Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth - Methodology of learning

As I have said for a number of times, Wolff's German logic is essentially a book on methodology. We have already seen how Wolff has tackled the methodology of mathematical sciences – especially syllogistics – experimental sciences – for instance, Wolff knew that correlation does not always imply causation – and even history.

But almost half of the book is aimed not so much for easing scientific or philosophical work, but for teaching skills necessary for university studies. Thus, we find Wolff giving hints about how to evaluate cognitive capacities in oneself and in others, how to evaluate new discoveries and writings, how to interpret books, how to dispute, and finally, how to develop one's powers of thinking. I shall not comment on these parts of the book extensively, but merely raise some interesting points:

1. Wolff makes the observation that if a person is to have an ability for making discoveries in some field of science, it is not enough that she knows the necessary data and that she knows the general rules of reasoning. In addition, the person must also have practice in how things are justified in that precise field. In a sense Wolff is here going against the idea that there is some unitary superscience, like physics is sometimes presented nowadays to be, with which problems in all fields of life could be solved. Wolff thus accepts that result of one science cannot be directly transferred to another, because one must also learn how things are justified in this second science.
   
   
2. When discussing the interpretation of Bible – a task probably very important at the time – Wolff announces the following hermeneutical principle: because the Bible is, by definition, a book made by a good understanding, we must assume that apparent contradictions etc. are caused by our incapacity of combining the correct meaning with the words used. Interestingly, Wolff notes that the same principle should be applied to all books that are ”made with understanding”. In principle Wolff is here assuming the so-called principle of charity that we should interpret a writing in the best manner that is possible.
   
   
3. Wolff also admits that all writings should be evaluated only according to the context of their own time. This is something that philosophy scholars sometimes forget to do.
   
   
4. Wolff thinks that it is more virtuous to leave disputations with people who clearly do not know anything about the issue talked about than to continue arguing with them. This is a very appropriate advice in the age of heated and polemical Internet discussions.

With these few words I wrap up Wolff's German logic. My next guest shall be finally someone else than Christian Wolff. Who is the philosopher? Well, he is probably so obscure that most of the readers wouldn't even know him if I revelaed his name, so I shall leave the revelation to the next time.

Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth - Levels of certainty

The Platonic analogy of a broken line is probably familiar to all philosophy enthusiasts. Just like the knowledge of shadows and pictures is to the knowledge of real things, so is the knowledge of the whole world of sense experience to the knowledge of the world of ideas, and how the knowledge of the sense experience is to the knowledge of the ideas, so within the knowledge of the ideas is the doxa to the episteme. I am not doing Plato-study here, so I won't consider more closely how e.g. the doxa is to be differentiated from the episteme. What is interesting here is the idea that certainty of knowledge comes in grades: for instance, that knowledge of sense experiences is unreliable compared to our knowledge of mathematical issues.

This idea was inherited by later philosophers and eventually also reached Germany, where Wolff finally translated the Platonian classification of the levels of certainty to German, although Wolff apparently left out the lowest rang of the Platonic ladder. Comparing to the sense experience Wolff speaks of Glauben, Platonic doxa is replaced by Wolff's Meinung and the highest level of episteme has been transformed into Wissenschaft. The three terms play an important role in the later German philosophy, so let's have a look at them in more detail.

The term Wissenschaft or science is already familiar to us. One might wonder why Greek episteme corresponds with science, when words like ”epistemology” suggest that the Greek original has something to do with knowledge in general. Yet, if we look at how Plato and Aristotle used the word, science or Latin scientia is a very apt translation. For instance, in Aristotle's Posterior analytics episteme refers to a deductive system of knowledge based on indubitable axioms and definitions. As we saw in the previous text, in Wolff this mathematical ideal of science has already been replaced by a more modern notion of science as based on both axioms (in mathematics) and reliable experiences (in experimental sciences).

The meaning of Wolff's Meinung or opinion is also easy to understand, although unlike the Greek original, Wolff appears to evalue opinion as the lowest cognitive state. Opinion is essentially a weaker version of science: ”If we assume definitions that appear to be possible and in inferences assume some axioms, which appear to be correct, although we have not yet demonstrated them, and which we cannot corrobarate through indubitable truths – then we arrive to opinions”. That is, opinions might be argued for, but they still lack the ultimate certainty of science based on incontrovertible truths: if I have an opinion of something, things might still be different than I think. Furthermore, opinions are more subjective than science, because my opinions might known to be false by another person. One might even think that one's opionions are scientifically certain, if one is not aware of how things are demonstrated in science.

