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.
