Hoffmann: Study of reason – Principles of deduction

After judgements, a study of different modes of proof follow – this has been the official order since someone decided to place Analytics after On Interpretation in Aristotelian corpus. Somewhat refreshingly, Hoffmann leaves the classification of various types of proof to the practical part of logic, where we shall find it later. In theoretical part, he instead concentrates on some general features of all proofs and especially on various principles followed in proving.

Just like in case of judgements or propositions, proofs are not just strings of propositions for Hoffmann, but instead, processes of human thinking, in which we feel ourselves forced to move from some judgements to a new one. Note that Hoffmann is not yet interested whether these proofs have any connection to truth or not – this is the topic of the next chapter. Indeed, he admits outright that some proofs lead to false conclusions, even if they are so seductive to follow. Yet, now it is time only to describe the very act of following a proof, not give rules for determining a validity of a proof.

What is interesting in proofs is that we feel forced to follow them, no matter what the matter, that is, the topic in question. If propositions assumed just are of certain structure, then we feel obliged to draw some conclusions of them. It is then the form of proofs, embodied in some grounding principles, which are the topic now – nowadays we might speak of rules of inference, and it is important that Hoffmann clearly separates them from mere axioms, which by itself or as mere propositions do not force us to make any conclusions.

While Hoffmann accepts a number of grounding principles, he thinks they can all be based on three most basic principles. The first one should amaze no one. What understanding finds the easiest to deal with, Hoffmann begins his introduction of the principle, are those thoughts that require least amount of analysis. Such thoughts are being and non-being, which understanding at one notices to be impossible to combine. This is the basis of the principle of non-contradiction. Just like with Wolffians, the principle is for Hoffmann ontological, while the corresponding logical principle ”nothing can be both true and false”is based on the ontological principle.

For Wolff, the principle of contradiction was not just ontological, but apparently covered all cases of incompatible properties, since he used ”wooden” and ”iron” as an example of contradictory characteristics. Hoffmann, on the contrary, notes that, while undoubtedly a rule that understanding follows, such denial of incompatible properties must be based on another principle – what is incompatible in understanding must be accepted as incompatible in reality. Third is the corresponding positive principle – what is necessarily conjoined in understanding must be accepted as conjoined in reality.

This is actually all that Hoffmann presents as a theory of proofs – except he uses these basic principles to derive some further principles. I shall leave most of them unmentioned and concentrate on the most interesting of them, that is, the principle of sufficient reason or ground.As we have seen earlier, Hoffmann doesn't buy the Leibnizian principle as it is, but makes an important emendation – actions started by an immediate use of a force that does not need any external determination do not need to have any sufficient reason. The principle of sufficient reason in its Hoffmanian guise is based on the principle of necessary conjoinment – we must always connect an event to some force starting it, thus, it must have begun by an active force just described or by a force determined by something external.

What Hoffmann is suggesting is that a sort of Humean account of causality as a string of events with no intrinsic connection is just so alien to human thinking that we cannot accept it. Of course, this does not yet prove whether this human way of thinking is true – that type of question we have to leave to the next post.

Syllogistic 102 revisited

Just like the the investigation of judgements, the investigation of syllogisms is much more extensive in Wolff's Latin than in his German logic. While German logic concentrated on the so-called first figure of syllogisms and noted that all other figures could be reduced to the first figure, in Latin logic Wolff goes through even the second and third figures of syllogism. As I have adequately dealt with all these figures, I shall not touch this topic anymore.

This still doesn't mean that I could leave the rest of this text blank. The three figures all have categorical judgements as premisses, thus, syllogisms of these figures could be called categorical. As categorical judgements are the simplest type of judgements, Wolff also calls them simple syllogisms. In addition, we might also have syllogisms with non-categorical premisses, which appropriately are then called complex syllogisms. Wolff goes on to note that complex syllogisms include at least hypothetical syllogisms, with one hypothetical judgement as a premiss, and disjunctive syllogisms, with one disjunctive judgement as a premiss. There is no indication that these two are all the types of complex syllogisms, although Kant will later appear to assume this.

