Logical division, Hoffmann says, shares some similarities with definition, since, in a sense, both are divisions – in definition, we divide a concept (say, humanity) to more abstract concept (rationality and animality). A third form of division is then mathematical division of quantities into smaller quantities. The peculiar sort of division we now investigate, on the other hand, is a division of genus into its species. Its importance lies in providing premisses for disjunctive deductions – if we know all the species of a genus, we know that a member of the genus must belong to some species.
Compared to his account of definition, Hoffmann's discourse on divisions is rather straightforward. Just like in case of definition, Hoffmann distinguishes between nominal and real division – in nominal division, the different classes have merely a common name, but in case of real division, they truly belong to same genus. A real division presupposes then, obviously, idea of genus as a whole and idea of species as separate from one another. Furthermore, division also requires a ground for distinguishing the species from another and also some element that is common to all the species – this latter element should then be something essential to genus, or otherwise there might be a species that wouldn't have this element. Species should also fill the genus in the sense that no other species can belong to the same genus. Finally, changing from one species to another should affect thing in question in some essential manner – thus, if a piece of iron from London changes into a piece of iron from France, nothing of consequence happens to iron and therefore iron from London and iron from Paris do not form a true division.
After division, Hoffmann turns his attention to judgements, and we can be equally quick with them also. Hoffmann counts various manners in which a judgement can be imperfect. There are external reasons, such as when the proposition is more restricted than it could be, that is, when we say ”for some x”, when we could as well say ”for all x” – in such a case, improving the proposition would require a change in the subject. There might also be internal reasons, such as ambiguity – improving such a proposition does not change the subject or predicate, but merely modifies the connection between them.
Hoffmann describes in more detail methods for improving ambiguous propositions. In many cases, all it takes is to clarify the concepts or ideas that form the judgement, which is something Hoffmann has already explained in theoretical part of his work. There is also the possibility that the ambiguity derives more from the manner of connection between ideas in judgement. Hoffmann especially mentions comparative judgements, such as ”A has more quality X than B”, where X is a quality that can appear in many shapes – for instance, because intelligence is something that can appear in many shapes, ”A is more intelligent than B” might mean e.g. that A does crossword puzzles better than B, makes calculations more reliably than B etc. In such cases, Hoffmann notes, we should state clearly in what manner A exceeds B.
So much for divisions and judgements, next time we shall see what Hoffmann has to say about deductions in his practical division of logic.