Moving on to more particular rules of heuristics, Bolzano states that the essential question is what problems it is meaningful to solve. This meaningfulness, he adds, consists of two elements, firstly, whether it is probable that we can solve them, and secondly, whether the solutions would be useful. Concerning the latter, Bolzano especially highlights questions concerning our duties.
In order to choose the most meaningful problems, Bolzano contends, we should have some understanding of what kinds of problems we could have. Thus, the rest of this part of his logic is concentrated on presenting a wide array of types of problems that could come up in many disciplines and strategies for solving them. Especially the first half of this discussion contains problems that are properly logical, and a number of them seem rather artificial and make the reader wonder if they would really come up so often in particular sciences.
The first problem is the analysis of representations: are they simple or complex, and if they are complex, from which representations and in what manner are they put together? Bolzano notes that answering these questions is very difficult, because most of our representations are not clear and therefore their potential parts are not intuited. Thus, he says, we commonly make the mistake and state that a complex representation is simple, just because it is so familiar to us from childhood. Then again, Bolzano adds, it is also very common to take simple representations as complex, especially when they are subordinated to other representations that we therefore consider their parts (Bolzano's favourite example is trying to define actuality in terms of possibility).
Besides analysis, a fundamental problem is to check whether a representation corresponds to any actual or non-actual objects. This task, he adds, should usually be preceded by analysis, because simple representations are always real in the sense that they are not contradictory and thus possibly refer to an object. On the other hand, Bolzano states, from a complex representation it is often simple to check whether they have any obvious contradictions and are thus imaginary.
Even if we have established a representation to be real, Bolzano explains, it might still not actually refer to an object. Thus, he notes, we should proceed to check whether the representation presupposes an existing object or not. If it would refer to a non-existing object, like a number, Bolzano thinks, the consistency of the representation is enough to establish that it has an object. On the other hand, if it would refer to an actual object, the existence of such an object has to be established from conceptual truths or experience.
A somewhat more difficult problem is to check whether a representation refers to a singular object or several. Bolzano notes that if we can divide the representation into two representations contradicting one another and referring to objects (like triangles can be divided into those having a straight angle and those not), the answer is obvious. If we cannot find such a division, he adds, we can only with more or less confidence determine that it is singular. Sometimes, Bolzano points out, the very form reveals it (e.g., when we speak of a sum of all B), and even if it does not have such a form, but its objects do not exist, the determinations contained in the representation might be enough to determine the unique object (e.g. even numbers between 6 and 8). Bolzano thinks it a more difficult problem, if we have a pure concept, the objects of which should be actual: if the uniqueness is not determined by form, Bolzano is certain only of the notion of God as referring to a single object; of the other concepts of this type, if experience has shown that they refer to at least one object, he thinks it probable that they refer to more than one.
Because science should be as exact as possible, Bolzano considers it an important problem, whether a representation could be simplified by removing unnecessary determinations. In order to do this, he explains, we need to first analyse the parts of the representation and check if it has objects or not: if it has objects, we should try dropping its determinations one by one and seeing if it still refers to all the same objects; if it does not have objects, we should vary its constituents and thus turn it into a representation with objects and then do the dropping of determinations.
Moving on to cases with several representations, the obvious problem is what relations their extensions have. Bolzano goes into great detail to discuss these relations, but ultimately they all boil down to the question whether some combination of these representations or their negations refer to objects: for instance, representations A and B are compatible, if representation “something that is both A and B” has objects.
A somewhat different problem is that of finding similarities between objects. Bolzano notes that in order that we may give an answer other than trivial one that they are all things, we must have some representations of them. The problem then becomes analysing the representations and finding common characteristics shared by all of them. Bolzano actually gives not that much instructions on how to do this, but suggests that we should practice this kind of problem by thinking and comparing different kinds of things.
A natural contrast to the previous problem is, of course, that of finding differences between objects, in other words, characteristics that belong to one object, but not to another. Again, Bolzano says, we need some representations for the objects, but this time such that they are mutually exclusive, so that the representation referring to one object does not refer to the other object. Again, Bolzano has not so much instructions for this problem, but he suggests analysing the representations, in the hope of finding some characteristic belonging to one representation and not to the other. He also points out that although a single characteristic might not be the point of difference, a number of them put together could be it.
