Frege, Gottlob (1848–1925)
FREGE, GOTTLOB
(1848–1925)
Life
After studying mathematics, physics, chemistry, and philosophy at the universities of Jena and Göttingen, the German mathematician, logician, and philosopher Gottlob Frege obtained his mathematical doctorate in Göttingen (1873) and his mathematical habilitation in Jena (1874). From 1874 to 1879 he taught mathematics at the University of Jena as a lecturer; in 1879 he was promoted to adjunct professor, and in 1896 to associate professor. Frege never obtained a full professorship. He retired from teaching in 1917 because of illness, becoming emeritus in 1918.
While he received little professional recognition during his lifetime, Frege is widely regarded in the early twentyfirst century as the greatest logician since Aristotle, one of the most profound philosophers of mathematics of all times, and a principal progenitor of analytic philosophy. His writing exhibits a level of rigor and precision that was not reached by other logicians until well after Frege's death.
Main Works
In the monograph Begriffsschrift (1879) Frege introduces his most powerful technical invention, nowadays known as predicate logic. In his second book, Die Grundlagen der Arithmetik (1884), he discusses the philosophical foundations of the notion of number and provides an informal argument to the effect that arithmetic is a part of logic (a thesis later known under the epithet logicism ). The pamphlet Funktion und Begriff (1891) is an elucidation of Frege's fundamental ontological distinction between functions (with concepts as a special case) and objects; certain difficulties with the views expressed therein are discussed in the essay "Über Begriff und Gegenstand" (1892). Frege's most celebrated achievement in the philosophy of language, the distinction between the sense and the reference of an expression, is expounded in his landmark essay "Über Sinn und Bedeutung" (1892). Grundgesetze der Arithmetik (volume 1, 1893; volume 2, 1903), his magnum opus, constitutes his abortive (because of Bertrand Arthur William Russell's antinomy) attempt at rigorously proving the logicist thesis. The essay "Der Gedanke: Eine logische Untersuchung" (1918) is a conceptual investigation of truth and that with respect to which the question of truth arises (called thoughts by Frege).
Frege's Logic
By replacing the traditional subjectpredicate analysis of judgments with the functionargument paradigm of mathematics and inventing the powerful quantifiervariable mechanism, Frege was able to overcome the limitations of Aristotelian syllogistics and created the first system of (higherorder) predicate logic. He thereby devised a formal logical language adequate for the formalization of mathematical propositions, especially through the possibility of expressing multiply general statements such as "for every prime number, there is a greater one."
The first presentation of his begriffsschrift (concept script—Frege's logical formula language) is contained in the 1879 monograph by the same name. At this time, the linguistic and philosophical underpinnings of begriffsschrift, as well as the description of the language itself, are still somewhat imprecise. There are, for instance, no formation rules given for the formulas of the language; functions seem to be identified with functional expressions; the meanings of the propositional connectives are specified in terms of assertion and denial rather than truth and falsity; and although Frege officially countenances only one inference rule, namely, modus ponens, he tacitly uses an instantiation rule for the universal quantifier as well. The first volume of Grundgesetze, however, presents a mature and amazingly rigorous version of the system, taking into account the various insights Frege had developed since the publication of Begriffsschrift. Unless otherwise noted, the following discussion pertains to this later system; for the time being, one should ignore the courseofvalues operator, which is discussed later on in connection with Russell's antinomy.
The primitive symbols of Frege's begriffsschrift are then those for equality, negation, the material conditional, and the first and higherorder universal quantifiers. In addition, there are gothic letters serving as bound variables (of first and higher orders), as well as Latin letters, whose role one would today characterize as that of free variables (again, of various orders). Disjunction, conjunction, and the existential quantifier are neither primitive, nor are they introduced as abbreviations, as would be customary today; rather, Frege notes that they can be simulated by means of the existing primitives.
Frege carefully distinguishes between basic laws (axioms) on the one hand, and inference rules on the other hand. With respect to a specified set of basic laws and rules of inference, he comes close to a rigorous definition of derivations in the predicate calculus.
The logical connectives, as well as the quantifiers, are taken to be denoting expressions, having as references the requisite truth functions and higherorder functions, respectively. Equality undergoes a radical change in interpretation between the time of Begriffsschrift and that of Grundgesetze. In the earlier system, assuming that the expression A refers to the object a, and the expression B to object b, Frege construes identities of the form A = B metalinguistically, taking them to mean that the expressions A and B are coreferential, rather than that a and b are the same object. In Grundgesetze, however, identity is conceived of as a binary relation between objects, much as is standard today (this change in interpretation is, incidentally, accompanied by a switch in notation from the triple bar ≡ to the now customary double bar =). Arguably, there is an analogous shift in the understanding of the universal quantifier; the formulations in Begriffsschrift suggest that it is to be interpreted substitutionally, whereas it is fairly clear in Grundgesetze that an objectual interpretation is intended. But the issue is difficult to judge, not only because the language of the earlier work is rather imprecise but also because it is not clear whether Frege was aware of the significance of the distinction between objectual and substitutional quantification.
Frege's perhaps most impressive achievement in pure logic is his celebrated definition (with the proof of its adequacy) of the ancestral (or transitive closure) R* of a binary relation R with the help of secondorder quantification, already contained in Begriffsschrift and central to the logicist enterprise. Informally, an object a bears the ancestral R* of a relation R to an object b if b can be reached from a in a finite (nonzero) number of R steps. That is, whenever there are objects a_{1}, a_{2}, …, a_{n} (n > 1) such that a_{1}Ra_{2}, a_{2}Ra_{3}, …, a_{n1}Ra_{n}, then a_{1} bears R* to a_{n}. For example, if R is the parenting relation (so that xRy holds if and only if x is a parent of y ), then R* is the ancestor relation (i.e., xR*y holds if and only if x is an ancestor of y ), because x is an ancestor of y if y is a child of x, or a child of a child of x, or a child of a child of a child of x, and so on. Frege's idea is to define R* from R as follows: a stands in the relation R* to b if and only if b has every property F such that (1) all objects to which a bears R have F, and (2) F is hereditary with respect to the relation R (meaning that, whenever something x has the property F, and x bears R to some y, then y also has F ). Note that this definition employs secondorder quantification (over all R hereditary properties F ).
