Introduction to my translation of Husserl's Logic and General Theory of Knowledge (Hua XXX)
By Claire Ortiz Hill
Logic and General Theory of Science is the translation of Edmund Husserl’s Logik und allgemeine Wissenschaftstheorie edited by Ursula Panzer under the auspices of the Husserl Archives in Cologne and published for the first time in 1996 as volume XXX of the Husserliana series. The main text of the volume is the final version of lecture courses on the philosophy of logic and science given by Husserl at the universities of Göttingen and Freiburg between 1910 and 1918. The course was initially entitled Logik als Theorie der Erkenntnis (Logic as Theory of Knowledge) and was taught in Göttingen for the first time during the Winter semester of 1910/11. According to Panzer’s introduction (p. XIX), looking back, Husserl characterized it as the final version of his Göttingen lectures on formal logic. He would teach it a second and third time during the Winter semesters of 1912/13 and 1914/15 under the title Logik und Einleitung in die Wissenschaftstlehre (Logic and Introduction to the Theory of Science) and a fourth time at Freiburg during the Winter semester of 1917/18 under the present title.
Panzer explains that these lectures were considerably reworked over the years, a fact considered to be of particular significance because those changes were made during a decisive period in the evolution of Husserl’s thought. She notes that Husserl wrote to Georg Misch on November 16, 1930 that, by the time his Ideas was published in 1913, he had already lost all the interest that formal logic and all real (reale) ontology had held for him in the face of a systematic grounding of a theory of transcendental subjectivity. And, of the very subjects making up much of Logic and General Theory of Science, he wrote to Hermann Weyl on October 10, 1918 that for twenty years a central part of his lectures had been the theory of functional judgments, of judgments with empty places, and the distinguishing of the different modes of this empty something, and in addition, the implementation of the fundamental distinctions between factual and formal ways of judgment, between proposition form and proposition (or judgment), proof- and theory-form and theory, as well as of the objective correlates associated with them. His concept of complete manifolds already acquired in the beginning of the 1890s had also, as had all the distinctions he had dealt with extensively in lessons in Göttingen, proved especially fruitful. But he added that, in spite of all that, and of all the work he had devoted to it, he had not found the time and serenity to carry that train of thought completely to the end, because it had had to be more important to him to develop his ideas about transcendental phenomenology.
Prepared for oral delivery in the classroom, these lectures are refreshingly lively and spontaneous, clearer, more explicit and readable than the books Husserl published during his lifetime. Appendices II and III convey the enthusiasm for logic and philosophy of science that he was intent upon injecting into his students by means of them. There he explains that through his lectures he was seeking to provide something entirely different from what could be learned from books, because lectures do not in general exist for the purpose of replacing books, or to be spoken books or excerpts from books. He warned students that the usual expositions of logic are dangerous for beginners, because they only too easily deaden their sense of genuine scientificity and cover up “the difficulties from them, like chasms covered over with greenery”. It was his desire instead to introduce them to the inmost essence of work in logic and critique of knowledge, to the nature of their problems and methods and prepare them to be able to derive benefit and excitement from reading important writings in the field, something which is achieved by laying bare the essential meaning of centuries of efforts as they relate to the state of things at the time and its insights, by imparting an understanding of the problems, goals, methods and a sense of the deeper meaning of the theoretical efforts “in which the greatness, grandeur, and force of philosophical science lies”, by teaching them to sense the inmost spirit of the intentions of logic and theory of knowledge and to feel profoundly concerned by the fact that the problems to be surmounted were of grave import for anyone interested in ultimate truth and philosophy.
The only justified task of the teacher, he declares, is to train beginners to philosophize. If ‘philosophy’ is the word for the highest aims of knowledge and the sciences directly oriented toward them, he suggests, then the discipline we want to devote ourselves to is in fact first philosophy, because it is a matter of investigations that must precede and be attended to before further philosophizing can be contemplated in earnest. He defines his mission as being one of raising students above the naïve standpoint and showing them “the way to the mysterious solitudes in which one day the sphinx of knowledge must unveil its riddle”. He says that he could also call it the way to the “mothers” of knowledge, to the essence-principles of knowledge in terms of its ultimate origins. Regarding these mothers, he says that he loves to recall Mephistopheles’ words in Goethe’s Faust: “Enthroned sublime in solitude are goddesses. Around them is no place, still less any time”. Mephistopheles, he reminds students, had described that solitude in a horrifying manner and tried to deter aspirants from venturing into those untrodden, and not to be trodden, places where they would see nothing “in interminably empty farness”, not hear their own footsteps, not find anything firm where they rested. Husserl, however, counseled his students to be unafraid and to respond with Faust, “Just keep on, we want to fathom it. In your nothing, I hope to find the universe”.
These lectures find Husserl enthusing that even though formal logic belongs to one of the oldest sciences of mankind, everything in it is evolving; the “ossified concepts and formulas of the tradition are coming alive again, moving, evolving” (§19b). In Alte und Neue Logik, taught in 1908/09, he speaks of what a delight it is to be alive and to share in striving after the greatness coming into being in those days, which are not, as often said, times of decadence, but rather the beginning of a truly great philosophical era in which age-old goals will finally be met at the cost of truly heroic strain from toil, and new, higher, goals will everywhere be held out. We in modern philosophy, he said, are no less than visionaries (Phantasten). We have the courage and determination of the highest goals, but we strive after them on the most reliable paths, those of patient, constant work.
When trying to answer questions that Husserl’s philosophy has raised, philosophers of logic, science, and mathematics often find that he seemed to assume his readers to be in possession of facts about the development of his thought that are not found in the writings published during his lifetime. By making available clear, explicit, and revealing discussions of important topics less thoroughly and clearly treated in those writings, the publication of his lecture courses is gradually providing really significant material for putting together a complete picture of his teachings on crucial matters. By providing material necessary for assuring that his ideas will one day have the impact on the history, philosophy, invention, and pedagogy of modern logic, set theory, and the foundations of mathematics that they should have had from the beginning, these lecture courses afford needed insight into how the father of phenomenology managed to exercise the profound philosophical impact he did on the times that gave birth to twentieth-century philosophy.
This particular volume is a very important link in the chain of ideas connecting Husserl’s Logical Investigations of the turn of the century to his Formal and Transcendental Logic of 1929 and helps to subvert many idées reçues about the development of his thought. For instance, it routs the widely held view that he lapsed back into psychologism after trouncing it in the earlier work. Indeed, these lectures show him still pursuing the course set out in that work well into the second decade of the century and displaying utter consistency with stands he had begun taking toward, for example, categoriality, analyticity, axiomatization, completeness, Platonism, idealism, empiricism, logicism, manifolds, mathematics, anti-psychologism, objective and subjective presentation, pure logic, dependent and independent meanings, during the 1890s and which were still pursued in his latest work, topics, moreover, that were of fundamental importance to the makers of twentieth-century philosophy in English-speaking countries and still are to their followers (ones enchanted by Michael Dummett’s interpretation of Gottlob Frege’s writings, for example).
