Having investigated how to define and divide concepts, Crusius turns next to the question what applied logic has to say about propositions. He notes that most perfections and imperfections that propositions could have do not require specific rules, since they would concern the whole logic. The only topics left to consider in this regard, Crusius says, are rules for avoiding obscurity in propositions and rules for analysing propositions into their constituent propositions.
Starting with obscurities in propositions, Crusius notes the most prominent reasons why a proposition is obscure. These reasons include ambiguity of words and concepts belonging to two different genera. Crusius exemplifies the latter with one of his pet peeves: when we speak of grounds of something, we might be talking of causes or of existential grounds.
A specific case of obscurity Crusius mentions involves words indicating some quantity, like great or best. Such words create obscurities, he explains, if we do not have any clear measure to which to compare them (thus, if I say that this is the best world, this is still obscure, since we do not know on what scale this goodness is measured on). Even more obscure the case becomes, if instead of one quantity we have several and we do not know what it is we are talking about (for instance, if someone is said to have good memory, it is unclear whether e.g. they are quick to memorise things or whether they retain for a long time what they have memorised).
Moving on to the second topic or logical analysis or exposition of propositions, Crusius means by this showing what other propositions are thought at the same time when a certain proposition is thought distinctly. He notices at once that logical analysis does not mean going through all propositions that can be deduced from the proposition-to-be-analysed, if they cannot be thought with this original proposition. This does not mean that the propositions found through analysis need to be explicitly observable in the analysed proposition, Crusius adds: for example, although it is not evident from the form of the proposition, all real definitions include an implicit statement that the subject of the proposition exists or is at least possible. Similarly imperatives include also implicit propositions about the will of the speaker and often even indications of moral necessity.
sunnuntai 14. kesäkuuta 2026
perjantai 5. kesäkuuta 2026
Crusius, Christian August: Road to certainty and reliability – Divisions
After definitions, Crusius moves on to divisions, in which a more extensive logical concept (genus) is separated into narrower logical concepts contained in it (species and in rare cases even individuals) adequately, in other words, in such a manner that all individuals of the genus belong to one and only one of the divided species. Crusius is clear to distinguish division from mere distinction of concepts in general, which could be applied also to concepts that are not logical (say, when distinguishing cause from effect). He also points out the obvious fact that the divided concept is more undetermined and the narrower concepts go through all the possible determinations that could be attached to the divided concept.
Just like definitions, Crusius divides divisions into nominal and real kinds. By nominal division, he means an account of different meanings an ambiguous word has. In addition to this definition, nominal divisions are not of that much interest to Crusius, who deals more with real divisions, where species or individuals contained in an abstract concept are separated from one another. Crusius notes that real divisions come with four important elements: firstly, we have divided and undetermined general concept, secondly, the narrower concepts, into which the general concept is divided or determined, thirdly, the difference between the different members of the division or the narrower concepts, and finally, the point of division in the undetermined concept, to which the narrower concepts or separated species relate as further determinations. Crusius states that this point of division can be an existential or causal abstraction, and in the former case, an external or internal abstraction.
Crusius points out many reasons why we need to make divisions, first of them being that division brings abstract concepts closer to practical applications. Furthermore, he adds, if we did not have a clear idea of a hierarchy of a general concept dividing into many species, we would have to make do with mere individuals, with no idea of their commonalities, thus being deprived of many important truths. Yet, the main use Crusius envisions for divisions lies in their application in disjunctive deductions.
Crusius goes through the most prominent classifications of real divisions. Thus, real divisions are either divisions of logical oppositions or divisions of real oppositions and also either divisions of contraries or divisions of contradictories. Furthermore, Crusius notes that if we compare divisions applied to the same general concept, these alternative divisions can be subordinated – one division applies the other division and then divides its members further – or coordinated, that is, made according to completely different schema.
Besides these classifications, Crusius hands out some rules, such that the point of division should optimally be essential to the divided general concept and that in choosing from different alternative divisions, we should consider what is the most useful of them. Finally, he shares some methods for finding divisions, such as following experiences of differences or going through possible determinations and causal connections of the general concept to be divided.
