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

Listing all the possible rules of deduction, Crusius thinks, is the only systematisation of the doctrine of deduction needed. Still, he will go through some prominent types of deduction, often just to note their name, but sometimes also to point out some interesting features of this type of deduction. He begins by dividing the whole field of deductions into existential and casual deductions, depending on the type of connection of ideas on which the deduction is based. Furthermore, he notes that many of the actual deductions we make are cryptic in the sense that the order in which we go through it in our thoughts differs from the order of the listed rule of deduction used in that deduction. I shall again mainly list out the types of deduction Crusius introduces and give further details only when he himself considers it important.

Verbal deductions

• Deduction of equivalence uses rule 3) to derive from the truth of falsity of a proposition the truth or falsity of a proposition differing from the first only by some contingent formality (e.g. order of the words: justice is a condition of healthy state, thus, healthy state has justice as its condition).
• Deduction of external abstraction uses rule 4) to derive from an essential relation of two concepts that the concepts can be applied in same contexts when adding some external abstraction (e.g. being learned implies being intelligent, thus, a learned person is an intelligent person).
• Objective deduction is a subspecies of the deduction of external abstraction, where the concept mentioned in premiss is regarded as an object of another thing (e.g. wisdom is useful => a person who despises wisdom despises something useful).

Hypothetical deduction

Two propositions called antecedent and consequent are connected to one another and it is deduced either that when one is true, the other is also, or that when the latter is false, the former is also. Crusius notes that hypothetical deduction is actually just a combination of two types of deduction:

• The first type is, Crusius thinks, actually one form of the deduction of external abstraction: one thing is a sign of another thing, thus, the existence of one thing signifies the existence of this other thing.
• In the other type, the connection is based on an ideal or even real causality between antecedent and consequent (e.g. if God is just, evil is not unpunished, but God is just, thus, evil is punished).
• Crusius notes that both types of hypothetical deduction divide again into two subspecies, depending on whether truth is derived from truth or falsity from falsity. The latter kind uses the rule 40): from a true proposition cannot follow anything false.

Some deductions using rules for logical subordination of concepts

• Deduction by induction uses the rule 21) to argue that something, which belongs to all individuals or species, belongs universally also to the genus. Induction divides into induction from species and induction from individuals.
• Dilemma uses the rule 20) to argue from the denial of all species or individuals to the denial of genus (e.g. if God would be variable in their decisions, they would not have known everything from eternity or they would not have considered many things correctly or they would have changed something against the rules of wisdom, but all three possibilities are absurd, thus, God is never variable in their decisions).

Disjunctive deduction

Crusius is careful to distinguish dilemma from disjunctive deduction, where the rules of contradiction are applied to a disjunctive premiss by denying one or several disjuncts. Crusius distinguishes three possible types of disjunctive deduction:

• Using rule 1) to argue from positing of one disjunct to the denial of others
• Using rule 2) to argue from denial of one disjunct to indeterminate positing of one of the remaining disjuncts
• Using rule 2) to argue from denial of all disjuncts, but one, to the positing of the remaining disjunct

Crusius notes that the disjunctive premiss can also be implicit, especially when the disjunction is obvious: then we are dealing with an immediate deduction of opposition. Crusius outlines two subtypes of it:

• Deduction of immediate opposition in predicate, where the implicit disjunctive premiss opposes a proposition to its contradictory opposite and the deduction goes straight from the truth of one to the falsity of the other, without even mentioning the quantity in the proposition (e.g. the world is finite, thus, it is not infinite).
• Immediate deduction of opposition of copula, where the implicit disjunctive premiss posits the possible determinations of copula against one another in regard to their quantity and quality, in other words, the premiss says that a predicate must hold of all, of none or only of some individuals of the subject.

Crusius distinguishes disjunctive deduction from deduction by force of opposition, which uses the rule 7) and starting from two adequate opposites, where one knows the definition one opposite and the commonalities of both, argues that in denying the differentiating element of one the differentiating element of the other subsists (e.g. truth and error are adequately opposed, truth is the relation of correspondence between thought and its object and error is also a relation of thought and its object, thus, in error thought and object do not correspond with one another). Crusius notes that this deduction has some preconditions:

• The two concepts must be adequate opposites, that is, there must be no other possibilities in their proximate genus, so that either they two opposites are contradictorily opposed, or if they are contraries, one must otherwise know or postulate that there are no other possibilities.
• We should know how much in common both things have and what belongs to their proximate genus.
• We should know the complete definition of the one opposite.