In addition to the topics just mentioned, philosophers trained in the Analytic tradition are presently especially well primed to appreciate a wealth of interesting insights published here, for example, those into (in alphabetical order): the word ‘all’, demonstratives, existential generalization, extensionality, formal logic, the foundations of arithmetic, functions and arguments, identity/equivalence/equality, imaginary numbers, logicism, mathematizing logic, modality, philosophical grammar, philosophy of language, quantification, relations, sense and meaning, set theory, states of affairs, theory of inference, theory of judgment, theory of probability, truth, wholes and parts. This is of particular significance because these subjects have long been the bailiwick of philosophers who, loath to inquire into their own history and to acquire the linguistic skills needed to study works unavailable in English, have long viewed Husserl’s work through a glass darkly.
In contrast, Husserl rarely mentions phenomenology here, to which there are only eleven references beginning in the last third of the course. Four of them appear in the appendices. Only in §69, the very last section of the work, where Husserl argues that a systematic phenomenology of consciousness and its consciousness-correlates is needed to solve problem of reason, is phenomenology considered somewhat closely. The word ‘transcendental’ is mentioned a mere fifteen times and is so only in connection with transcendental numbers, Kantian philosophy or to defend himself against charges that he was trying to penetrate into a mystical transcendental world by means of his own intellect.
The main text of this last version of the lecture course is divided into three sections. The first is devoted to defining and characterizing formal logic, the second to a systematic theory of meaning and judgment, and the third to the theory of science. It is followed by eighteen appendices of related material which Husserl himself had designated for insertion into specific places within the text, or individual pages which he had added to the bundle of papers making up the text of the course. All the texts published in the appendices are from Logik als Theorie der Erkenntnis 1910/11. Kant is the philosopher most cited here, followed closely behind by Brentano. Only Aristotle, Bolzano and Euclid merit more than a few mentions.
Husserl states the subject and purpose of these lectures in the first few minutes of the course. He says that it can be designated by one word: ‘understanding’, a synonym of ‘reason’ signifying the many mental activities and achievements which span all areas of the life of the mind and are familiar to everyone from experiences prior to any logic. Understanding, he tells students, governs all the sciences, technical arts and spheres of extra-theoretical life. Sciences are only truly sciences to the extent that the understanding governs their findings, gives them form. It is what sets norms for them, inquires into what is right, demonstrates error. It creates the unity of theoretical knowledge out of unconnected experiences, isolated convictions, presumptions, inferences. In all domains of nature, it unveils the inviolable system of laws that are called the theory of the particular field and it is owing to it that nature figures before the mind’s eye as ruled by laws. It casts light on the darkness of the inner life and the life of the will, judges beauty and ugliness, goodness and badness, appropriateness and inappropriateness.
An essentially new line of inquiry opens up, Husserl teaches, when we turn our gaze away from the things we know through the understanding to consider the ways in which it operates and inquire into the forms in which this takes place, which forms they imprint upon the contents thought by means of it, which norms they are bound to in so doing, and how reaching or missing the goal of truth essentially depends upon the observance or non-observance of these forms and norms. He cites John Locke who, in An Essay Concerning Human Understanding, compared the understanding to the eye, which while making all things knowable for us, does not perceive itself, and only with great difficulty can make itself its own object. He maintains that it is only in logic–whose goal it is to acquire systematically exhaustive, purely theoretical self-knowledge of the understanding’s ways of experiencing and of all the kinds and forms of the thought-contents specific to it–that the understanding reflectively investigates itself. So it is that the purpose of this course is to provide a glimpse of the vast science that this awareness opens up and of the great importance the attainment of this aim has for knowledge. In so doing, he seeks to introduce beginners to great and interesting areas of logic to the extent that they can lay claim to scientificity. The imperfect state of knowledge of the understanding, he declares in Appendix II, explains the imperfect development of philosophy and why dogmatic-didactic presentations of philosophy prove fruitless. There, he cites Kant’s words in the Critique of Pure Reason that one cannot learn philosophy, but only to philosophize.
Much of this work is devoted to an exploration of the realm of meaning, which Husserl credits Bolzano with having sighted. Meaning, he suggests, displays an inner structure which is amazingly regular in form and can be compared to that of a crystal. Just as crystals have their crystal form and conform to a crystal system, he theorizes, it is part of the essence of meanings that “they form fixed configurations into concrete meanings such that all meaning is bound, so to speak, to fixed crystal-configurations and only so crystallized can have concrete being”. According to him, every judgment is itself a crystal, a crystal configuration, a crystal structure which is the formal structure of its components. He judges the metaphor to be apt except for the fact that in the sphere of meaning every crystal must be from the fixed, coherent system; there is no amorphousness that the same matter can assume (§27b).
In §19a, Husserl importantly says that the Fourth Logical Investigation represented his first attempt to do justice to the theory of forms of meanings as a highly important underlying level of formal logic. In note 3 found at the end of the Fourth Logical Investigation of the second edition of the Logical Investigations, he had lamented the fact that logic still lacked a prime foundation, a scientifically rigorous, phenomenologically elucidated distinction between primitive meaning elements and structures and essence-laws germane to knowledge, because logicians had never scientifically formulated a purely logical theory of forms. He maintained that that was why the many theories of concept or of judgment had yielded so few tenable results. He had concluded §13 of the Fourth Logical Investigation of that edition writing that he hoped that the much improved study of the theory of forms of meanings that he had hinted at in a note at the end of that section of the first edition, and that he had expounded in his courses at Göttingen since 1901, would be made available to a wider public. According to Panzer’s introduction (p. XXVII) the more thorough presentation of the theory of meaning forms sketched in the Fourth Logical Investigation is published for the first time in this volume.
In these lectures, Husserl indeed further develops ideas set forth in the Fourth Logical Investigation. For example, §§20-21 find him teaching students that the difference between independent and dependent meanings is ideal, therefore a priori, and that pertaining to it is the wealth of a priori laws which he had called “purely” logical-grammatical in the Logical Investigations. These ideas about the distinction between independent and dependent meanings makes for interesting comparison with Frege’s and Russell’s thoughts on the same, something which I studied in “On Fundamental Differences between Dependent and Independent Meanings” and “Incomplete Symbols, Dependent Meanings, and Paradox”.
