Chapter 11: The Idea of Pure Logic

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The goal of chapter 11, according to Husserl is twofold: to provide a general sketch of the analyses to come, and to clarify the concept of the Idea of Pure Logic.

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§62 the unity of science. The interconnection of things and the interconnection of truths

The question is “what makes science science”? (p. 225). In the psychological-anthropological sense, science is a unity of cognitive acts and relevant external factors. If we bracket aside the psychological and anthropological, the unity of science requires an ideal interconnection that contextualizes the real. In order to do science, there must be an ideal interconnection from which arises an ideal validity that determines a real human act as a relevant scientific act or as an irrelevant one.

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Husserl calls this ‘ideal interconnection’ ‘objective interconnection’, and distinguishes between two meanings. First, objective interconnection can be interpreted as an interconnection of the things our thought-experiences are intentionally directed towards. These can be actual or possible thought-experiences. Husserl used the qualifier “intentionally”, which is an important and foundational term for phenomenology. We will have to wait for Husserl to define it.

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Second, objective interconnection can mean an interconnection of truths, “in which this unity of things comes to count objectively as being what it is.” (ibid). This definition will require some parsing. A thing exists, and in its existence it is determined to be this type of thing and not another kind. This determination of the thing, that it is thus and not something else, is a truth, which Husserl calls the “self-subsistent truth” that is the necessary correlate to the self-subsistent being. (p. 225-226).

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There are two clarifications now concerning the relation between these two meanings of objective interconnection. First, they are mutually inseparable, “given together a priori” (p. 225). Husserl qualifies this inseparability as “self-evident” (p. 226). The reason for this inseparability is due to the fundamental correlative relationship between truth and the thing which the truth is about. Second, this inseparability is not an identity. A distinction must be made between truths and the interconnections of truths from the actual existence of things and their interconnections. To establish this distinction, Husserl points out that there are truths which correspond to truths as their objects, but these truths of truths do not coincide with truths of things. An example could be the distinction between a truth of a logical law versus a truth about the color of a fish. Hence Husserl wrote:

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Nothing can be without being thus or thus determined, and that it is, and that it is thus and thus determined, is the self-subsistent truth which is the necessary correlate of the self-subsistent being. (p. 225-226).

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Husserl now clarifies that he is using the terms ‘objectivity’, ‘object’, ‘thing’, in the widest sense. An ‘object of knowledge’, for example, can refer to either ideal or real things.

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Both types of interconnections are two kinds of unity. Through abstraction we can separate these to unities. It is in knowledge that we can in thought separate them. Knowledge is likewise used in a broad meaning to cover both a simple act of knowing and “logically unified interconnections of truth” (p. 226).

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Knowledge in its strictest sense, however, is an inwardly evident judgment. We will now finally receive an elucidation of what Husserl means by inwardly evident. When a object is given in this manner, it is given in a “primordial fashion”. Before us appears the state of affairs, and within it, the object itself. The object is present as it truly is, as it actually is. It is “known to be such”, that we know the truth that it is thus and not something other. We can focus in on this truth of the object being thus determined. If we reflect on this individualized truth, it becomes ideationally abstracted so as to become the theme of our apprehension. The apprehended truth is the ideal correlate “of the transient subjective act of knowledge…” (p. 227) and not some possible act of knowing.

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Corresponding to the interconnections of knowledge is the interconnections of truths. To this realm belongs science, which when objectively understood is concerned with unified truth. The unity of a scientific field is the general correlation between the unity of objectivity and the unity of truth. There must be a unity amongst the types of objects investigated and their corresponding ideal correlates of truth interconnections as a unified whole.

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§63 Continuation. The unity of theory

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The question now is “what constitutes the unity of a science”? (p. 227).  Husserl notes two mistakes: first, not every systematizing of truths into a single association constitutes as a science, for some systematizations are external ones; second, there must be a unification of interconnected demonstrations. Science must have a systematization and a unity of interconnected demonstrations, but these two conditions, while necessary, are not sufficient. One must have a ground.