Just like with the three figures, Wolff is eager to show that we can simplify the variety of syllogisms. Now, I failed to mention last time that Wolff appears not to take disjunctive judgements in the form of the current propositional logic, that is, as a combination of propositions (p or q). Instead, he favours the notion of disjunctive judgements as a combination of concepts: A is B or C. Furthermore, while in modern propositional logic ”p or q” is meant to include the possibility that both p and q are true, Wolffian ”A is B orC” is clearly meant to indicate that A cannot be both B and C at the same time. Thus, a disjunctive syllogism is then of the form

A is B or C

A is / is not B

Then A is not / is C

Wolff points out that we could then understand the judgement ”A is B or C” as a combination of hypothetical judgements ”if A is B, then it is not C” and ”if A is not B, then it is C”. Clearly then disjunctive syllogisms can be reduced to hypothetical judgements.

Wolff continues by noting that an important part of hypothetical syllogisms, including all the disjunctive syllogisms, could be reduced to categorical syllogisms, namely, those in which both the antecedent and consequent of the hypothetical judgement have the same subject, in other words, if the syllogism is of the form:

If A is B, then A is C

A is B

Thus A is C

Wolff suggests that we could always read such a syllogism in the following form:

All Bs are Cs

A is B

Thus, A is C

Problem is that in the original syllogism the hypothetical might hold only for As. Consider the following deduction:

If a triangle is a figure with two equally large angles, then it is an equilateral triangle

This triangle is a figure with two equally large angles

Thus, it is an equilateral triangle

Clearly the corresponding categorical premiss ”all figures with two equally large angles are equilateral triangles” is false. A possible solution is to restrict the scope of the middle term in the following manner:

All As that are Bs are Cs

This A is B

Thus, A is C

Still, even if the reduction works with these hypothetical syllogisms, Wolff has to admit that it isn't so easy in those cases, where the hypothetical judgement doesn't have a subject shared by its antecedent and consequent.

Wolff also considers incomplete syllogisms, which Aristotle had in his Rhetoric called enthymemes. In practice, such entyhmemes are nothing but hidden syllogisms, where we leave some premisses implicit. Thus, if I deduce ”I am hungry, therefore I must eat” I am actually assuming the general premiss ”if someone is hungry, he must eat".

An important subdivision of enthymems is formed by so-called immediate syllogisms, which according to the tradition could be used to prove something without full syllogistic trappings: an example includes ”All As are Bs, thus, some As are Bs”. Wolff points out that we could add as a new premiss the tautology ”some As are As” and then we would have a normal categorical syllogism, albeit one with a tautology as a premiss.

Somewhat more surprising is Wolff's view that even inductions are enthymemes or deductions with implicit assumptions. Namely, if we conclude from the fact that certain individuals or species of things have a feature that this feature is shared by whole of their common genus, we have assumed a further premiss that what holds for an individual or a species holds also for superior genera. It appears problematic to suppose that induction could be deduction, but I would like to point out that it is not meant to be valid deduction, because one of the premisses might well be untrue – the inductive principle or possibility to generalise might be wrong either absolutely or under some circumstances.

I might finally note that Wolff describes a possibility to concatenate syllogisms to form chains of deduction or polysyllogisms and that the highest form of deduction or demonstration can use only axioms, definitions, indubitable experiences and previously proven propositions as premisses. Next time we shall see what all this has to do with truth.

Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth - Syllogistic 102: go figure

”That which belongs to all things of a kind must also belong to this that is of the same kind.”

”What is denied of a whole kind must also be denied of anyone of the same kind.”

These somewhat complex sentences Wolff calls the principles of syllogisms. They are supposedly not the final axioms of syllogistic, because Wolff thinks they are themselves based on the so-called principle of contradiction: a thing cannot both have and not have a characteristic.

The two principles could also be stated through three statements:

”A property C belongs or does not belong to all things of a kind B.”

”A is a thing of kind B.”

”Thus, the property C belongs or does not belong to A.”

or in a symbolic form:

B – C

A – B

Thus, A – C

Such a combination of three sentences is what has been traditionally called a syllogism. Actually all the sentences in a syllogism have their own traditional names. The first statement is known as a major proposition, or as Wolff calls it, an upper proposition (Ober-Satz), while the second statement is known as a minor proposition (in Wolff, Unter-Satz or lower proposition). A major proposition here characterises a certain kind or species and it often does describe a general law connecting two concepts. The minor proposition here states that a certain thing belongs to a certain kind and it often presents an example of a general species. The major and minor proposition together are called premises (in Wolff, Förder-Sätze or front propositions), while the third proposition is then the conclusion of the syllogism (in Wolff, Hinter-Satz or back proposition).