It is clear that, if b can be reached from a in a finite nonzero number of R steps, then Frege's definition correctly implies that aR*b, for if F is any property and b can be reached from a in one step, then by clause (1) of the definition b must have F, and if b can be reached from a by some number of R steps greater than 1, one must have passed through an object to which a bears R, and which thus has F by clause (1), and every further object through which one has passed, including the last object b, must have F by clause (2). On the contrary, if b cannot be reached from a in a finite nonzero number of R steps, then b lacks just that property of being reachable from a in a finite number of R steps (a property that fulfills conditions [1] and [2]). In modern notation Frege's formal definition is as follows:
aR*b : ↔ ∀ F ((∀x (aRx → Fx ) & ∀x ∀y (Fx & xRy → Fy )) → Fb ).
It should be noted, finally, that Frege did not regard the sentences of his begriffsschrift as mere forms, open to arbitrary interpretation. Rather, he took them to express definite thoughts (i.e., propositions). This is manifest in the presence of a special symbol, the vertical judgment stroke, whose occurrence before a begriffsschrift formula indicates that the formula's content is actually asserted (and not talked about or simply entertained without judgment as to truth and falsity). While Frege did discuss the formal character of logic in terms of preservation of consequence on substituting nonlogical expressions for others (witness his correspondence with David Hilbert and the 1906 essay series "Über die Grundlagen der Geometrie"), he showed little inclination to pursue such investigations himself. Frege also has little to say about the characterization of propositions as logical truths; there is no indication that he had anything like Alfred Tarski's modeltheoretic criterion in mind. He occasionally remarks that logical axioms are required to be "obvious," but generally takes it for granted that the specific basic laws he lays down are in fact logical truths.
Frege's Ontology and Philosophy of Language
Frege's mature ontology is characterized by the fundamental dichotomy between saturated entities or objects (Gegenstände ) on the one hand, and unsaturated entities or functions on the other hand. Functions are unsaturated or incomplete in the sense that they carry argument places that need to be filled; an object is anything that is not a function. Concepts are special functions, namely, functions whose values are always one of the two truthvalues: the True and the False (which Frege takes to be objects, as will be explained). The realm of functions is stratified: Unary functions mapping objects to objects are first level, unary functions mapping firstlevel functions to objects are second level (an instance being the concept denoted by the firstlevel existential quantifier, which maps every firstlevel concept under which some object falls to the True, and all other firstlevel concepts to the False), and so on. The stratification becomes more complicated with functions of more than one argument, since there exist, for instance, functions of two arguments with one argument place for unary firstlevel functions and one argument place for objects (an instance being the application function, which maps a unary firstlevel function f and an object a to the result f (a ) of applying f to a ), and so on.
The saturatedunsaturated dichotomy has, for Frege, a parallel in the linguistic realm. Singular terms, such as proper names and definite descriptions, are (linguistically) saturated (or complete) and refer to objects; predicate and functional expressions are incomplete and refer to functions. In determining the ontological status of certain entities Frege often proceeds by analyzing the expressions used to refer to them and takes the saturated or unsaturated nature of the expressions as a reliable guide to their ontological saturation status.
Now since the expression "the concept horse " is grammatically a singular term, Frege takes it to refer to an object, which commits him to the paradoxical claim that the concept horse is not a concept (compare to "Über Begriff und Gegenstand"). In an attempt to resolve this predicament Frege proposes that with every concept F is associated a certain (proxy) object that serves as the referent of "the concept F " (some commentators believe that Frege intended the extension of F to be this proxy object, but the interpretive issue remains contentious). There remains a fundamental problem, however, for on the one hand, objects and concepts belong to distinct ontological categories, so that no predicate can be meaningfully applied to both a concept and an object; but on the other hand, Frege's explanation of this categorial distinction requires him to use the predicates "is an object" and "is a concept" in just this way—as contrasting (nonempty) predicates that can be applied to the same items. This creates some famous difficulties, some of which are discussed in the essay "Über Begriff und Gegenstand," because singular terms such as "the concept horse " cannot, according to Frege, refer to concepts, but refer to certain (proxy) objects instead.
Frege's most famous invention is perhaps his distinction between the sense (Sinn ) and the reference (Bedeutung ) of a linguistic expression, first introduced in his short 1891 booklet Funktion und Begriff, and expounded in detail in the 1892 essay "Über Sinn und Bedeutung." In the case of a singular term its reference is the object denoted by the term, whereas its sense is determined by the way that object is presented through the expression (its mode of presentation). Frege conceives of complete (declarative) sentences, perhaps infelicitously, as peculiar singular terms, so that their references, the special logical objects the True and the False, respectively, are objects. The thought expressed by a sentence is then defined by Frege to be the sentence's sense. The sense of a sentence is thus the mode of presentation of its truthvalue; that is, on a natural reading, the sentence's truthconditions. In the case of incomplete expressions, such as predicates and functional expressions, the references are of course the corresponding unsaturated concepts and functions.
While not explicitly discussed in "Über Sinn und Bedeutung," it becomes clear from the FregeHusserl correspondence that Frege intended the notion of sense to apply to predicates as well. Scholarly discussion continues whether Frege considered the senses of unsaturated expressions to be functions, or whether he regarded all senses as objects (a stance suggested by the fact that every sense can be referred to by means of a singular nominal phrase of the form "the sense of the expression X"). In the essay "Der Gedanke" Frege expounds a Platonistic view of senses as inhabitants of a "third realm" of nonperceptible, objective entities, as opposed to the (perceptible) objects of the external world and the subjective contents (ideas) of humans' minds.
Frege was motivated to introduce the sensereference distinction to solve certain puzzles, chief among them (1) the apparent impossibility of informative identity statements and (2) the apparent failure of substitutivity in contexts of propositional attitudes. As for (1), Frege argued that the statements "the morning start is the evening star" and "the morning star is the morning star" obviously differ in cognitive value (Erkenntniswert ), which would be impossible if the object designated constituted the only meaning of a singular term. The sensereference distinction allows one to attribute different cognitive values to these identity statements if the senses of the terms flanking the identity sign differ, while still allowing the objects denoted to be one and the same.
Regarding (2), Frege noticed that the sentences "John believes that the morning star is a body illuminated by the sun" and "John believes that the evening star is a body illuminated by the sun" may have different truthvalues, although the one is obtained from the other by substitution of a coreferential term. He argued that, in contexts of propositional attitudes, expressions do not have their usual reference, but refer to their ordinary senses (which thus become their indirect references); then since "the morning star" and "the evening star" differ in ordinary sense, they are not, in the context at hand, coreferential, having distinct indirect references. Debate continues as to Frege's intentions concerning indirect senses of expressions, in particular whether iterated propositional attitude contexts give rise to an infinite hierarchy of indirect senses.
