This is a preprint version of the paper that appeared in definitive form in Husserl's Logical Investigations, Daniel O. Dahlstrom (ed.), Dordrecht: Kluwer, 2003, pp. 69-93. The published version should be consulted for all citations.

Incomplete Symbols, Dependent Meanings, and Paradox

By Claire Ortiz Hill

The subject of Husserl's Fourth Logical Investigation, namely the fundamental distinction between independent and dependent meanings that lies concealed behind inconspicuous grammatical distinctions like those between syncategorematic and categorematic expressions, complete and incomplete expressions, is a topic of prime importance for the understanding of major issues in twentieth century western philosophy.

Most philosophers would, however, view that as a most exaggerated claim. For the full implications of Husserl's distinction have yet to be drawn and confronted. Wishing to redress that situation, I systematically studied a number of the issues in my Rethinking Identity and Metaphysics, On the Foundations of Analytic Philosophy (Hill 1997) and in a paper on the set theoretical paradoxes and the paradoxes of modal logic. However, in those works I did not explicitly integrate Husserl's ideas into what I had to say.

I undertake to do that here. I do so in the spirit of J. N. Mohanty's interesting and perspicacious observation of many years ago that Husserl anticipated many recent investigations in logic and semantics in recent times and what "is more, a true understanding and appraisal of his logical studies is not possible except in light of the corresponding modern investigations" (Mohanty 1976, VII). By drawing connections between Husserl's, Gottlob Frege's and Bertrand Russell's theories about incomplete symbols, dependent meanings and the causes of the set-theoretical paradoxes, I hope to lay more groundwork for understanding and appraising Husserl's logical investigations and the contribution that they must eventually make to modern logic and semantics. To this end, I here integrate Husserl's unpublished, and little known, remarks on Frege's "Function and Concept" (Frege 1891) and "On Concept and Object" (Frege 1892), and on Russell's paradox into ideas developed in the Fourth Logical Investigation.

Terminological Considerations

Of course, when trying to understand and appraise the significance of Husserl's writings for modern philosophy of logic and philosophy mathematics, readers of English must still navigate their way around a number of terminological obstacles. For very much of what Husserl had to say to them still lies concealed behind misleading translations.

In this case, since the Fourth Logical Investigation is devoted to the study of a distinction between kinds of meaning, it is here necessary to a special look at the issues surrounding words like: 'Bedeutung', 'bedeutsam', 'bedeutend', 'Bedeutsamkeit', 'bedeutungslos', 'Bedeutungslosigkeit', 'Sinn', 'sinnvoll', 'sinnlos', 'Unsinn', 'unsinnig', 'Widersinn', and 'widersinnig'. Since Frege's theory of meaning is under study here too, it is also necessary to keep in mind: 1) that he had already made his famous distinction between Sinn and Bedeutung in all the writings discussed here, and 2) that Husserl made a point in the First Logical Investigation of saying that he himself used the words 'Sinn' and 'Bedeutung' as synonyms and thought it dubious to differentiate between their meanings by using 'Sinn' for meaning in his own sense and 'Bedeutung' for the objects expressed as Frege had proposed (LI I, § 15). On Husserl's personal copy of "Function and Concept", he marked the passage about distinguishing between sense and meaning and underlined the words 'Sinn' and 'Bedeutung' (Frege 1891, 29).

Husserl's choice not to distinguish between sense and meaning simplifies matters for us here. For in English we can speak of sense, nonsense, and senselessness. We do not, however, say senseful or unsenseful, and there is no reason to start doing that here when we can use the words 'meaningful' and 'meaningless'. Harder problems, though, surround the translation of the words 'Widersinn' and 'widersinnig', which play a central role in the arguments of this essay. Although Husserl used these words in a perfectly normal way, they do not translate as neatly into English as one may wish. The German word 'wider' means against, counter, contrary to, in opposition to. So a very literal translation of these words might be 'countersense' and 'countersensical' (which might not be English words at all). Some have chosen to translate them thus; others have chosen 'absurdity' and 'absurd'. Husserl himself used 'Absurdität' and 'absurd' as synonymous with 'Widersinn' and 'Widersinnig' (ex. LI I, § 19; LI IV, Introduction, §12).

'Widersinn' and 'widersinnig' may, however, be understood in the sense of paradox or contradiction and paradoxical, contradictory, illogical, which better suits the purposes of this essay. In that case, these words fall into the family of 'widersprechen' (to contradict), 'Widerspruch' (contradiction), and 'widersprechend' and 'widerspruchsvoll', two common German words meaning contradictory. Given the importance of these terms to my main argument, I have chosen to leave 'Widersinn', 'Widersinnigkeit', and 'widersinnig' in German.

Moving on, the kinds of meanings studied in the Fourth Logical Investigation lie hidden behind grammatical distinctions, like those between syncategorematic and categorematic expressions, geschlossenen and ungeschlossenen expressions. Now, J. N. Findlay used the words 'closed' and 'unclosed' to translate 'geschlossenen' and 'ungeschlossenen' (LI IV, Intro.), a choice that really leaves readers in the dark about exactly what basic distinction Husserl had in mind. Since 'geschlossen' and 'ungeschlossen' also mean complete and incomplete in English, by translating them in that way we can, however, connect Husserl's ideas with those of his contemporaries. Frege often used the term 'abgeschlossen' (ex. Frege 1891, 25, 31, 32; Frege 1892, 54, 55) to describe what was complete as opposed to what was incomplete (unvollständig), in need of completion (ergänzungsbedürftig) to make up a complete whole (ein vollständiges Ganzes) (ex. Frege 1891, 24, 25, 27, 31, 32). And this is perfectly consistent with choices of Husserl (ex. LI IV, §§ 4,10), who actually wanted to make much the same point as Russell later would when he wrote of the "great many sorts of incomplete symbols in logic" that "are sources of a great deal of confusion and false philosophy because people get misled by grammar" (Russell 1918, 253).

