that we want to remove all possible implicit restrictions. Similarly, Bolzano thinks, words like “every” or “the” do not really add anything to a representation: “humans are mammals”, “every human is a mammal” and “the human is a mammal” all mean the same.

Bolzano discusses further the features of the parts of representations, such as the fact that the parts can of course have their own parts. This is especially clear in a case where the parts are not representations, but propositions, like the representation of a creature which lives on Earth – “which lives on Earth” is a proposition with its own parts. Bolzano points out that in some cases the parts of representations connect one another immediately like not and being in not-being, while in other cases, the connection of parts requires mediation of other parts, like in the case of the creature which lives on Earth, where the part “which” connects a representation and a proposition into a whole. Bolzano also notes that in some cases, like the now many times mentioned creature which lives on Earth, the parts have an intrinsic order, with “creature” being in a sense earlier than or prior to the other parts – he is adamant that this order is not temporal, since representations do not exist. He also emphasises that not all parts of a representation need to have such an order, when a representation could be called just a sum of its parts, like “red and round”, which is the same representation as “round and red”.

We have already met representations with the form “A that is x”. Bolzano notes the language sometimes hides deceptively this form: the memory of Julius Scaliger means the memory which Julius Scaliger had. Similarly, constructions like righteous person can be easily turned into this form (a person that is righteous), but Bolzano suggests being cautious at times: painted fish is not a fish which is painted, but a painting, which represents a fish. Furthermore, he notes, an expression of the form “this A” can in some contexts mean “this which is A”. but in other contexts just the A (these assertions are the assertions, which we are currently engaged with.

Another common form Bolzano mentions is that of “something that is A”. He calls such a representation concrete, while the A is then an abstract representation of characteristic. Bolzano also notes that some representations are neither abstract nor concrete, like something, nothing, this A and Socrates. Bolzano also warns the reader that verbal expressions do not always reveal what representation is concrete, for instance, when we speak of animals, we actually mean something that has animality. Sometimes the same word can in some contexts refer even to concrete and in others to abstract representation, for instance, virtue usually means characteristics, but sometimes something that is virtuous.

Bolzano moves on to define simple representations as such that have no parts. It is at first problematic, whether any representation is simple, but Bolzano is convinced that finitely complex representations must have simple representations as parts. He also suggests that infinite wholes, like space, must also have simple parts, such as points in space. While representations thus have a lower limit, Bolzano thinks they cannot have any upper limit: to any representation could be added more and more things, like new characteristics (is the creature that lives on Earth a plant or an animal?) or even propositions (e.g. the truth that earthly creatures exist).

Bolzano notes that philosophers have sometimes insisted that representations should correspond to their objects. He has difficulties in even understanding what this correspondence means. It cannot mean that the representation should have the same number of parts as its object, because there are representations without any objects. Even if we restricted the notion of correspondence to representations with objects, we would still face the difficulty that some representations have propositions as their parts. Bolzano suggests the emendation that in these cases it would be parts of the proposition that would have to also correspond to their object. Yet, he quickly discards this solution, because in a representation like “land that has no mountains”, the concept of mountains should not correspond to anything in the land, which should be a land with no mountains.

Another possible interpretation of the notion of correspondence is that the parts of representation should correspond to the characteristics of the corresponding object. Yet, Bolzano points out, there are parts of representations that are not characteristics of its object, such as the already familiar something and which. Furthermore, he adds, there are characteristics of objects that are not part of the corresponding representation, for instance, because many objects simply have characteristics that we are not aware of. Indeed, Bolzano adds, in some cases there can be two different representations that have the same object, but different content, such as the notions of equilateral and equiangular triangle. There are even, he thinks, objects with an infinite number of characteristics (say, an irrational number, expressed as an infinite sum), although we certainly cannot think of all of them.

Bolzano’s latest arguments seem suspect, because they concern not representations in themselves, but only our thoughts of representations. Yet, he emphasises, the same is true even of representations in themselves, since although e.g. all equilateral triangles are also equiangular, equiangularity is not a part of the representation of an equilateral triangle: it is certainly not a part of the representation of a triangle – since not all triangles are equiangular – and not even a part of the representation of equilateral – this is obvious, once you think of an equilateral quadrangle or parallelogram that is not equiangular.

