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.