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1. For those who are curious, the following brieftable will give one just ahint as to some of the differencesbetween the modern notation and Frege’s:

Logical Notion	Modern Notation	Frege-style Notation
It is not the case that \(Fx\)	\(\neg Fx\)
If \(Fx\) then \(Gy\)	\(Fx\rightarrow Gy\)
Every \(x\) is such that \(Fx\)	\(\forall x Fx\)
Some \(x\) is such that \(Fx\)	\(\neg\forall x\neg Fx\),   i.e.,   \(\exists xFx\)
Every \(F\) is such that \(Fa\)	\(\forall F \, Fa\)
Some \(F\) is such that \(Fa\)	\(\neg\forall F\neg Fa,   i.e.,  \exists F \,Fa\)

So, for example, whereas a modern logician would symbolize the claim‘All \(A\)s are \(B\)s’ as:

\(\forall x(Ax\rightarrow Bx)\)

Frege would symbolize this claim as follows:

For a more complete description of Frege’s notation, see R.Cook’s Appendix (“How to ReadGrundgesetze”) inthe Ebert, Rossberg, and Wright translation of Frege 1893/1903, Reck& Awodey (2004, 26–34), Beaney (1997, Appendix 2), Furth(1967).

2. For those who would like astrictly faithful understanding of Frege’s definition of membershipinGg I, §34, a more exact representation wouldbe to use the definite description operator \(\iota\) and to castFrege’s definition as follows, where the symbol \(\in\) replacesFrege’s symbol \(\frown\) and where \(G\) is any unary functionsymbol:

\(x\in y \eqdef \iota\alpha\exists G (y\eqclose \epsilon G \, \amp\,G(x)\! =\! \alpha )\)

What’s being defined here is the value of the function \(x\in y\). AndFrege defines it as:the object \(\alpha\) such that thereexists a function \(G\) for which \(y\) is the course-of-values for\(G\) and \(\alpha\) is the value of \(G\) for the argument \(x\). Ifwe restrict this definition to those functions that are concepts, (a)the course of values \(y\) becomes an extension of a concept, and (b)the object \(\alpha\) in Frege’s definiens is a truth value and,indeed, signifies the truth-value The True when \(x\) falls under theconcept \(G\). So the claim \(G(x) = \alpha\) in Frege’s definitioncan be represented by the predication \(Gx\). That means the variable\(\alpha\) is no longer needed and so we can transform Frege’s termdefinition that identifies the value of the denoting term‘\(x\in y\)’ to a formula definition that gives truthconditions for the formula ‘\(x \in y\)’. We eliminate thevariable-binding description operator and the variable \(\alpha\) fromFrege’s definiens, so that the definiens in Frege’s definition nowbecomes:

\(x\in y \eqdef \exists G (y\eqclose \epsilon G \amp G(x))\)

In other words, \(x\) is an element of \(y\) if and only if \(y\) is the extension of some concept \(G\) such that \(Gx\). And this is the definition we’ve used in the text.

3. These models of second-order logic with a Comprehension Principle forConcepts are called ‘general models’ (as opposed to‘standard’ models in which the domain of concepts is takento be the power set of the domain of objects). These general modelsexploit the fact that there are only a denumerably infinite number ofconditions on objects expressible in the language and hence, only adenumerably infinite number of instances of comprehension. Thesegeneral models include in the domain of concepts only enough conceptsto make these instances of comprehension true. Thus, only adenumerably infinite number of concepts are required, even if thedomain of objects is denumerably infinite. So we emphasize that it isthe interaction of the Comprehension Principle for Concepts with Vbthat engenders the paradox.

4. It is important to note here that Frege’s definitions of themembership relation and the notion of equinumerosity require asecond-order language, since both definitions involve quantificationover concepts.

5. The reason this isweaker than Frege’s definition is that his definition ofequinumerosity requires any witness \(R\) to the equinumerosity ofF andG to be (globally) functional and one-to-one,where:

• \(R\) isfunctional \(\eqdef\) \(\forall x\forall y\forall z (Rxy \amp Rxz \to y=z)\)
• \(R\) isone-to-one \(\eqdef\) \(\forall x\forall y\forall z (Rxz \amp Ryz \to x=y)\)

That is, inGl (§§71, 72), Frege saysthatF andG are equinumerous just in case:

R is functional &R is one-to-one &
\(\forall x(Fx \to \exists y(Gy \amp Rxy)) \amp \forall y(Gy \to \exists x(Fx \amp Rxy)) \)

This is stronger than the definition of equinumerosity we’ve given inthe text. The definition we’ve given doesn’t require \(R\) to befunctional generally;R might fail to be functional withrespect to any non-\(F\) objects it might relate to other things. Forexample, ifa is a non-F object andbandc are distinct non-G objects, thenR canwitness the equinumerosity ofF andG in the sense that we’ve defined in the text eventhoughR relates the non-F objecta to both thenon-G objectsb andc. But that is ruled out byFrege’s definition.

