Bernard Bolzano: Study of science – Propositions about propositions

Bolzano has already mentioned the possibility of propositions about propositions, but he then left a fuller account of that notion to a time after the discussion of relations between propositions, since many propositions about propositions deal with just such relations. As he is now finally in a position to deal with the topic, Bolzano is especially interested in showing how to present such propositions in his official form of all propositions: “subject has a predicate”. Thus, for instance, the notion of compatibility of propositions would be fully expressed as “representation of a collection of certain representations, which put in place of i, j… in the propositions A, B, C… makes them all true, refers to some object” (the notion of incompatibility is, of course, expressed by a negation of this proposition). As he is well aware, we express such complex propositions in a much simpler manner and especially the notion of variable representations in the propositions is usually left implicit (for instance, we just say that the propositions can be true at the same time).

Bolzano goes quickly through all the various forms of compatibility and incompatibility, but we really need to look at just the notin of derivability, as it is the only one where he feels the need to give the full official form. It is really no wonder he refrains from doing it in other cases, since even this one is a mouthful: “representation of certain parts in A, B, C… M, N, O…, which are such parts that every arbitrary collection of representations, which put in their place makes the A, B, C… true, makes also the M, N, O… true, refers to objects”.

From the species of compatibility and incompatibility of propositions, Bolzano moves on to discuss the relation of complementarity, which is obviously important to disjunctions. We may quickly go through Bolzano's official structure for the basic notions in this regard, that is, the material complementarity – “representation of a true proposition among M, N, O… refers to some object” – and formal complementarity – “representation of a true proposition among a collection of propositions, whatever else is put in place of representations i, j…, following rule R, refers to some object” (it is not clear what this rule R refers to). Bolzano is as quick in presenting his official structure of propositions on probability – “relation of the set of all such representations that make the propositions A, B, C… true to the set of those that in addition to A, B, C… make also M true is m” – although he does add that we commonly say something to be probable, when this m is more than ½.

Bolzano's official structures for material and formal consequence are by now probably not surprising, former being “relation of truths M, N, O… to truths A, B, C… has the characteristic of a relation of consequence to its ground”, and the latter “the relation of propositions M, N, O… to the propositions A, B, C… has the characteristic that each collection of representations that put in the place of i, j… makes the A, B, C… true, makes also the M, N, O… into such truths that are the consequence of the former”. What is more interesting is his remark that we often speak of grounds and consequences, when we should actually speak of partial grounds and consequences. Furthermore, Bolzano defines cause as an existing object, the existence of which is a partial ground of some truth, and effect as an existing object, the existence of which is one consequence of an existence of a cause. He also points out that the existence of a cause implies the existence of an effect, but not necessarily the other way around, since an effect could have many possible causes. Thus, he defines condition as a full or partial cause, the existence of which is derivable from the existence of the effect.

Bernard Bolzano: Study of science – Relations between propositions

Moving on to relations of propositions, Bolzano begins by noting, just like with representations, that there are no two completely equal propositions in themselves. Again like with representations, he continues, propositions in themselves can still be similar in the sense they have so many commonalities that they can be easily confused with one another. Bolzano points out that propositions with similar representations are not necessarily similar and that similar propositions need not have very similar representations. For instance, “some As are B” is similar to “representation of an A that is also B has objectivity”, although the representations in them are not, and “Sun lights the Earth” and “Earth lights the Sun” have similar representations, but are not similar propositions.

Just like with representations, Bolzano states, we can compare the content of propositions, that is, their constituents. He points out that all propositions share at least the copula “have”. Thus, to make the definition sensible, Bolzano takes affinity of propositions to mean that they share at least one constituent beyond “have”. Clearly, he points out, such affinity can have degrees, and propositions with affine representations are affine, and affine propositions have some affine or same representations.