To describe what was complete as opposed to what was incomplete, Frege usually employed his own words 'gesättigt' and 'ungesättigt', which have always been translated by 'saturated' and 'unsaturated' (ex. Frege 1892, 38, 47 n., 54-55). Related to the German word for satiation, 'gesättigt' and 'ungesättigt' might however, just as well have been translated by 'filled' or 'unfilled', which more neatly fit in with Husserl's discourse in terms of meaning fulfillment, filling up (Ausfüllung) (ex. LI IV, § 10; compare Frege 1891, 25) or Erfüllung (LI IV, § 9), which Frege translators have translated by 'realized' (ex. Frege 1892, 49, 50), but which could just as well have been translated by 'fulfilled', which facilitates comparisons with Husserl's ideas.

The primitive, essential distinction between independent and dependent and meanings

Since the distinction between categorematic and syncategorematic expressions is a grammatical one, Husserl realized that it might at first sight seem that mere grammatical considerations lie behind it and that the structure of the expressions had no relation to the structure of their meaning. In that case, the syncategorematic words going into the making of the expressions would then always be completely meaningless (bedeutungslos), and only the whole expression would have meaning (nur dem gesamten Ausdruck kommt wahrhaft eine Bedeutung zu) (LI IV, §§ 4, 5).

Husserl, however, considered that not to be the case. He rather saw the completeness (Vollständigkeit) and incompleteness (Unvollständigkeit) of expressions as being emblematic of a more primitive, essential distinction between independent and dependent meanings. One must not merely distinguish between categorematic and syncategorematic expressions, he taught, but also between categorematic and syncategorematic meanings. He reasoned that only meaningful (bedeutsame) signs were referred to as expressions and that it was superficial to put syncategorematic parts of expressions on the same level as other, generally entirely meaningless (bedeutungslosen) parts of expressions like letters of the alphabet, sounds, syllables, prefixes, or suffixes that may, for example, only be part of the sensory apparatus of the expression (LI IV, §§ 4, 5, 7).

We are right, Husserl pointed out, to say that words like 'but' or 'father's' have a meaning, but the same cannot be said of 'bi'. All three stand in need of completion, but the need of completion (Ergänzungsbedürftigkeit) is essentially different in each case. In the one case, it only concerns the expression; in the other it not merely concerns the expression, but above all the thought. Whenever syncategorematic expressions function in a normal manner, i. e., whenever they appear in connection with an independent, complete (abgeschlossenen) expression, Husserl maintained, they always have a determinate meaning relation to the whole thought, are bearers of meaning for a certain dependent component of the thought and in this way contribute to the expression (LI IV, §§ 4, 5).

So, Husserl concluded, if one wants to demarcate the ambiguity surrounding the completeness and incompleteness of expressions and determine both its problematical significance and the inner reasons why certain expressions can stand on their own as discourse complete in itself (als abgeschlossene Reden) and others cannot, one must go back to the realm of meanings and detect the need for completion (Ergänzungsbedürftigkeit) inherent in certain "dependent" meanings. The laws governing this dependency are grounded a priori in the nature of the corresponding contents. Each kind of dependency is accompanied by a law stipulating that a content of the relevant kind, say a, can only be in connection with a whole W(a, b…n), where the signs b…n stand for specific kinds of contents. Clarifying what lies concealed behind such grammatical distinctions thus leads to an application of the general ontological distinction made in the Third Logical Investigation between dependent and independent objects to the realm of meaning (LI IV Introduction; § 7).

The laws of meaning

It was further Husserl's conviction that the primitive, essential distinction between dependent and independent meanings formed the necessary basis for discovering the essential categories of meaning in which were grounded a variety of essential laws of meaning whose business it was to distinguish sense (Sinn) from nonsense (Unsinn) by determining the a priori forms in accordance with which the meanings of the different meaning categories might combine into One meaning instead of producing chaotic nonsense (Unsinn) (LI IV, Introduction).

According to Husserl, any instance of dependent meaning was accompanied by an essential law that governed its need for completion (Ergänzungsbedürftigkeit) by other meanings and established the ways in which they might be connected together that ruled out. other remaining possible combinations that would yield just a jumble of meanings instead of One meaning. The impossibility of combining meanings in certain ways was not merely subjective, Husserl insisted. It did not merely lie in our actual incapacity to achieve unity. It was objective, ideal and grounded in the "nature", the pure essence of the realm of meaning. Meanings, he repeated over and over, were governed by a priori laws that regulated the ways in which they might be combined with new meanings, in which they might fit together and constitute meaningful (sinnvolle), coherent meanings (LI IV, § 10).

Mere combinations of words like 'a round or', 'king but or', or 'a man and is' are nonsensical, meaningless (unsinnig, sinnlos), Husserl emphasized. In such expressions, each word has a meaning, but their meanings do not combine to give a coherent meaning to the whole expression. They are meaningless, utterly incomprehensible. It is completely obvious that so combined no meaning exists, or can possibly exist, for them. On no account can they refer to any object. Moreover, not only is there not any question of reference to objects, but there is not any question of truth either. They break the laws about what can be meaningful. Meaning itself is missing. (LI IV, § 12)

To make his point Husserl took the coherent, meaningful expression 'this tree is green' and proposed formalizing the given meaning, the independent logical statement, to obtain the corresponding pure meaning form: 'this S is p'. It is clear that formalized in that way it can be interpreted in infinitely many ways, he explained. The statement 'this tree is green' can of course be transformed. We can put any noun or noun phrase in the place of 'S' and any adjective in the place of 'p'. We can say 'this gold', 'this algebraic number', 'this blue crow', etc. 'is green' and we will again obtain a coherent, meaningful (sinnvolle) meaning and an independent sentence of the form indicated. Such free exchange of expressions within a given category may yield false, dumb, or funny meanings, but it will necessarily yield coherent meanings. (LI IV, §10)

We are not, though, absolutely free in how we go about this, but are bound by strict limitations. Not just any meaning can be substituted for S or for p. Once meaning categories are violated, the coherency of the meaning is lost. As examples of the kind of transgression that he had in mind, Husserl gave: 'this reckless is green'; 'more intensive is round'; 'this house is equal'. We can, he noted, substitute 'horse' for 'similar' in the relational form 'a is similar to b', but we thus still obtain only a series of words, in which each word as such indeed has a sense, refers to a complete combination of meanings (vollständigen Sinneszusammenhang), but we do not have a coherent, complete meaning (geschlossenen Sinn) (LI IV, §10).