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“Scientific knowledge is, as such, grounded knowledge.” (p. 227). Knowledge can be qualified as ‘grounded’ only once we see the necessity of something to be thus, “of its being so and so”, (ibid).  ‘Necessity’ here refers to “law-governed validity of the states of affairs in question”, (ibid). The ground of some thing is a law governance that this thing is thus determined and cannot be otherwise. The following two propositions hence are equivalent: “The state of affairs as a matter of law is to see its truth as necessarily obtaining” and “to have knowledge of the ground of the states of affairs or of its truth”, (ibid).

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A general truth can hence be defined as “the explanatory ground of a law from which a class of necessary truths follows.” (p. 228). This leads us to a distinction between individual and general truths. Individual truths are assertions concerning individual, singular, actual existents. Hence these truths are contingent and presupposes certain conditions and circumstances. Since individual truths belong to what is actual, these are factual truths. Husserl then defines a ‘necessary existence’ to be that which is based on the interconnection of a fact with another according to a law, such that its existence rests upon the laws governing the interconnection.

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General truths, on the other hand, are free of assertions concerning individual, actual existents, and yet are a general law by which they follow; they “permit us to infer the possible existence of what is individual.” (ibid). The proof of general truths leads us to laws that are in essence unable to be proven further. These are called basic laws. Basic laws have an apodeictic necessity of their being so and so, such that they cannot be otherwise, and this being thus determined is in essence, intrinsic to the kind of object we are dealing with.

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Husserl defines the ‘unity of a systematically complete theory’ as the “systematic unity of the ideally closed sum of total laws resting on one basic legality as their final ground and arising out of it through systematic deduction”. In contrast, a “looser” definition of a theory is a deductive system that does not have its final grounds in the basic laws, but grounds that lead us to them.

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Husserl ends this section by noting that there are two different meanings to the term ‘ground’; one, of course, in how Husserl elucidated it in this section, the other following the traditional language of logic. As Husserl observed, “every explanatory interconnection is deductive, but not every deductive interconnection is explanatory. All grounds are premisses, but not all premisses are grounds.” (p. 229). In logic, we have the tendency to call every premise a ground from which the consequences flow. This is an equivocation since there are different kinds of premises. If a deduction obeys laws, then the conclusion will follow according to the law, but, as Husserl cautions, the conclusion need not follow from the laws. If something follows from the laws, then we obtain Husserl’s pregnant meaning of the term ‘ground’.

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§64 The essential and extra-essential principles that give science unity. Abstract, concrete and normative sciences

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The central question of this section is what constitutes as the unity of truths belonging to a single science? Husserl distinguishes between two types of principles of unity, essential and extra-essential.

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Scientific truths are essentially unified if their connections rest upon grounded knowledge. Such an essential unity of truths within a single science forms a unity of explanation that has as its goal the basic laws which are the principles of explanation. This unity in explanation is theoretical unity. A science is hence called a theoretical science when it is ideally closed, embracing all possible facts with its principles of explanation having a single basis. Theoretical sciences are nomological in that their unifying principle is law. Theoretical sciences are distinguished from both practical and normative ones.

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Several sciences, however, take into account external factors. There is a unity of the thing to be scientifically investigated. “One connects all the truths whose content relates to one and the same individual object, or to one and the same empirical genus.” (p. 230). These are called concrete sciences and include history, geography and astronomy. Concrete sciences are also descriptive, in that they are fixed by the type of object or class investigated. Now, since concrete sciences can have varying understandings and explanations of the empirical unities that lead to divergent theories, this type of unity is called extra-essential.

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Concrete sciences derive all of their theoretical content from nomological sciences. Nomological sciences are the ground and foundation upon which a concrete science can be built. There is hence a hierarchy, not in terms of value, but an epistemological one, where nomological sciences are required for the concrete sciences to pursue their own objects of inquiry. In doing such inquiry, concrete sciences tend towards the nomological in attempting to provide general explanatory theories to build science, since such unity in explanation is properly speaking the domain of nomological sciences.