Bolzano goes on to give further arguments for his position, now from a different angle. Simplicity, he says, is a characteristic of a simple representation, but it is clearly not part of this representation, since simple representations should have no parts. Furthermore, Bolzano adds, simple representations usually represent something and thus have the characteristic of being something, but this being something is again not their part for the same reason that they are simple and thus have no parts. Finally, if we have several representations, then the two representations differ, but this difference is not part of either representation.

Bolzano is again a pioneer on the issue, as most earlier logicians, in Bolzano’s opinion, apparently had very different ideas on what composition of representations means. Most of them had ascribed themselves to the opinion that parts of a representation form like a sum, forgetting parts like “something”, “which” and “is”. Furthermore they had often confused parts of a representation, on the one hand, with parts of its object, and on the other hand, with characteristics of representations that are not its parts. Indeed, Bolzano observes, these differences are what underlies the Kantian notions of analytical and synthetical judgement – analytical judgement merely points out some constituent parts of the subject representation, while synthetical judgement moves on from this to further characteristics of the subject – although Kant and his followers did not properly understand the difference.

Bernard Bolzano: Study of science – What are representations?

If the first part of Bolzano’s logic – the fundamental science – was rather an outlier in a logic book, the next part – the elementary science – seems at first sight much more traditional. Thus, Bolzano notes that since logic deals with the question of dividing science into groups of truths that are presented in scientific treatises, we must investigate the characteristics of truths. Before we can get into truths or true propositions, he continues, we must first deal with characteristics of propositions in general, and before that, we must deal with characteristics of parts of propositions, which he calls somewhat unexpectedly representations. Furthermore, Bolzano adds, we should also deal with a useful subset of propositions and truths that indicate that certain propositions can be derived from others – these are deductions. Thus, Bolzano’s elementary science, so named because it deals with the ultimate elements of scientific treatises, deals almost traditionally with representations, propositions, truths and deductions, with some of the nomenclature and the inclusion of truths the seemingly only peculiarities.

Starting with representations (Vorstellung), Bolzano notes that they have internal characteristics that require no comparison to anything else and external characteristics that arise from relations to other things. The latter he then divides into characteristics in relation to other representations and characteristics in relation to other things, like propositions. But before going into any of these, Bolzano states, we have to investigate the very notion of representations in themselves.

Bolzano begins by noting that he has used the term “representation” earlier, but only in what he takes to be its common sense or then in contexts where it is evident what is meant. Just like with propositions, he wants to distinguish representation in itself from this common meaning, which he calls a thought of representation or subjective representation. ; In comparison, Bolzano states, representation in itself could be called an objective representation and it refers to any constituent of a proposition in itself that is not yet itself a proposition.

Bolzano notes that his explanation of representation might not be enough, since the very notion of proposition in itself was somewhat mysterious, so he tries to give another explanation. He starts from representation in the common sense or the subjective representation – whenever we are perceiving, imagining or thinking something, without making any explicit judgements or assertions, we are representing, and since all of these activities suppose a subject who does these, they could be called subjective representations. As mental states, Bolzano adds, these subjective representations have actual existence.

Now, Bolzano continues, every subjective representation has a content or material and this material is the objective representation in itself. This representation in itself does not require any subject, but then again, it also does not exist. Because representation in itself is not connected to any particular subject, it can be thought by a number of different persons and still remain the same representation – this is why it is called objective, Bolzano explains. He even suggests that there are representations that are not thought by anyone beyond God, like the number of grapes in southern Italy this year.

Bolzano goes on to distinguish representation from several other notions. Although representation was called the material of subjective representation, he begins, it is definitely not the object to which subjective representation refers to: indeed, a representation can refer to many objects, like the representation of an ancient Greek philosopher refers to both Plato and Aristotle. Another clear distinction between the two, Bolzano adds, is that the representation in itself does never exist, but objects of representation can exist – then again, they do not always exist, for instance, if the representation in question is the notion of a proposition. He even fathoms that some representations do not have any object, giving as an example the square root of –1.