Just like definitions, Crusius divides divisions into nominal and real kinds. By nominal division, he means an account of different meanings an ambiguous word has. In addition to this definition, nominal divisions are not of that much interest to Crusius, who deals more with real divisions, where species or individuals contained in an abstract concept are separated from one another. Crusius notes that real divisions come with four important elements: firstly, we have divided and undetermined general concept, secondly, the narrower concepts, into which the general concept is divided or determined, thirdly, the difference between the different members of the division or the narrower concepts, and finally, the point of division in the undetermined concept, to which the narrower concepts or separated species relate as further determinations. Crusius states that this point of division can be an existential or causal abstraction, and in the former case, an external or internal abstraction.
Crusius points out many reasons why we need to make divisions, first of them being that division brings abstract concepts closer to practical applications. Furthermore, he adds, if we did not have a clear idea of a hierarchy of a general concept dividing into many species, we would have to make do with mere individuals, with no idea of their commonalities, thus being deprived of many important truths. Yet, the main use Crusius envisions for divisions lies in their application in disjunctive deductions.
Crusius goes through the most prominent classifications of real divisions. Thus, real divisions are either divisions of logical oppositions or divisions of real oppositions and also either divisions of contraries or divisions of contradictories. Furthermore, Crusius notes that if we compare divisions applied to the same general concept, these alternative divisions can be subordinated – one division applies the other division and then divides its members further – or coordinated, that is, made according to completely different schema.
Besides these classifications, Crusius hands out some rules, such that the point of division should optimally be essential to the divided general concept and that in choosing from different alternative divisions, we should consider what is the most useful of them. Finally, he shares some methods for finding divisions, such as following experiences of differences or going through possible determinations and causal connections of the general concept to be divided.
keskiviikko 3. kesäkuuta 2026
Crusius, Christian August: Road to certainty and reliability – Definitions
If in the previous chapter Crusius dealt with the application of sensations and experience, he now turns his attention toward abstract ideas, which due to their non-concreteness are prone to confusion and might even not correspond to anything existent. Thus, he argues, abstract concepts should always be defined. By definition Crusius means a method of making objects of an abstract concept distinguishable from other objects. Definition needs therefore not reveal the fundamental essence of a thing. Still, it is something more than mere description, which just collects some distinctly abstracted predicates of a thing, which still are not sufficient to distinguish the thing from others, except perhaps in some limited context.
Crusius goes on to indicate some basic rules about definitions. Definition, he says, should contain only concepts that belong to the defined object always and are thus logically essential to it. This means that the concepts contained in a definition are either genera or propria, which in case of definitions are usually called specific differences. Furthermore, Crusius continues, these logically essential concepts can be either essential as such to the object defined or just natural, which then have to be used sparingly and always with the appropriate restriction that they belong to the object regularly and are lacking only in extraordinary cases. Some of the logically essential concepts, he adds, are essential to the object in themselves, but others only when we suppose the existence of the world.
Parts of a definition, Crusius explains, must be more distinct than the defined object, in other words, obscure concepts should not be defined by other obscure concepts. Especially metaphors and circular definitions should be avoided. Even so, he admits, the parts of the definition need not be fully distinct, just more distinct than the object defined. Crusius adds also that a definition must be adequate in the sense that it contains all the same individuals as the defined concept and nothing more.
Crusius divides definitions according to their purpose: the purpose of a nominal definition is to explain what a word means through an abstract idea separated into its constituents, while the purpose of real definition is to transform an obscure or at least concrete concept into an abstractly distinct and adequate concept. A clear consequence of these explanations is that the same definition can be nominal and real, depending on its purpose.
Starting with nominal definitions, Crusius notes that they should be used to define something possible, because it would be of no use to have a word that refers to something that could not be thought. Indeed, he adds, nominal definitions should in general be useful, which implies that they should not give occasion to error and confusion and should conform as much as possible to ordinary language Crusius emphasises also that nominal definitions can be used only to deduce nominal conclusions, how a word can or must be used.
