Crusius, Christian August: Road to certainty and reliability – Rules of deduction

Crusius divides deductions or arguments into demonstrative deductions, where the premisses make us unable to think that the conclusion would not hold, and probable deductions, where the premisses make it more difficult to deny than to assume the conclusion. He also states that probable deductions differ from demonstrative only through their matter, while the forms of deductions are always demonstrative and thus always connected with the fundamental principles of deduction.

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 essence

29) 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 part

44) 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.

Crusius, Christian August: Road to certainty and reliability – The ultimate foundations of knowledge

After propositions Crusius proceeds in a rather conventional manner to deductions or arguments, where the truth of a conclusion is based on the assumed truth of premisses. He underlines that the conclusion of a deduction is not held to be true because of its content, but because of the specific relation it has to these premisses. Thus, Crusius suggests distinguishing this relation as the form of the deduction from its individual propositions (that is, premisses and the conclusion), which then work as the matter of this deduction, while the form provides the rule that the deduction follows. He also emphasises that although we can express deductions in words and usually communicate them to others through this medium, this use of words changes nothing essential in the proceedings and can therefore safely be ignored.

Before going on to classify the rules followed by different deductions, Crusius suggests investigating the first principles on which all these deductions are based on. He begins from the capacity of human understanding to think, combine and separate concepts, but notes that it has its natural limits: there are certain seeming concepts or their combinations or separations that we cannot think of. Since human souls strive naturally toward perfection, Crusius argues, and in case of understanding this implies a natural drive to truth, we should follow this natural disposition and accept as true (and respectively as false) what we cannot think as anything but true (and respectively as anything but false).

All our thoughts originally derive from external sensation, Crusius begins his study of these natural dispositions. External sensation provides us with concepts of objects, and as long as a distinct sensation continues, he underlines, we are forced to think of these objects as existing and present. Such an external sensation also makes our internal sensation active, and through this internal sensation, Crusius explains, we become conscious of characteristics, parts and circumstances of external sensations. Then again, he adds, such external sensations induce in us also other concepts that are not externally sensuous and that lead us to universal propositions that we should think as true.

It is no surprise that the first of these universal propositions Crusius mentions is the so-called principle of (non-)contradiction. He at once adds that this proposition as such is just identical, saying nothing more than that what is, is, and what is not, is not. Thus, Crusius argues, the principle of contradiction requires the establishment of further concepts, to which it can be applied, in order to make any use of it. The established concepts might not refer to anything real, he states, and then the applied principle of contradiction has a merely hypothetical reality. Hence, Crusius insists, we must know from elsewhere that the concept in question refers to something real.

Crusius suggests as a further rule for the establishment of concepts following the essence of our understanding: combining concepts that the sensations represent as combined or that we are necessitated to think as combined, because disappearance of one makes the other one vanish also, and similarly separating concepts that sensations represent as separated. Concepts established in this manner generate propositions that are not identical and form the positive core in our knowledge, Crusius thinks: every force is in some subject, all that is generated is generated by a sufficient cause, every substance exists somewhere and at some point in time etc.

Crusius insists that these propositions are not merely derived from the principle of contradiction, but adds that this does not mean that these propositions are not certainties: it is merely a question of how they are generated in our understanding and how we thus come to know them. Furthermore, it does not mean that we could not deny the opposites of these propositions through the principle of contradiction, but only that this principle is not sufficient for establishing the concepts involved. Finally, it does not mean that the principle of contradiction itself would be uncertain, but only that it is empty and thus not the only principle of human certainty. For example, Crusius explains, the principle of contradiction easily explains that every effect presupposes a cause, but only because by an effect is meant something generated by a cause and thus the concept of effect involves the concept of cause. Furthermore, he adds, the concept of effect still refers only to a hypothetical reality, since we do not know what things are effects: even seeing something being generated doesn’t tell us that it is an effect, since causeless generation is just absurd, but not contradictory.

The argument of the emptiness of the principle of contradiction is probably targeted against the Wolffian school, who were famous of basing intricate ontological truths on it (of course, it might well be, and I’d argue that it is so, that especially Wolff himself referred by the principle of contradiction to a stronger principle that is not identical, but this is beside the point here). Crusius still finds some reasons why anyone would hold the principle of contradiction to be the only principle of our knowledge. Firstly, he says, demonstration from the principle of contradiction seems comparatively easy, since otherwise we would have to use internal sensation and attention to account for the physical possibility or impossibility of our thoughts. Furthermore, Crusius thinks, the principle is the only fundamental proposition required for pure mathematics, and someone might want to extend the indubitable certainty of that discipline to the whole of knowledge. Finally, he concludes, we are more used to deducing from already presupposed concepts than searching for the ground of reality in the establishment of concepts.

The true fundamentals of all our deductions, for Crusius, are then three propositions: 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 adds at once the cautionary remark that while the first principle is indubitable – no understanding could think of any contradictions – the two others might mislead us, since due to our finite nature, we are sometimes incapable of thinking things that an infinite understanding can think. Especially if the two principles are used to contradict the very principle of contradiction, Crusius underlines, we should reject this use as unfounded. Furthermore, he adds, revelation by a higher form of understanding can show us truths we cannot comprehend: thus, we should accept such a revelation, especially if we can have at least a symbolic concept of what is described by the revelation. Crusius gives as an example the case of limits: we can never think anything existent without the notion of limit, but this does not mean that something unlimited couldn’t exist, just that we are too limited to think about it. Similarly, although we cannot think of two pieces of matter existing in the same place, this does not mean that God could not be omnipresent in every part of the universe.

Crusius proceeds to explain how the other true propositions can be generated from the three principles. Firstly, he begins, we can apply these highest principles to propositions discussing certain characteristics of objects that we cannot think without these characteristics: the connection between the object and the characteristic forms then an axiom. Then again, Crusius continues, we can also apply the three principles to such relations between propositions, where one proposition must be true if the other propositions are assumed to be true: this relation is inscribed in some rule of deduction or argumentation. Just like axioms are immediately connected to the three principles, the rules of deduction connect all the other true propositions to them, he concludes.

The justification of axioms and rules of deduction, Crusius insists, is an example of what he calls a subsumptive deduction, where the conclusion concerns a concept or an individual or a set of individuals contained in a concept that is the topic of the premisses. Yet, he clarifies, this does not mean that all deductions should be subsumptive, unlike the Aristotelian tradition had assumed. Indeed, he separates primary or formal deductions, which justify axioms and rules of deduction and are always subsumptive, from secondary or material deductions, which concern more specific topics and which might not be subsumptive. Thus, Crusius exemplifies, while we can subsumptively deduce from the principles the rule that if a cause brings about a whole, it brings about its essential parts, when the rule itself is used, the resulting deduction is not subsumptive. True, he admits, all deductions can be transformed into a subsumptive form, but this is just jugglery that hides the essential manifoldness of types of deduction.