Crusius, Christian August: Road to certainty and reliability – Probability

Until now, Crusius has dealt with demonstrations, where, he says, all alternatives are refuted as unthinkable, until only one remains. Now we are about to start the study of the method of probability, where a proposition is held to be more true than false or even fully certain, but alternatives or opposites can still be thought of. Crusius divides probability into three species: verisimilitude or what is more clearly possible and more to be assumed true than its opposite, reliability or what deserves to be taken as true in such a measure that one could act on it without further consideration, and moral certainty or what we consider to be undeniably certain, even though we can think of its opposite.

Crusius notes that reasons for taking something as probable can be cognitive in the sense of concerning just the characteristics of propositions and their relations toward other propositions, while some reasons concern also the connections of things with purposes. Starting with the merely cognitive reasons, Crusius considers the matter of probability, from which probable propositions are made, that is, logical possibilities or propositions that cannot be demonstrated true or false – he adds that actually even demonstrated propositions can be considered logical possibilities, if we ignore their demonstrations.

Crusius divides logical possibilities into perfect logical possibilities, where we understand how the predicate can be true of the subject, and imperfect logical possibilities, where we cannot understand this, but observe no contradiction in combining the predicate with the subject. Perfect logical possibilities have two levels, depending on whether we understand the possible connection of the subject and the predicate through an observation of an actual example or through mediation of other ideas. Crusius also distinguishes true logical possibilities from apparent or verbal possibilities, where at first sight the proposition seems to be irrefutable through demonstration, although further investigations show otherwise. Furthermore, he distinguishes logical possibility of propositions from metaphysical possibility of a thing that can be thought of, but might not exist.

How probable propositions can then come out of logical possibilities? Crusius notes that if we cannot prove a proposition, we are inclined to doubt its truth and even deny it. Then again, if the opposite of this proposition assumes even more without any proof, this makes us more inclined to reject the opposite. Then the original proposition is deemed probable. The fundamental essence of probability, Crusius says, is thus that a logical possibility assumes less without demonstration than its opposite. In other words, all probability is generated from the improbability of the opposite.

Crusius notes that sometimes the reason for assuming the probability of a proposition lies only in the subjective circumstances of the individual thinker: then we are dealing with subjective probability. Then again, the reason might also lie in the nature of things and the universal essence of reason and then we are dealing with objective probability. Thus, Crusius argues, subjective probability often vanishes with a more thorough investigation, for instance, when we find new alternatives we have not investigated, while an objective probability must remain constantly probable.

Crusius divides probability also to common and learned probability, depending on whether one only indistinctly feels the reasons for taking a proposition as probable or knows them distinctly. Furthermore, the learned probability has, he says, two different levels, depending on whether we can give the reasons of the probability only according to matter or whether we can evaluate it also according to form, that is, whether we can recognise the logical signs of probable and take them even to the highest grade of probability. By these logical signs of probable, Crusius appears to refer to rules, by which to determine whether something is probable. Just like with rules of deduction, I shall mostly just list them, using the six categories Crusius classifies them into.

Rules belonging to consideration of multifaceted possibility

1) The basic rule of this category: something that can occur in many different manners is more probable than something that can occur in less manners (e.g. because there are more ways to lose than to win a tournament, it is probable that our team will lose it)

2) If one of many possible alternatives must be true, the probability of one alternative is less, the greater the number of other possibilities (e.g. the more there are teams in a tournament, the less probable the win of our team is)

Crusius notes that both 1) and 2) work as stated, only if there are no other reasons that will change the scales between the alternatives, for instance, when a number of attempts is multiplied (e.g. if our team takes part in several tournaments, the probability it will win at least one of them increases).

Rule of consideration of chance coincidence of possibilities

3) A proposition that assumes the co-occurence of many possibilities by chance is improbable. Crusius notes that this rule does not work, if the proposition speaks of deliberately arranged coincidences. Furthermore, he admits that chance sometimes imitates foresight, if the number of attempts increases indefinitely.

