From subordination Hoffmann moves on to non-subordination, which holds on between pairs of ideas, one of which can be thought without the other. The most important subgroup of non-subordination is total diversity, in which both ideas have some aspect that is not shared by the other idea, just like body has features that soul doesn't and vice versa. Just like in many cases of subordination, diversity might also be just accidental or dependent on the peculiaties of the person thinking the ideas: for instance, one might not know that humans are animals and hence think that humanity and animality are diverse ideas.
At this point, Hoffmann introduced the notion of ”Punkt”, which we might perhaps properly translate as an aspect. Hoffmann's idea is that all subjects have various of such aspects, which can then be determined in different manners: say, a colour, a figure and speed would be such ”Punkts”. One of these aspects can always be determined only by one complete determination, thus, only subordinated ideas might determine the same aspect (thus, a strawberry can taste both sweet and strawberrish, but not salty). The notion of ”Punkt” is important, because it helps to divide diversity into two different classes. One of these classes is proper diversity, in which the two ideas are not connected, but can still exist in the same subject, because they do not determine the same aspect, like will and understanding. The more important type is opposition, in which the ideas either always exist in different subjects (like the notions of infinity and finity) or then determine the same aspect and therefore cannot exist at the same in same substances.
Hoffmann then goes on to classify different varieties of opposition: we have e.g. logical opposition, in which ideas are opposite, because they are different species of same genus, such as external and internal sensation, contradictories, one of which is always merely negative idea (visible and invisible), and contraries, both of which are determined ideas, like love and hate (note that with Hoffmann contraries can be contradictories, on the condition that the negative idea is something we can have a determinate idea of). Philosophically most interesting is perhaps the notion of causal or physical opposites, which are such that in addition to not existing in the same subject also have the tendency to cancel the other, if it happens to be just in its vicinity, like cold and warmth – this notion clearly resembles Kantian notion of real opposites.
Just as important as determining what types of opposites there are, it is also as important for Hoffmann to determine what is not opposed, although might seem to be. We might have difficulties to understand how some ideas can be combined in the same subject, like non-sensuality and cognition, but there still might be entities having both of these characteristics, just like God is supposed to know things without sensation. Similarly, one might have difficulties to understand how some entity could cause something, like how spirit could move material objects, but this doesn't necessarily mean that being a spirit would be opposed to being a cause of movement.
Hoffmann also notes that there are various levels of opposition between ideas, that is, they might cancel each other only partially, just like perpendicular and horizontal movement, which put together do not completely cancel one another and lead to a state of rest, but change into a diagonal movement. Some contrary ideas might have various intermediary stages, just like temperature can have many degrees between the extremes of cold and hot. Some contraries might be even said to exist in the same subject, if they merely cancel high degrees of the other contrary, just like vices merely cancel perfect, but not imperfect levels of virtue.
Just like there are various levels and types of subordination and non-subordination, Hoffmann thinks there are various levels between subordination and non-subordination. At the most extreme ends are essential connections and absolutely impossible connections, which are based on the very structure of the ideas. For instance, in case of essential connection, two ideas might necessarily exist together, like force and subject, or one idea might necessarily cause the other, like virtue causes good actions. Similarly, in case of absolutely impossible connections, two ideas might be unable to exist in the same subject, like roundness and squareness, or one of them might fail to cause another, like brute animals and speech.
Moving away from absolute impossibility we come at first to unnatural connections, which usually do not happen, but which might be effected by some third thing (obviously, at least God is meant here). At the other extreme are natural connections, which are something that occur, either because of the very structure of the ideas or because of some constant cause connecting them, unless some other thing hinders this connection, just like newborn humans usually have five fingers, unless some external causes hinders their development. At the very middle of this hierarchy, are then contingent connections, which occur sometimes, and merely possible connections, which are not impossible nor unnatural, but still just never happen to occur.
It is obvious that these levels of connection are modal notions, and Hoffmann is quick to note that he finds the use of such notions as necessity and possibility largely ambiguous and thus asks the reader to avoid those terms. It is not clear e.g. whether someone speaking of necessities means just essential existential connections or also essential causal connections or even merely natural connections. Furthermore, what is necessary and possible is more related to how we conceive things and what ideas we have. Thus, before trying to make any modal statements, one should carefully analyse what ideas one has. It is impossible to say what is e.g. necessary for human blood, unless one knows what one means by blood.
So much for relations of ideas, next time I shall take a look at what Hoffmann has to say about the clarity of ideas.