In the previous text I noted that syllogistic logic required only two divisions of judgements: to universal and particular and to affirmative and negative. Clearly then there are four different judgement types to consider: universal-affirmative (all As are Bs), particular-affirmative (some As are Bs), universal-negative (no As are Bs) and particular-negative (some As are not Bs). Now, Aristotle had painstaikingly investigated all the different possible combinations of two premisses and noted which combinations could be used as premisses of syllogisms. We need not bother with the details, but we may note that at least one premiss must be universal and affirmative.

Besides the judgements, the words or concepts in the syllogism have also traditional names. The subject of the conclusion, which in the example is also the subject of the minor proposition, is called the minor term (in Wolff, Förder-Glied or front term), and similarly predicate of the conclusion, which in the example is the predicate of the major proposition, is called the major term (in Wolff, Hinter-Glied or back term). The third concept, which, as it were, connects the minor and the major term, but vanishes when we come to the conclusion of the syllogism, is then called the middle term (Mittel-Glied).

In the example above, the middle term is in the middle of the syllogism in a very concrete sense, as it is the predicate of one and the subject of the other premiss. But we could also change the places of the three terms. For instance, we could place the middle term as the subject of both premisses:

B – C

B – A

Thus, A – C

Now, in this case the premisses tell that a certain species of objects is a common subspecies for two other species – and nothing else. Hence, the conclusion can at most be a particular judgement, some As (those that are Bs) are Cs. For instance, because bats are both mammals and flying animals, some mammals can fly.

We could also place the middle term as the predicate of both premisses:

C – B

A – B

Thus, A – C

In this case two affirmative premisses would tell that A and C share some predicate or are subspecies of the same genus. This does not by itself tell us anything new: two species of the same genus might have no common elements (like tigers and lions), but they also can have common elements (like teachers and writers, because a person can be both). More results are gained when one of the premisses is negative – one things has a predicate, the other does not, therefore, we cannot identify these things or even connect them in a judgement. Thus, because apples are plants, but bats are not, apples cannot be bats.

Aristotle classified the different syllogisms into three figures according to the three different positions the middle term could take. After Aristotle, people noted that there is actually a fourth possible figure. That is, we could reverse the positions of the minor and major terms in the first figure like this:

C – B

B – A

Thus, A – C.

Logicians quickly noted that the syllogisms of the fourth figure were not very helpful. It is then no wonder that some philosophers, like Wolff and even Aristotle himself, simply ignored it, and that Hegel mentioned it merely to make ridicule of the unnecessary complexity of syllogistic. Another piece of complexity one might also want to make fun of is the medieval invention of giving all the individual syllogisms a name of their own. Each of the four possible types of judgements was assigned its own vowel, and as every syllogism comes with three judgements, a name with just these three vowels was given to each syllogism. A famous example is Baroco, that is, a syllogism of the sort:

All gold is malleable,

But some people are not malleable,

So some people are not golden.

It seems unbelievable that one would try to argue for such an insignifanct conclusion with such complexities. And indeed, the name Baroco, or its modification, baroque, acquired later a meaning of unnecessary extravagancies. Indeed, as even Aristotle noted, all the other syllogisms could actually be based on the syllogisms of the first figure – and with his pragmatic nature Wolff instructs his students to ignore the other figures.

Despite the extravagant and unnecessary intricacy of syllogistic, we should not disvalue syllogistic completely. Syllogisms were the one form of argumentation by which from premisses known to be true one could infallibly deduce further truths. Of course, this infallibility is also based on knowing some truths beforehand: from false premisses syllogisms can produce both true and false conclusions. In the traditional terms, the truth of the premisses is what makes syllogisms into demonstrations. There have been various suggestions as to how one can find true premisses – we shall see how Wolff answers the question in the next text.

Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth - Syllogistic 101: the preliminaries

Ever since Aristotle's Posterior analytics, syllogistic logic had been a crucial part of philosophical methodology and at times methodology consisted of little else. There are at least two reasons why Aristotle thought syllogistic so important. Firstly, syllogistic was an improvement over Platonic dialectics, because it replaced individual arguments with a group of general schemes for constructing incontrovertible arguments. Secondly, the science most developed at the time, geometry, was easily converted into a syllogistic shape.

From Aristotle, the enthusiasism over syllogistic logic was transferred from one generation to another, and even when the fame of Aristotle dwindled, the syllogistic was still the core of the logic, and the only thing that truly threatened its position in methodology was the relatively young notion of experimental science. Thus, it is no wonder that Wolff is also obliged to give an account of syllogistic in his logic. Because the issue will undoubtedly appear in the future – Hegel at least loves the syllogism as a symbol – I shall expound in the following two blog texts syllogistic logic in more detail. Those who know the syllogistic by heart and those who are bored to death by formal logic can skip ahead.

Before going into syllogisms themselves, I shall say in this text something about their constituents. We have discussed concepts and words in previous texts, but the things between – that is, judgements – are still missing. Now, Wolff – and probably also many other logicians of the time – defines judgements in two manners. Firstly, in judging we supposedly think that a thing has or has not or could or couldn't have some characteristic: the thing is represented by the subject term and its characteristic by the predicate term, which Wolff calls respectively front and back terms (Förder – und Hinterglied). Secondly, the judgement is regarded as a combination of concepts, namely, the subject and the predicate.

The identification of these two definitions is problematic, because on a closer look they define two completely different things. As Husserl noted, thinking a combination of redness and ball or red ball is something else than thinking or considering the possibility or the fact that a ball is red – and as Frege would add, both are different from asserting that a ball is red. It is somewhat disturbing to think that quite a number of people had not noticed these what seem to be obvious platitudes.

A reasonable explanation for this apparent confusion is that Wolff and his fellow logicians had a different paradigm of judgement in mind. While the modern mathematical logic has taught to us to start from sentences like ”Mickey is a mouse”, where an individual is characterised in some manner, the Aristotelian tradition begun from sentences like ”Gold is malleable”. This is a case of a lawlike unification of two universal terms, and because of the lawlikeness, the assertion of their connection appears inevitable.

At least in case of Wolff, the explanation is made even more plausible by two facts. First fact is connected to a difference between universal and particular judgements, which Wolff equates with the difference between necessary or essential and contingent or accidental judgements. The equation itself is interesting, because it tells us something about Wolff's notion of alethic modalities: if all Xs are Ys then an X is essentially an Y, but if the connection between Xs and Ys is accidental, only some Xs can be Ys. Now, Wolff suggests that all the particular judgements can be turned into universal judgements by stating the conditions in which the particular connection of concepts is true: that is, if some Xs are Ys, then all Xs filling suitable conditions are Ys. What is important here is that Wolff clearly accepts that universal/necessary judgements are the norm to which all the other sort of judgements should be transformed.

Secondly, Wolff suggests that we are able to think or judge something through a given sentence only if the concepts combined in the sentence are distinctly known to conform with one another, while we are unable to think the sentence if the concepts are distinctly known to be contradictory; otherwise, we do not know whether we can think it or not. This might in itself sound completely harmless, but Wolff defines the conformity as the necessity of thinking one concept, when you think the other concept. In other words, in a true judgement two concepts must be necessarily or in a lawlike manner connected with one another – an accidental connection of characteristics is then not a true judgement.

A word on the classification of judgements. We have already seen Wolff divide judgements into universal or absolutely valid and particular or contextually valid. In addition, he mentions the division into affirmative (bekräftigende) and negative (verneinende) judgements. These two classifications are actually all that we need in syllogistics. Not only is Wolff then unaware of what Kant called infinite and singular judgements, but he and probably many other logicians fail to think the possibility of dividing judgements according to relation or modalities. Hypothetical and disjunctive judgements appear only in a place where Wolff shows how other deductions can be transformed into syllogistical form, while modalities are discussed in Wolff's ontology. When scholars then say that Kant merely assumed his category system from the logic of his time without any arguments, we might suspect that he actually just assumed the system, which was not based even in logic.