# Husserl and the Algebra of Logic: Husserl’s 1896 Lectures

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DOI: 10.1007/s10516-011-9166-8

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- Hartimo, M. Axiomathes (2012) 22: 121. doi:10.1007/s10516-011-9166-8

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## Abstract

In his 1896 lecture course on logic–reportedly a blueprint for the *Prolegomena to Pure Logic*–Husserl develops an explicit account of logic as an independent and purely theoretical discipline. According to Husserl, such a theory is needed for the foundations of logic (in a more general sense) to avoid psychologism in logic. The present paper shows that Husserl’s conception of logic (in a strict sense) belongs to the algebra of logic tradition. Husserl’s conception is modeled after arithmetic, and respectively logical inferences are viewed as analogical to arithmetical calculation. The paper ends with an examination of Husserl’s involvement with the key characters of the algebra of logic tradition. It is concluded that Ernst Schröder, but presumably also Hermann and Robert Grassmann influenced Husserl most in his turn away from psychologism.

### Keywords

Husserl’s 1896 lecturesIdea of logicAlgebra of logicPsychologismErnst SchröderHermann GrassmannRobert Grassmann## 1 Husserl’s Antipsychologism in the *Prolegomena*

In the *Prolegomena to pure logic* Husserl argues against any view of logic that does not have a foundation in an objective, formal, a priori, and theoretical discipline. According to Husserl, any view of logic without such foundation is psychologistic. His argument against this very specific form of psychologism consists of a negative and a positive part. The negative part comprising most of the book claims, at its most extreme, that without such a foundation relativism and skepticism will follow. Without such foundation there could be no science; nothing could count as knowledge or as truth. Thus the psychologistic views of logic are ultimately self-refuting. The book finishes with a positive part, in which Husserl describes the idea of pure logic, which sciences and logic in the more general sense should have as their foundation in order for us to avoid skepticism.

Husserl’s argument against *psychologism* in the *Prolegomena* is directed primarily against the conceptions of logic in which logic in the strict sense is not an objective discipline that is independent from empirical and psychological considerations. Husserl’s own *Philosophy of Arithmetic* (1891) falls prey to psychologism precisely because of this: in it Husserl advocates a view that logic is a technique, a *Kunstlehre*, a method of symbolic operations, not a theory. Husserl’s earlier view of logic is thus psychological according to his own understanding of the term (for an excellent discussion of the sense in which Husserl’s early work can be thought to be psychological, see Miller 1982, 19–23). Mohanty (2008, 63) has distinguished between various different sub-species of what he calls naturalistic psychologism: They are logical psychologism, that is, psychologism with regard to logic, psychologism with regard to the theory of numbers, psychologism in the theory of meaning, psychologism in the theory of truth, epistemological psychologism, and psychologism applied to metaphysics. Casted in these terms, Husserl’s aim in the *Prolegomena* is primarily a refutation of logical psychologism, which is inextricably interwoven with psychologism with regard to the theory of numbers^{1} and with psychologism in the theory of meaning, as well as truth. This is not to say that Husserl is not concerned with other species of psychologism elsewhere, but only that the main target of the argument of the *Prolegomena* is specifically logical psychologism.

Whatever term we use for Husserl’s antipsychologism in the *Prolegomena,* the idea of pure logic is the cornerstone of Husserl’s conception. It establishes the formal conditions for the sciences and truth. Husserl’s account of the idea of logic also forms the subject matter for the clarification of the essence of logic in the second volume of the investigations.

Much of the secondary literature on Husserl has been devoted to the question of possible influences behind Husserl’s rather radical arguments against psychologism in the *Prolegomena*. I do not want to go into that debate in any further detail except for making a remark that given the role of logic for Husserl’s argument, one cannot intelligibly address these influences without examining the development of Husserl’s conception of logic and mathematics. Here I agree with Mohanty who observes that: “Husserl’s rejection of the Brentanian psychologistic theory of numbers, as of mathematics and logic in general, is not due to the influence of Frege, but rather due to his evolving conception of the nature of mathematics and logic”(2008, 58).

