Abstract
Sections “Introduction: Hume’s Principle, Basic Law V and Cardinal Arithmetic” and “The Julius Caesar Problem in Grundlagen—A Brief Characterization” are peparatory. In Section “Analyticity”, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle, bearing in mind that with its analytic or non-analytic status the intended logical foundation of cardinal arithmetic stands or falls. Section “Thought Identity and Hume’s Principle” is concerned with the two criteria of thought identity that Frege states in 1906 and their application to Hume’s Principle. In Section “The Nature of Abstraction: A Critical Assessment of Grundlagen, §64”, I scrutinize Frege’s characterization of abstraction in Grundlagen, §64 and criticize in this context the currently widespread use of the terms “recarving” and “reconceptualization”. Section “Frege’s Proof of Hume’s Principle” is devoted to the formal details of Frege’s proof of Hume’s Principle. I begin by considering his proof sketch in Grundlagen and subsequently reconstruct in modern notation essential parts of the formal proof in Grundgesetze. In Section “Equinumerosity and Coextensiveness: Hume’s Principle and Basic Law V Again”, I discuss the criteria of identity embodied in Hume’s Principle and in Basic Law V, equinumerosity and coextensiveness. In Section “Julius Caesar and Cardinal Numbers—A Brief Comparison Between Grundlagen and Grundgesetze ”, I comment on the Julius Caesar problem arising from Hume’s Principle in Grundlagen and analyze the reasons for its absence in this form in Grundgesetze. I conclude with reflections on the introduction of the cardinals and the reals by abstraction in the context of Frege’s logicism.
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Notes
- 1.
In his letter to Russell of 28.7.1902, Frege makes some comments on abstraction principles and takes the transformation of the equivalence relation of geometrical similarity into an identity of shapes of, say, triangles as an example. He mentions that this is what Russell perhaps calls “définition par abstraction”. In Chapter XI “Definition of cardinal numbers” of his Principles of Mathematics (Russell 1903), Russell does apply the term “definition by abstraction”, but makes it clear that such a definition “suffers from an absolutely fatal formal defect: it does not show that only one object satisfies the definition. Thus instead of obtaining one common property of similar classes, which is the number of the classes in question, we obtain a class of such properties, with no means of deciding how many terms this class contains” (p. 114).
- 2.
Frege’s reaction to Russell’s discovery of a contradiction in his letter to Russell of 22.6.1902 speaks volumes with respect to the key role that Axiom V was intended to play in his logicist project. “I must give some further thought to the matter. It is all the more serious as the collapse of my Law V seems to undermine not only the foundations of my arithmetic, but the only possible foundation of arithmetic as such” (Frege 1976, p. 213).
- 3.
In Frege (1893, §10), Frege also identifies the True and the False with special value-ranges, but his motive for doing this is entirely different from that for the (projected) identification of numbers of any kind with value-ranges. In fact, the identification of the truth-values with their unit classes is intended to remove, in a first essential step, the referential indeterminacy of value-range names deriving from the semantic stipulation made in §3 and later to be enshrined in the formal version of Basic Law V.
- 4.
Recall that Frege defines the relation of equinumerosity in second-order logic in terms of one-to-one correlation.
- 5.
If Frege were to carry out his plan in Grundlagen, §104 to define, in a first step and after the fashion of the attempted contextual definition of the cardinality operator, fractions, irrational and complex numbers by using second-order or higher-order abstraction, he would face a whole family of Caesar problems each of which is supposed to be resolved by framing an appropriate explicit definition for these numbers in terms of extensions of concepts.
- 6.
In Begriffsschrift, Frege employs the terms “synthetic” and “analytic” only in two places, firstly in the course of discussing the notion of identity of content (“Inhaltsgleichheit”) in §8, and secondly when he comes to explain the nature and purpose of definitions framed in his concept-script in §24. Frege takes as an example the definition of a hereditary property in a series. He refers to it as formula (69). Since in formula (69) we do not acknowledge a judgeable content as true, but make a stipulation, (69) is “not a judgement; and consequently, to use a Kantian expression, also not a synthetic judgement” (Frege 1879, p. 56). Once the content of the definiens has been bestowed upon the definiendum the definition is immediately turned into an analytic judgement; for “we can only get out what was put into the new symbols in the first place” (p. 56). Thus, Frege uses here the term “analytic” along Kantian lines, namely as an equivalent for “epistemically trivial judgement”. It is true that according to Frege’s definition of “analytic truth” in Grundlagen, §3 the definitions turned into assertoric sentences are likewise to be regarded as analytic. Yet in contrast to Kant, he argues that there are analytic truths containing valuable extensions of our knowledge.