It is Glauben that is the most distant from its Platonic predecessor, pistis, but this just reflects the development of the Greek word. While for Plato pistis referred simply to sense experiences, even in Aristotle's Rhetoric pistis meant conviction and trust invoked by a good speaker, while in Pauline letters pistis refers to the first member of the triad ”faith, hope and love”. German Glauben means similarly both belief and faith.

Wolff's use of Glauben reflects Aristotle's rhetorical use of pistis: Wolff understands by Glauben the approval that is given to a statement because of a testimony of someone else. Yet, Wolff extends the role of such a conviction on a testimony from mere judicial matters. While opinion is only a sort of diluted version of science, Glauben is the counterpart of science. Remember that for Wolff science, at least in humans, deals only with possibilities, for instance, with what can be done: these are the things that can be demonstrated. What has actually happened, instead, is beyond scientific proof and we just have to believe the testimony of our own senses and of others, when it comes to such historical questions.

In addition to methodology of mathematican and experimental sciences, Wolff's logical work then also contains the rudiments of a methodology for history. An important element in these rudiments is to recognise how reliable a person describing some events is. Wolff suggests several rules of thumbs how one could decide e.g. whether a witness would have some reasons for lying about what has happened, but does not move further beyond such rules of thumbs.

Although Wolff appears not to use Glauben in the sense of religious faith, we might apply his definition also to faith. Then religion and faith would become intersubjective, communal issues. Having faith on certain religious dogmas would mean being convinced that the people ascribing to those dogmas are reliable witnesses who have no reason for lying on such matters and who are linked through a chain of equally reliable persons to an original witness who was there to actually see what the holy books describe.

Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth - Rationalism vs. empiricism

A common trope in philosophy text books is the supposed battle between rationalists and empiricists, in which the first wanted to base all knowledge on reason and the latter on experience and which was finally solved by Kant who discovered that knowledge was based on both reason and experience. It takes no great historian to discover that this simple tale of two battling schools with three great names on both sides is largely fictitious, not least because e.g. Leibniz did not form a common school with Descartes and Spinoza, but opposed the two in some issues even more than he opposed Locke, the only empiricist of note to have written at the time.

I am not sure who actually invented the fable of the two schools of philosophy, but the first signs of it is the already familiar Kantian tale of Locke as the intuitionist and Leibniz as the intellectualist and Kant himself as the necessary symbiosis of the two. But even after Kant this paradigm was not a given when interpreting the history of philosophy. For instance, Hegel distinguished empiricism from metaphysical school, which apparently included, in addition to the traditional rationalists, ancient philosophers like Plato and Aristotle. Even the separating principle of the two schools is not the same as in the separation of rationalists and empiricists. Metaphysical school, says Hegel, based science on common experiences and analysis of these experiences, while empiricists tried to base it on individual perceptions and then noted that no science of necessities and universalities could be based on them – this description fits not Locke, the paradigm empiricist, but characterises at most a caricature of Humean philosophy. Furthermore, Kantian philosophy is for Hegel not a symbiotic combination of the two schools, but more like a modification of empiricism in the sense that both disagree with metaphysical school about the possibility of certain kinds of knowledge.

In place of a strict division of two schools, various premodern philosophers form then more of a continuum of different standpoints from, say, Spinozan axiomatics as the ultimate in rationalism to Humean bundle of impressions without any necessary connection as the ultimate in empiricism. What I would now like to do is to see where in this continuum Wolff's philosophy fits in. One would expect that Wolff as the supposed follower of Leibniz would be closer to the rationalist end of the line. Yet, Wolff is distinctly aware that many sciences can be based only on experiences. Indeed, in addition to the method of syllogistic reasoning, Wolff tries to describe, however crudely, a method of experimentation, by which basic propositions could be discovered.

Wolff defines experience as something that can be known through perception. Note that he does not identify experience with perceptions. Instead, experience is in a sense more stable than a perception: while a perception might vary from one person to another, experiences are only such perceptions that we know to be capable of being at least in principle communicable to other persons. Thus, experiences are essentially intersubjective.