Mohanty further assumes that Hilbert and Cantor played an important role in Husserl’s development (ibid., 58). There are no signs of any kind of interaction between Husserl and Hilbert in the 1890s. The first signs of interaction between the two date to 1901 when Husserl received a position in Göttingen. Thus I find Mohanty’s assumption about Hilbert’s role somewhat far-fetched. However, Mohanty’s claim about Cantor is easy to accept: Cantor was Husserl’s colleague in Halle at the time, and they discussed for example Cantor’s view of Schröder (Schuhmann 1977, 52). Moreover, in his 1896 lectures Husserl explains Cantor’s results to his students (e.g., 2001a, p. 117). But what I want to emphasize here is that more than to Cantor, Husserl’s conception of logic is indebted to Ernst Schröder and other algebraists. Indeed, in the Foreword to the *Logical Investigations* when explaining the development of his views Husserl says so referring to the algebraic tradition as ‘mathematicizing logic’: “I then came to see in ‘mathematicizing logic’ a mathematics which was indeed free from quantity, while remaining nonetheless an indefeasible discipline having mathematical form and method, which in part dealt with the old syllogisms, in part with new forms of inference quite alien to tradition”(Husserl 2001b, 1–2).

According to Husserl’s own testimony (in the Foreword to the second edition of the *Logical Investigations*), the *Prolegomena* is a mere reworking of his lecture-courses given at Halle in the summer and autumn of 1896 (2001b, 5). It is not entirely obvious what he means by this, since he lectured on logic only in the summer of 1896 (Husserl 2001a, xi). However, the 1896 lecture course is certainly the best candidate for what Husserl refers to in the *Logical Investigations*. In it we can find the most detailed and explicit exposition of what Husserl means by logic understood as an independent, purely formal theory of inference. Moreover, contrary to most of Husserl’s later writings Husserl is relatively explicit about his sources. Husserl’s approach is a version of algebra of logic that has been earlier developed by for example Schröder, the Grassmann brothers, and Boole. Husserl refers explicitly and appraisingly also to Bolzano, Lotze, Twardowski, and Frege, and one can hear echoes from their work in Husserl’s own views. However, the references to these characters are all related to issues other than Husserl’s conception of logic in the strict sense as formal theory of inference.

## 2 Husserl’s Lectures on Logic 1896

*Wissenschaftslehre*for the study of the systematic orderings of truths and objects in the sciences. (He also uses the Bolzanian term

*Begründung*for the relationship between the truths.) The ordering is independent of any individual science. Indeed, in order to acquire an understanding of the

*Wissenschaftslehre*, the notion of pure logic has to be clarified (ibid., 19). Ultimately, Husserl’s explanation of the aim of the logic lectures has a rather Aristotelian flavor: the whole study is necessitated by our need to know:

When we raise ourselves to the abstract [realm] we give up the sensuous fullness of intuition, the freshly pulsing life of the individuals, in which we certainly immediately participate. But without abstraction, no concept, without concept, no law, without law no insight into the ground, no theory, no science, and without science no philosophy. The turning away from the green valleys and fields of life full of self-denial to the grey, leathery, sober theory is the only way to satisfy our highest and purest interest in knowledge (Husserl 2001a, 30)

^{2}^{,}^{3}

Criticizing psychologistic conceptions of logic Husserl makes it clear that by logic he means a pure theory of objective relationships that is independent from any methodological considerations. While Husserl’s over-all conception of the role of logic in the foundations of the sciences appears to be heavily indebted to Bolzano Husserl does not follow Bolzano in his conception of pure logic. As Stefania Centrone puts it “[a]lthough Husserl’s *Logikvorlesung* of 1896 follows in many respects the model of Bolzano’s *Wissenschaftslehre*, in the concluding section on the ‘doctrine of inferences’ its base is the algebra of classes of Boole-Schröder rather than Bolzano’s proof theory” (2010, 128). For short, Husserl wants to develop axiomatic theories of inferences analogously to the theories in arithmetic (Husserl 2001a, 30).