All philosophical thinking has its share of special terminology to which philosophers are obliged to adjust if they are ever to enter into the ideas propounded there. For example, Husserl taught that there are forms of subjective discourse which make no reference to the contingent empirical persons investigating and substantiating it, or for which reference to empirical-psychological facts is extra-essential. In the sciences, he held, it is not just a question of facts and meanings of statements, but it is also a question of insight into thinking itself with respect to its legitimacy.
In his discussions of these matters, he uses the term ‘noetics’, which will be unfamiliar to some readers. He characterizes noetics as a systematically formal theory of justification of knowledge which stands opposite to formal logic, analytic ontology and the theory of meaning, but is attached to them in the most intimate way. It is theory of science in the highest sense and also the discipline that makes the ultimate and highest fulfillment of epistemological needs possible, because only absolute knowledge can provide ultimate epistemological satisfaction. Part of the essence of all knowledge, he maintains, is that the Idea of the absolute “hovers” above it as its “guiding star”, so that if philosophy is to be the name for every kind of scientific investigation aspiring to serve the striving for absolute knowledge, then all logical disciplines, and noetics above all, deserve to be called philosophical disciplines (§§66-69).
It is well known that Franz Brentano taught Husserl that thinking has what he calls here the “obvious, and therefore wonderful” characteristic of being of intentional, of being thinking about something, which for him amounted to its having inherent meaning (§19d). He calls this the “miracle of consciousness” and considers that for the philosophically naïve, it seems most obvious that in subjective experiencing something can be intended which is not itself an experience, but lies beyond the experience, and that when it comes to such experiences subjects can be certain of the objective validity of their intending. However, he objects, this obviousness actually proves to be the “enigma of all enigmas”, the original one of which centers on the original fact that each experience called consciousness has intrinsic meaning (Appendix V).
Cognizant of the importance of the ambiguity, the two-sidedness, of the word ‘knowledge’, i.e., the fact that it may signify either the knowing process or what is known, as it is known, in §7 Husserl very succinctly defines what he came to call the “noematic” perspective as the turning toward what is known as such–toward the noema–as the essential counterpart to noetic reflections on the forms of knowing in pure generality from the standpoint of correctness.
In Ideas I, he was much more forthcoming about exactly what noemata are. Explaining the need to distinguish between the parts and phases of the intentional experience and the fact that it is consciousness of something, he explained there that,
Every intentional experience… is noetic, it is its essential nature to harbor in itself a “meaning” of some sort… and on the ground of this gift of meaning, and in harmony therewith, to develop further phases which through it become themselves “meaningful.” Such noetic phases include… the directing of the glance of the pure Ego upon the object “intended” by it in virtue of its gift of meaning, upon that which “it has in its mind as something meant”…. Corresponding… to the manifold data of the real (reellen) noetic content, there is a variety of data displayable in really pure… intuition, and in a correlative “noematic content,” or briefly “noema”…. Perception… has it noema, and at the base of which is its perceptual meaning… the perceived as such. Similarly, the recollection has its own remembered as such, precisely as it is “meant” and “consciously known” in it; so again, judging has its own judged as such, pleasure, the pleasing as such…. We must everywhere take the noematic correlate, which (in a very extended meaning of the term) is here referred to as “meaning” precisely as it lies “immanent” in the experience of perception, of judgment, of liking…. (§88)
Thinking as thinking about something, Husserl teaches in these lectures, has an immanent a priori constitution and from the noematic perspective has necessary types and forms in which it alone can acquire a relationship to objectivity, in which it can intend–mean–something and do so prior to any question as to whether the intending is valid or not. The nature and form of the intentionality of thinking is reflected in the nature and form of logical meaning, so that understanding the basic composition of logical meaning affords insight into the a priori essence of thinking and vice versa. The theory of forms of meanings thereby gives a theory of forms of thinking as that of logical meaning. So the theory of forms of logical meanings would also correlatively be a theory of forms of thinking with respect to its meaning-like essence, with respect to the possible forms of its intentionality. He says that studying the wonders of the intentionality of thinking is certainly a matter of the greatest interest and that one thereby goes about this by investigating the basic constitution of the realm of meaning prior to all questions of validity (§19c-d).
So these lectures contain ample information about Husserl’s theory of the essential parallelism obtaining between the various kinds of consciousness and the concept of linguistic meaning, thus supporting Dagfinn Føllesdal’s thesis that with the theory of noemata Husserl generalized the notion of linguistic meaning to the realm of all intentional acts. However, the noemata must not be too closely identified with Fregean linguistic meanings, because, as I pointed out in Word and Object in Husserl, Frege and Russell, for Husserl,
the logico-linguistic realm was but a stepping stone to an infinitely vaster realm of inquiry…. he undertook to do something that was radically different from anything Frege and his successors ever tried, or ever considered worth doing: he “plunged into the task of laying open the infinite field of transcendental experience” (CM, p. 31). Not just interested in the meaning of linguistic expressions, Husserl made it his life’s work to investigate as thoroughly and painstakingly as humanly possible the meaning conferred upon objects by the intentional acts of consciousness. To explain how the human mind confers meaning on its objects, Husserl posited the presence of structures analogous to the intensional “meanings” used by his contemporaries and predecessors in their discussions of the meanings of words. These structures were the noemata. 
Importantly, it was precisely during the years in which Husserl gave the lectures published here that he fully developed the noematic notion of meaning, which did not figure in the Logical Investigations. In his 1913 Foreword to the second edition of that work, he even signaled it as a “defect” of the First Logical Investigation that the essential ambiguity of ‘meaning’ as an Idea had been not stressed, because attention had not been paid there to the fundamental role of the distinction between the ‘noetic’ and ‘noematic’ in all fields of consciousness and to the parallelism obtaining between them, which was only fully expounded for the first time in Ideas I published in 1913. He regretted the fact that the Logical Investigations had unduly, one-sidedly emphasized the noetic concept of meaning and that the problem had not been understood and remedied until the end of the work, but considered that the distinction had nonetheless been implicit in many of the arguments of the Sixth Logical Investigation. Along these lines, Panzer stresses in her introduction that one must above all bear in mind that Husserl had had compelling reasons to change the title of his lecture course from Logik als Theorie der Erkenntnis (1910/11) to Logik und Einleitung in die Wissenschaftstlehre (1912/13, 1914/15), because it had become chiefly a matter of a systematic theory of forms of meanings from the noematic perspective as a basic part of an a priori, and primarily formal, theory of science, whereby the noetic theory of justification was for the time being expressly left to the side (pp. XV-XVI).