Bolzano goes on to define some types of affine propositions. Some of them, he says, can have the same subject or predicate or a contradictory subject or predicate. Bolzano particularly defines converted propositions as such that are otherwise completely same, but two representations have switched their places. Such switches can occur in different places of the propositions, but Bolzano defines as perfectly converted the propositions “what has a has b” and “what has b has a”; technically “a has b” and “b has a” are even more converted, he admits but it is so rare to find two characteristics that characterise one another. Bolzano defines as conceptually converted two propositions “representation of A that has the characteristic b has objectivity” and “representation of B that has the characteristic a has objectivity” – in other words, this is the familiar relation of “some As are Bs” and “some Bs are As”. He also mentions the traditional notion contraposition as the relation between contraposed “what has a has b” and contrapositing “what has not-b has not-a”.

Bolzano notes that the extensions of the propositions bring forth new relations. Still, he passes these quite quickly, since the extensions of the propositions were defined by the extension of the subjects of the propositions so that the relations really introduce nothing radically new. Bolzano also adds that similar relations could be defined in terms of extensions of predicates.

What Bolzano considers the most important relations of propositions come up with the use of variables, introduced with the notion of validity of propositions. What the notion of variables allows Bolzano to do is to introduce in modern logic very familiar analogies between propositional logic and algebra of sets (in Bolzano’s logic, relations of extensions of representations). For instance, he defines compatible propositions as such that, when taking the same representations in them as variable, some choice of these variables makes all the propositions true – similarly, incompatible propositions are then such that taking the same representations as variable, no choice of these variables makes all propositions true. Clearly the notions of compatibility and incompatibility of propositions are analogical to the compatibility and incompatibility of representations, and indeed, one could be defined in terms of the other – propositions are compatible, if the collections of representations making them true are compatible, and representations are compatible, if propositions saying that an object represented by one of these representations are compatible.

Yet, there is a catch in Bolzano’s definitions: the relations of compatibility and incompatibility depend on the choice of variables, so that propositions that are compatible with one choice could be incompatible with another choice. In practice, this dependence on the chosen variables does not usually affect that much of Bolzano’s discussion, just as long as the chosen variables are always the same.

Bolzano goes on to describe simple features of these new concepts, such as that if a proposition is compatible with another and this with a third, the first and the third might not be compatible (compatibility is not a transitive relation, we would say nowadays). What makes his discussion a bit more complex is that instead of propositions he is usually discussing collections of propositions, and as he notes a few times, what holds for individual propositions might not hold for collections of propositions (this is more evident with notions a bit more complex than just compatibility and incompatibility).

Bolzano shows special interest in propositions of the form “X has a” or “A has x”, where either the subject X or the predicate x is the only variable. Thus, he notes, propositions with predicates as variables – say, “A has x” and “B has y” – are compatible, just as long as their subjects refer to some objects: we just have to pick suitable predicates that fit with the kinds of objects. Similarly, propositions with the subjects as variables – “X has a” and “Y has b” – are compatible, just as long as their predicates are characteristics of something: we just have to pick suitable subjects that have these characteristics. Then again, Bolzano points out, the case is somewhat more complex, if the propositions had the same subject that was then taken as variable – for instance, with “X has a” and “X has b” – since such propositions are compatible if and only if the predicate representations are compatible or are characteristics of some shared objects.

So, what are the other similar notions discussed by Bolzano. Well, he defines derivability of a collection of propositions, in analogy with comprehension of representations, as a relation where the derivable collection of conclusions is always true, when the collection of compatible propositions or premisses from which it is derived is also. Here we find an example of the complexity involved with Bolzano speaking of collections of propositions, instead of just individual propositions. Thus, an individual proposition can never be derivable from both an individual proposition and its negation, Bolzano points out, unless it is formally true, that is, always true with this particular choice of variables. Then again, if a proposition is derivable from a collection of several propositions, it could be derivable, even if some or even all propositions would be replaced by their negations: for instance, conclusion “B and C are not equivalent” is derivable from premisses “every A is B” and “it is false that every A is C”, but also from their negations.

