Crusius thinks it is not necessary to name all possible types of deduction, since this has nothing to do with understanding why the deductions work. Instead, he will concentrate on the most important rules that govern deduction and only later mention some of the more prominent types of deductions that have been considered important enough for being given a name. Now, rules of deduction, Crusius continues, concern only conceptual connection of the very highest level of abstraction. While he admits three fundamental principles of knowledge, all of which could be used for establishing axioms – the principle of contradiction (nothing can both be and not be at the same time and in the same sense), the principle of inseparability (things that cannot be thought without one another cannot exist without one another) and the principle of incompatibility (things that cannot be thought together cannot exist together) – he thinks that only two of them can be used to establish rules of deduction: the principle of contradiction and a specific instance of the principle of inseparability, namely, the principle that everything has to have a sufficient cause.
Crusius begins from rules of deduction that he thinks can be derived merely from the principle of contradiction. I shall not describe in detail all these rules, but merely list them and point out some prominent features Crusius emphasises, if needs be:
The immediate rules of contradiction
1) No proposition can be at the same time true and false.2) Every proposition is either true or false. Crusius points out that 1) and 2) together imply that of two contradictory propositions, one is true, while the other is then false.
The rules of identity
3) Change in any arbitrary manner of thinking (e.g. change from active to passive voice) changes nothing in the truth or falsity of a proposition.4) A relation of two concepts based on their essence is not changed when the concepts are thought with different external abstractions.
5) What is true of something, when we think it through certain properties, is still true, even if we think it through other properties (e.g. same things are true of Venus, no matter whether we think it as the morning or the evening star).
Rules of diversity
6) If two objects cannot be distinguished in our thoughts in any manner, it is impossible to affirm the same predicate of the one and deny it of the other.7) If we have two different things and they can differ only in one aspect, they will differ in this aspect (otherwise they wouldn’t be different).
The rules of diversity, Crusius insists, lead us to what he calls the principle of sufficient foundation of knowledge: 8) it is not reasonable to assume something as true, if we do not have any sufficient reason for it. It is not at first clear why 8) should follow from 6) and 7), but his argument is that truth and falsity – admittedly very different things – must be distinguished through some criterion: either by our understanding immediately seeing what is true or by showing that what is to be taken as truth is connected to something we already know to be true.
Deduction from coordinated propositions: 9) if a proposition describes the only possible manner, in which other true propositions can be true at the same time, the first proposition is itself true.
Rules of deduction based on the relation of whole and parts
10) If a whole is posited, all its parts are also posited.11) If an essential part of something is lacking, it is not this thing. Crusius notes that this rule does not hold of mere natural parts, which could be replaced by something sufficiently similar without changing the essence of something.
12) If all parts are posited and combined in a manner appropriate to the whole, the whole is also posited, because in this manner the parts are equal to the whole.
13) What is in part is also in the whole. Crusius notes that we still might not be able to predicate the same thing of the part and of the whole (e.g. if there’s brown colour on a spot on a ball, there’s brown colour on the ball, but even if the spot is brown, the ball might not be brown, if it has spots of other colours also).
14) What belongs to all parts belongs in the same way to the whole, as long as we are not dealing with an abstraction of parts as parts. Thus, Crusius exemplifies, if all individual changes of a temporal series are contingent, the whole series is contingent, but although a tile on the floor is rectangular, the floor itself might not be. Furthermore, he emphasises, the rule can be applied only if the same thing belongs to all parts in the same manner and for the same reason: if all molecules of a salt cube can be dissolved in water, then all can be, but if all ingredients of medicine are poisonous, the medicine might not be, since the poisonous effects of different ingredients might be different and cancel one another.
15) What can be denied of all parts and is not an abstraction of whole as whole can be also denied of the whole.
Rules for logically subordinated concepts
16) What can be universally affirmed or denied of a subject can be similarly said of concepts logically subordinated by the subject (that is, individuals and species belonging to the subject).17) If an idea B is in logical subordination to another idea A and a third idea C is logically subordinated or opposed to the idea B, the idea C is logically subordinated or opposed to the idea A, at least particularly, but it is subordinated only if A and C are not two different species of B (otherwise, they would not share any individuals), and a sign for this is that the proposition where C is predicated of B is universal or if one can universally affirm of idea B the idea A.
18) If an idea is posited, so is also its proprium, same holds also of naturalia, but only in a limited manner.
19) If a relation toward an idea is posited, the same relation toward its proprium or genus must also be admitted, insofar as one regards the proprium or genus according to same manner of abstraction as in the original idea.
20) If all species are denied, then genus is denied, and if all individuals are denied, then species is denied.
21) What holds of all logical parts must hold of the whole, that is, what holds of all individuals must hold of species, and what holds of all species must hold of genus.
22) What belongs to species as actual belongs to genus as possible (e.g. if a human being can be learned, then an animal can be learned).
23) What belongs as possible to genus does not belong as possible to each species (e.g. an animal can be a monkey, but, for instance, a lion cannot be a monkey), but if we want to expressly take something that is possible for genus to be impossible for a determined species, we must prove it to be impossible for this species.