Husserl further instructs students that they will encounter a pure logic in a new sense through investigating what is stated in the stating, what is known in the knowing, as it is known, meant, in knowledge. In particular, they will first encounter “apophantic logic”, also a term which some readers might find mystifying and which Husserl variously defines as the logic of statements, the logic of asserting propositions, the pure logic of affirmative statements, the logic of the affirmative predicative proposition, which he considered to be essentially the logic Aristotle dealt with under the heading of Analytics. Aristotle, he notes, called judgment, the statement as such, ‘apophansis’, whose form is symbolized by ‘A is b’. It is a mere predication in which something is stated about something. For Husserl, pure arithmetic, the whole of formal mathematics or theory of manifolds was for essential reasons intertwined with the logic of affirmations and formed a higher tier of apophantics. He considered recognizing this and characterizing them so to be of the greatest importance philosophically (§§7, 9, 14, 15c, 19c).
In §66, he reminds students that in seeking to obtain the first idea of formal logic, and of apophantic logic to begin with, he had told them that sciences live on objectively in the form of writings in which scientific theories are given expression and that this continued existence is one of interrelated statements having their meanings in judgments and relationships of judgments referring to the objectivities of the scientific domain, for example, to numbers in arithmetic, to geometrical forms in geometry, to the things of the natural world and natural relationships in the natural sciences. He particularly examines, and criticizes, Kant’s distinction between analytic and synthetic judgments. Kant, he notes in §46c, had been the first to see the difference between the analytic and synthetic a priori and had rightly called it a classic distinction for transcendental philosophy, but he neither had a genuine concept of the analytical as determined by the conceptual sphere of the formal categories, nor had he understood the essence of formal logic. If, Husserl reasons, as Kant wished, we separate the logical a priori from everything empirical-methodological, we see that what is purely logical according to the tradition belongs exclusively in the apophantic sphere and that therefore formal logic to a certain extent sought to be apophantics. However, apart from Leibniz and some of those influenced by him, neither Kant nor anyone else had suspected that pure arithmetic and all the disciplines essentially related to it intrinsically belong together with the old formal logic. Kant’s concept of the analytical did not extend beyond his concept of apophantic logic. His definition of analytic judgments is therefore limited to categorical judgments and implies that every categorical judgment is analytic whose predicate concept is contained in the subject concept. So Husserl concludes that, however right Kant’s observation was, it blocked the way to the far more important realization of the fundamental separation of everything belonging to the realm of formal category from the sphere of the non-formal a priori, something which leads to a wholly unacceptable equation and equal treatment of arithmetic disciplines with other purely mathematical disciplines and the severing of both from what Kant called pure natural science.
Husserl explains here that by analytic truths in the broadest sense, he himself understands “analytic concept-truths, therefore all pure categorial truths, therefore, the entire pure mathesis, pure logic, then however, also their a priori and empirical individuations, therefore, the analytic necessities”. For him, the pure categorial concept-truths essentially belong together as a whole and form a single system of scientific disciplines to be dealt with under the broadest heading of formal logic, or analytics, or the mathesis universalis in Leibniz’ sense (§§45b, 47e, 50). In Appendix XV, he underscores the purely categorial nature of formal logic. Instead of formal logic, he tells his students, we can also say analytics or science of what is analytically knowable in general, the science that establishes and systematically substantiates analytic, categorial, laws.
Among the stated goals of Husserl’s late work Formal and Transcendental Logic was to redraw the boundary line between logic and mathematics in light of the new investigations into the foundations of mathematics and to examine the logical and epistemological issues such developments raise. This is something that the former student and assistant of Karl Weierstrass in Berlin, long-term friend of George Cantor in Halle, and member David Hilbert’s circle in Göttingen undertook to do in these lectures. In particular, he argues here that those who had mixed the roles of the philosopher and the mathematician had only succeeded in creating their own closed worlds. Arithmetic, algebra and analysis, he teaches, developed independently from philosophy and must remain independent. The theory of the analytic and traditional syllogistic logic, he maintains, is a piece of the pure mathematics of propositions and of predicates of possible subjects in general and as such is not the job of philosophers, but of mathematicians, who are the only competent engineers of deductive structures. If in the nineteenth century, mathematicians had also adopted the deductive theories of traditional syllogistic logic and gradually developed a mathesis of propositional, conceptual, and relational meanings in the spirit of the solely proper mathematical method, they had only laid hold of a field that was their rightful possession (§50).
So, according to him here, all arguing against mathematizing logic was symptomatic of a lack of understanding. With formal mathematics, he asserts, we do not actually enter into an essentially new domain, but are dealing with a field of pure concept-truths whose conceptual matter is inseparably linked to the original matter of the logic of meaning. Owing to the work of mathematizing logicians, he stresses, the disciplines of logical validity have reached a higher level of technical perfection in certain ways. They have seen the essential kinship between formal mathematics and formal logic and so have expanded the sphere of the exact mathematical disciplines to include the new ones of formal logic by carrying over to formal logic the same algebraic methods entirely suited to it. However, they have lacked a scientific understanding of thinking and so remain confused about the nature, meaning, and basic concepts of formal logic, and the idea of a theory of forms of meanings as a discipline that by its very nature is a comprehensive, difficult discipline prior to the disciplines of logical validity has also remained completely beyond their ken. The philosopher’s task is therefore one of providing a complementary reflection on the essence and meaning of the governing basic concepts and basic laws of deductive theories (§§50, 19a, d).
In these lectures, Husserl additionally spells out for his students how they might convince themselves that there is something in logic that is akin to what mathematicians have in mind when they speak of functions. In particular, he discusses a theory of functional judgments, which he specifically links to what mathematicians call a function, something which he considers that, without arriving at the full descriptive analysis of the kinds of judgment concerned, the sharp-witted Frege, had had the merit of recognizing in his article “Function and Concept”. The empty places, Husserl explains, are what mathematicians call arguments. Many empty places can occur in a judgment, so that the same judgment can have several places, or terms, of universality and several of particularity. These places can be pure empty places as, for example, “Something or other is red”, in which a nominal something figures in the subject position and is the bearer of a particular function, or universal as, for example, “Everything is red”. However, as a rule, the “something” is specified by a letter of the alphabet, as in the arithmetic example, a + b = b + a, where two terms function universally and are determined as the numbers a and b. Argument places are specifically nominal forms. He states as a principle that every nominal position in a judgment-form can become an argument place and take on the generality-forms in relation to it. Arguments or generality positions, terms of universality and particularity also occur in predication since nouns can occur there in many different ways. Everything formal, he stresses, is exclusively composed of terms of this kind. He considers the empty something to be of the greatest importance for the theory of meaning and all of formal logic. As what is specifically mathematical in the mathesis, specifically formal in formal logic, absolute emptiness of content was for him the hallmark of the formal logical. These ideas about functional propositions are expounded in §§26a-b, 32c, 40b-45a and Appendix XIII.