In the introduction to Grundlagen Frege enunciates "three fundamental principles" for his investigations. The first of these is an admonition to separate the logical from the psychological (a motif that runs through all of Frege's works); the third demands observance of the conceptobject distinction. But it is the second of these principles that has drawn most attention and interpretation: "never to ask for the meaning of a word in isolation, but only in the context of a proposition." Other (not obviously equivalent) formulations of the principle occur in sections 60, 62, and 106 of Grundlagen ; some authors take Frege to express a precursor of this principle in section 9 of Begriffsschrift, and some see an echo of it in Grundgesetze, volume 1, section 29.
The proper interpretation of the context principle continues to be contentious. While some philosophers regard it as being of the utmost importance to an understanding of Frege's philosophy, others view it as a rather illconceived and incoherent doctrine that he appears to have given up in later works. Those who take the context principle seriously mostly take it to claim some sort of epistemological priority of sentences (or perhaps the thoughts expressed by such) over subsentential linguistic items (or perhaps their senses). It is easy to see why one might have misgivings about such an interpretation; after all, it at least appears to conflict with another Fregean principle, namely, that of compositionality (according to which the sense/reference of a compound expression is determined by the senses/references of its constituent expressions), which he held in high regard throughout his life.
Frege's Philosophy of Mathematics
Frege was, first and foremost, a philosopher of mathematics. While he followed Immanuel Kant in taking the truths of (Euclidean) geometry to be synthetic and knowable a priori (forcefully defending this view against Hilbert's axiomatic method in geometry), he vigorously argued, against Kant, for the logicist thesis, that is, the claim that the arithmetic truths, presumably including real and complex analysis, are analytic. In comparing Frege's views with Kant's it is however important to keep in mind that Frege was operating with his own technical definitions of analyticity and syntheticity, which are not obviously equivalent to Kant's: According to Frege (Grundlagen §3), a mathematical truth is analytic if it is derivable by means of logical inference rules from the general logical laws (and definitions) alone, whereas it is synthetic if it cannot be proved without recourse to truths belonging to a particular area of knowledge. Thus, analyticity and syntheticity are, for Frege, logicoepistemic notions, while Kant took them to be part semantic (analytic judgments are those whose predicate is contained in the subject, they are true by virtue of the meanings of their terms) and part epistemic (synthetic judgments extend one's knowledge, analytic ones do not).
In the preface to Begriffsschrift Frege makes it clear that it was the question of the epistemic status of arithmetic truths that prompted him to develop his new logic. At this time, Frege still avoids outright endorsement of the logicist thesis, stating only that he intends to investigate how far one may get in arithmetic with logical inferences alone. But there can be little doubt that he already envisages a definite path along which the ultimate proof of logicism is to proceed. Thus, he notes in part 3 of this work that mathematical induction rests on the Begriffsschrift theorem that, if an object x bears the transitive closure R* of a binary relation R to an object y, and if x has a property F that is inherited along R, then y has F as well. It therefore seems clear that Frege already understood the possibility of logically proving the mathematical induction principle once the number 0 and the successor relation among natural numbers had been suitably defined, for the natural numbers could then be given as just those objects following 0 in the transitive closure of the successor relation.
By the time of Grundlagen the doctrine of logicism is firmly in place. Having vigorously criticized a selection of philosophical views about the notion of number (notably John Stuart Mill's empiricist and Kant's transcendentalist views), Frege, in the second part of that work, provides an informal, yet rigorous outline of how the reduction of arithmetic to logic may actually be carried out. He begins this endeavor by insisting that (1) ascriptions of number involve assertions about concepts and (2) the numbers themselves must be construed as objects. Frege argues for (1) by noting first that certain statements, like universal categoricals such as "all whales are mammals" and existential statements such as "there are books on the shelf," predicate something of concepts (rather than individuals). The first example statement is clearly not about any individual whale, but says of the concept whale that it is subsumed under the concept mammal ; the second example predicates nonemptiness of the concept book on the shelf. The point is even clearer with respect to negated existential statements; "there are no Venus moons" is obviously not about any moon of Venus (if the statement is true, there are none), but denies that something falls under the concept Venus moon. Indeed, Frege notes, saying that there are no Venus moons amounts to the same thing as ascribing the number zero to the concept Venus moon. And just as in these examples, the numerical statement "there are four books on the shelf" clearly does not predicate anything of any particular book; instead, it, too, is a statement about the concept book on the shelf.
The thesis that ascriptions of number are best understood, in analogy with these examples, as assertions about concepts, is further bolstered by the observation that everyday numerical statements invariably involve common nouns or predicates, which, according to Frege, refer to concepts. Moreover, faced with the fact that one may with equal justice say "there is one deck of cards on the table," "there are fiftytwo cards on the table," and "there are four suits of cards on the table," one is led to the recognition that there are different standards of unit involved in these assertions, and it seems perfectly natural to identify the respective concepts as these standards of unit. Thesis (2) is a consequence of Frege's view that the ontological category of an entity may be read off reliably from the linguistic category of expression that denotes the entity: According to Frege number terms typically appear as singular terms in natural languages, for example, as "the number of cards on the table" or "the number four." Furthermore, pure arithmetic number terms typically flank the equality symbol, positions that, in Frege's view, are reserved for singular terms. Hence, Frege concludes, numbers must be objects.
Thus on the one hand, numbers, qua properties of concepts, would seem to be (higherorder) concepts; yet on the other hand, they must be construed as objects. Frege solves this apparent difficulty by suggesting that attributive uses of number words, as in "Jupiter has four moons," can always be paraphrased away, as in "the number of moons of Jupiter is four" (or, even more explicitly, "the number belonging to the concept moon of Jupiter is four"). In the latter statement, Frege claims, the is must denote identity and cannot function merely as a copula, since four is a singular term, and singular terms cannot follow the is of predication. The paradigmatic ascription of number then has the form "the number belonging to F = x," where F represents a predicate and x a singular term. Thus, the number term only forms part of the (higherorder) property ascribed to the concept, so that the objectual nature of number and the attributive character of ascriptions of number are compatible after all.
Frege next identifies a constraint that his reconstruction of arithmetic will have to abide by. Of fundamental importance for arithmetic are judgments of recognition, that is, identities, and so the definitions of the numbertheoretic notions required for a proof of the logicist thesis must ensure that, in particular, identities of the form "the number belonging to F = the number belonging to G " receive the proper truth conditions. For this special type of identity statement, the truth conditions can readily be formulated in (dyadic secondorder) logical terms, namely, the number belonging to F is the same as the number belonging to G if and only if there exists a binary relation R that correlates the objects that are F oneone and onto with the objects that are G. Since Frege quotes a somewhat obscure passage from David Hume at this point in Grundlagen, the principle has, perhaps infelicitously, come to be known as Hume's principle (HP).