Conversions and contrapositions

Crusius defines conversion as a deduction where the place of the subject and the predicate is swapped in such a manner that because of the truth of the original proposition, the truth of the converted proposition must be admitted. Crusius notes the following types of conversion:

• Pure conversion, where the quality of the proposition (i.e. being affirmative or negative) is not changed and which uses the rule 25) to deduce from the relation of the subject to the predicate to the opposite relation of the predicate toward the subject. Crusius notes that the possibility of pure conversion is determined by the logical relation between the ideas. If the predicate belongs universally to the subject, it must be a genus or a proprium, and in the former case, the predicate is more extensive, and in the latter case, their extension is identical, and since the original proposition does not reveal which case we have, the converted proposition can have only uncertain particularity (e.g. all virtues are laudable, thus, at least something laudable is a virtue). One exception, Crusius insists, is formed of quantitative propositions where a magnitude is determined by another, where the predicate must be a proprium of the subject (e.g. if a line cuts another one perpendicularly, a right angle is generated, thus, if lines cross in right angles, they are perpendicular to one another). Crusius adds also the familiar rules that a particularly affirmative proposition can be converted particularly, a universally negative proposition can be converted universally and a particularly negative proposition cannot be converted.
• Pure conversion divides into conversion simpliciter, where the proposition is converted without a change of quantity, and conversion by accident, where the proposition is converted with change of quantity.
• Crusius divides conversion of quantitative propositions into simple conversion, where the subject remains as it is, and separating conversion, where the subject of the original proposition is composed of several characteristics, only of some of which convert to the predicate. For example, knowing that if two triangles fit in same apex and have parallel bases, they are similar, we can deduce by simple conversion that if two triangles are similar, they fit in the same apex so that they have parallel bases, or by separating conversion that if two triangles are similar and their bases are placed in parallel, they fit somewhere in one apex.
• Contraposition is, for Crusius, a conversion where the quality of the original proposition changes. Contrapositions of universal affirmative propositions, where the predicate is essential to the subject, use rule 11), (e.g. all humans have a reason, thus, what has no reason is not human). Then again, Crusius adds, negative propositions, whether universal or particular, can be contraposed into particular affirmative propositions by combining deduction of immediate opposition in predicate with a deduction of pure conversion, if the negative proposition is first turned into a contingently affirmative proposition (e.g. a swindler does not act correctly, thus, a swindler acts incorrectly, thus, some incorrectly acting people are swindlers).
• Crusius points out that contrapositions of universal affirmative propositions can in some cases, like conversion of quantitative propositions, be divided into a simple contraposition or a separating contraposition. Thus, from the proposition that with all humans, insofar as their mind is not too distracted or malicious, conscience has occasionally regrets, we can deduce with simple contraposition that if the conscience of some person has never regrets, they are too distracted or malicious, but also with separating contraposition that if the conscience of an unmalicious person never has regrets, they are too distracted.

Relative deductions

Relative deductions, Crusius says, use some of the rules 24)–27) and differ from conversions in that they involve more than merely the change of the places of subject and predicate. Crusius divides relative deduction into the following subtypes:

• Deduction by rule 24) from a relatum to the positing of its correlate (e.g. order that the God has created in the world is a means, thus, it has a purpose). Crusius notes this deduction presupposes first justifying that the relatum is truly a relatum.
• Deduction by rule 25 from a relation to an opposed relation, which involves more than just conversion (e.g. God is the creator of the world, thus, the world is created by God). Crusius notes that this type could also be seen a one form of the deduction of equivalence.
• Deduction by rule 26) that if an idea presupposes another and this other a third, the first presupposes the third (e.g. guilt presupposes law and law presupposes lawgiver, thus, guilt presupposes a lawgiver.
• Deduction by rule 27) from the similarity of a relation between A and B and between B and C to a relation between A and C, ignoring the grade of distance (e.g. the effect is later than the nearest cause and the nearest cause is later than the more distant, thus, effect is later than the more distant cause).

Crusius notes that we could define deductions from whole to parts and from parts to whole, which would use some of the rules 10), 12)–15), 18), 19), 22), 23), 41) and 42), but sees no benefit in classifying these deductions under one name.

Syllogisms

Crusius begins his study of syllogisms by again emphasising the need to respect the manifoldness of deductions instead of just reducing all of them into syllogisms. In fact, he is even against placing all syllogisms into one mold, stating that when logicians usually describe the characteristics of syllogisms, they often concentrate on just the syllogisms of the first figure, believing that it is enough to transform the other figures to the first figure.