Bolzano also defines the stronger relation of exact or adequate derivability, where none of the premisses or their constituents could be dropped without removing the relation of derivability. In this case, he points out, neither the conclusion nor any of the premisses can be formally true and no premiss is derivable from other premisses. While derivability is a transitive relation, exact derivability is not, although it has particular instances of transitivity. Thus, Bolzano can define composed derivability as a relation where the premisses of an exactly derivable conclusion are again exactly derivable from further premisses and the final conclusion is then also exactly derivable from these more ultimate premisses. Then again, he thinks, there are also cases of simple or non-composed derivability such as a proposition “all A are C” being derivable from “all A are B” and “all B are C”.

Further relations of this kind are equivalence (collections of propositions being derivable from one another), subordination (one-sided derivability), independence (collections of propositions being compatible, but not derivable from each other), exclusion (when all propositions of a collection are true, propositions of another collection are false), mutual exclusion (two collections excluding one another), contradiction (if all propositions in one mutually exclusive collection are false, all propositions in the other are true) and contrariety (mutually exclusive collections are not contradictories). None of these have anything surprising in them, so we can skip Bolzano’s discussion of them.

Bolzano moves on to discuss relations concerning not just what happens if propositions or their collections are true, but determining what propositions in a collection are true. Here, it can be a case of speaking about the truth and falsity of the propositions as such or as case of speaking about the truth and falsity of the propositions with some representations taken as variable – in the former case, the concepts are defined, Bolzano says, materially, and in the latter, formally, although most of the concepts make best sense in their formal meaning. The simplest example is when all propositions of a collection are true or all are false, but more interesting examples are discovered, when some, but not all propositions in a collection are true – Bolzano calls this a case of complementary propositions. Complementation, he points out, can be one-membered complementation or disjunction, so that only one proposition in a collection is true, and disjunction can be exact, when no proposition could be removed from the collection, without cancelling the feature that it has always at least one true proposition. Complementation might also be multimembered, if it has more than one true proposition, and in this case, it might even always have some exact number of true propositions. Finally, even if a collection of complementary propositions does not have any exact number of true propositions, its propositions might still be compatible, that is, all its propositions could be true in some cases. At the end of all this, Bolzano notes that all thus defined notions could also be conditional, that is, we could restrict our attention to such representations that make certain presuppositions also true.

Bolzano extends this notion of presuppositions to his previously defined idea of validity of propositions, calling it comparative validity and identifying it with the notion of probability. In principle, he suggests that we find out representations making the presuppositions true and then from these the particular representations making a proposition we are interested in true – the probability of this proposition is the number of the latter collection of representations divided with the number of the former collection. In practice, Bolzano admits, we have to make some limiting assumptions about the number of the cases, because the number of possible representations that could be used is infinite, due to all equivalent representations: basically, he restricts his whole discussion to rather simple cases of pulling balls out of a box. As always, Bolzano is interested to extend his discussion from individual proposition to collections of propositions, which he manages to do by proving that with suitable choice of variables and presuppositions, the probability of a collection of propositions is the product of the probabilities of the individual propositions

Bolzano discusses the relation of a proposition that is ground to a proposition that is its consequence. He notes that this relation does bear some affinity with the earlier notion of derivability, but is stronger: for instance, the proposition of weather getting warmer is derivable from the proposition of thermometer rising and vice versa, but only the former is the ground for the latter, not the other way around. Bolzano points out that the consequence relation can also be connected with the notion of probability. Thus, the presupposition of summer coming increases internally the probability of a proposition that thermometer rises, because the coming of summer is, together with other propositions, a ground for thermometer rising, while a weather report increases its probability only externally, because this report is no part of a ground of thermometer rising.