According to Husserl, the job of a science of meanings would be to construct meanings in accordance with essential laws, to discover the laws of combining meanings and transforming them and trace them back to a minimal number of independently elementary laws. It would be necessary first to identify the primitive meaning formations and investigate their inner structure in order to identify the pure meaning categories, which define the sense and extension of what is indeterminate (or what is precisely analogous in mathematics, the variable) in the laws, he maintained. (LI IV, §§ 10, 12, 13)

Changes in meaning

On his personal copy of Frege's "On Concept and Object", Husserl marked the sentence which reads: "Language has means of presenting now one, now another, part of the thought as the subject". And he tellingly underlined the word 'language', as if to underscore the fact that one could do with language things that logic did not allow (Frege 1892, 49). Section 11 of the Fourth Logical Investigation is, in fact, devoted to elucidating this very matter. In it Husserl studies the possibility of being led astray by the fact that meanings of each category, and even syncategorematic forms like 'and', may figure in the subject position otherwise reserved for substantival meanings. Taking a closer look, he noted, one sees that in the meaning modification process it happens that a meaning of another syntactical form, say adjectival or even just a mere form, may simply be transplanted into the subject position, as occurs, for example, in sentences of the kind: ''if' is a particle', ''and' is a dependent meaning'. The words are definitely in the subject position, but their meanings are not the same as they normally are.

That each word and each expression can generally be put into the subject position is not in itself surprising, Husserl confirms. What we need to look at, though, is not the composition of the words, but that of the meanings. For from a logical point of view, all change in meaning is to be judged logically abnormal. However, although logic's concern with identical, coherent meanings demands invariability of the meaning function, certain changes in meaning belong even to the normal grammatical state of each language. Within context of what is being said, the modified meaning is always easily understandable, and if the reasons behind the modification are of sweeping enough generality, if they are rooted, for example, in the general nature of the expression as such, or even in the pure essence of the meaning realm in itself, then the types of abnormality concerned recur everywhere and what is logically abnormal then appears to be grammatically sanctioned. (LI IV, § 11)

Turning to consider the suppositio materialis of the Scholastics, Husserl further noted that each expression, whether its normal meaning is categorematic or syncategorematic, can name itself. If we say ''the Earth is round' is a statement', it is not the meaning of the statement that is in the subject position, but the statement as such; judgment is passed not on the fact that the Earth is round, but on the statement sentence, and this sentence is functioning abnormally as its own name. If we say ''and' is a conjunction', we have not brought the meaning normally corresponding to the name into the subject position, but rather the independent meaning of the word 'and'. Relative to this abnormal meaning, 'and' is actually not syncategorematic expression, but a categorematic one. It names itself as a word. Something precisely analogous occurs, Husserl explained, when we say: ''and', 'but', 'bigger' are dependent meanings'. As a rule, we say: the meanings of the words 'and', 'but' and 'bigger' are dependent. Likewise, in the expression: ''man', 'table', 'horse' are concepts of things'. According to Husserl, it was not the concepts themselves that were here functioning in the subject position, but rather representations of these concepts (Vorstellungen dieser Begriffe). (LI IV, § 11)

In this section Husserl also discusses adjectives, which are normally predestined to play a predicative and attributive role. The adjective 'green', Husserl explains, functions normally with its "original", unmodified meaning in the sentence 'This tree is green' and remains unchanged in itself (in sich selbst) when we say 'this green tree'. Husserl maintains that this way of changing syntactical form as opposed to syntactical matter which, for example, takes place when a nominative meaning functioning as a subject changes to function as a object, or when a sentence functioning as the antecedent changes to function as the conclusion, is to be established first of all and is a main theme of the description of the permanent structures of the realm of meaning.

However, Husserl points out, what is adjectival can undergo still another modification in which the adjective not merely functions attributively in an nominative whole, but is itself used as a noun as in 'Green is a color' of 'Greenness is a difference in coloring'. In that case 'green' used in the original way and 'green' used as a subject obviously share an identical, common, abstract "core" (Kern), which has in each case taken on different forms, which are to be distinguished from their syntactical form. If the modification in the core form of the adjectival core content (of the core itself) has produced syntactical matter of the nominative type, then this noun enters into all the syntactical functions, even those that according to formal rules require nouns as syntactical matter. Further details on this matter, Husserl then concludes, belong to a systematic construction of his theory of meaning. (LI IV, § 11)

Logical laws as opposed to laws of meaning

Laws of meaning, according to Husserl, are not themselves logical laws, but rather supply pure logic with the possible forms of meaning whose formal truth or falsehood and reference to objects is determined by logical laws. Whereas laws of meaning serve to distinguish sense (Sinn) from nonsense (Unsinn) by providing pure logic with possible meaning forms, a priori forms of complex coherent, meaningful meanings, it is the job of logical laws to guard against formal or analytical Widersinn, formal absurdity, by dictating what it is that objects require to be consistent in purely formal terms. While laws of meaning distinguish what is sinnvoll from what is sinnlos, logical laws distinguish what is sinnvoll from what is widersinnig. (LI IV, Introduction; § 12)

Husserl was adamant about the differences between Sinn, Unsinn and Widersinn. What is widersinnig, he complained, is too often wrongly spoken of as being sinnlos. However, what is sinnlos, unsinnig, he insisted, has no meaning whatsoever and cannot have any. In contrast, what is widersinnig genuinely has a coherent meaning and can be determined to be true or false. What is widersinnig rightly belongs in the realm of the meaningful, constitutes a partial domain of what is sinnvoll. However, in contrast to what is sinnvoll, no object can correspond to what is widersinnig. The meaning is there, but no existing object can correspond to the existing meaning.