This is a good place to note that contrary to what is widely believed, Husserl did have a formal language. In §19d of these lectures, he argues that the rigorous carrying out of purely deductive theory requires one and the same method everywhere, namely, what is called the algebraic method, and that mathematicians are everywhere the technical experts qualified in deductive theory, whose development to technical perfection is everywhere the requirement of exact science. In his Logik, Vorlesung 1896 and his Logik, Vorlesung 1902/03, he set out the axioms, notation, and rules of inference for the conceptual and propositional calculus he advocated. In Alte und neue Logik, for example, he resorted to it to show that a propositional form has blank terms, empty terms, or variables, so that one can then say: Each system of values satisfying the propositional form or function F(αβ…) also satisfies the function F′ (αβ…) and that that satisfying signifies transforming ∏ αβ… F(αβ…) € F′ (αβ…) into a valid proposition, where ‘€’ is the sign for implication and ‘′’ marks the step from premise to conclusion. Like his contemporaries in Germany, Husserl adopted C. S. Peirce’s symbols for the universal and existential quantifiers ∏, ∑ which, unlike Frege’s, were widely used.
In §39, Husserl discusses the traditional talk of extensions of concepts according to which the totality of objects that are to be subsumed under each valid universal-concept are to belong to it as its extension. He taught that no pure concept has anything like an extension and that it is nonsense to say that for every concept a distinction is to be made between intension and extension. Indeed as shown by the remarks he wrote on his copy of Frege’s “Function and Concept” available at the Husserl Archives in Leuven, Husserl believed that there are extensionless concepts, that impossible, imaginary, absurd concepts are also concepts. It was in fact precisely the search to justify the use of apparently meaningless signs in calculations or deductive thought that had led him to develop his theory of manifolds as the third and highest level of pure logic. In the Prolegomena §70, he called his theory of complete manifolds the key to the only possible solution to how in the realm of numbers impossible, non-existent, meaningless concepts might be dealt with as real ones. In §72 of Ideas I, he wrote that his chief purpose in developing his theory of manifolds–which he likened to Hilbert’s axiom systems–was to find a theoretical solution to the problem of imaginary quantities.
Chapter 11 of this lecture course is devoted to a discussion of that theory of manifolds, which nicely complements the discussions of the same in Introduction to Logic and Theory of Knowledge (§19) and Formal and Transcendental Logic (Chapter 3). Here he teaches that since the procedure used is purely formal, and since not a single concept is used that does not arise out of the analytic sphere, his theory of manifolds, for him the highest level of mathematics, is the supreme consummation of analytics, the ultimate consummation of all purely categorial knowledge (§§57-59).
As is evident from the foregoing, Logic and General Theory of Science is replete with insights into matters that many philosophers have now been primed to appreciate out of enthusiasm for Frege’s ideas. It in fact takes readers back to the place where two main logical roads diverged during the early part of the twentieth century and affords a look down the one less traveled by. It invites phenomenologists and analytic philosophers alike to overcome pride and prejudice. Indeed, had not so many twentieth-century philosophers resolved to barricade themselves behind the walls of ideological prejudices, the kinds of insights expounded here could have altered philosophical landscape in English-speaking countries, so much so that history will eventually show that logic and philosophy would have followed a different, and better, course in the twentieth century had Husserl’s thoughts on these matters found their rightful place alongside the works of Frege, Russell, Carnap, Hilbert, Gödel, for example.
However, although these lectures are laced with pertinent lessons about matters that lived on to become the stuff of twentieth-century logic and philosophy of science, they at the same time draw clear epistemological and metaphysical lines between Husserl’s theories and those that dominated in English-speaking countries. This is particularly evident when one looks at the place Husserl accords to ideal entities here. Indeed, he instructs his students to adopt ontological views about the reality of essences, universals, Ideas, senses, meanings, concepts, attributes essential properties, modalities, propositions, intensions–anything hinting of the a priori–that are fundamentally antithetical to those that analytic philosophers have wanted to have. He plainly says that he completely means in absolute earnest that the recognition of ideal objects, or Ideas, as new kinds of atemporal, supraempirical objects is the pivotal point of all theory of knowledge and that he considers the proper grasp of them to be decisive for all further considerations. He contends that it is imperative to force people to concede once and for all that Ideas are genuine, actual objects by contrasting them with the empirical objects that they see in the natural attitude and alone are inclined to recognize as objects.
He suggests that people who want to know what Ideas are need only point to self-evident givens like the cardinal number series or to absolutely self-evident statements about members of the number series, which he claims everyone knows in a certain naïve way since they talk of numbers and do so in ideal ways. Only philosophers, he charges, shun Ideas. He contends that once one recognizes givens like the series of natural numbers as objectivities, one can only describe them in the way Plato did in his theory of Ideas, as eternal, selfsame, non-temporal and non-spatial, unmoved, unchangeable, etc.
He fully realized that what he was claiming was “very hotly combated as being mysticism and scholasticism”, that people trained in traditional philosophy instantly think Platonic Ideas and then such Platonic realism becomes associated with mysticism, Neo-Platonism and a magical view of nature, something as far removed as possible from genuine natural science. Philosophers then recall how this merged into scholastic realism during the Middle Ages. Anyone advocating giving ideal objects their due faces charges of being a reactionary, mystic, scholastic, the latter two being the strongest scientific terms of abuse of the time, in which formal logic is vilified as being empty scholasticism, and espousing idealism for a pure logic is left undefended (§§4, 5a, 8, 19a).
Indeed, on December 29, 1916, Göttingen philosopher Leonard Nelson wrote to David Hilbert that Husserl,
admittedly also originally came from the mathematical school, but… bit by bit turned more and more away from it and turned towards a school of mystical vision, whereby he also deadened the feel in his school for the demands and value of a specifically scientific method. He even goes so far, after his own lack of success with it, as to see a danger in methodological thinking and thinks that it would ruin philosophers for whom the truth only reveals itself in mystical vision. Even though Husserl himself remains protected by certain inhibitions from mystical degeneracy by virtue of strong ties to mathematics that he has not been able to cast off, one must unfortunately nonetheless note with horror that after the school as such had torn down the bridges to mathematics behind them, how unrestrainedly his students lapsed into every excess of Neo-platonic mysticism….