Having explained his motivation to provide the needed foundation for the sciences Husserl proceeds to a detailed criticism of the psychologistic conceptions of logic, and discusses the following issues: (1) whether logic is independent or dependent on some other discipline, and if the latter, then on what discipline (2) whether logic is an art or science, (3) whether it is a formal discipline, or whether it also deals with the content of knowledge, and (4) whether it is a demonstrative or empirical science (Husserl 2001a, 31). Many of his *Prolegomena* arguments against psychologism derive from here. If possible Husserl is even more pointed in his criticism about confusing the subjective and the objective than what he will later be in the *Prolegomena*. Already here he holds that relativism and skepticism are consequences of the subjective tendency of the psychological logic (ibid., 33–34). Like in the *Prolegomena* Husserl holds that logic should be independent, a science rather than a method, purely formal, and demonstrative (ibid., 32–45).

Before his systematic discussion of the lecture course Husserl emphasizes various important distinctions. The first distinction to be made is the one between subjective acts and the objective content. Husserl explains that the former is a real part of the act whereas the latter is not. The same objective content can be shared by a myriad of different subjective acts (ibid., 44–47). Next, Husserl distinguishes between the thoughts and the being, that is, the objective content of a sentence and the fact that something is so and so (ibid., 47–48). Husserl complains about the lack of an appropriate term for the objective contents. Bolzano called them “Vorstellungen an sich,” or objective Vorstellungen, which is the term Husserl also occasionally uses. Husserl’s discussion at this point shows unambiguous realism about concepts: he points out, for example, that Röntgen discovered, but did not *create* the concept of Röntgen ray (ibid., 49). The third distinction Husserl emphasizes here, but not so much in the *Prolegomena,* is the distinction between names and sentences. Corresponding to names there are objects and corresponding to sentences there are states of affairs.

Husserl then launches to the main part of the course. He follows the traditional order: the first part is dedicated to concepts, second to propositions (for the fear of psychologism he discusses propositions instead of judgments), and the third to inferences. In following I will briefly sketch the line of the argument in the first two parts, and then focus on the part on inferences, where Husserl explicitly formulates the theory of inference. The first two parts of the lecture-course are discussed briefly by Mohanty (2008, 52–57) and in more detail by Rollinger (2003). The theory of inference is discussed in detail by Centrone (2010), though she does not conclude anything on the basis of her otherwise excellent explications.

## 3 On Concepts

In the first part on concepts, echoing Frege, Husserl discusses issues such as reference and meaning. He also introduces the scholastic distinction between syncategorematic and categorematic expressions, and he uses the terminology of independent and non-independent parts of a whole (e.g. Husserl 2001a, 55–57). Husserl further distinguishes between relationships, combinations, series and manifolds (Husserl 2001a, §§22–23). He proceeds to discuss the concept of set, subordination, infinitely large cardinals and ordinals, etc. Following Bolzano and Cantor he defines a finitely large set as one that has no equally large proper subset. Husserl’s discussion of negation shows influence from scholastic logic, even though a reference to Bolzano as well as to Twardowski and Locke can be found here too. Husserl discusses also identity and equality, and finally equinumerosity. Two concepts have equally large extensions when there is a one-to-one correlation between them [es besteht gegenseitig eindeutige Korrespondenz], yet the extensions can be different (e.g., cardinals and ordinals). The extensions of two concepts can also be unequal so that one is larger than the other. However, Husserl adds that: “We cannot say of all concepts that they have equal or unequal extensions, because all concepts cannot be compared in this respect. We cannot compare the set of syllogisms and the one of triangles with each other (Bolzano); we are lacking a principle that would enable the mapping” (Husserl 2001a, 117). “Nicht von allen Begriffen können wir sagen, sie hätten gleiche oder ungleiche Weite, weil nicht alle Begriffe in dieser Hinsicht überhaupt vergleichbar sind. Wir können nicht die Menge von Syllogismen und die von Dreiecken miteinander vergleichen (Bolzano); es fehlt an einem Princip, das eindeutige Zuordnung ermöglichte” (Husserl 2001a, 117). This distinction between the concepts that can be compared to each other and those that cannot, appears to be related to the distinction between the conceptual and purely logical inferences Husserl establishes below.