Rules belonging to the consideration of more real possibility

4) The basic rule of this category: the more real possibility is probable. By a possibility being more real than another Crusius means that more of the causes and circumstances required for the existence of the possibility are already known to exist and can be presumed as existing. These causes and circumstances can also include signs of things, such as their effects, and partly even our thoughts of them, so that we can know at least their possibility through our understanding of them, and the more distinctly we understand the issues in question, the greater their possibility, which leads to further rules.

5) If we distinctly understand the way in which a predicate belongs to a subject in a possible proposition, the proposition is more probable than if we do understand this (e.g. a mechanical explanation of a physical phenomenon is more probable than an explanation through unknown forces).

6) If we already know an example where the possible proposition is actualised, but the same is not true of any of its alternatives, this proposition is more probable (e.g. if we know how a phrase is to be interpreted in one passage, it is probable that it will be interpreted in a similar manner in another passage).

7) The more we already know of the parts of what is assumed in a proposition or of the existence of their cause, the more probable this proposition is (e.g. if we can explain a phenomenon through known causes like air and salt, it is probable that no hidden causes are required to explain it).

8) If we know of only one alternative possibility and if it is improbable that we would have missed a possibility, then this possibility is probable. Crusius underlines that this rule leads to objective probability, only if we have so much experience of the issue or so much practice for this type of thinking that we can presuppose that it would be too great of a coincidence that only a single possibility would occur to us (thus, if we are not doctors, we should not say that the only cause of sickness we can come up with is the probable one).

9) If two things seem to be similar or dissimilar and it is improbable that we should find no reasons to suppose otherwise, their apparent similarity or dissimilarity is probably true.

Rules belonging to the expectation of similar cases or analogy

10) The basic rule of this category: what has been constantly encountered is also probable in all other and future cases, insofar as it is probably based on a common essence or a constant external cause. Crusius notes that the restriction of the conditional clause to probabilities is important, because otherwise we would be dealing with a demonstration. Furthermore, he adds, the rule is also based on another probability, that is, that we have experienced enough examples to draw the probable conclusion. Finally, Crusius notes, the conclusion cannot be made without the assumption of common essence or constant external cause (e.g. although I would have won all the poker games thus far, there is no guarantee that I will win the next one, since winning poker games is not part of my essence nor guaranteed by some constant cause).

11) If we have encountered something often, it is probable that we will encounter it many times in the future, as long as we cannot state any reason why it would be at least as possible that it would happen otherwise in other cases. Crusius notes that this probability is at least subjective, but it can be also objective, if there are good reasons to assume that it would be highly improbable that we would have experienced only examples justifying the probability.

12) What we have most often encountered in earlier examples is to be expected more than its alternative in any individual case.

13) What we have perceived to occur in one exemplary case occurs probably at least most times also in other cases of the same kind, insofar as it is probable that the similar cases have similar sufficient causes. Crusius notes again that if the conditioning clause would not be probable, we would be dealing with a demonstration. Furthermore, he points out that this rule presupposes the use of two other rules: we must use rule 8) to conclude that the sufficient cause we have envisioned is probably the only possible explanation, and we must use rule 9) to conclude that the cases are probably similar.

Rule or the consideration of conflict with known causes

14) What conflicts with existing sufficient causes is improbable.

Rules belonging to the correspondence with phenomena or circumstances

15) If a possible proposition corresponds with already known phenomena better than its alternatives, it is thereby probable. Crusius defines phenomena as something that is already otherwise known demonstratively or probably and that has a possible causal connection with what is assumed in a possible proposition, which is then called a hypothesis. Crusius divides phenomena into two kinds: mere phenomena that just possibly corresponds with the hypothesis and harmonious phenomena that agree also with one another and must therefore be derived from a single hypothesis.

16) Hypothesis based on two harmonic phenomena is more probable than alternative hypotheses based merely on a number of mere phenomena.

