After propositions, Crusius naturally progresses in his applied logic to proving, where a proposition (the conclusion) is held to be true because of its relation to other propositions assumed to be true (the premisses). This relation, he explains, is formed by a chain of deductions, and while the premisses could be called the matter of the proof, the deductive chains are similarly its form. Crusius points out that there is a clear hierarchy what can be proven from different kinds of propositions: thus, a nominal proposition about mere words cannot be used to derive ideal conclusions about possibilities, let alone real conclusions about the existence of things, while ideal propositions cannot be used to back up real conclusions. He also notes that proofs can have different purposes: sometimes we just want to see what follows from assuming some premisses as true – this could be called a hypothetical proof – while at other times we truly want to establish the truth of the conclusion – this is then an absolute proof.
Crusius goes through several ways to classify proofs, starting with whether the proofs use only the proposition of contradiction or whether they rely also on the two other fundamental propositions. Furthermore, he adds, proofs can be divided according to whether the method used is the demonstration or the method of probability. The first kind – demonstrative proofs – can then be combined with the classification of the first sentence, so that we gain the concepts of geometrical demonstration, which uses only the highest fundamental proposition, and disciplinary demonstration, which takes advantage also of the two other fundamental propositions.
Crusius also divides proofs in a priori, a posteriori and mixed proofs. Of these, he explains, a priori proof establishes not just that the conclusion is true, but also why it is true. A priori proof can also be what Crusius calls a hypothetical a priori proof, which tries to develop a conclusion implicitly contained in some premisses, or an absolute a priori proof, where the conclusion is intrinsically connected, but not implicitly connected with the premisses. The names of these subtypes is explained by hypothetical a priori proof being also hypothetical in the sense of merely explicating the premisses, while the absolute proof should establish truths through the mere fundamental propositions and axioms immediately dependent on them, such as those involving causal powers of things.
A posteriori proof, then, means for Crusius a proof which establishes only the truth of the conclusion without explaining it. As is to be expected, he includes all proofs dependent on experience in this class, but somewhat surprisingly also the so-called proofs ad absurdum, where a conclusion is derived by showing that the supposition of its falsity leads to absurd conclusions (of course, the latter do not explain why something is the case).
Crusius includes also a third kind of or mixed proof, which should be in some sense a priori and in another a posteriori. His paradigmatic example of such a proof is those found in geometry, where objects according to some defined concepts are postulated to exist and then the existence or characteristic of some objects is established. Such proofs are first and foremost a posteriori, since they do not really explain the conclusion – that’s why Crusius calls them mixed a posteriori proofs – but the method is still a priori in the sense that the justifying ground of the proof is comprehended from the mere idea of the proposition to be proved.
All proofs should prove the truth of their conclusions, Crusius says, but not all of them establish the necessity of the conclusion. A priori proofs, he states, always do this, but the degree of necessity depends on the necessity of their grounds: thus, Crusius notes, if we want to prove the existence of something absolutely necessary, it must be proven a priori from the properties of God. When it comes to other kinds of proof, he states, deductions ad absurdum and mixed proofs prove the necessity of their conclusions, while proofs from experiences prove only the truth of their conclusions.
Observing the purpose of proofs, Crusius notes that they either prove something from the standpoint of truth and try to establish some conclusion as true, or then they prove from the standpoint of humans, that is, indicate what a person must admit, if they accept something as true. Another point of division concerns the question whether the proof uses only the tools of understanding or whether it aims to justify some course of action and is thus dependent also on moral issues.
With these divisions out of the way. Crusius goes on to explicate some rules on how proving is to be effected. Thus, he says, we should begin with the preliminaries of making all the concepts distinct and analysing the required conclusion. If we want to prove the existence of something, Crusius underlines, we have to base the proof ultimately on experience, which is our only criterion of existence. This does not mean that we can use only a posteriori proofs for this, since we can also use an a priori proof to justify the existence of something from the existence of something else, for instance, through causal connections.
Proofs are based on premisses, but should the premisses then in turn be justified? Crusius notes that all true scholars are committed to linking all conclusions that we want to proof with the highest grounds of all thinking. Yet, he admits, in individual scholarly treatises such long chains of proof are cumbersome and we should, in case of proofs aimed for convincing certain persons, not go further than what our opponents consider acceptable, and in case of proofs aimed for establishing some truth, not go further than to certain postulates that require no further proof and to lemmas from other sciences.