## 4 On Propositions

In the second part on propositions Husserl discusses the formation of complex sentences from simple sentences (here he explicitly refers to Frege’s *Über Sinn und Bedeutung* [1892], cf. p. 134), categorical form of the sentences, extensional and intensional interpretation of the copula, negative categorical sentences, the meaning of negation, and quantification. Every now and then he wanders off to long discussions of primarily Sigwart and Brentano. Echoing e.g., Frege’s similar laments, Husserl’s own analysis of the difficulties is that the spoken language misleads in formulating the proper logic: “The last two lectures on the most influential German logicians have shown you what kinds of difficulties theoretical logic is fighting, how the grammatical incompleteness of the spoken expression as well as incompleteness of the research methods step into the way of elimination and fixing of the fundamental and primitive forms of sentences” (Husserl 2001a, 175). During the lectures Husserl comes to the following conclusion about the general affirmative sentences: They are not simple sentences, but they include an existential claim and a double negation, for example “All men are mortal” is a combination of two sentences, that there are men, and second, that there are no men that are not mortal (ibid., 181–182). This takes Husserl to discuss the concept of existence, in particular existence in mathematical statements such as “All triangles have three angles”. According to Husserl, the mathematical triangle certainly does not exist in real Wirklichkeit. “Nobody claims that there are real objects that correspond to rigorous mathematical concepts. However, we still can express general affirmative judgements about these non-existing triangles” (ibid., 183). After a closer analysis Husserl claims that the mathematical objects exist on the basis of definitions. Whether they exist in reality does not interest a mathematician (ibid., 183–184). “Wie wir es in der Geometrie mit geometrischen Existenzen zu tun haben, so in der Mythologie mit mythologischen” (ibid., 184). Husserl’s realism about concepts is in his 1896 lecture-course connected to structuralism about mathematical objects. The existence of mathematical objects is relative to the structure, which defines them. Husserl then goes on to a long discussion of Brentano and Sigwart on existence and truth, after which he concludes that existence and truth are concepts that are related to each other, and further indefinable (ibid., 220–221).

## 5 On Logical and Conceptual Inferences

Husserl’s lectures culminate in the third part of the lecture course on inferences. The theory of inferences provides us the conception of logic that is independent, a science rather than a method, purely formal, and demonstrative, as what Husserl called for in the beginning of the lecture course (Husserl 2001a, 32–45). It thus seems well justified to claim that the final part is the culmination of the lecture-course.^{4}

According to Husserl, inferences are typically understood in a very subjective sense. However, contrary to the tradition, Husserl emphasizes that the theory of inference is about objective relationships rather than about subjective acts of inferring. In its objective sense inference is in fact a causal proposition (kausaler Satz), and every causal proposition can be described as an inference (ibid., 234). The totality of all inferences is a closed but an infinite manifold of truths. It is impossible to enumerate all inferences (i.e., causal propositions), and to represent them in logic. However, logic deals with the laws that govern causal propositions independently of the specificity of the field to which they belong. Indeed, the logical inferences are purely formal; they depend only on the forms of the sentences in the premises and the conclusion (ibid., 235).

Husserl discusses also alogical (*alogische*), conceptual inferences as opposed to logical inferences. At this point he defines logical inferences by means of the idea of free variation: Logical inferences are the ones where the hypothetical truth is independent of the peculiarities of the matter, and will remain valid if the specifics are freely varied. For example “Wenn alle A B sind und S ein A ist, so ist auch S ein B”. The letters can be freely varied, which shows that the combination is logical. This is not the case in the following example: Hans is bigger than Kunz who is bigger than Wilhelm, thus Hans is bigger than Wilhelm. Here the inference relies on the specificity of the sentences. The names can be arbitrarily varied without changing the truth of the sentence. But the relationship between the sizes has sense (Sinn) only for whatever has a size. “a < b. b < c, therefore a < c” “does not belong to the general logical domain, it is restricted to the domain of quantities” (ibid., 239).^{5} Consequently, Husserl divides the relationships and combinations into two classes: into those that are generally logical, and those that are grounded on some other concepts, such as color, sound, spatial extension, time, etc. The relationship between concept and object, combination of individual objects to an Inbegriffe, combination of many predicates to a conjunctive predicate, combination of many sentences to conjunctive, disjunctive or hypothetical sentences are all logical. Examples of alogical combinations are before-after, left–right, combinations of vectors to a picture, of fields to a bodily construction, etc. Respectively the laws are divided into logical and alogical ones: “In logische Gesetze gehen nur logische Beziehungen und Verknüpfungen ein, sie konstituieren sich ausschlieβlich aus den allgemein logischen Kategorien und dem, was in ihnen gründet. In alogische Gesetze gehen auch andere Begriffe ein und sind aus ihnen nicht fortzuschaffen, etwa durch Verallgemeinerung”(ibid., 241). The rules of inferences are divided respectively. Given Husserl’s earlier distinction between manifolds that can be mapped to each other and those that cannot, one is tempted to think that the distinction between logical and conceptual inferences is based on that distinction. Thus conceptual inferences are the ones where the subject matter provides the principle that enables comparison between the manifolds. To be more specific, the subject matter provides the sort in respect of which the comparison is carried out.