Rules concerning relations
24) If a relatum is posited, its correlate must be assumed also. Crusius notes that such deductions have only a hypothetical power, as long as one has not at first proven that something is a relatum, for instance, if the world is an effect, it must have a cause, but we then still have to show that it is an effect.25) If an idea has a relation towards another, the other has the opposite relation to the first.
26) If an idea C presupposes another idea B and this a third idea A, C also presupposes A.
27) If in continuing relation the first member relates to second as the second to third, the first member relates in the same way to the third, as long as the distance between the related terms is not important (e.g. descendant of your descendant is also your descendant, but child of your child is not your child, because the concept of a child inherently involves the notion that it is an immediate descendant).
Rules concerning magnitudes
28) The more to something belongs such features, which describe a certain essence, the more one must ascribe to it that essence29) In the same measure as the number of similar parts or units in an integral whole increases or decreases, the whole also increases or diminishes
30) If two magnitudes increase or decrease in same proportion, they retain the earlier geometric relation, and if they are equal, the relation of equality remains also, if to both sides is added or from the is taken away the same
All the previous rules, Crusius insists, depend merely on the principle of contradiction. Principle 39), he thinks, does not, that is, the principle of sufficient cause: all that is generated has its sufficient efficient cause. Crusius thinks this implies that it is generated through an active force of some substance, which has been active and in which nothing is lacking that is required for generating the generated thing. Although the principle of sufficient cause cannot be deduced from the principle of contradiction, according to Crusius, it can be proven from the principle of inseparability, because our internal sensation shows that it is not possible for us to think a generated thing without asking for a cause, from which it is derived.
The principle of contradiction, due to its generality, Crusius thinks, applies to everything, thus, it can also be applied to the relation of causes and effects. Hence, he concludes, there must be rules of deduction derived from both the proposition of contradiction and the proposition of sufficient cause. I will continue listing them:
Rules for the inevitability of effects
32) If it is assumed that a sufficient cause is active and unhindered, the effect is generated inevitably. Crusius adds that when an active cause acts, in addition to active force and what is directly dependent on it there are also other things that have an influence in the effect only through their existence and that thus can be called existential grounds. These existential grounds use no special rules, since, on the one hand, they are mere circumstances of the efficient cause, making it sufficient, and if they are taken as independent existential grounds, they are covered by rule 8.33) What is not a free fundamental activity of freedom is generated from its efficient causes inevitably in such a manner that the efficient causes could have generated it with the assumed circumstances only in this manner and not otherwise. Crusius emphasises that if we are dealing with free fundamental activities, we can only deduce that the activity has a sufficient cause, but not that this cause determines the effect inevitably.
Rules for modalities involving causation
34) What can be causally and distinctly conceived as possible, when some causes are assumed to exist, is actually possible.
35) What can be understood as inevitable, when some causes are assumed to exist, holds inevitably, insofar as new causes do not hinder it.
Rules for similarities and dissimilarities of effects
36) Similar sufficient causes generate similar effects.
37) Dissimilar and still sufficient causes must be dissimilar in their effects, insofar as they do not act according to different laws, according to which they would differ more than according to mere direction and magnitude. Crusius explains that acting according to different laws means that the causes differ in the constitution of their internal activity, so that different causes could be directed to similar effects due to different internal essences. He adds that such essentially different laws are even necessary, because otherwise the infinite cause or God could not achieve with their omnipotence what creatures can do with their finite powers.
38) Adequate effects of opposed causes are equally opposed.
Rules for proportionality of causes and effects
39) Each effect is proportional to its sufficient cause, and as the sufficient cause increases or decreases in its causality, so does the effect.40) If a cause vanishes completely, what is connected to it as an effect vanishes also. Crusius emphasises we are speaking of a case where the cause is not replaced by an identical cause.
41) There cannot be more in the effect than in the cause
Rule 42) states that nothing false can follow from a true proposition. Crusius justifies the inclusion of this rule as one following from both the principle of contradiction and the principle of sufficient cause by saying that a falsity can be derived from a true proposition neither through the proposition of contradiction, as its ancillary effect, nor as an adequate effect of the true proposition. The former is immediately absurd, he says, while the latter would mean that truth or a correspondence of thoughts with the objects should make falsity or their non-correspondence possible, which means that an effect would contain more than its cause. Crusius adds that something true can follow from a false proposition, because a proposition is false when any circumstance in it is false, but other circumstances in it can be true and from these true propositions can be derived.
Rules about causing wholes and parts
43) A sufficient cause that generates a whole generates also a part44) What acts toward a part acts also toward the whole
Rules about effects of effects
45) If the cause generates something, it is also the cause of what is inevitably connected with the first effect. Crusius adds that the mediate effects still do not always belong to the intentions of the distant causes, because intentions depend on insight and wisdom of the acting cause.46) If the cause generates something, it is at least a cause of possibility of effects depending on the effect. Crusius notes again that the original causality is still always expressly directed toward causing this possibility, because this possibility might just be an inseparable ancillary circumstance of its causality.