Into the category of Widersinn, Husserl placed expressions like 'wooden iron', 'round square', 'all squares have five corners'. These are as respectable names and sentences as any, he maintained. They have meaning, but no object, no thing or fact such as is described by such expressions exists or can exist (LI IV, Introduction; § 12). In his unpublished notes on set theory, Husserl jotted down the following comments on the sentences: 'The present emperor of France is blond' and 'The present emperor of France is not blond'. Both, he noted, are objectless (gegenstandslos). 'The present emperor is blond' implies that France presently has a blond emperor and she has no emperor at all. The sentence is not valid, because it is objectless. A sentence is not valid, because it is objectless, in actual fact or owing to a contradiction (Widerspruchs) in the subject term. A sentence is not valid because it asserts something of its object that is not attributable to it. The sentence is widersinnig with regard to what is predicated of it. A sentence is not valid because it affirms something of its object, having valid existence, or recognized as valid, that is not generally attributable to objects of this sort, or not in fact attributable to this object itself (Husserl Ms A 1 35, 19a-b).

Husserl, Widersinn, and the Set-theoretical Paradoxes

Husserl never published his thoughts on the set-theoretical paradoxes that were the bane of Frege and a thorn in the side of Russell and of so many of their colleagues. Husserl did, however, leave behind a hundred pages of unpublished notes on set theory and, as has just been indicated, important points made there are of one piece with what he wrote in the Fourth Logical Investigation. Moreover, as will be seen below, they tie in with conclusions that Russell and Frege themselves came to regarding the causes of the set-theoretical paradoxes.

Remember that in 1902 Russell wrote to Frege of a contradiction derivable within the system of the Basic Laws of Arithmetic (Frege 1893). In his letter, Russell informed Frege that his idea that a function could also constitute the indefinite element was dubious because it led to the following contradiction: "Let w be the predicate of being a predicate which cannot be predicated of itself. Can w be predicated of itself? From either answer follows its contradictory. We must therefore conclude that w is not a predicate. Likewise, there is no class (as a whole) of those classes which as wholes, as not member of themselves. From this I conclude that in certain circumstances a set does not form a whole" (Frege 1980, 131).

In his unpublished notes on the set-theoretical paradoxes, Husserl plainly relegated them into the category of Widersinn. He affirms this over and over. For him, a set which contains itself as a element is widersinnig (Husserl Ms A 1 35, 12b). To the objection that there is no set that contains itself as an element, he maintained that one need merely respond that that is widersinnig (Husserl Ms A 1 35, 17a). In the case of the set-theoretical paradoxes, he wrote, if one is clear and distinct with respect to meaning (Sinn), one readily sees the Widersinn. The solution to the paradoxes lies in demonstrating the shift of meaning (Sinnverschiebung) that makes it that one is not immediately aware of the contradiction (Widersinn) and that once one perceives it one cannot indicate wherein it lies (Husserl Ms A 1 35, 12a).

All sets, Husserl reasoned in these notes, fall into one of two classes. Either they fall into class A of sets that contain themselves as an element, or they fall into class B of sets that do not. That being so, he asked to which of these two classes did the set PM of all sets that do not contain themselves as an element belong? By the law of the excluded middle, it would have to be one of the two. However, Husserl observed, one can show that neither or both of the possibilities must be valid. For were PM to belong to class A, that would mean that it contained itself and that would contradictory by definition. Were it fall into class B, PM would then not be the set of all sets of all Bs. (Husserl Ms A 1 35, 17a)

In actual fact, Husserl further noted, it proceeds from the paradox, if no conceptual shift is demonstrable, that a set of kind A or a set of kind B must be a Widersinn. Then the classification is widersinnig as well. He then looks at the alternative that both may not be the case and asks whether PM is not widersinnig. Since, he adds in parenthesis, a member has a completely secure meaning (Sinn), we have two cases: either concept A is widersinnig, or concept PM. Being mathematical, he adds having closed the parenthesis, the sense (Sinn) of the classification should be a classification of all possible sets in general, independently of questions of real existence having to do with real things of life instead of possible things generally, possible counted things, collected things. (Husserl Ms A 1 35, 17a)

Three pages later, we finding Husserl asking whether a set can contain itself as a partial set. If by partial set we understand any set which only contains members of M (as its members), he reasoned, then each set has itself as a partial set. However, if by partial set we understand only any set that only contains members of M among its members, but does not contain at least one member of M among its elements, then no set contains itself as a partial set. To his next question as to whether a set contains itself as a member, he answers that that could only naturally be valid of sets of sets (Husserl Ms A 1 35, 20a).

It is part of the idea (Idee) of the set to be a unit, he explains, as it were a whole comprising certain members as parts, but doing so in such a way that, vis-à-vis its members, it is something new which is first formed by them. A whole cannot be its own part. Just as it is contradictory (widersprechend) for a whole to be its own part at the same time, so it is contradictory (widersprechend) for a set for it to be its own member. From the fact that I can speak of all sets, it does not follow that the totality of sets can in return be looked upon as a set. All mathematico-logical operations performable with sets turn on the idea that sets can be looked upon as kinds of wholes, as new units, formations that are something new vis-à-vis their original members, so that out of these formations new units can then again be formed. (Husserl Ms A 1 35, 20b)

With respect to the set-theoretical paradoxes one can therefore say, Husserl affirms, that wherever mathematicians speak of sets, if the concept is to be a mathematical one, they must have a set essence in view. And whatever sets may then have as an essence, it is expressed with a relation that belongs to the essence, i.e., the relation between sets themselves and elements of a set. An essence relation makes it impossible for the members of the relation to be identical. Thus a set which contains itself as a element is widersinnig (Husserl Ms A 1 35, 12b).

Husserl's Reactions to Frege's Articles on Functions, Concepts, and Objects

For Husserl, we have seen, the study of syncategorematic and categorematic expressions, complete and incomplete expressions drew attention to the differences between dependent and independent meanings as well as the differences between laws of meaning that serve to distinguish meaningfulness from meaninglessness and logical laws that that serve to distinguish meaningfulness from Widersinn. In "Function and Concept" (Frege 1891) and "On Concept and Object" (Frege 1892), Frege too studied problems surrounding the completeness or incompleteness of expressions, meaningfulness, Widersinnigkeit, truth, falsehood and objective reference. Like Husserl, he affirmed that the distinctions to be drawn there were deeply grounded in the nature of the matter and of language (Frege 1891, 41; Frege 1892, 55).