A third type principle of scientific unity, and also another extra-essential one, are those sciences unified by an evaluative interest determined by a basic value or basic norm.

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§65 The question as to the ideal conditions of the possibility of science or of theory in general. A. The question as it relates to actual knowledge

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The question Husserl wrestles with now is what are the conditions for the possibility of theory in general? Theories consists of truths, which are interconnected through deductions, and as such, this leads to the questions of what is the conditions for truth in general and the conditions for deductive unity?

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This question is treated in two manners: A as it relates to actual knowledge, the content of §65, and B, as it relates to the content of knowledge, which forms the material for §66. The sense of the question can likewise be taken in two senses, psychological and ideal. The psychological sense, in general, raises the question of the possibility of any humans obtaining knowledge, and which Husserl stated he was ignoring.

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Husserl recalls a distinction he made in §32 (chapter 7) concerning ideal conditions. Ideal conditions are either noetic or logical. Noetic conditions have their conditions in the Idea of Knowledge as such, Knowledge is the a priori grounds for them.  Logical conditions, on the other hand, are grounded purely in the content of our knowledge (p. 233).

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It is worth now recalling our discussion from chapter 10. In that chapter, we saw that Husserl clearly rejected a Kantian interpretation of understanding and reason as a faculty of man (cf. §58). The question was whether Husserl rejected the concept of a faculty of reason in a more traditional sense. Man must certainly possess some kind of faculty or power in order to do science itself, and here Husserl clearly states thus. We read:

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It is evident a priori, as regards the former, that thinking subjects must be in general able to perform, e.g., all sorts of acts in which theoretical knowledge is made real. We must, in particular, as thinking beings, be able to see propositions as truths, and to see truths as consequences of other truths, and again to see laws as such, to see laws as explanatory grounds, and to see them as ultimate principles etc. (p. 233).

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In contrast to this human element in being able to do science, there exists inwardly evident truths that remain thus regardless of whether or not we have insight into them or not. They possess validity independent of the human constitution and his cognitive acts. This hence raises the question of the ideal conditions for the possibility of man’s knowledge of these a priori laws. A priori laws express the conditions for knowledge in general, which are the noetic conditions. The conditions for deductive and theoretical knowledge is the content of knowledge itself, this is their ground.

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§66 B. The question as it relates to the content of knowledge

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The questions concerning the possibility of knowledge has led us to basic laws with roots purely within the content of knowledge while being independent of any knowledge acts of a knowing person. The question concerning the possibility of knowledge can be raised in reference to theoretical knowledge, which is what Husserl has been focusing on, or in application to theory itself. Theory itself is a “certain ideal content of possible knowledge” (p. 234). Just as how a single truth can be known in many knowledge acts of different persons with a correspondence between the ideal content of truth and these knowledge acts; theory itself as an ideal, identical content, corresponding to many individual combinations of knowledge wherein the same theory is known, like how the same truth is known across different knowledge acts. Theory itself is hence made up of “purely ideal elements, of truths, and that in purely ideal forms, those of ground and consequent.” (ibid). Possibility is here defined as the “obtaining” of the concepts in question.

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The critical question now is “what therefore constitutes the ideal essence of theory as such?” (p. 235). What is clear is that the ideal concepts which delimit the possibility of theory as such must themselves be the measure by which any particular theory can be considered a theory. The more a particular theory deviates from this ideal basis for theory as such, the less it can be called a theory. The logical justification of a concept is achieved by a returning, a going back to the essence. To discuss the logical justification of a theory then, requires us to return to “the essence of its form”, to the ideal constituents of theory as such. The basic laws and concepts as the ideal constituents of theory as such regulate in a priori manner all particular theories. Husserl concludes this section with the following noteworthy summary and observation:

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To point to profounder insights and justifications, serves to bring out the supreme value of the theoretical investigations which help to solve our suggested problem. We are dealing with systematic theories which have their roots in the essence of theory, with an a priori, theoretical, nomological science which deals with the ideal essence of science as such, and which accordingly has parts relating to systematic theories whose empirical, anthropological aspect it excludes. (p. 236).