Bolzano also emphasises that representations are not words, since these are always sensuous objects, either sounds or written signs. Furthermore, he adds, some representations are not connected to any words, while other representations can be signified by many words. Bolzano notes that sometimes even propositions and judgements are taken as representations, e.g. when philosophers speak of true and false representations. Bolzano thinks it important to separate these notions, although representations can include propositions as parts: think, for instance, the representation of a conjecture that God is omnipotent.

Bolzano admits that the notion of representation in itself is strange, but necessary for logic. Although representation in itself does not exist, it can have what he calls Gegenständlichkeit (objectivity), by which he means simply that the representation is connected to Gegenstand or an object. In addition to this connection with reality, Bolzano argues, representations in themselves are a necessary assumption, if we want to speak of constituents of propositions and truths in themselves. He admits the name is not perfect, since representations suggests something mental, but no better is available: best option would be concept (Begriff), but Bolzano notes that this term is often reserved for a subset of representations (representations that are not intuitions).

Bolzano points out that just like in case of representation, there really isn’t any satisfactory definition of representation. True, he has managed to describe representations as constituents of propositions, but he also considers this a mere external characteristic of representation, not its essence. Even worse in this sense is the description of representation as a material of thoughts, since this ambiguous term might in another sense refer to the objects of thought.

As difficult as defining representation in itself, Bolzano finds defining subjective representation. He has described it as an appearance of some object in a thinking mind, but here the word “appearance” is just another word for representation, so the definition is actually circular. Bolzano considers other suggested definitions, but rejects all of them. Subjective representation is not an image, except in a metaphorical sense, but then it could be applied to other mental phenomena, since even chest pain could be called a metaphorical image of a heart attack. Subjective representation is also not a symbol referring to some thought or concept, and it also not a sign in the sense that smoke is a sign of fire.

Bolzano still considers whether other philosophers had any inkling of the representations in themselves, but finds nothing certain, although some of them had noticed that the same concepts could be thought by different persons. Yet, he does not want to be harsh with them, since even the easier notion of propositions in themselves had escaped them. Indeed, Bolzano says, representations have often meant something completely different, with the meaning varying from one author to another, and each of these uses have probably delineated important notions in other disciples. Yet, he concludes, the notion of non-propositional parts of propositions is important for logic, and since no better nomenclature is available, he will call these representations.

Bernard Bolzano: Study of science – Can we know any truths?

The second question of Bolzano’s fundamental science concerns knowledge or cognition (Erkenntniss), but before we can understand what it is, he says, we must first understand what judgements (Urteil) are. Judgement is, again, a term that Bolzano cannot give a definition of, so he suggests just that he is using it in the regular sense of the word. Since this is a rather meager explanation, he gives a further detail that judgements are the common element in e.g. decision, opining, belief and assent.

Bolzano goes on to indicate some further characteristics of judgements. Every judgement contains a proposition, and depending on whether the proposition is true or false, the judgement is correct or incorrect. As something belonging to the human mind, Bolzano thinks, judgement exists, but then again, its existence is not independent, but dependent on the person who makes the judgement. He emphasises that judgement is different from mere representation of proposition, which does not assert anything. Thus, Bolzano suggests, God judges every true proposition, but only represents false judgements.

What then does it mean to make a judgement? Bolzano observes that it is an activity that requires that we have first decided to consider representations, where this consideration has led to a confidence that some proposition is true. He notes that the force of this confidence can have different degrees and that we cannot decide what this degree is. If it happens that a proposition and its opposite seem equally probable, we cannot judge, but only doubt.

Knowledge or cognition, Bolzano defines, is a judgement that contains a true proposition: thus, all cognition involves judgement, but some judgements do not involve cognition. True, he admits, we sometimes speak of erroneous cognition, but then we mean that cognition is mingled with some errors. When Bolzano is thus asking whether we humans can know or cognise anything, he means to ask whether any of our judgements can be true. He clarifies that he is not taking a stance on whether children or mentally ill people can do this, but wants to know just whether his readers can know any truths.