To my knowledge, Husserl never referred to "On Concept and Object" in print. In a 1903 review, he did, however, fleetingly refer to "On Function and Concept" as a work by the "ingenious mathematician G. Frege… which unhappily has not found the attention that it deserves from professional logicians" (Husserl 1903, 247). Husserl did, though, own offprints of both articles and by marking passages and writing his comments in the margins, he left a valuable record of his keen interest in the ideas expressed there. Interestingly, he mainly, and almost exclusively, marked the very passages that deal with themes that went into the making of the Fourth Logical Investigation. Moreover, he signaled points of weakness in Frege's reasoning that both he and Russell ultimately concluded were the cause of the paradoxes derivable within Frege's system.

The matter of the completeness or incompleteness of expressions figures prominently in both of Frege's articles. "Statements in general", he informed readers in "Function and Object", "just like equations or inequalities or expressions in Analysis, can be imagined to be split up into two parts; one complete in itself (abgeschlossen), and the other in need of supplementation (ergänzungsbedürftig) or 'unsaturated'". In a passage that Husserl marked, Frege chose 'Caesar conquered Gaul' as an example. The second part, 'conquered Gaul', he explained, "is 'unsaturated' --it contains an empty place; only when this place is filled up (ausgefüllt) with a proper name, or with an expression that replaces a proper name, does a complete sense (abgeschlossener Sinn) appear" (Frege 1891, 31).

According to Frege, the predicable nature of concepts was conferred on them by a particular kind of incompleteness and dependency that they exhibited with regard to objects. For him concepts stood in need of supplementation and of completion, and he always cited their fundamentally incomplete nature as being what constituted the principal difference between them and objects. In Frege's special vocabulary "a concept is unsaturated in that it requires something to fall under it; hence it cannot exist on its own" (Frege 1980, 101). A concept word "contains a gap which is intended to receive a proper name" (Frege 1980, 55).

Not all of the parts of a thought may be complete (absgeschlossen), Frege reminded readers in "On Concept and Object". At least one must be in some way 'unsaturated' or predicative, or they would not stick together. The sense of the phrase 'the number 2', he explained, does not stick to that of the expression 'the concept prime number' without something binding them. In the sentence 'the number 2 falls under the concept prime number', this bond is contained in the words 'falls under', which need to be completed in two ways --by a subject and an accusative. It is only because their sense is 'unsaturated' in that way, Frege stresses, that they are capable of binding. Only when they have been supplemented in this twofold respect, he stressed, do we have a complete sense (abgeschlossenen Sinn), do we have a thought (Frege 1892, 54; also 46-48, 50; Frege 1891, 24, 25, 31; Frege 1979, 119-20; Frege 1980, 101, 141).

In these two articles, Frege was also particularly intent upon drawing analogies between concepts and what mathematicians call functions. In "Function and Concept", he explained that in a sentence like 'Caesar conquered Gaul', he called the meaning (Bedeutung) of the unsaturated portion the 'function' and Caesar the argument (Frege 1891, 31). On his personal copy of the article, Husserl underlined the word 'meaning' in that passage. On the page before, he had marked where Frege had written: "We thus see how closely that which is called a concept in logic is connected with what we call a function. Indeed, we may say at once: a concept is a function whose value is always a truth-value" (Frege 1891, 30).

In answer to the question as to what, once he had admitted objects without restriction as arguments and values of functions, it was he was calling an object, Frege answered that an "object is anything that is not a function, so that an expression for it does not bring any empty place" (Frege 1891, 32). Husserl underlined the word 'object' in that answer. On one of the first pages of "Function and Concept", he had written that the object of a concept would be something hard to express, but is easily represented in arithmetical notation (Frege 1891, 24).

On his copy of "Function and Concept", Husserl also marked where Frege had written: "I am concerned to show that the argument does not belong with a function, but goes together with the function to make up a complete whole (ein vollständiges Ganzes); for a function by itself must be called incomplete (unvollständig), in need of supplementation (ergänzungsbedürftig), or 'unsaturated'. And in this respect functions differ fundamentally from numbers" (Frege 1891, 24).

On his copy of "On Concept and Object" Husserl marked the statement that a "concept is the meaning (Bedeutung) of a predicate; an object is something that can never be the total meaning (Bedeutung) of predicate, but can be the meaning (Bedeutung) of the subject" (Frege 1892, 48), as well as Frege's declaration that the concept must always be distinguished from the object, even in cases like that of the concept word 'Venus', where just one object falls under the concept. 'Venus', Frege had stressed, could never be a proper predicate, although it can form part of a predicate. "The meaning of this word is thus something that can never occur as a concept, but only as an object" (Frege 1892, 44). Husserl further marked the paragraph in which Frege says that with his distinction between concept and object he had got hold of a distinction of the highest importance (Frege 1892, 54).

A link suggests itself between what Frege wrote about the incompleteness of concepts and functions and the completeness of objects in these articles and what Husserl would have to say about meaningfulness, meaninglessness, Widersinn, truth, falsehood and having an objective reference in the Fourth Logical Investigation. For Husserl marked the place where Frege wrote that if "we substitute 'Julius Caesar' for the proper name formed by the first six words of the sentence 'the concept square root of 4 is 'fulfilled' (erfüllt), we get a sentence that has a sense (Sinn) but is false; for that fulfillment (Erfülltsein)… can truthfully only be said of a quite special kind of objects, viz. Such as can be designated by proper names of the form 'the concept F'" (Frege 1892, 50). Frege had just written of the sentence 'there is Julius Caesar' that it is neither true nor false, but senseless (sinnlos), whereas the sentence 'there is a man whose name is Julius Caesar' has a sense (einen Sinn hat)" (Frege 1892, 50). Husserl marked Frege's conclusion that what the example showed "holds good generally; the behaviour of the concept is essentially predicative, even where something is being said about it; consequently it can be replaced there only by another concept, never by an object. Thus what is being said concerning a concept does not suit an object" (Frege 1892, 50).