In the lectures published here, Husserl defends himself against such charges by explaining that he was not adopting Ideas and classes of idealities out of some desire to penetrate into a mystical transcendental world by means of his own intellectual intuition, but rather for the same banal reason that he embraced things: because he saw them and in looking at them grasped them himself. He even compares ideal objects to ordinary stones found lying on the road. In response to charges that he was espousing scholasticism, he insisted that he was only asking for the intellectual integrity to allow what is prior to any theory, because it is the most evident of evident facts, to count as being exactly what it proclaims itself to be. If this is enough to have him called a scholastic, he protests, then that is all fine and good, and he asks whether it was not better to have integrity and be called a scholastic, or to lack integrity and be a modern empiricist! He says that he advocates integrity and does not fall flat on his face when labelled as a scholastic, because integrity stands the test of time (§§4, 8).
For those influenced by the Analytic tradition in philosophy who feel queasy about such talk of ideal entities, it is worthwhile to note here that in Russell’s article entitled “The Philosophical Implications of Mathematical Logic”, which is found translated in Husserl’s notes on set theory, Russell affirmed “that there is a priori and universal knowledge”, that “all knowledge which is obtained by reasoning, needs logical principles which are a priori and universal”. He further wrote that “it is necessary that there should be self-evident logical truths” and that these were “the truths which are the premises of pure mathematics as well as of the deductive elements in every demonstration on any subject whatever”. “Logic and mathematics force us, then”, Russell wrote, “to admit a kind of realism in the scholastic sense, that is to say, to admit that there is a world of universals and of truths which do not bear directly on such and such a particular existence. … We have immediate knowledge of an indefinite number of propositions about universals: this is an ultimate fact....”
As is all too well known, Husserlians and followers of the Frege-inspired Analytic school that dominated philosophy in English-speaking countries during the twentieth century have not in general spoken the same language. Fortunately, however, this translation appears at a time when the latter have heartily embraced Frege’s thoughts and concerns, many of which he shared with Husserl.
Both Husserl and Frege faced the same terminological confusions and they both fought their way through a terminological jungle to achieve conceptual clarity in spite of them. Though Frege’s choices are now more familiar to most English-speaking philosophers than Husserl’s are, they were often eccentric. As Russell wrote when he introduced Frege to the English-speaking world in Principles of Mathematics,
Frege is compelled, as I have been, to employ common words in technical senses which depart more or less from usage. As his departures are frequently different from mine, a difficulty arises as regards the translation of his terms. Some of these, to avoid confusion, I shall leave untranslated, since every English equivalent that I can think of has already been employed by me in a slightly different sense.
For example, of Frege’s famous distinction between Sinn and Bedeutung, Russell wrote “The distinction between meaning (Sinn) and indication (Bedeutung) is roughly, though not exactly, equivalent to my distinction between a concept as such and what the concept denotes”. For his part, in these lectures, Husserl explains his use of the term ‘nucleus’ in these terms,
the word concept is so ambiguous and in particular <is> also used so ambiguously in the field of formal logic itself that we cannot use it without thinking twice. In any event, it may be said that by means of my analysis, an extraordinarily important meaning of the word concept as “nucleus-content” has been scientifically defined. And at the same time, the general-nucleus narrows down the meaning of concept in a clear-cut way. It is indeed often said that generality is part of the essence of concepts. §25)
In the First Logical Investigation, he had written of how in the absence of fixed terminological landmarks, concepts run confusedly together and fundamental confusions arise, and he went on to defend his decision to use ‘sense’ and ‘meaning’ as synonyms as follows,
It is agreeable to have parallel, interchangeable terms in the case of this concept, particularly since the sense of the term ‘meaning’ is itself to be investigated. A further consideration is our ingrained tendency to use the two words as synonymous, a circumstance which makes it seem rather a dubious step if their meanings are differentiated, and if (as G. Frege has proposed) we use one for meaning in our sense, and the other for the objects expressed. To this we may add that both terms are exposed to the same equivocations, which we distinguished above in connection with the term ‘expression’, and to many more besides, and that this is so both in scientific and in ordinary speech. (§15)
In §42 of the Second Logical Investigation, he complained that the word ‘meaning’ was equivocal so that people did not hesitate to call the object of a presentation a ‘meaning,’ and to say the same of its ‘intension’, the sense of its name. He further noted that since a meaning is often called a concept, talk of concepts and objects of concepts is also ambiguous.
And this brings us to take a look at Husserl’s use of the notoriously hard to translate word ‘Vorstellung’, for which there is no satisfactory English equivalent. It has very frequently been translated as ‘idea’, ‘imagination’, or ‘representation’, all words charged with philosophical connotations that are not his. He addresses the problem in §8 stating, “One must, besides, also surely keep in mind the fact that the word idea (Idee) has taken on many meanings and that, especially in the parlance of English philosophy, it is an expression for subjective experiences, for presentations (Vorstellungen), that, however, Ideas in my Platonic sense are not presentations, but atemporal, supraempirical objectivities”. In these lectures, he in fact uses ‘Vorstellung’ ubiquitously in ways that are unusual in the English language. I have consistently translated ‘Vostellung’ as ‘presentation’ and can only ask readers to try to enter into his thought and divine his meaning.
Readers should also be aware that Husserl’s use of ‘Vorstellung’ in these lectures differs from his use of it in other periods of his career. In his “Psychological Studies in the Elements of Logic” of 1894, he wrote that he thought it was a good principle to avoid using a word as equivocal as ‘Vorstellung’ as much as possible. Students of his logic course of 1902/03 heard him complain that no psychological and logical term was laden with so many pernicious ambiguities as was ‘Vorstellung’. There he distinguished, as Frege had in Foundations of Arithmetic, between subjective Vorstellungen as psychological experiences and objective logical Vorstellungen, which he regarded as the “completely lost” distinction that Bolzano had for the first time identified as a “cornerstone of all genuinely pure logic”. In §5 of the introduction to the second volume of the Logical Investigations, he told readers that he should “have to raise fundamental questions as to the acts, or alternatively, the ideal meanings, which in logic pass under the name of ‘presentations’ (Vorstellungen)”. It was, he said, “important to clarify and prise apart the many concepts that the word ‘presentation’ has covered, concepts in which the psychological, the epistemological and the logical are utterly confused”. After discussing thirteen dangerous ambiguities associated with the word in §§44-45 of the Fifth Logical Investigation, he concluded that “However the notion of presentation is defined, it is universally seen as a pivotal concept, not only for psychology, but also for epistemology, and particularly for pure logic”.
In these lectures, Husserl speaks of how presentation may signify something psychological like the intuition or thought-presentation underlying the thought-act and having nothing to do with the theory of pure meanings, something like a mere neutral thought, or something different and mixed in with this. As an example, he gives the fact that that nouns are said to be expressions of mere presentations, but not complete judgments. And he recalls that in previous lectures on logic, he had defined a different, more important, concept of presentation. What he was presently calling nominal “syntagma”, he had called nominal presentation and had discerned as many kinds of presentations as there are syntagmas, thus making presentation the same as syntagma (§25).