The target of Bolzano is a full skeptic who does not accept any truths. Such an extreme skeptic, he says, could not even make judgements, because this would require considering something as true. Indeed, if they said that they doubted something, they would appear to know at least this doubt. A wary skeptic might avoid this trap by not even committing themselves to their own doubt. But if they say that they doubt this doubt, don’t they admit then this second-level doubt? This takes us to an infinite regression, but we can cut the escape of the skeptic short, Bolzano thinks, by pointing out that in all these discussions the skeptic constantly has representations and can know it.

Having found this certain truth, Bolzano continues, we can say with certainty that we know it. Indeed, the judgement that this judgement is a truth is a further truth and thus something we can know. Since we can clearly repeat this same procedure over and over again, there are innumerably many truths we can know.

Bolzano notes that the proofs he has just given satisfy the skeptic for a single moment, but a complete cure requires further convincing, which he puts in a form of dialogue, which begins by the skeptic denying that they can admit having representations, because they do not even know whether there is any I that would have them. Bolzano is convinced that the skeptic cannot really doubt their own existence, once they think about the issue, and even if they do not happen to consider their existence, they need not make any judgement about it in order to notice that they are having representations. The skeptic concedes this, but raises the further doubt that this is all just their peculiar judgement that is skewed to regard everything erroneously. Bolzano assures us that even such an erroneous person would be right in assuming that they have representations and do exist.

Skeptic moves on to a new level of attack by admitting that Bolzano has managed to show that seemingly good grounds justify the truth of some of the judgements, yet, the question is whether these grounds themselves are true. Bolzano answers that the grounds are immediately certain. Furthermore, he continues, although these grounds presuppose the truth of what is to be proved, knowing the truth of the conclusions does not presuppose knowing the truth of the grounds, just like our belief in seeing does not require belief in the existence of the eyes, although the existence of eyes can be proven from the fact we do see something.

The skeptic is still afraid of the possibility that the judgement about the representations is just an error. Bolzano suggests that the question is not anymore about being convinced of the judgement, because just attending to it makes the answer obvious and even being deceived presupposes having representations. Perhaps, he says, the question is that the skeptic does not understand why they are so convinced. Bolzano really doesn’t have any answer: our power of cognition is behind it, but we do not know how it works – then again, we only need to know that it does work.

The skeptic changes tack and questions the worth of knowing the existence of representations. Bolzano admits this is not much, but at least it assures us that our power of cognition is not completely faulty. Skeptic finds this hard to believe, since we cannot know whether our representations correspond to the objects in themselves, because we can’t compare representations with their objects. Bolzano finds this whole question deceiving: because the very idea of comparing representations with objects is irrational, the truth of our judgements cannot depend on this. Sometimes, he continues, we know our judgements to be true, because this is evident from the very concepts used: this happens especially in mathematics. At other times, our judgements are based on immediately certain intuitions, just like when I say that I see red. Finally, Bolzano concludes, sometimes our judgements concern causes of our intuitions, and these judgements are based on probable information about the possible effects of objects, received through constant occurrence of some types of intuitions together.

The skeptic is relentless and notes that despite all these cognitive tools, humans still manage to make many errors. Bolzano points out that immediately certain judgements cannot be doubted and neither can correct deductions. This leaves still probable arguments, and he admits they can lead to errors, but adds immediately that it is not reasonable to not assent to very probable judgements. The skeptic points out that in dreams and confused states our cognitive tools do not work. Bolzano thinks that even in these states the errors do not concern immediately certain judgements, but only consequences deduced from them: thus, although it is certain even in a fevered state that if I feel cold, I truly feel cold, it is wrong to conclude that this feeling is caused by coldness of the air surrounding us. Even so, he continues, these problems do not concern us in a healthy state. The skeptic makes the obvious retort that in a dream or confused state we often cannot recognise that we are in an abnormal situation. Crusius agrees, but thinks the important point is that in a healthy state we can distinguish it from abnormal state. The skeptic makes the final suggestion that perhaps we will have quite different experiences in the afterlife. Bolzano is sure of that, but asks the reader to focus on our current judgements that concern our current life.