Crusius divides real definitions into ideal definitions, which define only something possible that need not exist, and real definitions in the strict sense, which transform a concrete idea of an existing thing into an abstract and adequate idea. Beginning with ideal definitions, he notes that they are not to be confused with nominal definitions, which define only a name of something. Still, they are also not yet real definitions in the strict sense, because they cannot be used to deduce what really is or is not, but only hypothetical conclusions: what must be affirmed or denied when we presuppose the existence of objects corresponding to this concept. Of course, if we show that there exists something agreeing with the ideal definition, we have made it a real definition. Crusius also notes that in pure mathematics ideal definitions are always real definitions, since this discipline investigates abstract possible magnitudes with their properties and relations.
Real definitions in the strict sense, finally, are divided by Crusius into fundamental or first concepts and deduced or further definitions. By a first concept he means a definition that we can know without presupposing a proof from another definition. Crusius emphasises that a first concept is thus first only for us and need not be a natural ground for all other properties of a thing. Indeed, he adds, there could well be many first concepts, depending on what characteristics is our starting point for abstractions.
It depends on the thing in question where we should find their first concepts, Crusius explains. With physical, non-artificial entities (e.g. natural bodies and souls), the crucial question, according to him, is whether they are sensuous or not: with the former, the sensations provide us enough first concepts, while with the latter, we must just begin with the most undetermined idea. With artificial entities, the first concept is provided by their method of generation, and if they particularly are mechanical entities, the first concept is given by their immediate purpose and the structure required for this purpose. With moral entities (e.g. obligations and rights), Crusius notes, their first concept is defined by their purpose. Finally, he concludes, God and their activities can be defined only by symbolical concepts, indicating negatively what they are not, or pointing positively to their effects and to very indeterminate abstractions common to all things.
An important point for Crusius is that we should always demonstrate that a first concept does really refer to objects beyond our thought (if we couldn’t do that, it wouldn’t be a real definition, but only nominal or ideal). He does explain that in some cases we might not be speaking of physical, but of moral existence, in other words, that something should be the case (this is evidently what we should prove of moral entities). In any case, the proof can happen a posteriori from experience or a priori from connections between concepts or grounds.
The final type of real definition – that of further or deduced definitions – includes, Crusius says, many useless explanations that really add nothing to the first concept and even confuse what is meant with them. He delineates two useful kinds of deduced definitions, namely explanations of the essence of things, which serve to uncover the fundamental nature of the thing defined, and characteristic definitions, which help to distinguish individuals belonging to the defined class from others.
Starting with the explanations of the essence of things, Crusius notes that they might provide the physical ground – fundamental forces that constitute the defined thing – or the moral ground – explanation why the defined thing should or is allowed to exist (the traditional explanation why God allows the existence of sin is of this kind). In searching for these explanations, he adds, we might either start from a sensuously given first concept and then move to the grounds explaining these sensuous properties, or then we can start from an undetermined first concept of a non-sensuous or a moral thing and determine it in more detail.
Ending his discussion with characteristic definitions, Crusius explains that they are required, whenever the first concepts do not give us enough criteria for deciding how to apply these concepts. Examples of such characteristic definitions are criteria for deciding when something is true or false, criteria for deciding when some law applies, or criteria for recognising the existence of non-sensuous entities from their effects.
Crusius goes on to indicate some basic rules about definitions. Definition, he says, should contain only concepts that belong to the defined object always and are thus logically essential to it. This means that the concepts contained in a definition are either genera or propria, which in case of definitions are usually called specific differences. Furthermore, Crusius continues, these logically essential concepts can be either essential as such to the object defined or just natural, which then have to be used sparingly and always with the appropriate restriction that they belong to the object regularly and are lacking only in extraordinary cases. Some of the logically essential concepts, he adds, are essential to the object in themselves, but others only when we suppose the existence of the world.
Parts of a definition, Crusius explains, must be more distinct than the defined object, in other words, obscure concepts should not be defined by other obscure concepts. Especially metaphors and circular definitions should be avoided. Even so, he admits, the parts of the definition need not be fully distinct, just more distinct than the object defined. Crusius adds also that a definition must be adequate in the sense that it contains all the same individuals as the defined concept and nothing more.