## 6 Theory of Logical Inferences (Propositional Inferences)

To find the laws that govern the sentences in the inferences, Husserl proposes proceeding the way arithmeticians do. Curiously he raises the question of completeness: Initially arithmeticians had no conception about the completeness of their field. Arithmetic has developed through analysis of already used arithmetical laws and a systematic proof (of completeness) is missing: “How do the arithmeticians know that the combinations that they enumerate comprise all that are thinkable for numbers in general?” (Ibid., 243). Husserl’s answer is that we do not know for certain. Nevertheless, mathematics is an a priori science (ibid., 244).

The task in the theory of inference as well as in arithmetic is to build an axiomatic theory with which all thinkable problems could be solved in an ordered procedure. The difficulty is to show for every step that all the inferences themselves fall under laws that are already fixed as basic laws, and if this is not the case to show that the number of basic laws should be extended. Arithmetic gives us the model how this can be done (ibid., 245) Husserl then shows what one can do in arithmetic with the commutativity of addition: a + b=b + a. By means of substitution many other sentences can be formulated on its basis. It can also be used when solving arithmetical equations. This kind of procedure and grounding is not restricted only to arithmetic, but is possible wherever the domain is axiomatic (ibid., 246). Thus the practical nature of arithmetical procedure, namely calculating, must find its direct analogue in the formal discipline of inferences (ibid., 247). Calculation is an operation on signs and not on the concepts themselves. The grounding lies in the *arithmetica universalis*, in the algebra, “nicht viel anders als in der Ihnen praktisch bekannten numerischen Arithmetik”(ibid., 247). Husserl then proceeds to describe calculation in a manner very similar to the end of the *Philosophy of Arithmetic* claiming that “the thought operations and operations on signs are exactly parallel” (ibid., 248). There are no deductive inferences that would not have the corresponding combinations of signs. This way the pure game on signs leads always to propositions that can be interpreted as truths. (ibid., 248).

On the basis of the model given by arithmetic Husserl then explains what an ideal theory of inference looks like: It has primitive axioms that are independent from each other. Then there are propositions, theorems (Lehrsätze), that is, derived laws of inference. These derivations are themselves inferences or webs of inferences. But if we solve any such web into elementary inferences, then we come back to the first theorems and to only those inferences that fall under the axiomatic principles. The second proposition can have in its proof a form that is shown to be valid in the previous proposition. “Kurz, welchen Beweis man in der Theorie auch prüfen und analysieren mag, man wird immer in der Reihe der Axiome oder der vorher erwiesenen Gesetze solche finden, die ihn rechtfertigen”(ibid., 250).

Next Husserl discusses the general principles needed in proofs. Centrone (2010) provides a detailed analysis of them, so I will here only list them shortly.

- (i)
Inference from general to particular.

- (ii)
Modus ponens.

- (iii)
Rules of conjunction introduction and elimination.

- (iv)
Principle of distributivity of universal quantification over implication.

Husserl uses Schröder’s notation, according to which a conjunction between A and B is expressed by juxtaposition AB, disjunction by A + B, conditional A€B, negation with A_{0} or a long dash above the negated sentence letter, universal quantification Π, and existential quantification Σ.