Similarly, the definition we’ve given doesn’t require \(R\) to beone-to-one generally;R might fail to be one-to-one withrespect to any non-\(G\) objects to which other things areR-related. For example, ifa andb are distinctnon-F objects andc is a non-G object,thenR can witness the equinumerosity ofF andG in the sense that we’ve defined in the text eventhoughR relates both the non-F objectsaandb to the non-G objectc. But that is ruledout by Frege’s definition.

It seems that neither Frege’s derivation of Hume’s Principle fromBasic Law V nor his derivation of the Dedekind-Peano axioms fromHume’s Principle requireR to be functional and one-to-onegenerally. A relationR does indeed correlateFandG one-to-one if it satisfies the definition ofequinumerosity in the text while failing to be globally functional andone-to-one. So equinumerosity, as we’ve defined it, is sufficient topartition the domain of properties into equivalence classes, each ofwhich contains all and only those properties with the same number ofobjects falling under them. That is essential to the proof of Frege’sTheorem and hence the exegesis here is simplified.

6. Frege doesn’t call this principle ‘Hume’sPrinciple’ in his own writings. The label was instead introducedin Boolos (1987). Frege does cite Hume when he introduced thisprinciple inGl. InGl, §63, hequotes Hume’sTreatise (I, iii, 1):

When two numbers are so combined as that one has always an uniteanswering to every unite of the other, we pronounce them equal.

The idea in Hume does bear some resemblance to the principle Fregeconstructs, and so we shall continue to use Boolos’ label forthis principle.

7. We call this an implicit or contextual definition rather than anexplicit definition because the notation \(\#F\) can only beeliminated when it appears in a context of the form ‘\(\#F =\#G\)’. By contrast, an explicit definition would take theform:

\(\#F \eqdef\) the object \(x\) such that \(\phi(x,F)\),

where \(\phi (x,F)\) is some condition on \(x\) and \(F\). This wouldallow us to eliminate the \(\#F\) no matter what context it appearsin. We shall examine Frege’s attempt to give such a definitionmomentarily.

The reader might also find the following observation by W. Demopoulosuseful (from an early draft of Demopoulos and Clark 2005):

Frege’s contextual definition (i.e., Hume’s Principle) isnot ‘conservative’ over the language \(L = {0, S, N}\) ofsecond order arithmetic. (It is not conservative because it allows oneto prove statements that are otherwise unprovable using this languageand second-order logic alone. A proper, explicit definition onlyintroduces simplifying notation – the new theorems formulablewith the new notation introduced by an explicit definition would stillhave been provable had the new notation been eliminated in terms ofprimitive notation. As such, explicit definitions are conservative.)Indeed, the contextual definition allows for the proofbothof the infinity of the sequence of natural numbersand of theexistence of an infinite cardinal (which Frege called‘endlos’ inGl).

8. The reader might find the following observation in Demopoulos andClark 2005 (135–135) useful:

The characterization ‘Frege-Russell’ slurs over the factthat for Russell, the number associated with a set (concept of firstlevel) is an entity of higher type than the set itself. Beginning withindividuals – entities of lowest type – we proceed firstto sets of individuals and then to classes of such sets (correspondingto Frege’s concepts of second level). For Russell, numbers,being classes, are of higher type than sets. But for Frege,extensions, and therefore numbers, belong to the totality of objectswhatever the level of concept with which they are associated.Thus, while Russell and Frege both subscribe tosome versionof Hume’s Principle, their conceptions of the logical form ofthe cardinality operator, and therefore, that of the principle itself,are quite different: the operator is typeraising for Russell[since it takes us from a set to a class], and typeloweringfor Frege [since it takes a concept (set) to an object (individual)].This difference is fundamental, since it enables Frege to establish– on the basis of Hume’s principle – those of thePeano-Dedekind axioms of arithmetic which assert that the system ofnatural numbers is Dedekind infinite. By contrast, when thecardinality operator is type raising, Hume’s principle is ratherweak, allowing for models of every finite power.