In several places, Husserl showed himself at variance with Frege about matters of meaninglessness and truth value. For example, next to Frege's comment that for any argument x for which 'x+1' were meaningless (Bedeutungslos), the function x + 1= 10 would have no value and thus no truth value either, so that the concept: 'what gives the result 10 when increased by 1' would have no sharp boundaries, Husserl penned the objection that the function does not, however, do this to concepts first, but delimits their logical use. The call to work with possible concepts is generally justified, he acknowledged, but impossible concepts are also concepts. There we find the ambiguity again, Husserl continued writing. Having a value, he protested, surely does not mean having a truth value. Each function has eo ipso a value, or this not being the case, then value and truth value collapse together (Frege 1891, 33). Alongside Frege's statement "that we give the name 'the value of a function for an argument' to the result of completing the function with the argument", Husserl had commented that the definition was unclear and that value should surely be distinguished from truth value (Frege 1891, 25). And next to Frege's statement that the value of our function was a truth value, Husserl had noted that this was again not completely clear, --that each value of a function was necessarily true or false, existing or not existing (Frege 1891, 29). A few pages later he asked in the margins whether or not he could not speak of the truth or falsehood, meaning the existence of the capital of the number 4, whether he could not say that there was no capital of the number 4, as it appears absurd (absurde) concepts would not reckoned as concepts (Frege 1891, 31-32).

Frege's linguistic predicament

In "Function and Concept" and "On Concept and Object", Frege also, and most importantly, struggled with what he eventually came to call the "fatal tendency of language" to allow a concept word to be transformed into a proper name and so to come to be in a place for which it was unsuited (Frege 1979, 269-70; Frege 1980, 54-55). The essential differences between concepts and objects are, we find him explaining in letters, "covered up in our word languages" (Frege 1980, 55) where "the two merge into each other" (Frege 1980, 100) and "the sharpness of the difference is somewhat blurred, in that what were originally proper names (e.g., 'moon'), can become concept words, and what were originally concept words (e.g., 'God') can become proper names" (Frege 1980, 92).

By "a kind of necessity of language", he confessed in "On Concept and Object", his expressions, taken literally, sometimes missed his thought, so that he mentioned an object, when what he intended was a concept (Frege 1892, 54). Indeed, although he firmly believed that concepts were fundamentally different from objects, by the early 1890s he had found that in "logical discussions one quite often needs to say something about a concept and to express this in the form usual for such predication" (Frege 1892, 46). When "we say that the concept horse is not a concept, whereas, e.g., the city of Berlin is a city, and the volcano Vesuvius is a volcano", he famously wrote in "On Concept and Object", "we are confronted by an awkwardness of language, which… cannot be avoided" (Frege 1892, 46).

"Language is here in a predicament that justifies the departure from custom", he concluded. Owing to its predicative nature, the concept as such cannot play the part of the subject, so "it must be converted into an object, or, more precisely, an object must go proxy for it" (Frege 1892, 46). In a note to that claim, Frege wrote: "A similar thing happens when we say as regards the sentence 'this rose is red': The grammatical predicate 'is red' belongs to the subject 'this rose'. Here the words 'The grammatical predicate "is red"' are not a grammatical predicate, but a subject. By the very act of explicitly calling it a predicate, we deprive it of this property" (Frege 1892, 46 n.†). Husserl marked this note. As we have seen, he marked Frege's sentence: "Language has means of presenting now one, now another, part of the thought as the subject" and underlined the word 'language' (Frege 1892, 49).

Frege was additionally aware that "over the question what it is that is called a function in Analysis, we come up against the same obstacle; and on thorough examination one will find that the obstacle is grounded in the matter itself and in the nature of our language" (in der Sache selbst und in der Natur unserer Sprache begründet) (Frege 1892, 55). Functions do, he found, and sometimes have to, take other functions as their arguments. When that is the case, however, they are then fundamentally different from functions whose arguments are objects and cannot be anything else, he insisted. Just "as functions are fundamentally different from objects, so also functions whose arguments are and must be functions are fundamentally different from functions whose arguments are objects and cannot be anything else"(Frege 1891, 38). Frege called the latter first-level functions and the former second-level functions and distinguished between first-level and second-level concepts in the same way (Frege 1891, 38). "Function and Concept" closes with the affirmation that the difference between first-level and second level functions discussed there "is not made arbitrarily, but is deeply grounded in the nature of the matter" (nicht willkürlich gemacht, sondern in der Natur der Sache tief begründet) (Frege 1891, 41).

On his copy of "On Concept and Object", Husserl marked the sentence: "The relation of an object to a first-level concept that it falls under is different from the (admittedly similar) relation of a first-level to a second-level concept" (Frege 1892, 50). He also marked the line where Frege had written in the paragraph before: "I do not want to say it is false to say concerning an object what is said here concerning a concept; I want to say that is impossible, it is senseless (sinnlos) to do so" (Frege 1892, 50). Here Husserl tellingly underlined the word 'sinnlos' on his personal copy.

During the 1890s, Frege convinced himself that he could overcome this linguistic predicament by creating a "logical law", his Basic Law V governing the behavior of graphs of functions and extensions of concepts. As he explained in a sentence that Husserl marked in "Function and Concept": "Graphs of functions are objects, whereas functions themselves are not… Extensions of concepts likewise are objects, although concepts themselves are not" (Frege 1891, 32). Frege knew well that what he wished to sanction through Law V was "forbidden by the basic difference between first and second level relations" (Frege 1979, 182), but he temporarily convinced himself that, though an actual proof could "scarcely be furnished" and "an unprovable law" would have to be assumed, a transformation might "take place, in which concepts correspond to extensions of concepts…" (Frege 1979, 182). "Of course," he knew by 1906, "it isn't as self-evident as one would wish for a law of logic. And if it was possible for there to be doubts previously, these doubts have been reinforced by the shock the law has sustained from Russell's paradox" (Frege 1979, 182). (It is worth noting here that Husserl put a question mark where Frege had written in "Function and Concept" that the possibility of regarding an equality holding generally between values of functions as an equality is not demonstrable and must be taken to be a fundamental law of logic (Frege 1891, 26)).

