After judgements, a study of different modes of proof follow – this has been the official order since someone decided to place Analytics after On Interpretation in Aristotelian corpus. Somewhat refreshingly, Hoffmann leaves the classification of various types of proof to the practical part of logic, where we shall find it later. In theoretical part, he instead concentrates on some general features of all proofs and especially on various principles followed in proving.
Just like in case of judgements or propositions, proofs are not just strings of propositions for Hoffmann, but instead, processes of human thinking, in which we feel ourselves forced to move from some judgements to a new one. Note that Hoffmann is not yet interested whether these proofs have any connection to truth or not – this is the topic of the next chapter. Indeed, he admits outright that some proofs lead to false conclusions, even if they are so seductive to follow. Yet, now it is time only to describe the very act of following a proof, not give rules for determining a validity of a proof.
What is interesting in proofs is that we feel forced to follow them, no matter what the matter, that is, the topic in question. If propositions assumed just are of certain structure, then we feel obliged to draw some conclusions of them. It is then the form of proofs, embodied in some grounding principles, which are the topic now – nowadays we might speak of rules of inference, and it is important that Hoffmann clearly separates them from mere axioms, which by itself or as mere propositions do not force us to make any conclusions.
While Hoffmann accepts a number of grounding principles, he thinks they can all be based on three most basic principles. The first one should amaze no one. What understanding finds the easiest to deal with, Hoffmann begins his introduction of the principle, are those thoughts that require least amount of analysis. Such thoughts are being and non-being, which understanding at one notices to be impossible to combine. This is the basis of the principle of non-contradiction. Just like with Wolffians, the principle is for Hoffmann ontological, while the corresponding logical principle ”nothing can be both true and false”is based on the ontological principle.
For Wolff, the principle of contradiction was not just ontological, but apparently covered all cases of incompatible properties, since he used ”wooden” and ”iron” as an example of contradictory characteristics. Hoffmann, on the contrary, notes that, while undoubtedly a rule that understanding follows, such denial of incompatible properties must be based on another principle – what is incompatible in understanding must be accepted as incompatible in reality. Third is the corresponding positive principle – what is necessarily conjoined in understanding must be accepted as conjoined in reality.
This is actually all that Hoffmann presents as a theory of proofs – except he uses these basic principles to derive some further principles. I shall leave most of them unmentioned and concentrate on the most interesting of them, that is, the principle of sufficient reason or ground.As we have seen earlier, Hoffmann doesn't buy the Leibnizian principle as it is, but makes an important emendation – actions started by an immediate use of a force that does not need any external determination do not need to have any sufficient reason. The principle of sufficient reason in its Hoffmanian guise is based on the principle of necessary conjoinment – we must always connect an event to some force starting it, thus, it must have begun by an active force just described or by a force determined by something external.
What Hoffmann is suggesting is that a sort of Humean account of causality as a string of events with no intrinsic connection is just so alien to human thinking that we cannot accept it. Of course, this does not yet prove whether this human way of thinking is true – that type of question we have to leave to the next post.