Frege rejects HP as a definition of "the number belonging to F " on the grounds that it fails to specify truth conditions for contexts of the form "the number belonging to F = x," where x is a term that does not have the form "the number belonging to G," for example, when x is an individual variable. (This objection is now usually referred to as the Caesar problem—somewhat inaccurately, as Frege uses Julius Caesar as an example in arguing against a slightly different proposal for a definition). Some commentators maintain that Frege's only point in bringing up this objection is to show how HP is inadequate as a definition of number as described earlier. Other commentators see Frege as struggling here to arrive at adequacy conditions for the introduction of new sortal concepts into a language. On such a reading, however, it is difficult to see why Frege was not troubled by the obvious analogous problem arising for extensions of concepts in the Grundgesetze.
In any case Frege proposes an explicit definition of "the number belonging to F " that in effect amounts to taking this number to be the equivalence class of F under the equivalence relation of equinumerosity (which is explained in terms of the existence of a oneone and onto correlation): the number belonging to F, Frege stipulates, is the extension of the concept "concept equinumerous with F." Frege relies on a naive understanding of the notion of extension (later, in Grundgesetze, extensions themselves would be governed by an axiom that was to prove fatal for Frege's project). Frege then defines an object a to be a (cardinal) number if there exists a concept F such that a is the number belonging to F.
From the explicit definition of the number belonging to a concept, Frege proceeds to show that HP becomes derivable by means of pure logic and defines 0 as the number belonging to the concept "is an object not identical with itself" and 1 as the number belonging to the concept "is an object identical with 0." The successor relation among cardinal numbers is defined as follows: n succeeds m if n is the number belonging to some concept F under which some object a falls, and m is the number belonging to the concept "is an object falling under F, but not identical to a." Without proof Frege mentions the theorems that every number has at most one successor and one predecessor, and that every number except 0 succeeds some number. Making use of his definition of the ancestral (transitive closure) of a binary relation (as developed in Begriffsschrift ), he defines the finite or natural numbers as those objects standing to 0 in the transitive reflexive closure of the successor relation, that is, informally, as those numbers than can be reached from 0 by taking successors finitely many times. Frege observes that this definition allows for a rather straightforward proof of the mathematical induction principle for natural numbers.
At this point, he has effectively recovered all the axioms of (secondorder) Peano arithmetic from his definitions, except the one requiring every natural number to have a successor. Frege sketches a proof for this remaining axiom, which ultimately consists in showing by means of induction that, for any natural number n, the number belonging to the concept "object to which n bears the transitive reflexive closure of the successor relation" (i.e., informally, "natural number being less than or equal to n ") succeeds n (a fully detailed proof is carried out in Grundgesetze, although it is not entirely clear whether this is the same proof Frege intended in Grundlagen ).
While the exposition of Grundlagen is entirely informal, Grundgesetze, which Frege hoped to be the final word on the logical nature of arithmetic, carries out the earlier sketch with full rigor, containing pages and pages of formal deductions in begriffsschrift notation. The crucial element added in Grundgesetze is the rigorous treatment of extensions of concepts (more precisely, of coursesofvalues of functions, of which concept extensions are a special case). These are governed by Frege's basic law V, whose special case for concepts says that the extensions of concepts F and G coincide if and only if the same objects fall under F as fall under G. The use of extensions allows for the technique of typelowering: Firstlevel concepts can be simulated by their extensions, secondlevel concepts H can be simulated by the firstlevel concepts under which fall precisely the extensions of concepts falling under H, and so on. Frege makes extensive use of this technique; in particular, instead of defining the number belonging to F as the extension of the secondlevel concept "concept equinumerous with F," he is now able to take numbers to be extensions of firstlevel concepts. Otherwise, he follows the sketch of Grundlagen closely.
As Russell pointed out in a letter to Frege in 1902, the theory expounded in Grundgesetze is inconsistent, since it allows for the derivation of Russell's antinomy: Letting R be the firstlevel concept "x is the extension of some concept under which x does not fall," and r its extension, it follows easily from Frege's rules of inference, together with basic law V, that r both does and does not fall under R. Frege immediately realized that the antinomy threatened to undermine his life's work. While the second volume of Grundgesetze was in press, he hastily devised a quick fix that has come to be known as Frege's way out and added an appendix to the book, expressing both confidence that the revised system would prove capable of reconstructing arithmetic and worries about the philosophical underpinning of his revised basic law V. Frege's way out proved not to be a way out, since it was inconsistent with the existence of more than one object. The genesis of the antinomy in Frege's system is by now well understood; it arises through interplay of two principles that are individually consistent, namely, basic law V as mentioned earlier and impredicative secondorder comprehension (roughly, statements to the effect that there exists a concept with a certain property, where that property is itself specified with the help of quantification over concepts); Frege's system with basic law V but only predicative instances of comprehension is now known to be consistent, but too weak to allow for a reconstruction of substantial mathematics.
Frege's work on the logical foundation of real analysis remained fragmentary; the second volume of Grundgesetze contains only preliminary definitions and theorems. Presumably he had planned a third volume, which, however, never appeared. Toward the end of his life, Frege seems to have abandoned logicism altogether, suggesting that arithmetic was instead based entirely on geometry, and hence synthetic, as Kant had held. His ideas on how such a claim might be proved were, however, never worked out.
NeoFregeanism
Frege himself, and generations of philosophers and logicians after him, considered the mathematical content of Grundlagen and Grundgesetze largely obsolete because of the inconsistency of Frege's theory of extensions of concepts. In the 1980s, however, it began to be recognized that Frege had indeed hit on an exciting fact: If one takes the framework of Frege's theory to be essentially secondorder predicate logic and adopts HP (with a primitive operator "the number belonging to," attaching to concept expressions) as an axiom, all of secondorder Peano arithmetic becomes derivable, using the exact definitions and proofs employed by Frege (who used the explicit definition of "the number of F" only to prove HP from it, obtaining all further results directly from HP). This fact has become known as Frege's theorem. Importantly, it was soon observed that Frege arithmetic (i.e., full axiomatic secondorder logic plus HP) is consistent, in contradistinction to the system of Grundgesetze (indeed, consistent relative to secondorder Peano arithmetic).