Frege's Initial Reactions to Russell's Paradox

Russell's paradox about the class of all classes that are not members of themselves ultimately led both Frege and Russell to many of Husserl's conclusions about what can and cannot be done in logic. For understanding the logical behavior of incomplete symbols and dependent meanings came to play a preeminent role in the efforts of those two makers of modern logic and semantics to grasp the causes of the set-theoretical paradoxes and to avoid them (Hill 1997, 20, 22, 40-41, 69-70, 77-82, 87-90, 91-110, 145-47, 149-51).

When first confronted with Russell's paradox in 1902, Frege responded that it seemed to indicate that his Basic Law V was false and that his explanations did not suffice to secure a meaning for his combinations of signs in all cases (Frege 1980, 132). True to his convictions that the fundamental differences between predicates and objects are inviolate and founded in the deep nature of the matter, another of his first reactions upon receiving the bad news had been to write to Russell that his expression 'A predicate is predicated of itself' did not seem exact to him (Frege) and that he would rather say 'A concept is predicated of its own extension'. A predicate, Frege explained to Russell, was as a rule a first-level function which required an object as argument and which could not therefore have itself as argument (subject) (Frege 1980, 132-33). The problem with the proposition 'A function never takes the place of a subject', he told Russell, had already been dealt with in 'On Concept and Object'. It was "only an apparent one occasioned by the inexactness of the linguistic expression; for the words 'function' and 'concept' should properly speaking be rejected. Logically, they should be the names of second-level functions; but they present themselves linguistically as names of first-level functions. It is therefore not surprising that we run into difficulties in using them" (Frege 1980, 141).

To express ourselves precisely, Frege further informed Russell, our only choice would be to talk about words or signs. The proposition '3 is a prime number', he told him, could be analyzed into two essentially different parts: '3' and 'is a prime number'. The first part was complete (abgeschlossen) in itself, but the second stood in need of completion (ergänzungsbedürftig), and the same was to be said for the proposition '4 is a square number'. It makes sense, Frege maintained, "to fit together (sinnvoll zusammenfügen) the complete (abgeschlossenen) part of the first proposition with that part of the second proposition which is in need of completion (ergänzungsbedürftigen) (that the proposition is false is a different matter); but it makes no sense to fit together (sinnvoll zusammenfügen) the two complete (abgeschlossenen) parts; they will not hold together; and it makes just as little sense (ebensowenig sinnvoll) to put 'is a square number' in place of '3' in the first proposition" (Frege 1980, 142).

"This difference between the signs", Frege continued, "must correspond to a difference in the realm of meanings; although it is not possible to speak of it without turning what is in need of completion (Ergänzungsbedürftige) into something complete (Abgeschlossenes) and thus falsifying the real situation. We already do this when we speak of 'the meaning of "is a square number"'. Yet the words 'is a square number' are not meaningless (bedeutungslos). The analysis of the proposition corresponds to an analysis of the thought, and this in turn to something in the realm of meanings" (Frege 1980, 141-42).

In the appendix to the 1903 volume of Basic Laws, Frege still concluded that there was "nothing left but to regard extensions of concepts or classes, as objects in the full and proper sense of the word" (Frege 1903, 217) and that there was "nothing 'unsaturated' or predicative about classes that would characterize them as functions, concepts, or relations. What we usually consider as a name of a class… has rather the nature of a proper name; it cannot occur predicatively, but can occur as the grammatical subject of a singular proposition" (Frege 1903, 215).

Frege's Ultimate Conclusions

It was ultimately this very propensity of language to undermine the reliability of thinking by forming apparent proper names to which no objects correspond that Frege blamed for having "dealt the death blow" to his set theory (Frege 1979, 269). When asked shortly before his death to write on the paradoxes of set theory in order to provide further justification for his belief that it was untenable, he replied that the paradoxes of set theory arise because a concept is connected with something that is called the set which appears to be determined by the concept --and determined as an object (Frege 1980, 54). "The essence of the procedure which leads to the thicket of contradictions", he said, could be summed up as follows:

The objects that fall under F are regarded as a whole, as an object and designated by the name 'set of Fs'. This is inadmissible because of the essential difference between concept and object, which is indeed quite covered up in our word languages…. Because of its need for completion (Ergänzungsbedürftigkeit), (unsaturatedness, predicative nature), a concept word is unsaturated, i.e., it contains a gap which is intended to receive a proper name.… Through such saturation or completion (Ergänzung) there arises a proposition whose subject is the proper name and whose predicate is the concept word…. In such a proposition, concept word and proper name occupy essentially different places, and it is obvious that a proper name will not fit into the placed intended for the concept word. Confusion is bound to arise if a concept word, as a result of its transformation into a proper name comes to be in a place for which it is unsuited (Frege 1980, 55).

"One feature of language that threatens to undermine the reliability of thinking", he wrote that same year, "is its tendency to form proper names to which no objects correspond". The paradoxes of set theory arise, he explained then, because a concept is connected with something called a set, which appears to be determined as an object. One thus thinks of the objects falling under the concept as combined into a whole, which is construed as an object and designated by a proper name. Such an expression appears to designate an object, but there is no object for which this phrase could be a linguistically appropriate designation, he came to realize. The expression is surely a designation of a concept and thus could not be more different from a proper name (Frege 1979, 269-70). The "difficulties which this idiosyncrasy of language entangles us in are incalculable", Frege warned then (Frege 1979, 270). Experience had shown him "how easily this can get one into a morass" (Frege 1980, 55). He even went so far as to suggest that one "must set up a warning sign visible from afar: let no one imagine that he can transform a concept into an object" (Frege 1979, 55).

Russell on Incomplete Symbols and the Set-theoretical paradoxes

As for Russell, one of the things that his struggle with the paradox derivable within Frege's system taught him was that it "is important, if you want to understand the analysis of the world, or the analysis of facts, or if you want to have any idea what there really is in the world, to realize how much of what there is in phraseology is of the nature of incomplete symbols" (Russell 1918, 253). In particular, he learned through it that if a word or phrase that is devoid of meaning when separated from its context is wrongly assumed to have an independent meaning, we then get what may be called false abstractions, pseudo objects, and paradoxes and contradictions are apt to result (Russell 1906c, 165). Indeed, Russell came to believe that "if we assume, as Frege does, that the class is an entity, we cannot well escape the contradiction about the class of classes which are not members of themselves" (Russell 1906c, 171) and that it resulted from that contradiction that it "must under all circumstances be meaningless (not false) to suppose a class a member of itself or not a member of itself…" (Russell 1919, 185).