The final relation Bolzano considers is that of an answer to a question – or actually we have here many relations to consider. We may speak of an answer fitting a question, which is simply the truth desired in the question, if it is not absurd and has one. Furthermore, Bolzano notes, we can also define an answer given to a question as any proposition that someone suggests as the required truth. Such an answer can be correct or fit the question or incorrect, he adds, and while a correct answer must be a true proposition, incorrect can be true or false. A determined question, Bolzano points out, can have only one answer, or at least just many equivalent answers, while there are innumerably many incorrect answers, even for absurd questions. He also defines solution as an answer to a problem in the restricted sense: such a solution, at least if it is a correct one, will consist of rules, that is, of propositions expressing how a certain undertaking for a determined goal is to be effected.

Bernard Bolzano: Study of science – Different types of propositions

From general characteristics of all or most propositions, Bolzano moves to characteristics that could be used to classify different propositions. He begins by noting that no propositions are truly simple, because they all have subject, copula and predicate. Still, he points out, we can differentiate between simple propositions, where both the subject and the predicate are simple (copula is, of course, always simple “have”), and complex propositions, where at least one of the subject and the predicate is not simple.

Another of Bolzano's divisions concerns the question, whether some of the representations in a proposition are intuitions. Thus, he calls a proposition with nothing but pure concepts a conceptual proposition (conceptual truth, if it is a true proposition) and a proposition with at least one intuition an empirical proposition.

Bolzano notes that in his suggested form of propositions, the predicate is always abstract. Then again, depending on whether the subject is abstract or concrete, we could speak of abstract and concrete propositions. Bolzano points out that some representations are not abstract nor concrete (e.g. pure intuitions), so if such happens to be the subject of a proposition, this proposition is then not abstract nor concrete.

Bolzano lists types of propositions involving notions of collections, which due to their simplicity we need not go through in detail. Thus, he mentions e.g. collective and distributive propositions, with collective and distributive representations as their subjects (“ A, B and C together…” and “each of A, B and C…” being exemplary forms) which differ from propositions with collective and distributive predicates (both have the form “A has b and c”, but in one the predicate is understood collectively, in the other distributively). We might still mention Bolzano’s assertion of equality (“characteristic m belongs to objects A, B, C… together”), assertion of difference (“relation of A, B, C… to F, G, H… is such that the first have a characteristic m, the latter don't”) and determination (“characteristic m belongs exclusively to A, B, C…).

Moving on to propositions with negative representations, Bolzano thinks that since completely negative representations represent no object and thus no characteristics, they cannot appear as subject or predicate in any propositions. On the other hand, he continues, “something that is no A” can, so that we could at least have propositions with negative subjects.

Bolzano notes that often negation is said to be attached to copula, but he thinks it is actually the predicate that is actually then negative. True, he admits, language does often attach negation to the verb, but so it does with probability or necessity, which, according to him, do not characterise the copula, but the whole proposition. Thus, Bolzano concludes, negation seemingly attached to copula is actually attached to the whole proposition and indicates that the proposition is false. He justifies this statement with the example of a proposition with many objects in the subject: “all A do not have b” does not mean that for each A, they would not have b, but only that it is not true that all A have b. Even if the subject is singular object, we can say “A has no b”, which indicates a lack of characteristic b, that is, a characteristic of not-b, which Bolzano then takes as the defining moment of negative propositions, in difference from affirmative propositions.

Bolzano notes an interesting connection between negative and collective propositions. While propositions with collective and distributive predicates are equivalent when affirmative, they are not so when negative: “A has not the collection of characteristics b, c …” means not the same as “A has not any of characteristics b, c…”, because in the former case it still can have some of the characteristics in the collection. Generally all types of propositions have their affirmative and negative version, and if there’s nothing remarkable about their difference, I shall not mention them, even if Bolzano does.

From the standpoint of logic itself, Bolzano notes, it is remarkable that some propositions handle either representations or other propositions. If a proposition deals with a representation, he points out, its subject must then be a representation of a representation. A simple example is a proposition stating that a representation has – or has not – objects. Indeed, all the notions dealt in Bolzano's discussion of representations have their own corresponding propositions – for instance, we might assert a representation to be general or to be comprehend by another representation – and I will also skip this part of Bolzano's account as rather simple repetition.

