In the philosophy of language, the distinction between concept and object is attributable to the German philosopher Gottlob Frege. Overview[edit]. According to Frege, any sentence that expresses a singular thought consists Frege, G. ” On Concept and Object”, originally published as “Ueber Begriff und Gegenstand” in. Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. He is .. Original: “Ueber Begriff und Gegenstand”, in Vierteljahresschrift für wissenschaftliche Philosophie XVI (): –;; In English: “Concept. Download Citation on ResearchGate | Kerry und frege über begriff und gegenstand 1 | After describing the philosophical background of Kerry’s work, an account.

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From to he taught mathematics at the University of Jena as a lecturer; in he was promoted to adjunct professor, and in to associate professor. Frege never obtained a full professorship. He retired from teaching in because of illness, becoming emeritus in While he received little professional recognition during his lifetime, Frege is widely regarded in the early twenty-first 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. In the monograph Begriffsschrift Frege introduces his most powerful technical invention, nowadays known as predicate logic.

In his second book, Die Grundlagen der Arithmetikhe 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.

Grundgesetze der Arithmetik volume 1, ; volume 2,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” is a conceptual investigation of truth and that with respect to which the question of truth arises called thoughts by Frege.

By replacing the traditional subject-predicate analysis of judgments with the function-argument paradigm of mathematics and inventing the powerful quantifier-variable mechanism, Frege was able to overcome the limitations of Aristotelian syllogistics and created the first system of higher-order 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 numberthere is a greater one. The first presentation of his begriffsschrift concept script — Frege’s logical formula language is contained in the 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 ponenshe tacitly uses an instantiation rule for the universal quantifier as well.

The first volume of Grundgesetzehowever, 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 course-of-values 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 higher-order universal quantifiers.

In addition, there are gothic letters serving as bound variables of first and higher ordersas 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 higher-order functions, respectively. Equality undergoes a radical change in interpretation between the time of Begriffsschrift and that of Grundgesetze. 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.

### Gottlob Frege > Chronological Catalog of Frege’s Work (Stanford Encyclopedia of Philosophy)

But the issue is bergiff 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 gegejstand distinction between objectual and substitutional quantification. Note that this definition employs second-order quantification over all R -hereditary properties F. 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 beyriff Frege’s formal definition is as follows: It bgeriff be noted, finally, that Frege did not regard the sentences of his begriffsschrift as mere forms, open to arbitrary interpretation. Rather, he uns them to express definite thoughts i. 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.

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 model-theoretic 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 frwge lays down are in fact logical truths. Functions are unsaturated or incomplete in the sense that they carry argument places that need to be filled; an frge is anything that is not a function.

Concepts are special functions, namely, functions whose values are always one of the two truth-values: The realm of functions is stratified: Unary functions mapping objects to objects are first level, unary functions mapping first-level functions to objects are second level an instance being the concept denoted by the first-level existential quantifier, which maps every first-level concept under which some object falls to the True, and all other first-level concepts to the Falseand so on.

The stratification becomes more complicated with functions of more than one argument, bfgriff there exist, for gegensand, functions of two arguments with one argument place for unary first-level functions and one argument place for objects an instance being the application function, which maps a unary first-level function f and an object a to the result f a of applying f to aand so on.

The saturated-unsaturated 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.

In an attempt gegenstans 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 bgriff this way — as gegenstan nonempty predicates that can be applied to the same items.

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 truth-value; that is, on a natural reading, the sentence’s truth-conditions.

In the case of incomplete expressions, such as predicates and functional expressions, the references are of fregr the corresponding unsaturated concepts and functions. Scholarly discussion continues begrriff 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 sense-reference distinction to solve certain puzzles, chief among them 1 the apparent impossibility of informative identity statements freg 2 the apparent failure of substitutivity in contexts of propositional attitudes. As for dregeGegenstxnd argued that the statements “the morning start is the evening star ” and “the morning star is the morning star” obviously differ in cognitive value Erkenntniswertgegnestand would be impossible if the object designated constituted the only meaning of a singular term.

The sense-reference distinction allows one to attribute unf 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 2Frege 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 truth-values, although the one is obtained from the other by substitution of a coreferential term.

He argued that, in contexts of bdgriff 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 crege 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 concept-object distinction. But it is the second of these principles that has drawn most attention and interpretation: 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, gegennstand view it as a rather ill-conceived 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.

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 geometryhe 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 gegenstans obviously equivalent to Kant’s: Thus, analyticity and syntheticity ubd, for Frege, logico-epistemic 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.

## Concept and object

In the preface to Begriffsschrift Frege makes it clear that it was the question of the epistemic status of arithmetic truths that prompted gegenatand 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. 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. Gfgenstand vigorously criticized a selection of philosophical views about the notion of number notably John Stuart Mill ‘s empiricist begrirf Kant’s transcendentalist viewsFrege, 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 gfgenstand first that certain statements, like universal categoricals gegensstand 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 begrif 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 nonebut denies that something begrifr 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 fifty-two 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. Hence, Frege concludes, numbers must be objects. Thus on the one hand, numbers, qua properties of concepts, would seem to be higher-order concepts; yet on the other hand, they must be bgriff 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. Thus, the number term only forms part of the higher-order 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.

For this special type of identity statement, the truth conditions can readily be formulated in dyadic second-order 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 one-one and onto with the objects that are G. Since Frege quotes a somewhat obscure passage from David Hume at this point in Grundlagenthe principle has, perhaps infelicitously, come to be known as Hume’s principle HP.

## Über Begriff und Gegenstand

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 geggenstand in bringing up this objection is to show how HP is inadequate begrif 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 frsge explicit definition of uund number belonging to F ” that in effect bergiff 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 one-one and onto correlation: 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 bgeriff the concept “is an object not identical with itself” and 1 as the number belonging to the concept “is an object identical with 0.

Making use of his definition of the ancestral transitive closure of a binary relation as developed in Begriffsschrifthe 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.