"When we say that a number of objects all have a certain property," Russell saw, "we naturally suppose that the property is a definite object, which can be considered apart from any or all of the objects which have, or may be supposed to have the property in question. We also naturally suppose that the objects which have the property form a class, and that the class is in some sense a new single entity, distinct, in general, from each member of the class. Both these natural suppositions can be proved, by arguments so short and simple that they scarcely admit a possibility of error, to be at any rate not universally true" (Russell 1906b, 163-64).

So he came to teach that "there are no such things as classes and relations and functions as entities, and that the habit of talking of them is merely a convenient abbreviation" (Russell 1906b, 145). Symbols for functions did not have a meaning in themselves, he saw, but required some supplementation in order to acquire a complete meaning (Russell 1910, 225); such incomplete symbols were entirely devoid of meaning by themselves and only became significant as parts of appropriate propositions (Russell 1906c, 170). Standing alone, a propositional function was "a mere schema, a mere shell, an empty receptacle for meaning, not something already significant" (Russell 1919, 157). A function was not a definite object, but essentially a mere ambiguity awaiting determination. For it to occur significantly, it had to receive the necessary determination, had to occur in such a way that the ambiguity disappeared, and a wholly unambiguous statement resulted (Russell 1910, 229-30).

Once Russell had concluded that it was "natural to suppose that classes are merely linguistic or symbolic abbreviations", the important thing for him to do became "merely to provide a mode of interpreting the ordinary statements about classes without assuming that classes are entities" (Russell 1906a, 200). So he set to work to devise a way to escape the paradoxes "by the fact that classes are now not single entities… and are only parts of significant phrases, without being themselves significant in isolation" (Russell 1906a, 210).

Analogies that he was able to draw between the logical behavior of descriptions and classes as incomplete symbols and the success that he had with his 1905 theory of definite descriptions suggested to him a way in which classes could be analyzed away in much the same way as descriptions could. Thus he developed a procedure for analyzing away classes modeled after his way of handling descriptions as incomplete symbols with his theory of definite descriptions. He judged this move to be a major breakthrough in his efforts to unlock the mystery of contradictions arising from treating incomplete symbols as if they stood for objects.

Analyzed in his new way "all the formal properties that you desire of classes, all their formal uses in mathematics, can be obtained without supposing… that a proposition in which symbolically a class occurs, does in fact contain a constituent corresponding to that symbol, and when rightly analysed that symbol will disappear, in the same sort of way as descriptions disappear when the propositions are rightly analysed in which they occur" (Russell 1919, 266). This adaptation of his theory of descriptions to the theory of classes was became Russell's way of sweeping away some of his most onerous problems, but the whole story is much too long to tell here. I have endeavored to tell it elsewhere (Hill 1997).


In this essay I have discussed some of the main themes of Husserl's Fourth Logical Investigation and shown that important lessons that Frege and Russell learned from the contradictions derivable within system of the Basic Laws of Arithmetic turned out to be compatible with Husserl's ideas. All three actually came to many of the same conclusions about the inviolability of the laws governing the use of complete and incomplete expressions, dependent and independent meanings.

Of the three, Frege was the first to draw attention to the problems in "Function and Concept" and "On Concept and Object". However, silencing his doubts, he took logical law into his own hands; even as he published those two articles, he was developing the logical system of Basic Laws of Arithmetic that violated the very logical principles that he was proclaiming inviolable in his articles. By assuming "an unprovable law", he tried to force symbols standing for incomplete meanings to obey the same formal rules of identity that symbols for complete meanings do. By so mandating that a function could take another function as an object, however, he broke the law and suffered the contradiction about the class of all classes that are not members of themselves as a consequence. At least that is the way he saw it. Something deeper and logically prior to Basic Law V was at work that caused its failure. And what was it if not an a priori law of the kind Husserl wrote about in the Fourth Logical Investigation?

For the sake of those who feel queasy about finding Russell and Husserl discussed in the same place, or about any talk of self-evident a priori laws, it is important to note here that in Russell's article on the philosophical implications of mathematical logic that is found translated in Husserl's notes on set theory, Russell affirmed "that there is a priori and universal knowledge" (Russell 1911, p. 292). He wrote there that "all knowledge which is obtained by reasoning, needs logical principles which are a priori and universal"(Russell 1911, p. 292). He further declared 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" (Russell 1911, p. 292). "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..." (Russell 1911, p. 293).

Here I have tried to lay the foundations for the further discussion of what I have called a topic of prime importance for the understanding of some major issues in twentieth century western philosophy. Of course really big questions that remain are: Were Husserl, Frege, and Russell, who were after all as important philosophers as any that the twentieth century ever produced, right about these inviolable laws whose existence they believed that they had discerned? How responsible really was Frege's violation of the "deep nature of the matter" for the paradoxes? And most importantly, what does that have to do with us now?

I am of the conviction that a close and thorough investigation of the matter, of the kind I undertook in Rethinking Identity and Metaphysics (Hill 1997b) and my paper on connections between the set-theoretical paradoxes and the paradoxes of modal and intensional logics (see note 1) will show that it is precisely the kind of errors about incomplete symbols and dependent meanings that Husserl, Frege and Russell discerned that have been generating the failures of substitutivity, identity, reference problems, failures of existential generalization and so on that have done so much to heat up many philosophical discussions since their time.

Modern logic has yet to draw the full implications of Frege's and Russell's struggle with the paradoxes derivable in Frege's system has for the "classical" logic that was defended with such passion by so many during the twentieth century. The way that that logic took hold in English speaking countries and the virulent attitude that its proponents displayed towards modal and intensional logics effectively dissimulated many of the most important lessons that the set-theoretical paradoxes hold for philosophy. For intensions are dependent meanings and that one fact is the key to the opening of a great big can of worms for modern logic, semantics and analytic philosophy.


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