Crusius himself distinguishes the first figure of syllogisms by calling them subsumption deductions: the meaning of the name is provided by the fact that they use the rule 16), based on the relation of subsumption between the subjects and the predicates of the various propositions. He does understand why subsumption deductions have been preferred over other types, since the basis of our reasoning or all the primary or formal deductions are subsumption deductions, and furthermore, all other deductions can be brought into the form of a subsumption deduction through the following steps:

1. transform illogical abstractions into logical abstractions with relative deductions
2. use other deductions, like deductions of conversion, equivalence and external abstraction, to transform the propositions into an appropriate form
3. if necessary, add identical propositions and transform them into logical propositions
4. make the rule used in the original deduction into the first premiss of the new subsumption deduction.

Despite this possibility, Crusius still thinks that other types of deduction are of value, because they reveal other types of abstraction beyond the logical one. If our understanding could make only subsumption deductions, he argues, our deductions would have a mere hypothetical validity, since we wouldn’t have in our possession the real grounds implicit in the rules of other deductions.

Unlike subsumption deductions, Crusius insists, other syllogistic figures use the rule 17). His account of these figures follows, for the most part, the usual routes. The most important differences lie in his treatment of the fourth figure. Crusius rejects the so-called Galenic syllogisms, since he thinks they provide nothing useful for us, because they are so complicated that our understanding never really uses them. Instead, Crusius includes in the fourth syllogistic figure deductions from the first premiss that concept B is a predicate with a relation of subordination to concept A and the second premiss that the concept C can be affirmed or denied of the concept B universally to the conclusion that C must be affirmed or denied of A with the same universality. One might think that this type of deduction differs not so much from subsumption deductions, requiring only the swapping of the places of the premisses for its reduction. Crusius explains his choice by noting that this fourth figure, unlike subsumption deductions, can be always applied to disjunctive propositions. Crusius mentions two types of such applications:

• We can abstract from all the disjuncts together a common concept and predicate it of the subject of the other premiss (e.g futile desires reach their object or not, and when they do, they make us lose our piece of mind, because the achievement of one desire leads to further desires, but if they do not, they also make us lose our piece of mind, because the inability to achieve what we desire generates pain, thus, futile desires always make us lose our piece of mind).
• We can abstract from each disjunct particularly something and predicate disjunctively these abstractions from the subject of the other premiss (e.g. if the futile desires make us act, they either reach their object or not, and if they do, they produce more futile desires, and if they don’t, they cause pain, thus, if futile desires make us act, they cause either further futile desires or they cause pain). This type of fourth figure, Crusius thinks, cannot be reduced to a subsumption figure, because these cannot have disjunctive subjects in their premisses.

Comparative deductions

Comparative deductions use rules 28) and 39), and often also rule 30), determining magnitude of something from the magnitude of its essence or sufficient reason. Crusius specifies following types of comparative deduction:

• Simple comparative deduction argues that because a concept forms the essence or the sufficient cause of another, their magnitudes are also proportional (e.g. the essence of virtue consists in the correspondence of a moral state of a person with the law, thus, the greater the correspondence, the greater the virtue; or another example: the sufficient cause of the magnitude of angle of reflection is the magnitude of angle of incidence, thus, the greater the one, the greater the other).
• In complex or applied comparative deduction, the magnitude used as an epistemic foundation for another magnitude is determined more precisely and from this is deduced the determination of the other. This can happen in two ways, first of which is to determine the magnitudes only by their relation to a third thing (e.g. the stronger the motivation, the more certain are the actions based on them; motivations are stronger in in true virtue, based on an obedience to God, than in mere apparent virtue, based only on self-love; thus, actions based on true virtue happen more certainly than those based on mere apparent virtue).
• In second kind of applied comparative deduction, which Crusius calls deduction from greater to smaller or from smaller to greater, or in some cases, from equal to equal (e.g. healthy person has greater capacity to avoid being tired than an one afflicted with sickness; even a healthy person will feel tired when putting an effort to meditation; a sick person will be even more certainly tired in meditating).

Some mathematical deductions

• Arithmetical deduction uses the rule 29) to argue that an integral whole increases or decreases just like the number of similar parts increases or decreases (e.g. 2 + 3 = 5, 5 + 4 = 9, thus, 2 + 3 + 4 = 9). Crusius notes that this deduction cannot be presented as a syllogism, because there is no subsumption, but all three propositions are complex relative propositions and the idea of equality is their predicate.
• Common algebraic deduction uses the rule 30) arguing from two magnitudes increasing or decreasing in the same proportion that their previous relation remains. The name of the deduction, Crusius explains, is chosen because in algebra this deduction is used in finding the unknown in equations (e.g. it is known that x + 1= 2y – 2 – subtracting 1 from both sides gives x = 2y – 3; furthermore, it is known that x – 1 = y + 1, thus, adding 1 to both sides, x = y + 2 => 2y – 3 = y + 2 = x; adding 3, 2y = y + 5, subtracting y, y = 5 => x = y + 2 = 7).

