Before entering the logic proper, Crusius introduces the reader to the most important terms used, with a full acknowledgement that they will be properly explained only within the logic itself. In effect, Crusius is presenting a preliminary division of the kinds of thoughts present in logic, and indeed, in any science, where by thoughts he means activities of understanding for representing something, no matter whether only in understanding or also e.g. in words. This division, he says, can be done either according to the internal distinction of their essence or according to the purpose for which they are introduced.
Starting with the internal distinction of the thoughts, Crusius notes that they either define a certain condition and activity of will or represent certain effects of understanding. He is quick with the first type, mentioning only that it includes at least explanations of our intentions. Thoughts representing effects of understanding, on the other hand, he divides into concepts, propositions and proofs. Of these, Crusius says, concepts include definitions that are abstract concepts that can distinguish something from everything else. He divides definitions into nominal definitions, which define a word for the sake of determining its meaning, and real definitions, which define a thing that is first thought in an unanalysed concept, which we want to change in such a manner that we can distinguish its parts with conscious abstraction and thud distinguish the thing from all others. Such real definitions, Crusius notes, are either first concepts, proof of which presupposes no other definition of thing and can thus be primary in regard to our knowledge, or deduced definitions.
Moving on to propositions, Crusius divides them according to their ground into arbitrary and real propositions. Arbitrary propositions, he explains, are not meant to indicate a truth independent of our will, but to show what one assumes to be or wants to be observed as true, like when one states that a circle is to have 360 degrees. A real proposition, on the hand, ascribes to things something that truly belongs to them and should not depend on our arbitrary choice, such as when we state that every circle has a middle point.
As one kind of real proposition Crusius indicates postulates, which must be supposed to be true, although they cannot be proven with regard to all conditions. He divides postulates into postulates according to truth and postulates according to humans. The first type of postulate, Crusius explains, contains in itself something false or uncertain that is still so insignificant that it can be disregarded or that plays no role in the consequences of the postulate, such as when we e.g. determine the meridian of a city only approximately. Postulate according to humans, on the other hand, does not contain in itself anything false, but is of such kind that it cannot really be proven otherwise, but by everyone perceiving and judging it by themselves, for instance, when certain experiential propositions are assumed to be universal. Crusius states that unlike divine knowledge, all human knowledge is ultimately based on postulates, which implies that there are certain questions the humans cannot be certain about. He emphasises that we shouldn’t postulate willy-nilly just anything, thus, that the right to postulate something must be proven carefully. He also notes that something can be postulated, although it might as well be provable.
Compared to postulates, Crusius divides propositions provable according to all conditions to propositions requiring their own particular proofs and to propositions, the truth of which is once and for all shown in logic. He includes in the latter type, for instance, experiences and immediate propositions or axioms. By experience Crusius means a proposition, in which the combination of subject and predicate or antecedent and consequence is perceived immediately through the sensation. He provides many different ways to divide experiences. Firstly, an experience can be an internal experience, which is perceived through internal sensation, or an external experience, which is perceived through external sensation. Then again, an experience can be a regular experience, where not just the combination of subject and predicate is sensed, but also the subject and the predicate themselves are something sensuous, or a reflective experience, where the combination of subject and predicate is sensed, but the subject and the predicate are abstract. Finally, an experience is either a pure experience or a mixed experience, where a proposition is connected with an experience through a deduction that is easy and requires nothing else, but assuming experience.
By an axiom or an immediate proposition Crusius means a proposition, in which is immediately perceived such a relation of subject and predicate that if we want to deny the latter, the subject cannot be thought anymore. He explains that an axiom differs from an internal experience, because in axiom we perceive through sensation only the connection of subject and predicate, since denying the connection would go against truth, but we still can think the concept of subject in itself without thinking the predicate. Thus, something might be an axiom, even if it is deduced through mere axioms, and furthermore, individual persons might require proofs of axioms, although by nature this axiom might not necessarily require any further proof.
Crusius divides axioms into three classes. Firstly, an axiom might be an axiom of identity, where denial of predicate would cause a contradiction. Secondly, it might be an axiom of causality, where a sufficient cause is connected to its nearest effect, without which the cause cannot be thought of. Finally, it might be an axiom of inseparable related concepts, where one simply perceives an impossibility to think the subject with the denial of predicate, although predicate is not the effect of the subject and denial of predicate does not cancel the content of subject and thus produces no contradiction.