Syntagmas in turn are defined as: the syntactical “stuff”, which is “always given in syntactical form that lends it the specific thought-function in the proposition unit and overall meaning in general”. He says that he had found it necessary to introduce an artificial word that could be used when there was a danger of becoming ensnared in the myriad ambiguities surrounding the words ‘presentation’ and ‘concept’ and adds that if the Idea of presentation is to be appealed to in the realm of meaning, then the natural choice for the meaning of “presentational content” as distinct from the objectivity presented is the nucleus (§24).
Husserl takes care to distinguish the concept of syntagma from that of nucleus by pointing out that the former had been defined as concerning that which “is identical, which stands out as the same noun in a different predicative function, or as the same predicate–but in a different function−sometimes as actual predicate, sometimes as determining attribute”. He explains that it was by contrasting presentations such as “similar” and “similarity”, “redness” and “red” that he had been led to the concept of the full-nucleus. He had observed that primitive presentations of different syntactical categories coincide in terms of a content, have the same nucleus, which could be formed nominatively, non-nominatively, adjectivally, relationally. In comparison, however, he saw that the syntagmatic category clings to the syntagma. Changing the function does not change the same noun, but makes it stand out as such. There can be both something identical and something different in syntagmas of different category, and it is precisely the nucleus that remains the same. The syntagmas “similarity” and “similar”, or “redness” and “red” enjoy the same nucleus. Comparing them we find that syntagmas of a different category have an essence-nucleus. They differ in their nucleus-form, which is what forms the pure nucleus into the syntagma of the particular category (§25, Appendix VIII).
Every language also has its share of recalcitrant terms which for one reason or another frustrate translators’ efforts to capture their precise meaning. In the case of the German language, translators must cope with the fact that it delights in inventing compound words. Husserl makes lavish use of his freedom to do so, and most of his creations are not found in dictionaries. I have very often had to resort to hyphenations which, though not elegant in English, are nevertheless easily understandable.
In reading these lectures devoted to the theory of science, it is naturally important to keep in mind that the English word ‘science’ and the various words derived from it are narrower in meaning than the German word for science, ‘Wissenschaft’, and the words that are derived from it are. It helps to remember that these words contain the noun ‘Wissen’, meaning knowledge, or the verb ‘wissen’, to know. And, it is good to keep in mind that the English word ‘science’ has its roots in the Latin word scientia, meaning science, and scire, to know. In §60, Husserl sheds some light on his particular use of the term in these lectures. There he explains that taking up the Idea of the theory of science that had served as a guide from the beginning of the course, then
as the formal theory of meaning and ontology described, logic is the first manifestation of this Idea. Knowing (Wissen) in the sense of science (Wissenschaft) is thinking or thought-state-of-mind that refers back to thinking. Corresponding to thinking is something thought, and so corresponding to every science is a system of judgments in my meaning-theoretical sense, a system of postulated truths and probabilities, and these refer to objects and states-of-affairs. The science of meanings in general, of truths, possibilities, probabilities in general, of objects in general in absolutely pure, formal universality, yields a system of absolute truths to which every science is obviously bound, and which are prior in terms of validity to every science in general−as already to every judgment in general.
Since Husserl here takes up the questions of logical grammar of the Fourth Logical Investigation, it is also important to note that, as he used them, the words ‘Widersinn’ and ‘widersinnig’ do not translate neatly into philosophical English. The German word ‘wider’ means against, counter, contrary to, in opposition to. So, some have chosen to translate ‘Widersinn’ and ‘widersinnig’ as ‘countersense’ and ‘countersensical’. Husserl himself used ‘Absurdität’ and ‘absurd’ as synonyms for ‘Widersinn’ and ‘Widersinnig’ (ex. Logical Investigation I, §19; Logical Investigation IV, Introduction, §12). These words may, however, also be understood in the sense of paradox or contradiction and paradoxical, contradictory, or illogical. In that case, they fall into the family of ‘widersprechen’ (to contradict), ‘Widerspruch’ (contradiction), and ‘widersprechend’ and ‘widerspruchsvoll’, two common German words meaning contradictory. Given the difficulties and the importance that Husserl accorded to these concepts, I have chosen to leave ‘Widersinn’, ‘Widersinnigkeit’, and ‘widersinnig’ in German. ‘Unsinn’ can be translated as ‘nonsense’ and ‘unsinnig’ as ‘nonsensical’, but I have chosen to leave them in German where Husserl talked about Widersinnigkeit.
In relation to his theory of manifolds, in §57 of these lectures, Husserl points out that in the concrete spheres, the formal limits imposed by forbidding the use of any factual Widersinn, any widersinnigen, imaginary concepts, acts as a constraint and hindrance in deductively theorizing work, but that the marvelous thing about manifolds is that they free us from such prohibitions and explain why by passing through the imaginary, what is meaningless, must lead, not to meaningless, but to true results. In speaking of sets in §39, he notes that even inferential thinking, makes use of widersinnigen presentations to some degree. As an example, he points out that, although the set-presentation, the totality-presentation, is for the most part also realized by mathematicians to begin with, it is nevertheless of no use for argumentation because it involves a Widersinn. Holding fast to the meaning of totality, he explains, a totality of triangles, a totality of numbers, is not graspable intuitively and so cannot exist either. Here he was taking up an issue he had tackled at the end of Chapter 11 of Philosophy of Arithmetic, where he noted that despite the absurdity of the idea, analogies fostered a tendency to transpose the idea of constructing a collection for infinite sets, thereby creating what he called a kind of “imaginary” concept whose anti-logical nature was harmless in everyday contexts precisely because its inherent contradictoriness was never obvious in life. This was, he explained there, the case when “All S” was treated as a closed set. However, he warned, the situation changes when this imaginary construct is actually carried over into reasoning and influences judgments. It is clear, he concluded, that from a strictly logical point of view we must not ascribe anything more to the concept of infinite sets than is actually logically permissible, and above all not the absurd idea of constructing the actual set.
The German word ‘Evidenz’ is also without a good equivalent in English. In §30e of Introduction to Logic and Theory of Knowledge, Lectures 1906/07, Husserl called ‘Evidenz’ “a word for the fact that, as noeticians affirm and prove, there is a difference between acts that not only think that something is thus and thus, but are fully certain and aware, in the manner of perspicacious seeing, of this being and being thus. Therefore, the thing, the state of affairs is given in insight”. Evidenz, he maintained, was nothing other than the quality of givenness understood in a comprehensive way and not just limited to the being of individual real things. We face the problem of Evidenz when we come to understanding the correlation of consciousness and object that concerns all consciousness–even dreaming, hallucinatory, erring consciousness–and we then ask the closely interrelated questions as to how we come by the existence of any object in itself at all, how we know that any object at all exists in reality, where and when an object is truly given to us, how we know that an object is given and what it means for an object to be given to us. In §67b of the lectures published here, he teaches that Evidenz is what necessarily assures us that we are in the possession of truth. He points out that we can judge without seeing, blindly, routinely, and so on, but only if we judge insightfully does our judging have objective cognitive value. If we were not capable of Evidenz, he maintains, then no talk of truth and science would make any rational sense.