After this long detour, the skeptic accepts that they do know some truths, but still asks for a criterion to recognise truths. Bolzano suggests as a criterion that a judgement is reliable, if it is ascertained always when tested. The skeptic still raises the possibility that their memory might be in error, but Bolzano finds this very improbable.

Bolzano notes in the end that these kinds of investigations are usually not included in logic. Instead, he continues, their place is often taken by the so-called laws of thinking, such as the law of contradiction. Bolzano notes that often these laws concern things in general and not judgements, making them more fit for ontology than logic.

Bernard Bolzano: Study of science – Are there truths in themselves?

The purpose of fundamental science, the first part of Bolzano’s logic, is to remove all doubt that prevents any use of human reason. This task, he says, has two parts: first, we have to show that there are what he calls truths in themselves, and then, we have to show that humans can know at least some of these truths. Bolzano notes that one might doubt the meaningfulness of both of these tasks, since true skeptics wouldn’t even believe in the existence of other people and wouldn’t then even listen to any of their arguments. Bolzano notes that even if we cannot save such extreme doubters, we can convince people who are in danger of becoming skeptics. Furthermore, he adds, even the most stubborn skeptics live unskeptically and thus have the opportunity to be at least internally convinced, even if they refuse to admit this.

As a preliminary to the first task, Bolzano introduces the notion of proposition in itself (Satz an sich). He does not explain the phrase immediately, but only through comparison with other types of propositions. In other words, Bolzano says that uttered propositions are spoken phrases indicating something that must be either true or false, while a thought proposition is such that is not spoken, but only thought by anyone. Yet, he adds, propositions need not be said or thought at all, and if we ignore the question whether they are or not, we are dealing with propositions in themselves.

Bolzano is at some pains to explain why he can use the German word Satz for the notion he is describing: Satz is etymologically related to the verb Setzen, which implies that there is some person “setting up” this proposition. He explains that the etymology should not be taken literally here, just as a mathematical root of an equation is not at all like a root of a plant. Besides, he states, the concept is needed, and any other possible designation, like judgement (Urteil), would point even more to a thinker behind it. Indeed, Bolzano emphasises again and again that proposition is nothing anyone needs to be thinking (although it can be). This implies, he notes, that propositions in themselves do not exist, although thought of a proposition can exist. Then again, Bolzano points out, propositions in themselves can still concern thoughts (think of a proposition like “I am thinking myself”).

Bolzano is assured that previous logicians have at least implicitly used the concept of proposition in itself: for instance, they have admitted that the order of propositions in syllogisms is irrelevant, which would not be true, if they described the order of thinking and not relations of abstract propositions. True, he admits,they have not spoken of propositions, but judgements, mostly because many of them supposed that the phrase Satz referred only to a subclass of judgements, namely, assertions. Bolzano notes that even the suggested other types, such as questions, can be also seen as assertions: question just is an assertion saying something of the form “I ask this and this”.

Bolzano admits that his description of proposition is no true definition. His excuse is that no proper definition is simply available. The best historical alternative – that it is something that can be true or false – is not a classical definition, according to Bolzano, because it contains a disjunction. Other suggested definitions, he notes, have often concerned thoughts of propositions or then they have assumed the concept to be defined.

Bolzano has introduced the notion of a proposition in itself only to explain the further notion of truth. Words “true” and “truth”, he says, can mean many things, but the most appropriate is that truth is a characteristic of certain propositions in themselves, whether they are asserted or thought or not. Sometimes, Bolzano continues, we speak of truths, when we mean these propositions that have this characteristic. An even further deviation is to speak of true thoughts or judgements that contain true propositions or of collections of propositions or judgements. The least appropriate meaning, Bolzano thinks, is that of speaking of e.g. true friends, where we are referring to an object that truly is what it is described to be.