With such basic rules established, Crusius moves on to discuss fallacies, which he quickly admits is a topic, the various types of which need not be learned – and although he then proceeds to list various types of fallacies, this listing shows almost nothing of interest, so we can skip it over. What is more interesting is the notion of colliding or conflicting proofs, which Crusius has inherited from Hoffman and which bears striking resemblance with the so-called antinomies of Kant. In effect, Crusius defines colliding proofs as a case where we have various proofs that seem trustworthy by their form and matter and have the necessary signs for truth, but these proofs also lead to conclusions that cannot all be true at the same time. The colliding proofs are not fallacies as such, because proofs with similar properties are reliable in some contexts, but not just in this particular case.
The first question is, Crusius notes, how colliding proofs are possible: how can a proof have all the signs of being true, without being so always? One problem here is, Crusius explains, that something cannot be both true and false at the same time. His answer is to distinguish objective from subjective truth: objectively considered, nothing can be both true and false at the same time, but what is objectively false can still be subjectively true, that is, have signs of truth, but only in a restricted sense. Another problem is that colliding proofs seem to refute our capacity to know what is true, if we cannot rely on signs of truth. Crusius points out that these signs of truth do work, but with restrictions.
What then are the particular reasons for the appearance of colliding proofs? One special reason Crusius picks out are the latter two of his fundamental principles and all propositions based on them, which are reliable within their limits, but lack the completely universal validity of the law of contradiction. Thus, he explains in more detail, although these principles hold of finite things, they might have exceptions with infinite, which is not just different from finite things by some quantitative degree, but by its essence. Furthermore, Crusius adds, even if these principles hold of things in the known parts of the world, we do not know whether things are different in other parts of this world or at least in another possible world. Crusius concludes by assuring the reader that God must have had a good reason for making these principles work only within these restrictions, and any errors are just an unavoidable side effect. Besides, he adds, the collisions are a clear criterion, by which we know we have overreached the limits of our knowledge.
Crusius adds that not all collisions indicate fundamental restrictions of our knowledge, but some of them are generated by following principles that are not yet fully determined, so that we are not aware of all restrictions for their application. This happens, he suggests, when we are dealing with complex cases, where it is too hard to think all details at once and we have to just take the most usual details and abstract a universal rule from them, for instance, when in moral questions we cannot evaluate all factors, but have to assume a universal rule to decide at least something. Furthermore, Crusius notes, we might also just not have ever dealt earlier with cases where the restrictions of these principles become obvious: he gives as an example the development of Christian philosophy, where tenets of the religion have required the introduction of new restrictions to old principles followed universally by pre-Christian philosophers. Finally, Crusius points out the particular case of causal deduction, where we might still lack the full knowledge of some object and its influence, so that we are forced to be satisfied with what we have experienced and presume that things will follow the same course in the future.
Moving on to the collisions themselves, Crusius notes that they can involve formal principles used as signs for possibility, existence or connection of things. Such formal collisions happen especially in metaphysics, he adds, and they are the closest to the later Kantian antinomies. Crusius reminds the reader that these collisions always involve something more than just the law of contradiction, which allows no exceptions. Thus, the colliding proofs use principles that show something to be incomprehensible for us, although this incomprehensibility might just reveal the restrictions of the human mind. Crusius especially emphasises the force of obligation to decide, which of two seemingly equal proofs is to be accepted: for instance, he suggests, although the notion of freedom involves much that goes beyond our comprehension, we should accept it, because only freedom can explain the existence of divine law and moral guilt.
Crusius also thinks that the law of sufficient cause can never contradict the law of contradiction and proofs based on the first law are thus to be preferred over other proofs. For example, he explains, although the notion of creation of the world seems incomprehensible to us, we must accept it, because the world is such that it requires some ultimate causes.The final case of formal collision Crusius investigates involves using an otherwise reliable principle that in a particular context would force us to remove certain features of a thing, without which it could not be thought. For instance, he exemplifies, if we assume that a thing A can be divided infinitely, an actual infinite division would mean removing such intrinsic features of A, like its extension.
Just like some collisions involve just formal principles, Crusius explains, other collisions concern only the material propositions used in the proofs. Such collisions, he notes, occur when investigating efficient causes, especially in questions of morality. Crusius goes to great lengths to establish how to solve such moral dilemmas. To put it short, it all hinges on first analysing what justification each of the colliding proofs is based on and then finding out some principles of higher kind, on which to test the justifications of each proof. Even with this method, Crusius admits, moral collisions are often left undecided, because no satisfactory higher ground can be found.
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