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§67 The tasks of pure logic. First: the fixing of the pure categories of meaning, the pure categories of objects and their law-governed combinations

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Over the next three subsections Husserl is going to outline his program for investigating pure logic. First, the investigation of pure logic must clarify the primitive, core concepts, those which ‘constitute the Idea of unified theory” or are necessarily connected through ideal laws (p. 236-7). We are interested in investigating the form of theory, which requires stripping away concepts with “concepts of concepts” to reveal the pure, ideal core; concepts such as Concept, Proposition, and Truth. These likewise require investigations into the elementary connective forms, such as conjunction, disjunction, and hypothetical. Other related categories are those such as Object, State of Affairs, Unity, Plurality, Number, Relation, and Connection, which are ideally related to the previously mentioned concepts. These concepts are “the pure, the formal objective categories.” (p. 237). We must be clear that these concepts are, as pure, formal categories, independent of any particular piece of knowledge, even though all knowledge presuppose them and are ordered by them. Our task is the phenomenological origin of these concepts, by which Husserl means “insight into the essence” of these concepts, (p. 238). We are seeking clear, fixed, unambiguous meanings of these concepts.

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§68 Secondly: the laws and theories which have their grounds in these categories

The second area of research is the pursuit of the laws that provide the objective validity of the formal structures developed. In reference to the interconnection of knowledge, the truth and falsity of meanings as such, and in reference to the interconnection of things, the “being and not being of objects as such…” (p. 238). These laws will have the widest universality which “span all possible meanings and objects” that every science and theory must obey to be valid, (p. 239).

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§69 Thirdly: the theory of the possible forms of theories or the pure theory of manifolds

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Once the Idea of Theory is laid out, this science of sciences will “point beyond itself” to a “completing science”, a science that will in a priori fashion informs us of the forms of possible theories, all regulated according to the laws of Theory as such. The task is hence shifted from investigating Theory as such to possible theories.

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§70 Elucidation of the Idea of a pure theory of manifolds

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The title of §69 calls the theory of possible forms of theories the pure theory of manifolds, yet this section never mentioned what a manifold is. Now Husserl briefly describes what he means by manifolds. The theory of manifolds is a mathematical concept which inspired Husserl, and he utilizes in his investigation of the Idea of Theory itself. Mathematics during the second half of the 19th century saw several revolutions, due to several luminous visionaries such as Cantor and Riemann. One of the goals of this revolution was being able to explain the general foundation of all mathematics. When this task is turned to numbers, this inquiry became set theory, and when turned to space, the branch of mathematics called topology. In either case, several particular theories and concepts are generalized into overall reaching theories. Topology seeks to develop generalized concepts of space that apply to different shapes and geometries, i.e., Euclidean geometry, non-Euclidean geometries, dimensions greater than three, etc. Today, set theory forms the basis for all mathematics.

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It is with these dazzling new insights into mathematics and its formalization across its branches that inspired Husserl to apply the same general movement and process to generalize all particular theories under the general theory of theories, i.e. logic. Hence, Husserl writes:

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The most general Idea of a Theory of Manifolds is to be a science which definitely works out the form of the essential types of possible theories or fields of theory, and investigates their legal relations with one another. All actual theories are then specializations or singularizations of corresponding forms of theory, just as all theoretically worked-over fields of knowledge are individual manifolds. (p. 241-242).