Moving on to propositions about propositions, Bolzano notes that at this point he can deal only with propositions about characteristics of propositions, since their relations haven't been considered yet. In fact, he at this point mentions only the affirmation of A, saying that a proposition A has truth, and the corresponding negation of A, which says that proposition A has no truth.

Previous types of propositions appear in all sciences, Bolzano continues, but there are some that are useful, although they appear only in some sciences. His first example is the existential proposition that asserts or denies the existence of something.

An important type of propositions for Bolzano is formed by those describing mental phenomena, that is, effects caused by the soul, either within the soul itself or outside it. Two of such phenomena are already familiar to us, namely, subjective representations and. judgements. In addition to these, Bolzano mentions sensations of convenience or inconvenience, accompanying representations, wishes or desires, caused by judgements that certain objects would cause a sensation of convenience, volitions, which differ from desires by concerning what we should do, no matter if it is inconvenient, and actions, which are changes caused by volition on our soul or on certain other substances, primarily our organs and through them surrounding objects. All of these phenomena, Bolzano suggests, come with their own propositions, which may or may not assert the person e.g. sensing something.

An important subtype of propositions concerning mental phenomena is formed by propositions with the concept described by the German word Sollen (what ought to or should be done). Bolzano notes that the concept of Sollen properly applies only to actions or actually to decisions of will: each decision has a characteristic it should have. Thus, he defines an ethically good decision as such that is as it should be, whether it is a duty or just commendable decision, and the corresponding proposition he then calls an ethical proposition. On the other hand, Bolzano notes, propositions like “it should rain” use the concept improperly, expressing merely the uncertainty of what is asserted. A related notion is that what we may or are allowed to do, which can be defined as not something we should not do, and the corresponding proposition Bolzano calls an assertion of permission.

Another important subtype of propositions involving mental phenomena Bolzano considers are propositions about desires or problems (something is desired to be done). He is especially interested in problems that assert a wish for truth with certain characteristics, that is, in questions. More precisely, Bolzano explains, questions do not ask for truths in themselves (as he has said many times, these do not exist), but their appearance in mind as a thought or their linguistic expressions. He lists some subspecies of questions, such as questions for truth or falsity of a given proposition (e.g. is it true that God exists), questions about predicates for given subjects (e.g. what are the characteristics of triangles) and questions about subject for given predicates (e.g. who is the tallest person on Earth). An important subtype for sciences is formed of practical or technical questions, which Bolzano also calls problems in a stricter sense and which require truths describing how to reach a certain goal. Some questions, he notes, are determined in the sense that they correspond to only one or several equivalent truths, some are undetermined or correspond to many non-equivalent truths, while others are impossible or imaginary in the sense that they demand characteristics that are not to be found in any truth.

Bolzano goes on to mention some quite basic divisions of propositions: they can have a subject that represents no object or a subject that represents at least one object, and if latter, they can have only one object or many and maybe even infinite objects. He also points out the division that has already been mentioned many times, at least covertly, namely, that of true and false propositions.

Furthermore, Bolzano again reminds the reader that truth or falsity is an unchanging feature of a proposition, and if we appear to speak of propositions changing truth value, we are actually speaking of linguistic expressions that can change the proposition they signify, if they include words like now or this. He develops this idea by suggesting that we could think of some parts of propositions as variables, where the change of the variable part could generate propositions with different features – proposition referring to no object could become a proposition referring to some object or a true proposition could become false. If we determined the possible results of the variable, Bolzano notes, we could then measure how many of these variations are true and how many false. Thus, he defines validity of a proposition as a relation of the number of its true variations to the number of all variations: of course, this proportion is dependent on what parts are chosen as being variable. Bolzabo then defines a universally valid or formally true proposition as having the validity of 1, while a universally invalid or formally false proposition has then the validity of 0.