Causal deductions

Causal deductions argue according to rules handling causes and effects to a combination between a cause and its effect. Crusius notes that comparative judgements are no causal deductions, although they use one of the rules for causes and effects, because comparative deductions are used only for determining magnitude and they merely assume the causal connection of things. The ultimate foundation of all causal deductions, he thinks, are the principles of sufficient cause and contradiction, the latter insofar as it is applied to causes and effects. Crusius finds the following kinds of causal deduction:

• In a deduction of perfect causal abstraction, effect is understood from its cause through mere immediate propositions. These divide into further subtypes, first of which is causal deduction of perfect possibility that shows using the rule 34) that certain effects are possible, when assuming certain causes. This type of deduction, Crusius explains, is applicable in cases where the effect depends fully or partly on freely acting causes, whereby one can only search for sufficient motives and other grounds of possibility.
• The second subtype of the deduction of perfect causal abstraction are causal deductions from determining causes, using rules 33) or 35), where the effect is inevitable with the presupposed causes. This subtype divides further into hypothetical causal deductions from determining causes, where the effect must follow if the posited causes are present and no new causes or obstructions appear, and absolute causal deductions from determining cause, where it is presupposed that nothing can obstruct the causes.
• Another way to divide causal deductions of perfect causal abstraction, Crusius adds, is their simplicity or complexity. Simple deduction of causal abstraction abstracts through immediate propositions from the only represented acting cause of a substance its consequences (e.g. in a compressed elastic substance there is a striving to expand that is obstructed, thus, if the obstacle is taken away, the substance actually expands).
• Complex causal deduction explains an effect from the nature of several causes taken together. In order to be valid, it requires a distinct concept of the effect to be explained, description of one or several causes together with their activities, explanation of the nature of the object and its influence, and if necessary, the derivation of the nearest consequences of each through axioms and the rule of causal deduction. In addition, the effect to be explained should be able to be abstracted from all of these together as an immediate consequence in the final proposition of the deduction (e.g. the effect to be explained is how writing with feather happens; by writing one understands drawing certain lines on the paper that are after this distinguished by their colour from the other parts of the paper; the efficient causes are ink that is liquid and heavy, and feather that has certain shape and that is directed by the hand of the writer; the paper as the object must be level and have enough glue, so that it won’t break down; deduction: if the feather is put down and the fissures are pressed slightly from one another, a part of ink flows to the paper underneath, and since the paper is unbroken, the outdrawn ink remains on it, thus, in the very same order as the feather is moved are generated certain lines on the paper, which due to the outdrawn ink differ by their colour from the rest of the surface).
• Crusius also divides deductions of perfect causal abstraction into affirmative and negative. Affirmative deductions of perfect causal abstraction show that a certain effect is understandable from certain causes, while negative deductions of perfect causal abstraction show that this is not so. Negative deductions might argue that the given circumstances do not yet explain the effect as possible or unavoidable or they might even argue that the effect is not even possible in the given circumstances.
• In addition to deductions of perfect causal abstraction, there are also deductions of imperfect causal abstraction, where effect is not understood from its causes through mere immediate propositions, but a combination between cause and effect is still shown. Such a deduction can again be either affirmative or negative. In an affirmative imperfect causal deduction one argues from effect in general to the existence of a cause, or one argues something that belongs to possibility of an effect and judges that such is present in the cause by rule (human thinks, thus, there is a power to think in the human), or one argues that where precisely this cause is again present, precisely this effect must also follow.
• A negative deduction of imperfect causal abstraction either argues according to rule 37) from the dissimilarity of sufficient causes to dissimilarity of effects or conversely according to rule 36) from dissimilarity of effects to dissimilarity in their sufficient causes, or one argues according to rule 38) from the contrariness of causes to the contrariness of their adequate effects (e.g. fear and courage conflict one another, and courage makes one accustomed to adventure thus fear hinders the tendency to adventure).

Practical deduction

Practical deductions evaluate whether given means are in fact means for the given goal. Crusius points out that practical deductions do not use particular rules and are not distinguished from other deductions by their form, but only by their matter. Still, they need to be mentioned separately in logic, he argues, because if they are not particularly explained, certain common errors concerning them do not become evident.

Crusius notes that in speaking of practical deductions we should especially study the mediating causes, which are used by a person for reaching their goal and are equipped with capacities that can wholly or partly generate the goal. These mediating causes or means should be real grounds for the goal and they must be in the power of the acting person, at least so far that the person can direct the means to generate the goal. Sometimes the direction of the means requires the constant activity of the person (for instance, when a person reads books to become learned in a subject), but in other cases the person needs to just trigger the means, but not sustain them.

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.