Crusius also mentions hypothetical or merely seeming axioms. Such hypothetical axioms are generated by just arbitrarily concocting a definition or by joining such concepts, no necessity to combine which is immediately sensed. Such hypothetical axioms clearly cannot be assumed to be real axioms or even true propositions without proof. Thus, Crusius notes, if axioms are extended to include all propositions understandable from a definition, we must divide them into absolute or natural axioms and hypothetical axioms, because if this distinction is not made, anyone could prove anything by choosing suitable axioms.
Crusius notes that propositions requiring their own particular proofs could be divided into theorems, which are proven from many propositions and concepts together, and consequences or corollaries, which are proven from a single proposition assumed to be true. Then again, he continues, often to the definition of theorems is added the requirement that they must be very notable or not easily understandable and thus in need of a detailed proof, and then the propositions that would otherwise be theorems, but that are not significant enough are classed together with corollaries. Crusius suggests calling theorems in the first sense theorems by essence, while theorems in the second sense would be theorems by use. Furthermore, he mentions problems that are propositions explaining how a certain goal thought earlier in an undetermined idea should be determined. Such a problem, Crusius explains, is in its essence a theorem, but usually problems are distinguished from theorems.
Crusius explains that theorems and corollaries are not characterised by demonstrative method, because some sciences also use probable proofs, although mathematics by its nature uses always demonstrative methods. Thus, theorems of philosophy must be divided into demonstrative and probable theorems. One type of a probable theorem, according to Crusius is hypothesis, by which he means a proposition that is at first posited only as possible and then justified as probable by showing that it corresponds to actual conditions.
Crusius points out still further kinds of propositions. These include rules, which show a certain way to act, questions, which state something without affirming or denying it and posit a goal to decide whether it should be affirmed or denied, divisions, which aim to represent all possible determinations that an undetermined concept can have according to some aspect, and lemmas, which are borrowed from another science or from another part of same science to the current topic as grounds for proving certain things.
After concepts and propositions, Crusius turns to proofs, which he defines as distinct representations of a connection of a proposition with one or several other propositions that are assumed to be true, so that from this connection one knows that with the position of the truth of the other proposition also the first must be admitted as true. He notes that a proof concerns either the truth of a judgement of understanding or the reasonableness of one’s method, and this latter type he calls justification.
Furthermore, Crusius divides proofs into a priori proofs, where truth of a conclusion is derived from a ground that makes it true in such a manner that from this ground one understands not just that, but also why the conclusion is true, a posteriori proof, where one derives the truth of conclusion from a ground that only shows that the conclusion is true, but not why it is so, and mixed proofs, which share properties of both a priori and a posteriori proofs. He also divides a priori proofs into hypothetical proofs, where conclusion is contained in its justifying ground as a part, so that when the ground is thought distinctly, the conclusion is found and also assumed with the ground, and absolute proofs, where conclusion is not contained in its justifying ground, but only understood as a consequence inseparably connected with the ground. On the other hand, Crusius notes that a posteriori proofs prove either also the necessity or only the truth of the conclusion.
Crusius suggests further ways to divide proofs. Firstly, he says, proofs are either proofs according to truth, where the grounds of the proof are taken as true, or proofs according to a human, which show only that someone must admit something as true, because they assume certain propositions as true. Furthermore, Crusius notes that the method of proof is either demonstration or probability. Of the two methods, demonstration shows that the opposite of the conclusion cannot be thought at all, showing either that the opposite contradicts itself or that it cannot be thought due to the essence of our understanding. On the other hand, the method of probability shows that although the opposite can be thought, it cannot be assumed here as true. Crusius notes that probability is either common or infinite, so that the propositions shown by method of probability are either fully certain or reliable. Finally, he points out that usually proofs follow the propositions to be proved, and if they precede these propositions, they are called deductions.
At the very end of the chapter Crusius finally mentions the other way to divide thoughts, that is, according to their purpose. In this regard, he notes, thoughts are presented either because of objective or subjective causes. Presenting thoughts because of objective causes, Crusius explains, means that the nature of the topic in itself demands this, since the topic couldn’t otherwise be known distinctly and determinately enough, even by a trained understanding, or it could not be sufficiently proven or significant aspects of it would be ignored. According to him, all definitions of thoughts fall under this class.
Presenting thoughts because of subjective causes, Crusius continues, means that they are presented only because of subjective conditions, although the study of the topic in an abstract sense does not demand them, assuming that the reader of the study has been acquainted with science and its practical application. Thoughts presented in such a manner, he says, are called remarks or scholia. Scholia can then be divided into explanatory remarks, which improve the knowledge of the topic, for instance, by making it easier to understand, giving it a more secure foundation or giving insight how to use it, and into remarks that serve merely the enjoyment of the reader. If a scholium is of significant size, Crusius notes, it is called a digression.
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