It is still being debated whether, and to what extent, these discoveries have any bearing on the validity of the logicist thesis (restricted to arithmetic proper). While no one has seriously suggested that HP could be regarded as a principle of logic, some argue that it nevertheless enjoys some privileged epistemological status akin to analyticity, the principle being, in some sense, "analytic of " number. There are, however, serious difficulties in defending Frege arithmetic as being analytic. To start with, there is the familiar problem about the status of secondorder logic itself, quite independently of HP. But even granting that secondorder logic may count as logic in the requisite sense, further objections apply to HP. First, the principle is not ontologically innocent, since it requires the firstorder domain to be infinite, which is usually taken to be incompatible with analyticity. Second, any attempt to ground a privileged logical status of HP on its logical form (of an abstraction principle) runs afoul of the "bad company objection": There are abstraction principles of the same general logical form as HP that are inconsistent (such as Frege's basic law V). What is more, there are abstraction principles (like Boolos's parity principle) that hold only in finite domains, which makes them incompatible with HP, and hence it cannot be the logical form of an abstraction principle alone that could make HP analytic. Research on abstraction principles has increased significantly as a consequence of this discussion, as has work on the general logical and mathematical features of Frege's systems.
Frege's Influence
Through his publications, as well as through personal correspondence, Frege exerted a profound influence on Russell, who appears to have been the first major thinker to appreciate Frege's achievements in logic. Russell took over the logicist torch from Frege, and although Alfred North Whitehead and Russell's Principia Mathematica differs in many ways from Frege's work (it is much wider in scope, considerably less rigorous, and, in view of Russell's antinomy, takes a different approach to classes), it is clearly also heavily influenced by Frege (e.g., in imposing a structure of levels, or types, on the underlying ontology, and in the definition of number, nowadays often referred to as the FregeRussell definition of cardinal number). It is known that Russell had read "Über Sinn und Bedeutung" and at least parts of Grundgesetze when he developed his celebrated theory of descriptions; and while there is no direct evidence for such a claim, it seems plausible to assume that Frege's discussion of definite descriptions in these works (especially the fully worked out formal theory of Grundgesetze ) provided a helpful foil for Russell's own theory.
The degree to which Frege influenced Edmund Husserl is a more contentious matter. It is known that Husserl read all of Frege's major works and that the two corresponded extensively (except in the aftermath of Frege's rather hostile review [1894] of Husserl's Philosophie der Arithmetik [1891]). It seems fair to say that Frege (in particular, through the aforementioned review, as well as the preface to the first volume of Grundgesetze ) is at least partly responsible for Husserl's antipsychologistic turn.
While Frege met neither Russell nor Husserl in person, he did have personal interactions with both Rudolf Carnap and Ludwig Josef Johann Wittgenstein. As a student, Carnap enrolled in various classes on begriffsschrift taught by Frege in Jena between 1910 and 1914; surely it was Frege who instilled in Carnap the idea that mathematics was reducible to logic, a view that was to become central to the Vienna Circle's philosophy. More generally, Frege shaped Carnap's whole attitude toward philosophy. After his immigration to the United States, Carnap, with Alonzo Church, was instrumental in keeping Fregean ideas in logic alive in the United States (where they came to flourish, for instance, in the work in semantics of David Kaplan and Richard Montague). Wittgenstein first visited Frege in Jena in 1911, and then at least two more times, in 1912 and 1913, while he was Russell's student in Cambridge. In addition, the two corresponded rather extensively from 1911 to 1920; it is clear from this correspondence that Frege and Wittgenstein thought highly of each other (the end of the correspondence is marked by an exchange of rather critical remarks by Frege on the Tractatus and by Wittgenstein on "Der Gedanke"). Fregean themes pervade the work of both the early and the late Wittgenstein, and it appears that Wittgenstein's intellectual respect for Frege never subsided.
In spite of this illustrious group of correspondents, Frege was for many years regarded as a somewhat obscure and ultimately failed predecessor of Russell's, possibly because few philosophers fully acknowledged Frege's influence on them (of course, the extent of this influence may not have been clear to them at the time). In the 1930s Heinrich Scholz and his school in Münster, Germany, rediscovered Frege and began work on an edition of his works, but that never materialized. The situation changed somewhat in the wake of John Langshaw Austin's English translation of the Grundlagen, which appeared in 1950; Frege was read, at that time, mainly as a philosopher of language, and as such influenced, among others, the British philosopher Peter Geach. The originality and independence of Frege's work (especially from Russell's), as well as his important role as a progenitor of analytic philosophy, was brought to prominence through the writings of Michael Dummett in the 1970s, who was himself heavily influenced by Frege's methodology and interests. In the United States, besides those mentioned earlier, Donald Davidson's work also revived discussion of Fregean themes. Crispin Wright's neologicism, especially as subsequently articulated and criticized by George S. Boolos and others, caused a veritable renaissance of interest in Frege's logical and mathematical work, beginning in the 1980s and continuing to this day.
See also Analytic and Synthetic Statements; Analyticity; Aristotle; Austin, John Langshaw; Carnap, Rudolf; Categories; Church, Alonzo; Davidson, Donald; Dummett, Michael Anthony Eardley; Geometry; Hilbert, David; Hume, David; Husserl, Edmund; Identity; Kant, Immanuel; Kaplan, David; Logic, History of; Logical Positivism; Mathematics, Foundations of; Mill, John Stuart; Montague, Richard; Peano, Giuseppe; Propositions; Russell, Bertrand Arthur William; Scholz, Heinrich; Whitehead, Alfred North; Wittgenstein, Ludwig Josef Johann.
Bibliography
abbreviations
CP
Collected Papers on Mathematics, Logic, and Philosophy. Translated by Max Black et. al; edited by Brian McGuinness. New York: Blackwell, 1984.
PW
Posthumous Writings. Translated by Peter Long and Roger White; edited by Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach. Chicago: University of Chicago Press, 1979.
TPW
Translations from the Philosophical Writings of Gottlob Frege. 3rd ed, edited by Peter Geach and Max Black. Totowa, NJ: Rowman & Littlefield, 1980.
FR
The Frege Reader, edited by Michael Beaney. Cambridge, MA: Blackwell, 1997.
KS
Kleine Schriften, edited by Ignacio Angelelli. Hildesheim, Germany: Georg Olms, 1967.
FBB
Funktion, Begriff, Bedeutung: Fünf logische Studien, edited by Günther Patzig. Göttingen, Germany: Vandenhoeck and Ruprecht, 1962.
LU
Logische Untersuchungen, edited by Günther Patzig. Göttingen, Germany: Vandenhoeck and Ruprecht, 1966.
works by frege
Ueber eine geometrische Darstellung der imaginären Gebilde in der Ebene. InauguralDissertation der philosophischen Facultät zu Göttingen zur Erlangung der Doctorwürde vorgelegt von G. Frege aus Wismar. Jena: A. Neuenhahn, 1873. Reprinted in KS, tr. as "On a Geometrical Representation of Imaginary Forms in the Plane," in CP, pp. 1–55.
Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen. Dissertation zur Erlangung der Venia Docendi bei der Philosophischen Facultät in Jena von Dr. Gottlob Frege (1874). Reprinted in KS, tr. as "Methods of Calculation based on an Extension of the Concept of Quantity [Magnitude]," in CP, pp. 56–92.
Review of H. Seeger, Die Elemente der Arithmetik, für den Schulunterricht bearbeitet. Jenaer Literaturzeitung 1 (46) (1874): 722. Reprinted in KS, tr. as "Review of H. Seeger, Die Elemente der Arithmetik," in CP, pp. 93–94.
Review of A. v. Gall and Ed. Winter, die analytische Geometrie des Punktes und der Geraden und ihre Anwendung auf Aufgaben. Jenaer Literaturzeitung 4 (9) (1877): 133–134. Reprinted in KS, tr. as "Review of A. von Gall and E. Winter, Die analytische Geometrie des Punktes und der Geraden und ihre Anwendung auf Aufgaben," in CP, pp. 95–97.
Review of J. Thomae, Sammlung von Formeln welche bei Anwendung der elliptischen und Rosenhain'schen Funktionen gebraucht werden. Jenaer Literaturzeitung 4 (30) (1877): 472. Reprinted in KS, tr. as "Review of J. Thomae, Sammlung von Formeln, welche bei Anwendung der elliptischen und Rosenhainschen Funktionen gebraucht werden," in CP, p. 98.
"Über eine Weise, die Gestalt eines Dreiecks als complexe Grösse aufzufassen," Jenaische Zeitschrift für Naturwissenschaft 12 (1878) Supplement, p. XVIII. Reprinted in KS, tr. as "Lecture on a Way of Conceiving the Shape of a Triangle as a Complex Quantity," in CP, pp. 99–100.
Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: L. Nebert, 1879. Reprinted in Conceptual Notation and Related Articles, tr. and ed. by T. W. Bynum (OUP, 1972), pp. 101–203. Also in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 ed. J. van Heijenoort, tr. S. BauerMengelberg (Harvard University Press, 1967), pp. 5–82. §1–12 in TPW, pp. 1–20, and in FR, pp. 47–78.
"Anwendungen der Begriffsschrift," Jenaische Zeitschrift für Naturwissenschaft 13 (1879) Supplement II, pp. 29–33. Reproduced in Begriffsschrift und andere Aufsätze, ed. I. Angelelli (Hildesheim: Georg Olms, 1964), tr. as "Applications of the 'Conceptual Notation,'" (1879), in Conceptual Notation and Related Articles, tr. and ed. by T. W. Bynum (OUP, 1972), pp. 204–208.
Review of Hoppe, Lehrbuch der analytischen Geometrie, Deutsche Literaturzeitung 1 (6) (1880). Reprinted in KS, tr. as "Review of Hoppe, Lehrbuch der analytischen Geometrie I," in CP, pp. 101–102.
"Ueber die wissenschaftliche Berechtigung einer Begriffsschrift," Zeitschrift für Philosophie und philosophische Kritik 81 (1882), pp. 48–56. Reprinted in KS, tr. as "On the Scientific Justification of a Conceptual Notation," in Conceptual Notation and Related Articles, tr. and ed. by T. W. Bynum (OUP, 1972), pp. 83–89, and tr. by J. M. Bartlett as "On the Scientific Justification of a Conceptual Notation," Mind 73 (1964), pp. 155–160.
"Über den Zweck der Begriffsschrift," Jenaische Zeitschrift für Naturwissenschaft 16 (1883) Supplement, pp. 1–10. Reprinted in Begriffsschrift und andere Aufsätze, ed. I. Angelelli (Hildesheim: Georg Olms, 1964), tr. as "On the Aim of the 'Conceptual Notation,'" in Conceptual Notation and Related Articles, tr. and ed. by T. W. Bynum (OUP, 1972), pp. 90–100. Also tr. by V. Dudman as "On the Purpose of the Begriffsschrift," The Australasian Journal of Philosophy 46 (1968), pp. 89–97.
"Geometrie der Punktpaare in der Ebene," Jenaische Zeitschrift für Naturwissenschaft 17 (1884) Supplement, pp. 98–102. Reprinted in KS, tr. as "Lecture on the Geometry of Pairs of Points in the Plane," in CP, pp. 103–107.
Die Grundlagen der Arithmetik, eine logischmathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner, 1884; reprints Breslau: M. & H. Marcus, 1934, Hildesheim: G. Olms, 1961, and Darmstadt: Wissenschaftliche Buchgesellschaft, 1961. Tr. as The Foundations of Arithmetic by J. L. Austin, with German text, 2nd edition (Blackwell, 1953). §55–91, 106–109 also tr. M. S. Mahoney, in Philosophy of Mathematics: Selected Readings, 2nd edition, eds. P. Benacerraf and H. Putnam (CUP, 1983) pp. 130–159. Introduction, §1–4, 45–69, 87–91, 104–9 with summaries of remaining sections also in FR, pp. 84–129. Also reprinted as Die Grundlagen der Arithmetik, German centenary critical edition, ed. C. Thiel (Hamburg: Felix Meiner, 1986).
Review of H. Cohen: Das Princip der InfinitesimalMethode und seine Geschichte, Zeitschrift für Philosophie und philosophische Kritik 87 (1885), pp. 324–329. Reprinted in KS, tr. as "Review of H. Cohen, Das Prinzip der InfinitesimalMethode und seine Geschichte," in CP, pp. 108–111.
"Erwiderung," Deutsche Literaturzeitung 6 (28) (1885): 1030. Reprinted in KS, tr. as "Reply to Cantor's Review of Grundlagen der Arithmetik," in CP, p. 122.
"Über formale Theorien der Arithmetik," Jenaische Zeitschrift für Naturwissenschaft 19 (1886) Supplement, pp. 94–104. Reprinted in KS, tr. as "On Formal Theories of Arithmetic," in CP, pp. 112–121.
"Über das Trägheitsgesetz," Zeitschrift für Philosophie und philosophische Kritik 98 (1891). Reprinted in KS, tr. as "On the Law of Inertia," in CP, pp. 123–136, and tr. by R. Rand as "About the Law of Inertia" in Synthese 13 (1961), pp. 350–363.
Function und Begriff. Vortrag, gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaft für Medicin und Naturwissenschaft. Jena: H. Pohle, 1891. Reprinted in FBB. Tr. as "Function and Concept," in TPW, pp. 21–41, also in CP, pp. 137–156, and FR, pp. 151–171.