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Something to keep in mind here is that there appears to be a fluidity in Husserl’s concept of manifold. He will apply the term ‘manifold’ to the generalized objects of both topology and set theory, though strictly speaking of mathematics today, the two concepts, the topological and set theoretic, need to be distinguished. Manifolds also appear, in Husserl’s project in application to the theory of theories, as particular theories, whose generalization as the theory of theories becomes the theory of manifolds. The theory of manifolds in this sense now undertakes a philosophical meaning, given that ‘manifold’ is now used in application to all theories; as Husserl stated: “all theoretically worked-over fields of knowledge are individual manifolds” (ibid).

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A rather interesting comment Husserl made is the following. Once this project of generalizing mathematics is complete,

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It would be easy to show that a knowledge of the true intentions of such theories, as pure categorial forms of theory, would banish all metaphysical fog and all mysticism from the mathematical investigations in question. (ibid).

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Given that Husserl here is arguing for essences and forms of theories, deeply metaphysical terms themselves with Platonic connotations, what does Husserl mean by “metaphysical fog” and “mysticism”? This is a peculiar statement. Cantor, whom Husserl was familiar with, spent a lot of time wrestling with the philosophical and theological implications of his set theory, wherein he argued for the Augustinian view, which is again deeply Platonic. Does Husserl mean here the overthrow of the Aristotelian philosophy of mathematics? For this was certainly one of the great insights of 19th century mathematics. At any rate, Husserl’s comments here seem odd given that he is essentially presenting a new Platonism (as the early school of phenomenology interpreted his views, thinkers such as Roman Ingarden, Dietrich von Hildebrand, Edith Stein, and others).

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§71 Division of labour. The achievement of the mathematicians and that of the philosophers

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This section is dedicated to outlining in brief the role of the mathematician and of the philosopher in the study of pure logic. Both have a stake in its investigations. Mathematicians, according to Husserl, have already worked out some elements of pure logic through formal mathematics. In this day and age, mathematicians have come to take over logic, and in doing so, have advanced logic to staggering degrees while bypassing traditional prejudices. No one can forbid the mathematician from using his craft to further develop the laws and relations of pure logic. Here Husserl is quite strong in his judgment on the importance of the mathematician for the study of logic.

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Only if one is ignorant of the modern science of mathematics, particularly of formal mathematics, and measures it by the standards of Euclid or Adam Riese, can one remain stuck in the common prejudice that the essence of mathematics lies in number and quantity. It is not the mathematician, but the philosopher, who oversteps his legitimate sphere when he attacks ‘mathematicizing’ theories of logic, and refuses to hand over his temporary foster-children to their natural parents. (p. 244).

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Mathematics is inseparable from logic, for in Husserl’s view, only mathematics can provide us with a theory that is systemically closed and complete. The mathematician assumes the role as an “ingenious technician”, for he is not the pure theoretician, but the one who constructs the formal interconnections. It is the philosopher who provides the epistemological ground for the work of the mathematician in their common pursuit of pure logic.

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§72 Broadening the Idea of pure logic. The pure theory of probability as a pure theory of empirical knowledge

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The empirical sciences are not exact. The question is what are the ideal conditions of the empirical sciences in general. Empirical sciences cannot be reduced to the theoretical, for they include facts about objects. The whole of optics, for example, is not the math behind it, but requires taking into account the actual optics themselves and their properties. Empirical sciences have basic laws that are probable, but can never be established through insight to be apodictically certain. They are probable. These general observations led Husserl to then claim that the a priori basis for the empirical sciences is the Idea of Probability, and not the Idea of Theory. Probability hence represents a second great foundation for the development of logic as a technology.

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The question that arises then, is what is the role of the Idea of Theory, which is the theory of theory? For now it appears that there is a theory not dependent upon the Idea of Theory. I think Husserl would say something along the lines of the Idea of Theory is a requirement for the investigations into the Idea of Probability, but that probability represents its own category; and hence why Husserl states that probability is included within pure logic if we broaden its meaning, (p. 247).

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At  this point we have finished the Prolegomena, and can now begin Logical Investigations.