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.