"Über Sinn und Bedeutung," Zeitschrift für Philosophie und philosophische Kritik, NF 100 (1892): 25–50. Reprinted in FBB and KS. Tr. by H. Feigl as "On Sense and Nominatum," in H. Feigl and W. Sellars, eds., Readings in Philosophical Analysis (New York: AppletonCenturyCroft 1949), as "On Sense and Reference," in TPW, pp. 56–78, also in CP, pp. 157–77, and The Philosophical Review 57 (1948), pp. 207–230, and as "On Sinn and Bedeutung" in FR, pp. 151–71.
"Ueber Begriff und Gegenstand," Vierteljahrsschrift für wissenschaftliche Philosophie 16 (1892): 192–205. Reprinted in FBB and KS. Tr. as "On concept and Object," in TPW, pp. 42–55, also in PW, pp. 87–117, in CP, pp. 182–94, Mind 60 (1951): 168–180 and in FR, pp. 181–193.
Review of Georg Cantor: Zur Lehre vom Transfinitien. Gesammelte Abhandlungen aus der Zeitschrift für Philosophie und philosophische Kritik. Erste Abteilung. Zeitschrift für Philosophie und philosophische Kritik 100 (1892): 269–272. Reprinted in KS, tr. as "Review of Cantor's Zur Lehre vom Transfinitien," in CP, pp. 178–81.
Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet. Jena: H. Pohle, Band I: 1893, Band II: 1903. Repr. together, Hildesheim: Georg Olms, 1962, 1998; and Darmstadt: Wissenschaftliche Buchgesellschaft, 1962. Preface, introduction and §1–52 of Vol. I tr. as The Basic Laws of Arithmetic: Exposition of the System, tr. and ed. by M. Furth (University of California Press, 1964). Selections from both vols. also tr. in TPW, and in FR, pp. 194–233, 258–289. Selections also tr. by J. Stachelroth and P. Jourdain as "A Formal System of Logic and Mathematics" in Readings on Logic, eds. I. Copi and J. Gould (New York: Macmillan, 1964).
Review of Dr. E. G. Husserl: Philosophie der Arithmetik. Psychologische und logische Untersuchung. Zeitschrift für Philosophie und philosophische Kritik 103 (1894): 313–332. Reprinted in KS, tr. as "Review of E. G. Husserl, Philosophie der Arithmetik I," in CP, pp. 195–209. Extracts in TPW, pp. 79–85, and FR, pp. 224–226.
"Kritische Beleuchtung einiger Punkte in E. Schöders Vorlesungen über die Algebra der Logik," Archiv für systematische Philosophie 1 (1895): 433–456. Reprinted in LU, also in KS, tr. as "A Critical Elucidation of Some Points in E. Schröder, Vorlesungen über die Algebra der Logik," in CP, pp. 210–228, also in TPW, pp. 86–106.
"Le nombre entier," Revue de Métaphysique et de Morale 3 (1895): 73–78. Reprinted in KS, tr. as "Whole Numbers," in CP, pp. 229–233.
"Lettera del sig. G. Frege all'Editore," Revue de Mathématiques (Rivista di Matematica) 6 (1896–1899): 53–59. Reprinted in Giuseppe Peano. Opere scelte, II (Rome: Cremonese, 1958), also in KS.
"Über die Begriffsschrift des Herrn Peano und meine eigene," Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. MathematischPhysische Klasse 48 (1897): 361–378. Reprinted in FBB, tr. as "On Mr. Peano's Conceptual Notation and My Own," in CP, pp. 234–248.
Über die Zahlen des Herrn H. Schubert. Jena: H. Pohle, 1899. Reprinted in LU, also in KS, tr. as "On Mr. H. Schubert's Numbers," in CP, pp. 249–72.
"Über die Grundlagen der Geometrie." Jahresbericht der Deutschen MathematikerVereinigung 12 (1903): 319–324. Reprinted in KS, tr. in "On the Foundations of Geometry: First Series," in CP, pp. 273–284, also in On the Foundations of Geometry and Formal Theories of Arithmetic, tr. by EikeHenner W. Kluge (Yale University Press, 1971), pp. 22–37, and as "The Foundations of Geometry," The Philosophical Review 69 (1960), pp. 3–17.
"Über die Grundlagen der Geometrie II." Jahresbericht der Deutschen MathematikerVereinigung 12 (1903): 368–375. Reprinted in KS, tr. in "On the Foundations of Geometry: First Series," in CP, pp. 273–284, also in On the Foundations of Geometry and Formal Theories of Arithmetic, tr. by EikeHenner W. Kluge (Yale University Press, 1971), pp. 22–37, and as "The Foundations of Geometry," The Philosophical Review 69 (1960), pp. 3–17.
"Was ist eine Funktion?" In: Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburstage, 20. Feb. 1904 (Leipzig: J. A. Barth, 1904), pp. 656–666. Reprinted in FBB and KS, tr. as "What is a Function?" in TPW, pp. 107–16, also in CP, pp. 285–92.
"Über die Grundlagen der Geometrie I." Jahresbericht der Deutschen MathematikerVereinigung 15 (1906): 293–309. Reprinted in KS, tr. in "On the Foundations of Geometry: Second Series," in CP, pp. 293–340, also in On the Foundations of Geometry and Formal Theories of Arithmetic, tr. by EikeHenner W. Kluge (Yale University Press, 1971), pp. 49–112.
"Über die Grundlagen der Geometrie (Fortsetzung) II." Jahresbericht der Deutschen MathematikerVereinigung 15 (1906): 377–403. Reprinted in KS, tr. in "On the Foundations of Geometry: Second Series," in CP, pp. 293–340, also in On the Foundations of Geometry and Formal Theories of Arithmetic, tr. by EikeHenner W. Kluge (Yale University Press, 1971), pp. 49–112.
"Über die Grundlagen der Geometrie (Schluß) III," Jahresbericht der Deutschen MathematikerVereinigung 15 (1906): 423–430. Reprinted in KS, tr. in "On the Foundations of Geometry: Second Series," in CP, pp. 293–340, also in On the Foundations of Geometry and Formal Theories of Arithmetic, tr. by EikeHenner W. Kluge (Yale University Press, 1971), pp. 49–112.
"Antwort auf die Ferienplauderei des Herrn Thomae." Jahresbericht der Deutschen MathematikerVereinigung 15 (1906): 586–590. Reprinted in KS, tr. as "Reply to Mr. Thomae's Holiday Causerie," in CP, pp. 341–345.