Husserl then formalizes the whole theory (with slightly modified principles) by giving several logical axioms and derives by means of the principles several theorems. Centrone lists them all and also discusses in detail four of Husserl’s proofs (see Centrone 2010, 128–141). What Centrone does not mention is that in the end of his exposition Husserl also stipulates that *XX* ≡ *X* and *X* + *X* ≡ *X* (2001a, 261). These are Boole’s laws of inner weaving and inner joining that were advocated by Robert Grassmann in his *Formenlehre* (1872). These distinguish the algebraic system of logic from arithmetic. As I will explain below, Robert Grassmann emphasized the importance of these two laws to Husserl in two letters in November 1895 and February 1896. In both of the letters he accused Schröder to be utterly unscientific for not having them (Husserl 1994a, 160, 163–164).

In the end of his lecture course Husserl starts elaborating on the theory of conceptual inferences (see §66), but he never gives a full or explicit account of it. In it the basic form of judgement is “Γεa” which means that a certain object has a predicate a. The theory of conceptual inferences amounts to a theory of classes, parallel to the theory of logical inferences. In other words, Husserl suggests “conceptual” versions of the same principles and axioms as in his theory of logical inferences (Centrone 2010, 141–146). The theory of conceptual inferences appears to aspire to be a typed version of purely logical inferences.

To summarize, the 1896 lectures show that Husserl has already acquired many insights that will later resurface in the *Logical Investigations*. These include his criticism of psychologism, many insights about meanings and objects, and the theory of wholes and parts. Moreover, Husserl maintains a similar view of the theory of inferences in his 1902 lecture course on logic, which he held in Göttingen 2 years after the publication of the *Logical Investigations.*

The 1896 lecture-course shows also a work in progress; many times Husserl contradicts his own earlier views and wanders off to examine views about which he does not seem to be sure what to say. While many pieces, or better, moments of Husserl’s *Logical Investigations* are already in place, the distinction between grammar and logic proper has not yet been developed. Likewise, there is no clear notion of categorial intuition, and accordingly Husserl’s conception of truth appears to be unfinished. For example, Husserl defines the *anschaulichen Vorstellungen* to be subjective. The distinction between intuitive and conceptual according to him does not belong to logic but has something to do with subjective versus objective (2001a, 76), which is a distinction between psychological and logical. In the *Logical Investigations* Husserl will also abandon the traditional order of discussing logic starting from concepts, then moving to propositions and in the end focusing on inferences.

All kinds of influences can be detected in the 1896 lecture course. Its overall aim to be *Wissenschaftslehre* sounds very Bolzanian indeed. Husserl has later explained that his study of Lotze’s conception of Plato’s theory of ideas is what inspired him to accept Bolzano’s otherwise peculiar views. Husserl also uses the method of free variation in obtaining concepts. He owes the method of variation to Bolzano and in a way to Lotze as well (Beyer 1996; Centrone 2010). Bolzano and Twardowski also were proponents of the study of parts and wholes, which Husserl applies everywhere throughout the lecture-course. Husserl also refers to Frege’s *Über Sinn und Bedeutung* in his discussion of sentences functioning as non-independent parts of a presentation (p. 134). However, if we keep in mind Husserl’s main argument against psychologism in the *Prolegomena*, we should focus on Husserl’s conception of logic in the strict sense, i.e., as the purely formal, demonstrative theory instead of the discussions that some might categorize as belonging to philosophy of language. Husserl’s conception of logic follows Schröder’s *Vorlesungen*. Also the notation that he uses belongs to the algebra of logic tradition.

## 7 Husserl and the Algebraists of Logic

Since the evolvement of Husserl’s conception of logic and mathematics led him to abandon his earlier conception of logic, I will here briefly discuss Husserl’s relationship with the key figures of the algebra of logic tradition trying to trace possible sources for Husserl’s view.