- 7.
In Schirn (2016b), I analyze in detail Frege’s use of “analytic” in the period 1879–1892.
- 8.
See the controversy between Boolos (1997) and Wright (1999) concerning the analyticity or non-analyticity of Hume’s Principle. In a sense, one might say that their different views result from the distinct notions of analyticity which underlie their arguments; see Schirn (2006). See in this connection also Ebert (2008) on the dispute between the neo-Fregean (represented by Wright) and the “epistemic rejectionist” (represented by Boolos). Boolos jettisons the key idea of neo-Fregeanism, namely that Hume’s Principle and certain other abstraction principles are analytic truths, despite the fact that they involve specific ontological commitments. Ebert calls Boolos’s position epistemic rejectionism, since Boolos refuses to accept Wright’s idea that Hume’s Principle can be known on analytic grounds, or as Wright has also put it: that our knowledge of number theory may be regarded as deriving a priori from Hume’s Principle.
- 9.
See Frege’s argument for rejecting contextual definitions in Frege (1903, §66).
- 10.
It is clear that for Frege the self-evidence of a truth cannot serve as a general criterion of analyticity. On the one hand, he grants that there are non-evident sentences which are analytic truths, such as, for example, the equation “125,664 + 37,863 = 163,527”, provided that the logicist programme has been successfully carried out for cardinal arithmetic. On the other hand, Frege acknowledges the existence of self-evident, but non-analytic truths, such as the axioms of Euclidean geometry. For a true statement “a = b” to be analytic in Frege’s sense, the identity of the sense(s) of “a” and “b” is a sufficient condition, but it is not a necessary one.
- 11.
For a detailed discussion of this issue see Schirn (2016a, Sects. 5 and 6). To avoid misunderstanding here, let me mention that in Grundlagen Frege did not yet construe declarative sentences as truth-value names, as names of the True or the False, and hence as a special kind of (complex) proper names that can appropriately flank “=” on both sides. Yet if Hume’s Principle is considered to be a statement of the form “a ↔ b”, the argument above for its epistemic triviality, if “a” and “b” have the same judgeable content or express the same thought, would equally apply.
- 12.
For details see Schirn (2016c).
- 13.
See also Dummett’s comment on CRIT 1 in Dummett (1981, p. 324). It remains unclear why the view involved in this criterion, namely that two analytically equivalent sentences express the same thought, should be irreconcilable with Frege’s ideas about sense as stated in other writings. Dummett owes us a plausible explanation why there should be a serious conflict.
- 14.
Ebert (2008) argues in detail that Hume’s Principle involves the existence of infinitely many objects only when it is combined with additional metaphysical assumptions concerning the existence of properties. He claims that in an Aristotelian universe, where there are no empty concepts, Hume’s Principle will never inflate an originally finite domain to an infinite one.
- 15.
In Grundlagen, §64, Frege uses the word “zerspalten”.
- 16.
Argument-places of the first kind are suitable for the insertion of proper names; argument-places of the second kind are suitable for the insertion of monadic first-level function-names; and argument-places of the third kind are suitable for the insertion of dyadic first-level function-names; cf. Frege (1893, §23).
- 17.
Not to be confused with a similar distinction drawn by Dummett in Dummett (1981, pp. 333 f.).
- 18.
In Schirn (2016c, d), I introduce the term “truth-value name”, which Frege does not use; as far as I can see, he only uses the term “name of a truth-value”. By a truth-value name I understand a function-value name that has the syntactic structure of a declarative sentence, and hence does not only refer to one of the two truth-values, but also expresses a thought. By contrast, the value-range name “
” or the definite description “
” do not express a thought, although, due to Frege’s stipulations in Grundgesetze, both names refer to the True. They are names of a truth-value, but they are not truth-value names qua sentences. Both names express a complex, non-propositional sense. - 19.
When in Grundlagen, §64 Frege says that the judgement “The straight line a is parallel to the straight line b” can be construed as an equation, he chooses an infelicitous way of phrasing. This judgement qua sentence (note his use of quotation marks) can never be construed as an equation in a strict sense, although it can be transformed into an equation. The ensuing statement “We replace the symbol || with the more general symbol =, by distributing the particular content of the former symbol to a and to b” is rather metaphorical and far from being clear. The linguistic operation of replacing a symbol with a more general symbol has nothing to do with abstraction. And what is it to mean precisely that the particular content of “=” is distributed between a and b? By the way, replacing the symbol “||” by “=” in “a || b” would be permissible at least from a syntactic point of view. The matter stands differently in the case of Hume’s Principle. The sign for the second level relation of equinumerosity cannot literally be replaced by “=”.