17) If we must count the phenomena backing the hypothesis up, we must divide the phenomena into smaller phenomena that can exist without other phenomena (thus, phenomena that are intrinsically linked to one another must be thought as a single phenomenon).

18) When counting harmonic phenomena, the correspondence of the phenomena creates a new phenomenon backing up the hypothesis (in other words, adding another harmonic phenomenon to a known phenomenon already raises the sum of the phenomena into three, because we have a) the original phenomenon, b) the new phenomenon and c) the correspondence between the phenomena).

Crusius points out that not all phenomena are of equal strength, because some of them may have internal weaknesses that cancel their power to prove things. He identifies three possibilities that can create weaknesses: the matter of a phenomenon might be uncertain, the possible causal connection between the hypothesis and the phenomenon might be uncertain, and the phenomenon might concur as well with other possible hypotheses.  Hypotheses, Crusius continues, can also have their weaknesses or difficulties, if they conflict with actual circumstances. In order to avoid difficulties, he explains, we can sometimes add subsidiary hypotheses. Such an addition makes the hypothesis a more complex possibility and thus sometimes also more difficult to prove.

Having gone through the various rules for deciding that something is probable, Crusius notes that they can be applied either to individual or to universal propositions: in the latter case, the probable proposition is called a presumption. Thus, Crusius divides all probabilities into presumption probabilities, which are proven by subsumption to a presumption that has been proved earlier, simple correspondence probabilities, which are known through a correspondence with the phenomena, and mixed cases involving both kinds of proofs.

An important type of presumptions are logical presumptions, which are the most general types of presumptions related to the first five of the six rules of probability:

1. presumption or rarity (we presume improbable what happens only in rare cases),
2. presumption of miraculous chance (we presume improbable what involves a chance coincidence of many possibilities)
3. presumption of more real possibility (we presume probable what is most real) and its subtypes, presumption of sufficient cause (we presume that an effect will follow an unhindered sufficient cause), presumption of unapparent cause (in issues we are experts, we presume improbable what has no causes) and presumption of deficient power in given cause (we presume improbable what the well-known given causes have no power to produce)
4. presumption of analogy of perpetual and frequent (we presume probable what we have encountered perpetually or frequently) and the presumption of uncommon, divided into presumption of fully uncommon (we presume improbable what we have never witnessed) and the presumption of rare (we presume improbable what we have rarely witnessed), and 
5. presumption of repugnant or unnatural (we presume improbable what is opposed to existent and sufficient causes).

Crusius still goes through methods for deciding what to choose when different rules of probability contradict one another – in essence, these methods rely ultimately on the very notion of probability. I shall already move on to the next subject, that is, the additional weight of non-cognitive issues or purposes and goals in deciding what is probable. Crusius notes that this new weight might be derived either from obligations involving prudence (do this if you want to achieve that) or from obligations of moral law (do this in order to follow the will of your ruler). In the first case, he explains, it might be prudent for us to act according to what is probable, while in the second case, we might be obligated to act according to what is probable. This is especially true when the probability is what Crusius calls infinitely large, when we can be certain that any future evidence will just reveal examples justifying its probability.

Crusius suggests that God specifically intended the humans to have only probable knowledge of certain key truths. Firstly, he explains, this is due to essential limitations of human nature, but secondly, God also wanted us to fulfil the duty of trusting God with less than perfect knowledge of the things. Indeed, Crusius thinks, even our belief in the veracity of demonstrations is often based on probability, especially if the demonstrations are very complex: we might have made an error following the demonstration once, but it is improbable that we have gone through its details many times and always found it convincing. In fact, he concludes, we often find probable arguments more convincing than demonstrations, since a single fault in our line of reasoning will make demonstration faulty, while probable arguments depend on an accumulation of many facts speaking for the intended conclusion, so that even if one supposed fact is revealed to be faulty, the whole argument does not stand on it alone.