Related to this notion of validity, Bolzano points out that no proposition is formally true or false, if all its representations are taken as variables. Still, he thinks, it is of interest if there is at least one representation, such that taking it as variable, the proposition is formally true or false (for instance, “a human that is evil deserves no praise”, when the representation “human” is taken as variable). Bolzano decides to call such propositions analytical, while a synthetical proposition is then such that it has no representation that could be taken as variable so that the proposition would then be formally true or false. He then goes on to list some examples of very general analytical truths: identical or tautological proposition “A is A” – or as Bolzano prefers to say, “A has a” – “A that is B is A”, “A that is B is B” and “every object is either B or not-B”. He points out that all the examples he just listed were such that they are formally true, if all but logical parts (whatever that means – Bolzano admits that the notion is hazy) are taken as variables and decides to call them logically analytical propositions.

Another feature related to the notion of variables in propositions is what Bolzano calls a conversion of a proposition, where two representations within the same proposition change their place. If such replacing does not change the truth value of the proposition (for instance, if it is true both that Titus loves Cajus and that Cajus loves Titus), he calls the proposition convertible or reciprocable proposition. Bolzano then defines analytically reciprocable proposition as such where the changed representations could be anything without changing truth or falsehood, like in the proposition “A that is B is A”.

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

Having described various characteristics and relations of representations in great detail, Bolzano turns next to certain combinations of representations or propositions. He reminds the reader that we are again talking about propositions in themselves, not our thoughts of propositions or judgements. In Bolzano’s opinion, he has sufficiently justified the notion of a proposition in itself in his fundamental theory, so he can right away move to discussing their further characteristics. Just like with representations, he will begin with general characteristics of all or most of propositions, then move on to speak about different types of internal characteristics propositions and finally deal with different relations propositions have to one another. Somewhat differently from representations, Bolzano will then consider propositions involving such relations and in the last section propositions that have a linguistic expression which makes it hard to analyse their constituents.

Starting with the characteristics of all propositions, Bolzano begins with the most obvious one that propositions in themselves do not exist unlike our thoughts or assertions about propositions. He adds another quite familiar characteristic that a proposition is composed of representations, which makes it then sensible to define a content of a proposition as the sum of all its parts. What is perhaps not as evident is that propositions can be used to make indefinitely more new propositions, Bolzano notes: we can make representations out of propositions or of collections of them, and when something is said of this representation, we formulate a new proposition with the original ones as its constituents.

Bolzano is dedicated to the idea that every proposition is either true or false. True, he admits, a verbal expression might be true in one sense and false in another or its truth could be completely indeterminate, but this is not true of propositions in themselves. Furthermore, as Bolzano has pointed out earlier, every proposition is fixed to a particular time, so that it is either always true or always false, that is, incapable of changing its truth or falsity.

Bolzano states that at least all true propositions, and if not all, at least many of the others also, concern some object. The representation of this object, he continues, is then a constituent of the proposition, namely, its subject. Furthermore, Bolzano adds, if not all, at least many propositions assert some characteristic about this object, and the representation of this characteristic is the constituent of the proposition called the predicate. In addition to the subject and the predicate, he concludes, a proposition must still contain a third representation connecting the predicate to the subject or the copula, which Bolzano always takes to be the representation of having.

“A has b”, where A represents an object and b its abstracted characteristic, is for Bolzano a general formula of propositions. He admits that all propositions do not appear to fit into this formula, but thinks that on a closer analysis they can be seen to follow it. Thus, Bolzano argues, every proposition contains a verb, which always contains, according to him, the representation of having. For instance, “A works” can be analysed into “A is working” and this again further into “A has the characteristic of working”. Generally, Bolzano says, every proposition of the form “A is B”, where B means a concrete object with some characteristic b, can be turned into a form “A has b”. This is true, he insists, even of propositions with seemingly simple verbs like “A should”, “A acts”, “A wants” and “A senses”, which he turns into propositions “A has an obligation”, “A has activity”, “A has a want” and “A has a sensation”. A peculiar case is that of the propositions of the form “A is”, but Bolzano thinks they could be transformed into “A has existence”.