A problem combined of the two previous is that of finding a characteristic shared by all given objects, but differentiating it from another group of objects or even from all other objects. As we might guess from the two previous problems, Bolzano notes that to even begin solving the problem, we need two sets of representations: in other words, to find out similarities within one group of objects, we need representations for each of the objects, and to differentiate this group from another group, we need another set of representations, the two sets being mutually exclusive. In fact, Bolzano points out that we can even weaken these conditions and assume only that we have one set of representations exclusive to one group objects and another set covering both groups, since representations required for the second group can be defined in terms of the common representations and of the negations of the representations in the first group.
Bolzano moves on tackle a group of problems that involve finding a representation with a certain relation to a given group of objects, whether it is a case of this representation covering at least these objects, covering these objects and also more objects, covering only a part of these objects, covering exactly these objects, covering less than the whole group, but more than a given part of it or even covering a part of these objects together with some other objects. Especially the more complex problems feel very contrived, although they do include the problem of dividing a group of objects into a number of mutually exclusive representations. As with the problems of finding similarities and differences, Bolzano states, we require some representations that cover these objects (and in some cases, excluding all other objects), and indeed, pretty much all of them hinge on finding some similarities between these given representations.
Bolzano moves on to suggest problems involving judgements or propositions. Just like with representations, the fundamental problem is to analyse the judgements into their constituents. From these constituents, he continues, we can determine most of the internal characteristics of a judgement. The only exception, Bolzano points out, is the question whether the judgement is analytical or synthetical, which we must test by varying the parts of the judgement and seeing whether it remains true or false.
Just like with representations, Bolzano says, one problem concerns the relation of different propositions. The peculiarity of his logic is that many of these relations are defined in terms of variable constituents of propositions. Thus. to see whether two propositions are compatible, we need to see whether we can choose such values of variables that make both of them true. In some cases, Bolzano points out, such tests would have to be done infinitely many times, for instance, in case of derivability, which is defined as the derivable proposition being true with all values that make its premisses true. Bolzano notes that establishing such derivability must then be done by formal considerations used in the doctrine of inferences or by means of concrete sciences.
An obvious problem involving propositions is to test whether they are true. Bolzano notes that a necessary preliminary for this task is to decide whether the proposition in question is purely conceptual or empirical. In the former case, he thinks, a true proposition would be either self-evident or based on some self-evident truths. In the latter case, Bolzano continues, only the falsity of the proposition can be established conceptually, that is, by showing that it contradicts some conceptual truth. If we want to confidently assert the truth of an empirical proposition, he notes, we must first show that its subject term refers to some object and then establish that there is no perception that contradicts the proposition and that many of our perceptions agree with it in the sense of being derivable from it. All of these questions, Bolzano states, can usually be answered only with probable answers, while in some cases we may even know that the question cannot be settled through mere empirical means.
Bolzano defines proof, firstly, as anything used for directing the attention of a thinker in order to generate in their mind a judgement that they did not assent to before or at least not with the same degree of confidence. In this sense, proofs can be even sensuous objects and changes, but Bolzano gives a more restricted definition of proofs, covering only sets of judgements or propositions that mediate our judgements. An obvious problem is to test how convincing a proof for a proposition is, especially from the standpoint of the thinker searching for truth (Bolzano saves the question of how convincing a proof is from the standpoint of a learner to another place). This problem boils down to two questions: can we assume all our premisses (even the implicit ones) with sufficient degree of confidence and is the supposed conclusion derivable from these propositions or from a part of them or do they at least provide for the conclusion a degree of probability over doubtfulness? Bolzano concentrates his attention especially on the second question, pointing out signs how to recognise traditional fallacies in proofs.
After these rather logical questions, Bolzano moves on to deal with more substantial problems involving grounds of propositions. In fact, these are so fundamental questions that they deserve a post of their own.
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