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.
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