The algebra of logic originates in George Boole (1815–1864) in his *Mathematical Analysis of Logic* (1847). His aim was to provide an algorithmic alternative to the traditional Aristotelian approach. Logical theories, according to the algebra of logic tradition, were thought to have a similar structure as the ordinary algebra. Instead of classifying different kinds of syllogisms, the aim was to represent logic by means of a deductive theory. While Boole is the finder of the algebra of logic tradition, Ernst Schröder was the most important representative of the tradition in Germany. Schröder’s work in turn was greatly influenced by the brothers Hermann and Robert Grassmann and also by Peirce’s logic of relations (Peckhaus 1996, 2004a). Hermann Grassmann (1809–1877) is the older and more original of the brothers, famous for his *Ausdehnungslehre* (1844, 1861). He worked in collaboration with his brother Robert Grassmann (1815–1901) who published a book called *Formenlehre* (1872), which presents a system close to that of Boole’s. Volker Peckhaus has shown that Schröder was first acquainted with the essentials of the algebra of logic through Grassmann’s *Formenlehre* and not through Boole (Peckhaus 1996, 225).

Husserl was familiar with the Grassmanns’ work already in the 1880s. Hermann Grassmann’s son and Robert Grassmann’s nephew, Hermann Grassmann Jr, was Husserl’s good friend. They had gone to school together in Leipzig, and Hermann Grassmann Jr. gave his father’s *Ausdehnungslehre* (1844) to Husserl as a gift already before Husserl went to Berlin to study with Weierstrass. Hermann Grassmann Jr. worked as a teacher of mathematics in Halle at the time Husserl was there. He was a family friend and spent for example Christmas often with the Husserls. Hermann Grassmann Jr. also asked his uncle Robert Grassmann to give advice to his philosopher-mathematician friend. Consequently Robert Grassmann sent two letters to Edmund Husserl in November 1895 and February 1896.^{6} In his letters Robert Grassmann, besides himself and his brother, advocates Aristotle as the only sharp logician. Curiously, in both of the letters Robert Grassmann complains that Schröder allows in his logic formulas such as a − a = 0, and a/a = 1, while these should not be permissible in logic. In logic a + a=a and aa = a should hold (Husserl 1994a, 160, 164). In other words, Robert Grassmann claims that Schröder did not maintain Boole’s laws of inner joining and inner weaving that we saw above Husserl included in his logic lectures of 1896.

While it is possible that Husserl’s interest in the algebra of logic tradition was influenced directly by the Grassmanns, it appears likelier that Husserl’s adherence to the tradition was mainly due to Schröder. In 1891 Husserl wrote a published review of the first volume of Ernst Schröder’s *Vorlesungen über die Algebra der Logik*. In it Husserl criticizes the “logical side” of Schröder's presentation but praises its “technical side”. According to him, “[t]hrough the invention of new methods, and the improvement of methods previously given, he documents that excellence as a mathematician which science has long recognized in him. And in as much as in the calculus we have a substantial enrichment of the old logic, the laudable algorithmic results of the author turn out to the advantage of that discipline itself, and in a manner which I certainly would not like to see underestimated” (Husserl 1994c, 71). After this Husserl engaged in a published debate about intensional and extensional logic with Andreas Heinrich Voigt, in which the applicability of the algebra of logic tradition to intensions was debated. Moreover, Husserl and Schröder sent their works to each other in 1898. In Husserl’s *Briefwechsel* we can find a letter from Schröder who complains that he has not had time to comment on Husserl’s very interesting piece of writing and in which he sent his article *On Pasigraphy* to Husserl. Husserl apparently already had the German version of the paper “Über Pasigraphie, ihren gegenwärtigen Stand und die pasigraphische Bewegung in Italien” published in the *Verhandlungen des ersten internationalen Mathematiker*-*Kongresses in Zürich vom 9.*–*11. August 1897* (Husserl 1994b, 245).

Husserl also studied the British algebraists. He lectured on their work in his 1895 lectures “Über die neueren Forschungen zur deduktiven Logik”. In those lectures, Husserl traces the roots of the general deductive conception of logic to Leibniz and Lambert. He discusses then William Rowan Hamilton, Augustus de Morgan, and finally George Boole in great detail. Thus, it is also possible that Husserl’s conception of logic was directly influenced by Boole as well.