Despite this stability, experiences deal still only with individual things and might even be deceptional, because human perceptions are not always reliable. Yet, Wolff admits that true universal propositions could be based on experiences. The method for this universalisation is careful experimentation: one varies the situation and so tries to determine the conditions in which the experienced phenomena appears.

Wolff's methodoloy of experiences appears surprisingly empiricist. Still, he is not a pure-bred Lockean, although he does mention latter's work favourably at the beginning of his logic. Wolff does accept also the possibility of substantial knowledge being based on self-evident or analytic axioms, as we already saw in his treatment of mathematics. In other words, analytical propositions are not empty or tautologies according to Wolff. Wolff's justification of the substantiality of such propositions is characteristically pragmatic and even pragmatist. An axiom or a definition can be informative, because it might help us to determine postulates, that is, self-evident ways to affect things. Thus, because we know what a circle is, we know also how to produce one, if suitable materials are given. Even logic is not for Wolff a mere formal system but a helpful tool as a methodology of science.

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.

Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth - Words and concepts

After Wittgenstein and the linguistic turn in the philosophy it is hard to remember that there was a time when there was no philosophy of language to speak of, at least beside few passing remarks. What little discussion of language there was, happened usually within another discipline, like logic or psychology. Thus, in Wolff's German logic, we find a whole chapter dedicated on the issue of words.

We have already seen Wolff telling how words are one means for representing things and thus could be used as one possible type of concepts in addition to mental images. A definition of words was still lacking, but Wolff is quick to provide us with one: words are signs for our thoughts that we use in communicating our thoughts to other people.

The role of communication is important in Wolff's definition. While the analytic philosophy of language began from semantic considerations, Wolff's starting point would be pragmatics, that is, the use of language in a concrete context of human communication. Thus, Wolff begins from the question of how a person can understand what the other is saying: firstly, Wolff says, the speaking person must think with the word a certain concept and the listener must think the exact same concept through the same word.

More important than the actual definition of mutual understanding is that the meaning of the word or the concept connected with it is then based on this universal communicability and it is not just assumed that words have some meaning. Thus, the semantics does not just float about without any anchor to the actual communication. I think that in analytic philosophy it was only Paul Grice who first thought of doing this – time would have been saved, if Frege and Russell had read some Wolff.

A consistent and subject-independent meaning is then something that is not instantly given. Instead, the meaning of the words must be decided in an interaction between speakers. This pragmatic nature of semantics leaves room for possible misunderstandings and arguments over the meaning of the words.

But it is not just other people's understanding of the words used that a speaker may fail to connect with. In addition, a speaker might connect no concepts with the words she pronounces. This lack of meaning is made possible by the difference between the phonological form of the word and the reference of the word. Thus, a person can well know the word in the sense that she has heard people using it and knows how to pronounce it, but she might not know what the word conveys, just like a theologician who has heard the word ”trinity” a lot during his studies and has so learned to use the word regularly, although he has no clear concept of the reference of the word – although Wolff is quick to admit that someone else might have this clear concept.

The most crucial lack in Wolffian philosophy of language is that Wolff provides us with no theory of how meaning of the words is carried into the meaning of sentences. He merely makes the analogy that as concepts are to words, such are Urtheil or judgements as unions of concepts to Satz or sentences as unions of words, and even here Wolff appears to ignore the distinction at times.

Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth - Clear and distinct concepts

Last time we saw that thoughts were for Wolff effects in soul connected with a consciouness of oneself and that sensations or perceptions as consciouness of things were a subspecies of thoughts. A concept, then, is according to Wolff also something in thought, namely, a representation (Vorstellung) of something in thoughts. Wolff's definition thus points out two essential characteristics of a concept: it is a mental event and not, say, an abstract ontological structure, and it is connected to an object that it represents.

Otherwise Wolff allows for quite a variability in concepts. The object of the concept or something represented (Sache) might be either concrete or abstract. Furthermore, the means of representation might differ e.g. from concrete images to mere words. Hence, both a mental image of the Sun and a verbal explanation of virtue fulfil the criteria of Wolffian concepts.