Crusius divides definitions according to their purpose: the purpose of a nominal definition is to explain what a word means through an abstract idea separated into its constituents, while the purpose of real definition is to transform an obscure or at least concrete concept into an abstractly distinct and adequate concept. A clear consequence of these explanations is that the same definition can be nominal and real, depending on its purpose.
Starting with nominal definitions, Crusius notes that they should be used to define something possible, because it would be of no use to have a word that refers to something that could not be thought. Indeed, he adds, nominal definitions should in general be useful, which implies that they should not give occasion to error and confusion and should conform as much as possible to ordinary language Crusius emphasises also that nominal definitions can be used only to deduce nominal conclusions, how a word can or must be used.
Crusius divides real definitions into ideal definitions, which define only something possible that need not exist, and real definitions in the strict sense, which transform a concrete idea of an existing thing into an abstract and adequate idea. Beginning with ideal definitions, he notes that they are not to be confused with nominal definitions, which define only a name of something. Still, they are also not yet real definitions in the strict sense, because they cannot be used to deduce what really is or is not, but only hypothetical conclusions: what must be affirmed or denied when we presuppose the existence of objects corresponding to this concept. Of course, if we show that there exists something agreeing with the ideal definition, we have made it a real definition. Crusius also notes that in pure mathematics ideal definitions are always real definitions, since this discipline investigates abstract possible magnitudes with their properties and relations.
Real definitions in the strict sense, finally, are divided by Crusius into fundamental or first concepts and deduced or further definitions. By a first concept he means a definition that we can know without presupposing a proof from another definition. Crusius emphasises that a first concept is thus first only for us and need not be a natural ground for all other properties of a thing. Indeed, he adds, there could well be many first concepts, depending on what characteristics is our starting point for abstractions.
It depends on the thing in question where we should find their first concepts, Crusius explains. With physical, non-artificial entities (e.g. natural bodies and souls), the crucial question, according to him, is whether they are sensuous or not: with the former, the sensations provide us enough first concepts, while with the latter, we must just begin with the most undetermined idea. With artificial entities, the first concept is provided by their method of generation, and if they particularly are mechanical entities, the first concept is given by their immediate purpose and the structure required for this purpose. With moral entities (e.g. obligations and rights), Crusius notes, their first concept is defined by their purpose. Finally, he concludes, God and their activities can be defined only by symbolical concepts, indicating negatively what they are not, or pointing positively to their effects and to very indeterminate abstractions common to all things.
An important point for Crusius is that we should always demonstrate that a first concept does really refer to objects beyond our thought (if we couldn’t do that, it wouldn’t be a real definition, but only nominal or ideal). He does explain that in some cases we might not be speaking of physical, but of moral existence, in other words, that something should be the case (this is evidently what we should prove of moral entities). In any case, the proof can happen a posteriori from experience or a priori from connections between concepts or grounds.
The final type of real definition – that of further or deduced definitions – includes, Crusius says, many useless explanations that really add nothing to the first concept and even confuse what is meant with them. He delineates two useful kinds of deduced definitions, namely explanations of the essence of things, which serve to uncover the fundamental nature of the thing defined, and characteristic definitions, which help to distinguish individuals belonging to the defined class from others.
Starting with the explanations of the essence of things, Crusius notes that they might provide the physical ground – fundamental forces that constitute the defined thing – or the moral ground – explanation why the defined thing should or is allowed to exist (the traditional explanation why God allows the existence of sin is of this kind). In searching for these explanations, he adds, we might either start from a sensuously given first concept and then move to the grounds explaining these sensuous properties, or then we can start from an undetermined first concept of a non-sensuous or a moral thing and determine it in more detail.
Ending his discussion with characteristic definitions, Crusius explains that they are required, whenever the first concepts do not give us enough criteria for deciding how to apply these concepts. Examples of such characteristic definitions are criteria for deciding when something is true or false, criteria for deciding when some law applies, or criteria for recognising the existence of non-sensuous entities from their effects.
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