- 20.
According to Frege, identity is a first-level relation.
- 21.
Parsons (1997, p. 270) seems to accept Wright’s idea that Fregean abstraction effects a reconceptualization for the case of first-order principles. In those cases, he says, “it seems that what we are doing is simply individuating the objects we have in a coarser way, one might say carving up the domain, or a part of it, a little differently.” This is very vague and thus far from being clearer than Wright’s characterization. Fine (1998, p. 532) introduces the term “definition by reconceptualization” and says (but does not further explain) that it rests on the idea that new senses may emerge from a reanalysis of a given sense. He further claims that the idea derives from Grundlagen, §§63–64. However, as I have argued, Frege’s attempted contextual definitions via abstraction cannot sensibly be described in such a way that new senses emerge from a reanalysis of a given sense.
- 22.
- 23.
Note that the third-order abstraction principle above applies in Grundlagen only to second-level concepts (and their extensions), not to monadic second-level functions in general. I assume that around 1884 Frege did not yet consider extensions of concepts and extensions of relations as special cases of what after 1891 he calls value-ranges of one-place functions and value ranges of two-place functions (double value-ranges). Thanks to his device of “level-reduction”, he is able to confine himself to the introduction of value-ranges of first-level functions. First-level functions which appear as arguments of second-level functions are represented by their value-ranges, “though of course not in such a way that they give up their places to them, for that is impossible” (Frege 1893, §34).
- 24.
Frege uses “function” here and I have respected his choice. However, strictly speaking, it is only the function-name that has an argument-place or argument-places that are suitable to take names of the appropriate syntactic category. Plainly, in a function one cannot replace a letter by another one.
- 25.
As Frege puts it, “the relation in which one member of the cardinal number series stands to that immediately following it” is defined in Frege (1893), §43 (Def. H) as follows:
$$ \begin{aligned} {\text{f}}& := {\text{VR}}\alpha {\text{VR}}\varepsilon (\neg \forall {\mathfrak{u}}\forall {\mathfrak{a}}({\text{N}}({\text{VR}}\varepsilon (\neg (\varepsilon \in {\mathfrak{u}} \to \varepsilon = a))) \\ & \quad = \varepsilon \to (a \in {\mathfrak{u}} \to \neg {\text{N}}{\mathfrak{u}} = \alpha ))) .\\ \end{aligned} $$From Def. H one can easily derive
$$ \begin{aligned} & e \in (a \in \text{f}) \to (\neg \forall {\mathfrak{u}}\forall a({\text{N}}({\text{VR}}\varepsilon (\neg (\varepsilon \in {\mathfrak{u}} \to \varepsilon = a))) = \varepsilon \\ & \quad \to (a \in {\mathfrak{u}} \to \neg {\text{N}}{\mathfrak{u}} = a)). \end{aligned} $$What needs to be proved regarding the single-valuedness of f is therefore
$$ \begin{aligned} & \neg \forall {\mathfrak{u}}\forall a({\text{N}}({\text{VR}}\varepsilon (\neg (\varepsilon \in {\mathfrak{u}} \to \varepsilon = a))) = \varepsilon \to (a \in {\mathfrak{u}} \to \neg {\text{N}}{\mathfrak{u}} = d)) \\ & \quad \to \neg \forall {\mathfrak{u}}\forall a({\text{N}}({\text{VR}}\varepsilon (\neg (\varepsilon \in {\mathfrak{u}} \to \varepsilon = a))) = \varepsilon \to (a \in {\mathfrak{u}} \to \neg {\text{N}}{\mathfrak{u}} = a). \\ \end{aligned} $$This is a formula which, by repeated application of contraposition and the introduction of German letters, emerges from
$$ \begin{aligned} & {\text{N}}({\text{VR}}\varepsilon (\neg (\varepsilon \in {\text{v}} \to \varepsilon = c))) = \varepsilon \to (c \in {v} \to {\text{Nv}} = d \\ & \quad \to {\text{N}}({\text{VR}}\varepsilon (\neg (\varepsilon \in {\mathfrak{u}} \to \varepsilon = b))) = \varepsilon \to (b \in u \to ({\text{N}}u = a \to d = a))))). \\ \end{aligned} $$The latter formula can be derived from
$$ \begin{aligned} & c \in v \to ({\text{N}}({\text{VR}}\varepsilon (\neg (\varepsilon \in u) \to \varepsilon = b))) = {\text{N}}({\text{VR}}\varepsilon (\neg (\varepsilon \in {\text{v}} \to \varepsilon = c))) \\ & \quad \to (b \in u \to {\text{N}}u = {\text{Nv}})). \\ \end{aligned} $$It is at this point of his proof of the single-valuedness of f that Frege expressly invokes Hume’s Principle. According to Hume’s Principle, the last-mentioned derivation can be carried out by showing that there is a relation which maps the u-concept into the v-concept and whose converse maps the v-concept into the u-concept. And that there is indeed a relation which maps the \( {\text{VR}}\varepsilon (\neg (\varepsilon \in \to \varepsilon = b)) \)-concept into the \( {\text{VR}}\varepsilon (\neg (\varepsilon \in v \to \varepsilon = c)) \)-concept follows from the identity of the cardinal number belonging to the first concept with that belonging to the second, which of course must still be proved.