There are still quite difficult looking propositions, such as the traditional hypothetical and disjunctive propositions, and Bolzano is willing to ignore them for now and to deal with them later. Instead, he notes that the supposed inflections of the verb according to its person, number or gender are just a feature of natural language, which has the habit of repeating information, just in case the listener or the reader misses something, and do not concern the copula. Similarly, Bolzano says, the temporal determination of verbs – even when dealing with timeless entities, like numbers – is just another feature of the natural language, and in a proposition it is actually the subject that is determined by time (in other words, it always represented an object at some time – or then a timeless entity). Similar questions concern such additional determinations as “often”, “rarely” and “probably”, which he understands to relate the whole proposition to our capacity of knowledge or to other propositions. Then again, negation Bolzano considers to be a determination of the predicate (in other words, “that picture is not beautiful” means actually “that picture has a lack of beauty”).

Bolzano has now managed to define the concept of proposition through the concept of representation: it is a combination of two arbitrary representations through the concept of having, where having is the copula, that which has is the subject and that which is had is the predicate. Yet, he adds, there is nothing to guarantee that a representation is a simpler concept than proposition, since we could also define representations as constituents of propositions.

Bolzano has already mentioned that most and at least all true propositions speak of some objects. This makes it sensible to define, like with representations, the extension of proposition and to distinguish it from the mere quantity of this extension. In fact, Bolzano states, the extension of a proposition is simply the same as the extension of its subject, and indeed, the subject points out all the objects the proposition is about. On the contrary, extension of the predicate, he adds, is not the same as the extension of the proposition and it is quite indeterminate, which portion of this extension is handled by the proposition: for instance, “Caius has understanding” does not mean that Caius has every sort of understanding, but only that he has the kind of understanding he has, and which kind, is uncertain.

Bernard Bolzano: Study of science – Kinds of characteristics

The final section of Bolzano’s study of representations concerns relations representations have to other things, although Bolzano notes many of them have to be left later, when these other things are treated specifically. Still, he begins by defining a correct representation of an object simply as a representation that truly is a representation of that object and similarly an incorrect representation of an object as one that is not. Bolzano points out that correctness and incorrectness are only external characteristics or relations of representation to a certain object, so that we cannot speak of correct or incorrect representations in isolation, although he does make the exception that imaginary representations can be always called incorrect, because they are not representations of any object.

Bolzano continues by noting that a correct representation of an object can be complete in the sense that any characteristics of the object can be derived from the proposition stating that the object is represented by the representation and truths recounting the characteristics of the representation – other representations are then incomplete. He explains further that a complete representation does not need to contain all the characteristics of the object as its constituents – indeed, he adds, there would be an infinity of such characteristics. Bolzano points out that a representation of an individual object is always a complete representation: otherwise, some characteristics of the object could not be derived from this representation, so that we could have another object without these characteristics fitting the representation, which would then not be a representation of just this individual object. This means, he argues, that all intuitions are complete representations.

Furthermore, Bolzano notes, if there are no two actually existing objects with the exact same characteristics, all complete representations must be singular representations, because a representation with multiple objects would have to leave some of their individual characteristics undetermined. He also adds that just like with the notion of correctness, we cannot say from a representation by itself whether it is complete, but this requires comparison with the object: the same representation can be complete for one and incomplete for another object. Bolzano will go on to note the same thing for nearly all notions studied in this chapter, and indeed, it is rather obvious, since he is dealing with notions inherently dependent on relations of representations to other things.

Bolzano moves on to discuss especially notions related to characteristics. Characteristics as such are, of course, not representations, but features of objects, thus, he underlines in many places that we are here actually speaking of representations of characteristics: despite the importance of difference for explaining the inclusion of these notions in a discussion of representations, we can usually just ignore it and just speak of characteristics directly.