Just like there are propositions in themselves, Bolzano says, there are truths in themselves or objective truths, that is, truths no matter whether anyone says or thinks it. Just like propositions in themselves, he thinks, truths in themselves do not exist, except when thought by someone. Bolzano does admit that metaphysically speaking, God does know all truths, but this does not lie in the very concept of truth: truth in itself differs from a known truth. Furthermore, he continues, truth differs from certainty, which is a property of judgements, and from existence, although truths can refer to something existent. Interesting is the relation of truth to thinkability and knowability. Bolzano notes that all truths are thinkable, but not everything thinkable is true. Even more, he adds, all truths are knowable and everything knowable is true, but the concepts are still different, because knowledge and thus also knowability have degrees, but truth does not.

Bolzano could not define propositions in themselves, but he suggests we can define truths in themselves. Propositions always have a subject or a topic, of which they figuratively say or predicate something. Propositions are true, Bolzano underlines, if this subject actually has what the proposition predicates of it. The only weak point in this definition, he says, is the word “actually”, which in this context means the same as “truly”. Still, Bolzano thinks, this is no problem, since we can do without this word: proposition is true means that proposition predicates of its subject what the subject has.

Bolzano considers several alternative definitions of truth, dismissing quickly the so-called metaphysical definition, equating truth with existence. A more interesting definition is that of truth as correspondence between thought or representation with its object. Bolzano cannot, of course, accept this definition, because it speaks only of thoughts or representations of truths. Furthermore, he says, no one has really been able to explain what this correspondence is supposed to be. If it is meant to say just that a representation represents its objects, well, Bolzano thinks, this is what representations always do. If it means that representations within a proposition have the same relation to one another as their objects, this cannot be literally true, he points out, because e.g. a representation of God is not the cause of a representation of the world.

Further suggested definitions of truth Bolzano finds even less convincing. Truth cannot be just universal validity, since every person does not know every truth. Furthermore, truth cannot be defined as agreement with the rules of thinking, because these rules are either defined in terms presuming the notion of truth or then the definition also includes probable propositions that are still not true. Finally, truth is not defined by permanence, which is at most a sign of truth, not its essence.

Bolzano criticises attempts to extend the notion of truth. Firstly, he is not fond of the concept of subjective truth or of truth relative to a person, since we already have notions like opinion. Similarly, Bolzano forbids the idea of a formal truth, which at best means something like non-contradictoriness, which should not be confused with truth.

With all these preliminaries taken care of, Bolzano can finally move to his actual task, that is, proving that there are truths in themselves. This does not mean, he underlines again, that we should prove that such truths exist, but only that at least one proposition in itself is true – or to put it in other terms – that the proposition “no proposition is true” is not true. This, Bolzano quickly notes, is evident because “no proposition is true” contradicts itself: if it were true, it would itself not be true. Thus, there must be at least one truth. Even further, Bolzano points out, the same proof can be applied again. Say that we know there to be a certain number n of truths. Well, if we pick out these n truths and consider the proposition “no proposition beside these specific truths is true”, we note again that this proposition contradicts itself and that there are more – and indeed, infinitely more – true propositions.

Bolzano notes that a hardcore skeptic might not be impressed with this proof. They would object that if they are to be convinced by this proof, they must already suppose that they have a capacity to know truth, thus already presupposing that there are truths in themselves. Furthermore, Bolzano continues with the skeptic’s objections, the proof assumes the premiss that “no proposition is true” is a proposition in itself, thus assuming another truth before we showed that there are any truths. Bolzano is not afraid of these objections. Firstly, he admits that the person convinced of the proof must have a capacity to know truths, but they themselves need not explicitly have this as an opinion. Secondly, Bolzano agrees with the skeptic that the mentioned premiss is true and goes even so far as to suggest that its truth is immediately convincing. Yet, he adds at once, this is no problem, but another proof for what we set out to demonstrate. The method Bolzano used was chosen just because it so forcefully showed the self-contradictoriness of all skepticism, but this does not mean that there aren’t other ways to do the same thing.