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.

Hoffmann: Study of reason – Defining definition

One important result of the theoretical part of Hoffmann's logic was to understand the importance of the clarity of our ideas, but also to note the ambiguity of what clarity means – it is quite a different matter to have sensational clarity than analytical clarity. Now, Hoffmann notes that it is either quite easy to make our ideas sensationally clear – if we have forgotten what an apple looks like, we just have to go out and see an apple – or then it is quite impossible – we cannot sense things like courage, for instance. The case of analytical clarity is more intricate, and definitions, or combinations of abstractions resulting from analysis, are the primary tool for gaining it.

Hoffmann's definition of definition starts from an actual explanation of how definitions are formed – first, we analyse our ideas, then we combine the analysed abstractions in order to see how our original ideas consist of certain features or to find completely new ideas. He declares that this is far more satisfying way to define definitions than just describing them as revealing the essence of things – such a definition does not yet tell how we can discover the structure of any essence. Only slightly better is to characterise definition as a combination of genus and differentia, which actually says merely that all things share properties with other things, but also differ from them. Its main fault lies in suggesting that all definitions must have such a structure, although one might also define an idea as a common element in several genera (just like humans can be defined as belonging to both genus of animals and to genus of rational entities).

Equally erroneous, Hoffmann thinks, are the usual ways of differentiating between nominal and real definitions. Nominal definitions cannot be mere explanations of words compared to real definitions as explanations of things, since in explaining how a word is used, one is also explaining what sort of thing one is speaking of. Furthermore, nominal definitions are not defined by consisting of mere sensuous ideas, since we can well have nominal definitions of e.g. character traits. Most importantly, Hoffmann denies the validity of equating real definitions with generative definitions, since one and the same thing might be generated in many ways, and one and the same method of generation might produce many different kinds of things.

Instead, Hoffmann thinks that the division between nominal and real definition lies in difference between possibility and actuality. Nominal definitions are such that define mere ideas, and all they need is general coherence. Real definitions, on the other hand, should refer to things beyond mere ideas, and thus, when announcing a real definition, one should take care that the definition defines something that truly exists.

Rest of Hoffmann's tale of definitions concentrates then on real definitions. The most important division of them consists of what Hoffmann calls first concepts. The idea behind this notion is somewhat complex. Hoffmann thinks that all real definitions should be justified through something. One possibility is to justify them through other concepts and their real definitions, but obviously this route cannot go on indefinitely. Thus, at some point we must come to concepts, definitions of which have to be justified through things themselves, and these then are the first concepts.

Such concepts might describe individual properties of things, but also combinations of such properties or essences, which might be essences of either naturally or artificially produced things and which might be either necessary combinations (like three angles with three sides) or contingent (like heaviness with gold). In general, first concepts divide into five classes. There are relative essences, consisting of mere ideal relations, mathematical essences, consisting of mere quantitative properties, existential essences, consisting of existentially connected properties, physical essences, consisting of causes and effects, and moral essences, consisting of means and purposes.

All these essences have different ways to be defined, Hoffmann remarks. Relations can be defined only through the properties of what is related, while mathematical essences can be defined either through their method of generation or through their sensuous properties. Definitions of physical essences depend on whether the things in question are natural or artificial: natural physical things might be defined by their method of generation, their various sensuous properties and causal powers and their relations to other things, while artificial physical things are defined by their structure and their purpose. While moral essences in general should be defined just in case of purposes and means, especially in case of rights and obligations one must also consider conditions in which those rights and obligations can be actualized.

Finally, existential essences can be defined through various means. Firstly, they can be defined through sensuous changes affected by them – for instance, substance is something that subsists by itself, that is, that we can see to exist in various places, not bounded to another thing. Secondly, they can be defined through their inexisting parts or abstractions – for instance, a real thing can be defined through its abstracted properties of a) being thinkable and b) existing outside thinking. Finally, they can be defined by explaining their method of abstraction – for instance, extension is that which is left of a spatial thing, when we abstract from its forces and from the substrate behind them.

Hoffmann also considers whether one needs some further essences, notably in metaphysics or logic. In case of metaphysics, Hoffmann can just note that all essences handled in it, fall into some already dealt cases. Same holds in logic, where e.g. a concept of subject is relational and concept of deduction is causal.

