Hoffmann's idea of demonstrating truth of something is based on the notion of closing off possibilities – when we can show that alternative accounts are against some principle of reason, we can be sure that the remaining account is the true one. In some cases, we cannot do this either for a proposition or for its contradictory. In that case, we can just conclude that both propositions are possible.
This possibility in question is logical, and as one might have guessed, Hoffmann thinks it is just one type of possibility. A more formal notion is verbal possibility, which just means that the words used in a proposition refer to some ideas. On the other hand, a more substantial notion is metaphysical possibility, by which Hoffmann means lack of contradictions, while even more substantial is physical possibility, which means capacity to physically actualise content of some proposition.
Getting back to logical possibilities, they are lacking in the sense that they come with no way to justify them. Indeed, this lack of justification makes it natural for us to reject them and thus they can be called internally improbable. As one can clearly see, if proposition and its opposite are both considered just possible, both of them are internally improbable. A more interesting notion of improbability is relative improbability, which is improbability arising from comparison of proposition with its contradictory. Probability is then defined as a counterpart to relative improbability: if we consider proposition to be more likely to affirm than its opposite, although both are possible, the proposition is probable.
Just like demonstration must be based according to Hoffmann to some principles, so must argumentation through probabilities. The basic idea behind argumentation of probabilities is that a proposition always implies or involves a number of conditions that the world must satisfy. Some of these conditions could be accepted without any ado, but others require more justification. The more a proposition involves conditions that require justification, the less probable it is.
One interesting question is whether Hoffmann meant us to read these characterisations of probability objectively or subjectively. The answer is that he actually had both possibilities in mind. Probabilities might be just subjective, if our incapacity to justify the seemingly improbable propositions is just based on our lack of experience or on general limitations of human cognition. If we can show that neither is the case, we can conclude that the probabilities are objective.
It might appear that Hoffmann's principle of probability is difficult to apply in concrete cases. Yet, just like in case of demonstration, the highest principle implies a number of more particular principles that are easier to use. Thus, we know that the less possibilities an event has for occurring, the less probable it is. This means, among other things, that a single possibility is more probable than a combination of many independent single possibilities and that a more indeterminate possibility is more probable than a more determinate possibility.
Hoffmann is satisfied with mere general rules, but notes that there are many kinds of probability, each having their distinct rules. As one could guess in case of Hoffmann, the probability could be about causal or existential propositions. Causal probability can be physical, that is, concern reasoning either from causes to effects or from effects to causes. On the other hand, it may also be political probability, which concerns reasoning from the means a person uses to the ends he strives to attain, or moral-practical probability, which concerns reasoning from given ends to means required for those ends.
Two kinds of existential probability concern things past (historical probability of what has happened) and things in future (whether something will happen in these conditions). In addition, Hoffman points out a third class relating to signs. These signs might be some concrete things, for instance, when a diplomat tries to determine what a representative of foreign nation means by his expression. Yet, in most cases the signs are words. One is either trying to determine the meaning of words in general, in critique, or then the meaning of words in a particular text, in hermeneutics.
The importance of emphasising these different types of probability lies in the distinct presumptions made in each field. Presumption, Hoffman defines, is a proposition taken as probable in some particular field of knowledge. The presumptions are valuable, because due to their probability they can be used as premisses in probable reasoning. Particularly, if some proposition is in conflict with such a presumption, its contradictory will be more probable. Hoffmann enumerates a number of possible forms of presumptions: we might, for instance, think something is probable, because its absence is rarity or because there is no cause to suggest otherwise – or even that this presumption is accepted by reliable authorities.
Hoffmann goes to some lengths to describe how probabilities could be quantified. In general he delineates two alternative possibilities, arithmetical and geometrical. In arithmetical quantification of probability one chooses some arbitrary unit of probability and compares other probabilities to it, while in geometric quantification of probability they are compared to the totality of completely certain proposition.
This is as far as I will go with Hoffmann's Vernunftlehre, although he still does have couple of interesting things to say about the forms of method (analytical, synthetical and analytic-synthetical) and their various subtypes (for instance, mathematical synthetical method differs in Hoffmann's eyes very much from other types of synthetical method, because it alone cannot be used to justify existence assumptions). Instead, I am going to make a comprehensive estimate of Hoffmann's life work.
Because most of Hoffmann's shorter writings had been written against Leibniz, Wolff and Wolffian school and even his masterpiece, Vernunftlehre, contained many explicit and implicit criticisms of their positions, it seems especially interesting to consider what are the actual differences between Wolff's and Hoffmann's positions. Clearly, Wolff and Hoffmann disagreed especially in metaphysical questions – e.g. Hoffmann thought Leibnizian idea of pre-established harmony to be ridiculous, while Wolff suggested it was the best possible hypothesis about body-soul interaction. But what is especially interesting is the question whether Hoffmann's logical works are any different from Wolff's logic.
It is easier to begin with similarities and on basis of the common elements to find the specific differences characterising Wolff's and Hoffmann's notion of logic. First of all, it is clear that both Wolff and Hoffmann understand logic not just as a description of a formal structure of thinking, but as a methodology of scientific research. But the two philosophers differ in their beliefs concerning the unity of this methodology. Wolff strives to give a unified methodology of sciences, and in cases where he admits the existence of many methodologies (e.g. historical and philosophical methods, or demonstration of truth and argumentation for probability), he is keen to suggest that one of them is ideal. Hoffmann, on the other hand, is more aware of the differences between disciplines and their methodologies – mathematical reasoning differs from physical and moral reasoning.
Wolff is not a hard-headed rationalist trying to spin everything out of empty definitions, which is often the caricature applied for him, but instead, he used a more mixed methodology, in which empirically discovered premisses play an important role. Hoffmann's methodology is similar, when it comes to empirical matters, but his acceptance of a variety of methodologies allows far more tools of securing knowledge – all reasoning cannot be reduced to syllogisms, Hoffmann insists.
Both Wolff and Hoffmann also accepted that all human methodologies have their proper limits and that especially divine affairs lie beyond the ken of human understanding. Yet, the reasons for their acceptance were somewhat different. For Wolff, it is more of a quantitative question – human understanding just cannot regard all the infinite facets of the actual world, let alone all the possible worlds or the infinite mind of God. In Hoffmann's eyes, there are more essential reasons, why human mind cannot understand some things – it has to follow the agreement of its ideas and shun from conflicts between ideas, but it might well be that these are merely laws of human thinking, which might lead even to contradictions if taken to extremes. This brings us to the greatest difference.
What is missing in Wolff's methodological works is a deep consideration of the very capacity to know – he just assumes the psychological make-up of human mind and proceeds to state what are the best ways to gather knowledge for such a mind. It is characteristic that Wolff relegates the question of truth to the applied part of logic. With Hoffmann, on the other hand, this question takes the center stage. He is quite aware that justifying our capacity to know the truth is quite difficult and requires a completely different methodology from other sciences – if we would dare, we could call him a transcendental philosopher before Kant.