- 26.
Thus, following Frege’s strategy in Grundlagen, we have the choice between “\( N_{x} F( x ) = N_{x} G( x ) \leftrightarrow Eq_{x} ( {F( x ),G( x )} ) \)” (or its definitional variant “\( Ext\varphi (Eq_{x} (\varphi(x),F(x))) = Ext\varphi (Eq_{x} (\varphi(x), \) \( G(x))) \leftrightarrow Eq_{x} (F(x),G(x)) \)”) and “\( Ext\varphi (Eq_{x} (\varphi (x),F(x))) = Ext\varphi (Eq_{x} (\varphi(x),G(x))) \) \( \leftrightarrow \forall \varphi (Eq_{x} (\varphi(x),F(x)) \leftrightarrow Eq_{x} (\varphi(x),G(x))) \)”, whenever we wish to establish whether the number that belongs to F(x) is equal to the number that belongs to G(x).
- 27.
See Heck (1997, p. 283 f.).
- 28.
The term “Relation” is Frege’s shorthand expression for “Umfang einer Beziehung” (“extension of a relation”). Thus, in his logic Relationen (in English, I use “Relations” with a capital “R”) are value-ranges of two-place (first-level) functions whose value, for every pair of admissible arguments (objects), is either the True or the False. Note that in Frege (1903), Frege did not yet set up a definition of the reals. He only informs us about the way how he wants to define them.
- 29.
- 30.
Hale (2000) pursues the aim of providing an informal axiomatic characterization of quantitative domains, on the basis of which it will be possible to introduce the real numbers by means of an appropriate abstraction principle. Following this idea, he introduces objects—cuts—corresponding to cut-properties by the following abstraction principle:
$$ {\text{Cut}}:\# F = \# G \leftrightarrow \forall a(Fa \leftrightarrow Ga) $$where F, G are any cut-properties on RN+ and a ranges over RN+.
Informally, a cut-property is a non–empty property whose extension is a proper subset of RN+ and which is downwards closed and has no greatest instance (cf. p. 411). Like Hume’s Principle and Axiom V, Cut is a second-order abstraction principle. Yet unlike the former two principles, Cut is a restricted abstraction principle: the domain for the abstraction embraces only cut-properties on a certain specified underlying domain of objects. See Hale’s discussion of cut-abstraction on p. 414 ff. If I am right, then Cut would not have been an option for Frege had he thought about the prospects of saving the intended purely logical foundation of analysis in the aftermath of Russell’s Paradox. First, to use it as a contextual definition is ruled out from the outset. Second, in the light of Frege’s conception of primitive laws of logic, Cut can hardly claim to be such a law. Hence, it could not be laid down as a logical axiom in the theory of real numbers. Moreover, I doubt that Cut could be shown to be analytic in any plausible sense of analyticity. (See also Wright’s reflections concerning a possible extension of the neo-logicist programme to analysis in Wright 1997.) It seems to me that the neo-Fregeans have pulled out almost all the stops to make neo-logicism palatable and attractive. Ingenious and thought-provoking as some of their work undoubtedly is, I do not think that the approach deserves to be called (neo-)logicism in any well-established sense of logicism.
” or the definite description “
” do not express a thought, although, due to Frege’s stipulations in Grundgesetze, both names refer to the True. They are names of a truth-value, but they are not truth-value names qua sentences. Both names express a complex, non-propositional sense.References
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Acknowledgments
My thanks go to the editor of this volume, Sorin Costreie, for inviting me to write an essay for it and his interest in my work on Frege. Special thanks are due to Otávio Bueno for interesting discussion and his inspiring enthusiasm about this collection.
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Schirn, M. (2016). On the Nature, Status, and Proof of Hume’s Principle in Frege’s Logicist Project. In: Costreie, S. (eds) Early Analytic Philosophy - New Perspectives on the Tradition. The Western Ontario Series in Philosophy of Science, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-24214-9_4
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