What I find interesting is that Wolff appears to nominate perceptions as the primary causes of our concepts. Sure, Wolff does recognise other ways to generate new concepts, such as abstraction or variation of characteristics in known concepts. Yet, all these other methods appear to demand that we already have some concepts from which to produce new concepts, while only perceptions can create concepts without the help of old concepts. Wolff leaves hence no room for the so-called innate ideas, which Locke had famously argued against. I shall probably comment on this when I shall discuss the relationship of Wolff's philosophy to empiricism.

Wolff does not just define, but also classifies concepts into a hierarchy of more and more perfect kinds of concepts. Wolff's classification was probably not original, but based on the tradition of logic preceding him. Still, Wolff was at least the first to translate these terms to German. As the German logicians used the essentially same classification and even Hegel comments on it, I shall introduce Wolffian hierarchy in more detail.

Descartes had spoken of clearness and distinctness as criteria for the reliability of perceptions, but as far as I know, he had never properly explained what he meant by these terms. Well, Wolff does that for him. Clear (klar) concepts are for Wolff such that they allow us to recognise things that they represent. If a concept is worthless for this purpose, it is obscure (dunckel). Thus, Wolff suggests, if we have seen a plant in a garden, but we cannot say whether we have seen that same plant in another place, our concept of the plant is obscure. Although the difference between obscure and clear concepts appears simple, Wolff suggests that there is actually a continuum of possible levels of clearness. Thus, we might be able to distinguish the forementioned plant from a dandelion, but not from a rose.

In a distinct (deutlich) concept, the level of clarity is so high that we can state what Wolff calls Merckmahle, by which the thing represented by the concpt is recognised. Not all clear concepts are distinct, which is proved by the case of colours. Primary colours are undoubtedly quite clear concepts for anyone with a normal vision, as we have no problem of separating e.g. blue from red. Still, we cannot really say what distinguishes blue from red, apart from one being blue and other red.

The notion of Merckmahle is somewhat undistinct itself, although it was widely used in the later German logical tradition and even by Kant. Yes, we do know that they help to distinguish things, but it is unclear what they are. Now, Wolff says in passing that Merckmahle are nothing more than new concepts. Thus, a distinct concept is such that can be distinguished from other concepts through yet another concept, just as we can distinguish a triangle from other polygons through the concept of three that characterises the number of the angles of a triangle. In other words, a distinct concept can be defined.

The definition in question might be either nominal or real. We have earlier seen how Wolff distinguishes the two: nominal definitions merely analyse the meaning of a phrase, while real definitions tell how a thing described by such a phrase can be generated. It is probably the latter sort of definition Wolff is referring to, when he indicates microscopes as a tool for making our concepts distinct -  microscopes cannot be used to analyse the nominal meaning of a word, but they might come in handy, when we want to know what a thing is made of.

The further stages in the hierarchy of concepts merely add more Merckmahle to a concept. In a full (ausführlich) concept, the given Merckmahle allow us to recognise the thing represented in all possible cases. That is, a mere distinct concept characterises a thing, but a full concept will truly define and identify it. Finally, in perfect (vollständig) concepts even the Merckmahle are clear and distinct concepts, that is, even the components of the definition can be further defined.

Wolff is keen to advertise that his writings consists of perfect concepts, while the concepts of his followers are often not even full – like Cartesian concept of matter, which fails to distinguish matter from mere space – and in some cases they are outright obscure. Wolff also provides a number of examples of full concepts. Some of them are rather amusing, like the concept of rain as many drops of water that fall side by side and one after another from a cloud through air. Others are nowadays quite controversial, like Wolff's concept of marriage as a union between a man and a woman for the sake of conceiving and raising children.

Now, it is not difficult to see that just like the notion of clarity, the notion of perfectness comes also in grades, because we can always ask whether the constituents of the definition can also be defined. Thus, a concept that is perfect enough for mathematical purposes might still require more analysis from the viewpoint of ontology. The natural question is then whether the perfectness has any limit, that is, whether there are concepts that can only be clear, but not distinct. Wolff suggests that there must be, in case of both nominal and real definitions. In nominal definitions the limit is reached when we find words that are undefinable, for instance, when we cannot explain anymore what green is, except by pointing out green objects. In case of real definitions, on the other hand, the limit is reached when we find things that cannot be generated – in effect, for Wolff, God.