First concepts serve as a beginning of definition, and Hoffmann characterises all further forms of real definition also through their purpose in cognition. He also notices that some definitions might actually have various purposes and thus fall into more than one kind. Furthermore, he mentions fascinatingly that some definitions, what he calls ignoble, serve no purpose at all – unfortunately, he provides no example of such an intriguing class.

The two true classes of real definition, which are not first concepts, are characteristic definitions, which help to distinguish things, and causal definitions, which help to explain sensuous properties of a thing. The two classes overlap one another, as Hoffmann already implied. Starting from causal definitions, it is not so much the existence, but properties of things that are explained by them – the definition begins from the essence of a thing and thus can be used a premiss to explain why the thing has this or that property. This is a very wide understanding of causality and could be applied also e.g. to mathematical things.

Characteristic definition, then, might actually be also a causal definition – by showing the essence of a thing, we also make it possible to distinguish it from other things. This sort of characteristic definition Hoffmann calls a priori, but he also accepts a posteriori characteristic definitions, which are clearly non-causal – in these definitions, we distinguish a thing through some conditions we find it in.

This concludes Hoffmann's theory of definitions. Next, he will handle divisions.

Hoffmann: Study of reason – Words and ideas

Until now we have seen Hoffmann investigate merely ideas abstracted from other ideas, but this is a good time to reverse the process and start to combine ideas back. Hoffmann notes that when we are doing this combination for abstract ideas, we are still using our judgement – remember that combinations of non-abstract ideas were effected by ingenuity.

Hoffmann's division of the classifications of these combinations are rather familiar: we have combined ideas, propositions or judgements and deductions or proofs. Intriguing is his manner of explaining the difference between ideas like ”blue house” and propositions like ”house is blue”, which, he says, depends only on the mode of cognition: in the former case one is thinking merely about the combination, while in the second case one is also taking note of the existence of such a combination. Hoffmann is then one of the few philosophers before Frege and Husserl to note the importance of assertion in judgements, which are then something else than mere combinations of concepts.

What I am especially investigating in this post is Hoffman's notion of combined ideas, and I shall look at judgements and proofs in later posts. In fact, Hoffmann has here rather little to say about combined ideas. Sometimes ideas are at least partially subordinated, that is, they refer to some common entities. Then combining two ideas simply means finding an idea that shares with properties of both and that refers to these entities that belong to the extension of both ideas – just like red balls are a subset of both balls and red things. Such a combination Hoffmann calls definition, well knowing that it is more something that could be defined, instead of a literal definition.

Sometimes ideas are not subordinated at all, but completely diverse. Then we cannot combine them in a similar manner as subordinated ideas. We might still combine them under another idea, that is, as divisions of a more extensive class, like shirts and socks can be combined as a division of some type of clothes.

An important aspect of combinations of ideas is relating them to individuals. Hoffmann describes the notion of an individual through various means. Firstly, he notes that individuals can exist only once in the whole universe. This does not really help define individuals logically, but Hoffmann has another suggestion: individuals are the smallest parts of logical wholes. The second definition might sound rather weird, but it does have its point. A peculiar feature of logical parthood is that it is a transitive relation, that is, every logical part of a logical part is also a logical part of the original – subspecies of a species is also a logical part of the respective genus. All parts of an individual, though, are not its logical parts – parts of Aristotle cannot be human beings, but are instead things like feet and hair.

Now ideas of individuals can be related to abstract ideas in various ways. An idea of an individual can be a proximate individual of an abstract idea, that is, it might add to the abstraction only individuality, just like the idea of this triangle is related to the notion of triangularity. Then again, thought of an individual might have lot more in it, just like the idea of Aristotle contains something else than individuality and notion of being a scholar. Finally, an idea of an individual might have less content than some abstract idea, i.e. when we think that some individual figure is a triangle.

A special question is then the possible relation of combinations of at least partially subordinated ideas or definitions with actual individuals. We do know there are at least possible individuals connected with these combinations, since the ideas combined are at least partially subordinated. Sometimes the combination is real in a quite strong fashion, when one of the ideas is a ground for the other, such as when rationality is strongly dependent on the animality of something. Even here one can have various levels of naturalness. Rational animal or human being is in a sense most natural combination, since we cannot add any idea that would add to the essential features of the individual referred to with this combination of ideas. Then again, we might have a combined idea of such a level, that the combination of features is not similarly complete, for instance, on level of animality in general.