Bolzano begins simply by defining a correct characteristic as such that belongs to an object corresponding to a certain representation (other characteristics are, of course, incorrect). A more interesting notion is that of an essential characteristic, which he defines as characteristic that is correct for an object of a pure concept (other characteristics are then inessential). Evidently, the less objects there are in the concept chosen, the more the object has essential characteristics. Thus, Bolzano argues, for a singular and therefore complete concept, all the characteristics of the object are essential. He also suggests that if a characteristic can be expressed by a pure concept, it will be essential for some pure concept (for instance, the concept of something that has this characteristic).

Bolzano defines as absolutely proper or exclusive characteristic as one that belongs to this object and no other and contextually proper or exclusive characteristic as one that belongs to this object and no other in some kind, to which the object belongs. Characteristics that do not fit even the definition of contextually exclusive characteristic are then common or shared characteristics. Bolzano notes that for an exclusive characteristic, the representation “something that has this characteristic” is a complete representation of the object, since it represents only this object.

Bolzano goes on to point out that absolutely or contextually exclusive characteristics can be used for recognising the object that has them and could then be called its distinguishing feature (Kennzeichen). Such a distinguishing sign can then consist of several characteristics, which he calls marks, defining them as characteristics that at leas in connection with others are suitable for recognising an object. If a collection of such marks are enough for recognising the object, Bolzano states, they should be called sufficient marks, while otherwise they are object: if a single mark is sufficient by itself, it is then also a distinguishing sign.

Bolzano also divides marks into immediate and mediate marks, where a mediate mark is essentially a mark of a mark: for instance, speaking is an immediate mark of reason and thus a mediate mark of humanity, he explains. Furthermore, Bolzano defines affirmative mark as a characteristic that does not belong to all things of a kind, but to some of them, and negative mark or condition as a characteristic that belongs to all things of a kind, but not exclusively to them. The meaning of these definitions lies in that a presence of positive mark can always reveal that an object belongs to the kind, but its absence does not guarantee that an object does not belong to it, while conversely the absence of negative mark shows that an object does not belong to a kind, while its presence does not guarantee that an object belongs to the kind. In this sense, distinguishing signs of some kind can then be called both affirmative and negative.

Bolzano defines original or constitutive characteristic as such that its representation is already a constituent in the representation describing the object – other characteristics of the object are then called derived or consecutive. He notes that if the representation in question is a pure concept, an original characteristic is essential to that concept. Then again, Bolzano assures the reader, an original characteristic need not be internal, because some representation of an object can concern only its relations: Bolzan’s example is the representation of the middlemost rose on his window.

Bolzano considers the question whether an original characteristic can be used as a distinguishing sign of the object. He notes that if we are dealing with an original characteristic connected to a general representation, the answer is negative, since then the characteristic cannot be exclusive to the object in question. Then again, if we are dealing with a singular representation, Bolzano ponders, the question hinges on whether any of the original characteristics could be used to derive the others. He ends the discussion of original characteristics by pointing out that although representation of each characteristic of an object belongs to constituents composing the representation of the object, we cannot say that each constituent of its representation is original or even expresses characteristic of the object (a simple example is a representation of not-red that has red as its constituent, which still is not characteristic of the object).

Bolzano ends the discussion of representations with the notion of a difference between two objects by defining it as a characteristic that belongs to one, but not to the other object. Like all characteristics, he notes, differences can be divided into internal and external, essential and inessential, shared and proper, and original and derived. Furthermore, Bolzano differentiates a numeric difference between individual things from a specific or generic difference between kinds and quantitative differences involving difference of quantities from all other, which he calls qualitative differences.

Bolzano also considers the question whether all two objects (that is, not just two representations of the same object) differ by some characteristic. He notes the obvious answer that the characteristic of being this object and not the other does the trick, but then specifies the question to ask for internal differences and not mere relations. Bolzano himself believes, like Leibniz, that this is true of actually existing objects. He argues for this by asking the reader to picture two completely identical objects: in order to be identical, they would have to be surrounded by exactly identical things and experience the exact same events – otherwise, these things and events could change their features from one another – which all is extremely improbable. In any case, Bolzano concludes, the Leibnizian principle seems to be true about at least most things.

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.