Jean Van Heijenoort notoriously opposed the algebraic tradition to Frege’s concept-script. The algebraic tradition from Leibniz to Boole, De Morgan, and Jevons, according to him, copies mathematics “too closely, and often artificially” whereas Frege “freed logic from an artificial connection with mathematics but at the same time prepared a deeper interrelation between these two sciences” (1967a, p. vi). Frege is celebrated for having discovered quantification theory that Boole did not have (1967b). However, as we saw above, Husserl included a theory of quantification into his approach. In so doing he is again following Schröder who introduced Σ and Π into his 1891 *Vorlesung*. Within the algebra of logic tradition quantification was developed independently of Frege by Charles S. Peirce and Peirce’s student Oscar Howard Mitchell, whom also Schröder credits in his *Vorlesung* (Peckhaus 2004b). The later representatives of the algebra of logic tradition include Whitehead, C. I. Lewis, Löwenheim, and Tarski (Peckhaus 2004a, 558).

Thus, we may conclude that Husserl obviously adhered to general tenets of the algebra of logic tradition. However, it is more difficult to single out whom in particular Husserl is indebted to. Schröder’s influence on Husserl appears greatest and most direct, but through Schröder and/or directly, Husserl was probably also influenced by the Grassmanns, Boole, and Peirce. Accordingly, when considering the question of who influenced Husserl in his development towards an antipsychologistic view of logic these characters are the ones we should mention. Husserl read their works, interacted with them, and learned from them. Husserl also used their results in developing his own view of logic as a theoretical discipline, a view that represents the algebra of logic tradition.

It should be noted that in the *Logical Investigations* Husserl takes an additional step towards even a more general account of logic: he discusses an overall theory of theories in the context of which individual theories should be examined and compared to each other. On this occasion he mentions Cantor, Riemann, Grassmann, Hamilton, and Lie to be developing something of the sort (2001b, §§69, 70). Thus the algebraic theory of inference he maintains in his logic lectures appears to be embedded into a more general framework. Husserl’s idea is that in it the individual theories could be related to and compared with each other.

Husserl writes in 1895, 1896 (Husserl 2001a, 241), and in the *Prolegomena* (§21) that arithmetic and logic should not be distinguished. His formulation from 1895 is “Und so werden wir uns mit der zunächst wohl befremdlichen Auffassung Lotzes befreunden müssen, dass die Arithmetik nur ein relativ selbständiges und von alters her besonders hoch entwickeltes Stück der Logik sei”(ibid., 271–272). This should not be understood in the sense of Fregean logicism as a claim that arithmetic should be reduced to logic, but rather in the sense that for Husserl logic has an algebraic form that can also be found in arithmetic.

Die sinnliche Fülle der Anschauung, das frische pulsierende Leben des Individuellen, das unserer unmittelbaren Teilnahme sicher ist, geben wir freilich auf, wenn wir uns zum Abstrakten erheben. Aber ohne Abstraktion kein Begriff, ohne Begriff kein Gesetz, ohne Gesetz keine Einsicht aus dem Grunde, keine Theorie, keine Wissenschaft, und ohne Wissenschaft keine Philosophie. Die entsagungsvolle Abwendung von den grünen Tälern und Gefilden des Lebens zur grauen, ledernen, nüchternen Theorie ist eben das einzige Mittel, um unsere höchsten und reinsten Erkenntnisinteressen zu befördern.

Rollinger (2003) argues against Mohanty’s view that “[i]t would certainly be amiss to say, along with one of the most prominent commentators [Mohanty] on Husserl’s work in logic, that the later motifs of his transcendentalism, such as the view that ‘scientific objectivities are idealization of the life-world,’ were already present in his early logic”(p. 200). Given the quoted passage, Mohanty’s conception appears rather well grounded.

For some reason Mohanty 2008 does not discuss the third part on inferences at all. His discussion of the 1896 lectures is only about the parts on concepts and sentences (2008, 52–57).

In German: “gehört nicht in das allgemein logische Gebiet, er ist eingeschränkt auf das Gebiet der Gröβen”(2001a, 239).

Husserl also owned many books by Robert Grassmann: *Formelbuch der Denklehre*, *Die Logik und die andern logischen Wissenshaften*, *Formelbuch der Formenlehre oder Mathematik* and all the published parts of the *Das Gebäude der Wissenschaften* (Husserl 1994a, 159). However, most of them are not cut open. Only the “Denklehre” (the first part, second half of ‘*Gebäude des Wissens’*) is completely opened, of the other volumes only the preface is cut open (I owe this information to Carlo Ierna at the Husserl Archives, Leuven).