Combination might also be more arbitrary, if neither of the ideas were a ground for the other, although in some cases like these one idea might be grounded on the genus of the other idea. For instance, in a black hat, blackness is not an essential possibility of hats, but it might be an essential possibility of the material of the hat.

In the case of the black hat there is some real entity corresponding to the combined idea. Black hat is also something that can be physically produced, while some combinations corresponds to entities that exist only through human decisions, like kings. Even more arbitrary would be cases where the combination corresponds only to some ideal entities, like star signs.

All the natural and especially the arbitrary combinations multiply the number of ideas indefinitely. One thus needs a new means for thinking these multifarious ideas in a quick manner – words are used to that effect, Hoffmann notes, but instantly adds that words by themselves are not enough and that we must be able to think something without words, in order to avoid an infinite regress. Hoffmann knows that it is quite arbitrary which words refer to which ideas, but hopes that the hierarchy of words would somehow mirror the hierarchy of ideas.

So much for Hoffmann's notes on simple combinations of ideas and words. Next time I shall take a look at his classification of judgements.

Defining existence

I've already discussed the essentials of Wolff's theory of language when studying his German logic. What I especially emphasised at that point was the role of language in communication – since we cannot transmit our thoughts directly through telepathy, we have to use words that just suggest thoughts of certain concepts. In addition to this secondary role of being a surrogate of mental images, Wolff now also mentions a more substantial task of words as crucial for more abstract thought. Words are necessary for representing general terms that we cannot imagine, such as virtue, number or existence, and so, we might say, in these case words just are concepts.

The importance of words for abstract and general thoughts is especially clear in case of definitions, which often are the only way to produce certain abstract thoughts. Definition is for Wolff an essentially linguistic phenomenon, that is, it must be something either spoken or written. Definition should represent a certain concept, but not just any concept – it must be complete and determinate in the sense that through it we come to know enough characteristics for distinguishing and identifying the object of definition.

In effect, then, definition helps to make our concepts more clear and distinct. Before hearing the definition, we might have just an obscure idea of a concept referred to by a term to be defined. Definition then returns this originally obscure term to at least clear terms, that is, terms, the corresponding concept of which refers to objects we can distinguish from other objects. We might note how Wolff is here touching on the question known as the paradox of analysis: either an analysis of a term leads to an identical proposition or tautology and is thus useless or it leads to a non-identical proposition and is thus wrong. Wolff would admit that definitions are identical propositions, but would note that it is not the definition as such, but the whole process of defining and so clarifying our concepts that is important.

Definitions should then lead us to terms clearer than terms to be defined. This is actually one of the formal requirements of a good definition, according to Wolff. In addition, one should not use in definition the term to be defined or another term requiring the definable term for its own definition, because this would leave the original term as obscure as it was before.

Wolff also suggests that terms referring to modes of things cannot be used in definition: definition should be something that helps us discern things constantly, but modes are variable and therefore offer no reliable method for distinguishing things. For instance, suppose we have learned to identify a certain species of rabbits by the colour it has in summer. We still wouldn't be able to recognise it in the winter, if it happened to change its colour according to seasons. Instead, we would have to be able to recognise the rabbit through such properties it has constantly, that is, its essence and attributes. Note that these properties might be constant dispositions, provided that we can generate conditions in which these dispositions are activated. Thus, we can recognise different gemstones by the hues they present in certain lighting, because we can take the gem to be tested in the required lighting.

The ideal form of definition of a species of objects would, firstly, indicate the nearest genus to which the species belongs and then note how this particular species differs from other species belonging to that same genus. In essence, this is the classical mode of definition through genus and so-called specific difference, familiar especially from the traditional classification of animal species.

In speaking of definitions, Wolff of course mentions also the important difference between nominal and real definitions. What is important in this distinction is to note that Wolff did not uncritically assume that one could gain insightful theories just by putting words together: this would boil down to mere generation of nominal definitions, which do not imply anything about the actuality or even possibility of things so defined. Instead, one should aim to produce real definitions, which do allow us to say something about this and an important part of which is formed by genetic definitions telling how something can be generated.

So much for definitions, next time I shall be speaking of judgements.