‘Geist’ is another troublesome German word. ‘Mind’ (the faculty of reasoning and understanding) is a proper, though imperfect, translation for it in philosophical contexts, and this is particularly so here, because Husserl is mainly talking about logic and philosophy of science and his lecture course is meant to be a critique of reason. I have used it and the corresponding adjective ‘mental’ (of or pertaining to the mind) when it is a matter of the psycho-physical reality of human beings. However, ‘Geist’ also translates as ‘spirit’ and in these lectures it and its adjective ‘spiritual’ are appropriate translations when speaking of supra-individual, immaterial, abstract realities, such as, for example the communal spirit, the world of the spirit. I have also used it for animals where it is a question of “an animating or vital principle held to give life to physical organisms”. My dictionary also defines ‘spirit’ as meaning ghost, the third person of the Trinity, fairy, sprite, elf, angel, and demon.
It is likewise very difficult to know how to translate ‘Gemüt’, which does simply mean mind, but which refers to the emotive, affective dimension, in contrast to the Verstand (understanding) and Wille (will). Dictionaries typically give ‘mind’, ‘soul’, ‘temperament’, ‘feeling’, ‘spirit’, ‘heart’ as translations for ‘Gemüt’, but most of these words overlap in very complicated ways with other German philosophical terms such as ‘Seele’ and ‘Geist’. To convey the meaning of ‘Gemüt’ I have usually appealed to expressions using the adjective ‘inner’.
As the years fly by, I grow increasingly grateful for the truly generous, steadfast support I have received from senior colleagues, who for decades, year after year, always made themselves available to help me. So once again I must thank, in alphabetical order: Paul Gochet†, Ivor Grattan-Guinness†, Jaakko Hintikka†, Ruth Barcan Marcus†, and Dallas Willard†. To the above list I must add my spiritual director of 24 years Jacques Sommet s.j.†, Barry Smith, my first philosophy teacher Roger Schmidt, my last philosophy teacher Maurice Clavelin, and Bernd Magnus†, who in 1970 agreed to direct my Senior Honors Thesis on Husserl and in 1977 oriented me toward work on Husserl and contemporary logic. Then one thing led to another…. Thank you.
This translation was made possible by a fellowship from the National Endowment for the Humanities in Washington, D. C. I am very grateful to it for its support. It is dedicated to the memory of my precursor Dallas Willard.
 Cited Hua XXX, p. XXIII, n. 4.
 Cited Hua XXX, p. XXIII n. 1.
 Johann Wolfgang von Goethe, Faust, Part II, Act I.
 Husserl, Edmund, Alte und neue Logik, Vorlesung 1908/09, Dordrecht, Kluwer, 2003, p. 6.
 Immanuel Kant, Critique of Pure Reason, II. Transcendental Doctrine of Method, Chapter 3, The Architectonic of Pure Reason.
 “On Fundamental Differences between Dependent and Independent Meanings”, Axiomathes, An International Journal in Ontology and Cognitive Systems 20: 2-3, online since May 29, 2010, 313-32, (DOI 10.1007/s10516-010-9104-1). “Incomplete Symbols, Dependent Meanings, and Paradoxes”, in Husserl's Logical Investigations, Daniel O. Dahlstrom (ed.), Dordrecht, Kluwer, 2003, 69-93. Both papers are anthologized in Claire Ortiz Hill and Jairo José da Silva, The Road Not Taken, On Husserl’s Philosophy of Logic and Mathematics, London, College Publications, 2013.
 Dagfinn Føllesdal, “Husserl’s Notion of Noema”, Journal of Philosophy 66 (1969), 680-87.
 Claire Ortiz Hill, Word and Object in Husserl, Frege, and Russell, the Roots of Twentieth Century Philosophy, Athens, Ohio University Press, p. 30. The page reference in the text is to Husserl’s Cartesian Meditations, The Hague, Martinus Nijhoff, 1973, p. 31.
 See his “Foreword II”, Logical Investigations vol. I, London, Routledge and Kegan Paul, p. 48.
 In this regard, it interesting to compare the theory of noetics expounded in these lectures with his ideas about the “subjective ideal conditions” making possible the operations of thinking in §§32, 64-65 of his “Prolegomena to Pure Logic”, the first volume of his Logical Investigations, with his §§25-33 Introduction to Logic and Theory of Knowledge, Lectures 1906/07; and with §94 of Ideas I, where he speaks of noesis and noema in the sphere of judgment.
 See also Claire Ortiz Hill, “Husserl and Frege on Functions”, Husserl and Analytic Philosophy, Guillermo Rosado Haddock (ed.), Berlin, de Gruyter, 2016, pp. 89-117.
 Edmund Husserl, Logik, Vorlesung 1896, Dordrecht, Kluwer, 2001, pp. 272-73 and his Logik, Vorlesung 1902/03, Dordrecht, Kluwer, 2001, pp. 231, 239-49.
 Op. cit., Husserl, Alte und neue Logik, Vorlesung 1908/09, p. 213.
 Translated in my book with Jairo da Silva, The Road Not Taken, On Husserl’s Philosophy of Logic and Mathematics cited above, pp. 390-91.
 Published in German in G. E. Rosado Haddock (ed.), Husserl and Analytic Philosophy, Berlin, de Gruyter, 2016, pp. 289-319. Unfortunately, Russell’s text is cited (p. 317), but not reproduced there.
 Bertrand Russell, “The Philosophical Implications of Mathematical Logic”, Essays in Analysis, London, Allen & Unwin, 1973, pp. 292-93.
 This is one of the main subjects of my Word and Object in Husserl, Frege and Russell cited above.
 Bertrand Russell, Principles of Mathematics, London, Norton, 1903, p. 501.
 Ibid., p. 502.
 Edmund Husserl, “Psychological Studies in the Elements of Logic”, Early Writings in the Philosophy of Logic and Mathematics, Dordrecht, Kluwer, 1994, p. 146.
 Op. cit., Husserl, Logik, Vorlesung 1902/03, p. 82.
 Gottlob Frege, Foundations of Arithmetic, Blackwell, Oxford, 1884 (1986), §27n.
 Op. cit., Husserl, Logik, Vorlesung 1902/03, p. 56.