"Die Unmöglichkeit der Thomaeschen formalen Arithmetik aufs Neue nachgewiesen." Jahresbericht der Deutschen MathematikerVereinigung 17 (1908): 52–55. Reprinted in KS, tr. as "Renewed Proof of the Impossibility of Mr. Thomae's Formal Arithmetic," in CP, pp. 346–350.
Notes to Jourdain, Philip E. B., "The Development of the Theories of Mathematical Logic and the Principles of Mathematics." Quarterly Journal of Pure and Applied Mathematics XLIII (1912): 237–269.
"Der Gedanke: Eine logische Untersuchung." Beiträge zur Philosophie des deutschen Idealismus 1 (1918): 58–77. Reprinted in LU and KS, tr. as "Thought," Part I of Logical Investigations, ed. P. T. Geach, tr. P. T. Geach and R. Stoothoff (Blackwell, 1977), included in CP, pp. 351–372, and in FR, pp. 325–345. Also as "The Thought: A Logical Inquiry," in Mind 65 (1956): 289–311, tr. by A. M. and Marcelle Quinton.
"Die Verneinung. Eine logische Untersuchung." Beiträge zur Philosophie des deutschen Idealismus 1 (1918): 143–157. Reprinted in LU and KS, tr. as "Negation," Part II of Logical Investigations, ed. P. T. Geach, tr. P. T. Geach and R. Stoothoff (Blackwell, 1977), included in CP, pp. 373–389, and in FR, pp. 346–361.
"Logische Untersuchungen. Dritter Teil: Gedankengefüge." Beiträge zur Philosophie des deutschen Idealismus 3 (1923): 36–51. Reprinted in LU, KS, tr. as "Compound Thoughts," Part III of Logical Investigations, ed. P. T. Geach, tr. P. T. Geach and R. Stoothoff (Blackwell, 1977), included in CP, pp. 390–406, and Mind 72 (1963): 1–17.
unpublished works by frege
Wissenschaftlicher Briefwechsel. Edited by Gottfried Gabriel et al. Hamburg, Germany: Felix Meiner, 1976, abridged for English edition by Brian McGuinness and translated by Hans Kaal as Philosophical and Mathematical Correspondence. Chicago: University Press of Chicago, 1980.
works about frege
Antonelli, G. Aldo, and Robert May. "Frege's New Science." Notre Dame Journal of Formal Logic 41 (3) (2000): 242–270.
Beaney, Michael, and Erich H. Reck, eds. Gottlob Frege: Critical Assessments of Leading Philosophers. 4 vols. London: Routledge, 2005.
Blanchette, Patricia. "Frege and Hilbert on Consistency." Journal of Philosophy 93 (1996): 317–336.
Boolos, George S. Logic, Logic, and Logic, edited by Richard Jeffrey. Cambridge, MA: Harvard University Press, 1998.
Burge, Tyler. "Frege on Knowing the Foundations." Mind 107 (1998): 305–347.
Burgess, John P. Fixing Frege. Princeton, NJ: Princeton University Press, 2005.
Demopoulos, William, ed. Frege's Philosophy of Mathematics. Cambridge, MA: Harvard University Press, 1995.
Demopoulos, William. "The Philosophical Basis of Our Knowledge of Number." Noûs 32 (1998): 481–503.
Dummett, Michael. Frege and Other Philosophers. New York: Oxford University Press, 1991.
Dummett, Michael. Frege: Philosophy of Language. 2nd ed. Cambridge, MA: Harvard University Press, 1981.
Dummett, Michael. Frege: Philosophy of Mathematics. Cambridge, MA: Harvard University Press, 1991.
Dummett, Michael. The Interpretation of Frege's Philosophy. Cambridge, MA: Harvard University Press, 1981.
Ferreira, Fernando, and Kai F. Wehmeier. "On the Consistency of the (11 CA Fragment of Frege's Grundgesetze." Journal of Philosophical Logic 31 (2002): 301–311.
Goldfarb, Warren. "Frege's Conception of Logic." In Future Pasts: The Analytic Tradition in Twentieth Century Philosophy, edited by Juliet Floyd and Sanford Shieh. New York: Oxford University Press, 2001.
Hale, Bob, and Crispin Wright. The Reason's Proper Study: Essays towards a NeoFregean Philosophy of Mathematics. New York: Oxford University Press, 2001.
Heck, Richard G., Jr. "The Consistency of Predicative Fragments of Frege's Grundgesetze der Arithmetik." History and Philosophy of Logic 17 (1996): 209–220.
Heck, Richard G., Jr., ed. Language, Thought, and Logic: Essays in Honour of Michael Dummett. New York: Oxford University Press, 1997.
Heijenoort, Jean van. "Logic as Calculus and Logic as Language." Synthese 17 (1967): 324–330.
Hodes, Harold. "Logicism and the Ontological Commitments of Arithmetic." Journal of Philosophy 81 (3) (1984): 123–149.
McFarlane, John. "Frege, Kant, and the Logic in Logicism." Philosophical Review 111 (2002): 25–65.
Quine, Willard Van Orman. "On Frege's Way Out." Mind 64 (1955): 145–159.
Reck, Erich H., ed. From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy. New York: Oxford University Press, 2002.
Resnik, Michael D. Frege and the Philosophy of Mathematics. Ithaca, NY: Cornell University Press, 1980.
Ricketts, Thomas. "Frege's 1906 Foray into Metalogic." Philosophical Topics 25 (2) (1997): 169–187.
Sluga, Hans D. Gottlob Frege. Boston: Routledge and Kegan Paul, 1980.
Stepanians, Markus S. Frege und Husserl über Urteilen und Denken. Paderborn, Germany: Schöningh, 1998.
Tait, William W., ed. Early Analytic Philosophy—Frege, Russell, Wittgenstein: Essays in Honor of Leonard Linsky. Chicago: Open Court, 1997.
Tappenden, Jamie. "Metatheory and Mathematical Practice in Frege." Philosophical Topics 25 (2) (1997): 213–264.
Thiel, Christian. Sense and Reference in Frege's Logic. Translated by T. J. Blakeley. Dordrecht, Netherlands: D. Reidel, 1968.
Weiner, Joan. Frege in Perspective. Ithaca, NY: Cornell University Press, 1990.
Wright, Crispin. Frege's Conception of Numbers as Objects. Aberdeen, Scotland: Aberdeen University Press, 1983.
Zalta, Edward N. "Frege's Logic, Theorem, and Foundations for Arithmetic." In The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta. Stanford, CA: Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University, 2005.
Zalta, Edward N. "Gottlob Frege." In The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta. Stanford, CA: Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University, 2005.
Kai F. Wehmeier (2005)
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