9. The higher-order version of the Law of Extensions asserts that aconcept G is a member of the extension of the second-order conceptconcept equinumerous to F iff \(G\) is equinumerous to \(F\).If we temporarily suppose that we can have higher-order\(\lambda\)-expressions of the form \([\lambda H \, H\apprxclose F]\),then we could represent the extension of the second-order concept justdescribed as:

\(\epsilon[\lambda H\, H\apprxclose F]\)

Then, the higher-order law of extensions would be formalizable asfollows:

\(G \in \epsilon[\lambda H\, H\apprxclose F] \equivwide G\apprxcloseF\)

This principle is used implicitly on several occasions in thederivation of Hume’s Principle inGl. Thosereaders who read the material on the derivation of Hume’sPrinciple inGg will see that this principle getsreformulated as the Lemma to the Proof of Hume’s Principle.

10. Strictly speaking, we should represent this concept as follows:

\([\lambda z \, [\lambda y \, Ayp]z \amp z\neq r]\)

But we have applied the following instance of \(\lambda\)-Conversionto the first conjunct within the matrix of the \(\lambda\)-expression:

\([\lambda y \, Ayp]z\equiv Azp\)

We thereby simplify the entire expression to:

\([\lambda z \, Azp\amp z\neq r]\)

11. The Facts numbered 3, 4, 5, and 6 correspond to Theorems 123, 124,128, and 129, respectively, inGg I. Facts 1, 5, and7 correspond to Propositions 91, 84, and 98, respectively, in Part IIIofBegr.

12. Readers interested in seeing how our theorems correspond toFrege’s will find it instructive to see how Fact 3 correspondsto Frege’s Theorem 123, which is (Gg. I, p.138):

First, let’s simplify the notation for relations inFrege’s theorem 123. Since our notation \(Rxy\) corresponds tohis

and our notation \(R^{*}(x,y)\) corresponds to his:

we may substitute them into Frege’s theorem, to get a formulathat looks like this:

Now if we eliminate Frege’s notation for universally quantifiedconditional claims, we get:

And if we use normal variables instead of the Fraktur variables andreplace \(F(x)\) by \(Fx\), we get:

Consider the 3rd line from the top in the above. Since \(\forallx(Fx\rightarrow\forall y(Rxy \rightarrow Fy))\) is equivalent to\(\forall x\forall y(Fx\rightarrow (Rxy \rightarrow Fy))\), which isequivalent to \(\forall x\forall y(Rxy\rightarrow (Fx \rightarrowFy))\), we may use our abbreviation of the latter formula, namely\(\mathit{Her}(F,R)\), to get:

Translating the above conditional into modern notation yields:

\(R^{*}(a,b)\rightarrow(\mathit{Her}(F,R)\rightarrow (\forallz(Raz\rightarrow Fz)\rightarrow (Fb)))\)

Now if we gather the 3 antecedents of the conditionals into aconjunction, we get the equivalent formula:

\([R^{*}(a,b)\amp \mathit{Her}(F,R)\amp \forall z(Raz\rightarrowFz)]\rightarrow Fb\)

Finally, if we switch the 2nd and 3rd conjuncts in the antecedent, andreplace ‘\(a\)’ by ‘\(x\)’ and replace‘\(b\)’ by ‘\(y\)’, we get:

\([R^{*}(x,y)\amp \forall z(Rxz\rightarrow Fz)\amp\mathit{Her}(F,R)]\rightarrow Fy\)

And this is our Fact 3.

13. Facts 2, 3, 6, and 7 correspond to Theorems 132, 134, 141, and 144,respectively, inGg I.

14. A relation \(R\) is one-to-one (‘R is 1-1’) justin case it satisfies the following condition:

\(Rxz\amp Ryz\rightarrow x=y\)

So Fact 8 in the text is a fact about the weak ancestral whenever therelation R in question is 1-1. We shall prove that the Predecessorrelation is 1-1 in the third subsection of Section 5. Then Fact 8 andthe fact that Predecessor is 1-1 will both play a crucial role in theproof that every number has a successor.

To prove Fact 8, assume that \(R^{*}(a,b), Rcb\) and that \(R\) is1-1. We want to show \(R^{+}(a,c)\). Now by Fact 5 concerning the weakancestral, we know that it follows from \(R^{*}(a,b)\) that \(\existsz[R^{+}(a,z)\amp Rzb]\). So call an arbitrary such object‘\(d\)’. So we know \(R^{+}(a,d)\amp Rdb\). Now since\(R\) is 1-1, it follows from \(Rdb\) and \(Rcb\), that \(c=d\). So,\(R^{+}(a,c)\), which is what we had to show.

15. See the work by Wright cited in the Bibliography for a defense ofsomething like this position. Wright justifies this position onFregean grounds by appealing to Frege’s Context Principle, whichasserts that a word has no meaning (reference) except in the contextof a proposition (truth).

16. See Rosen 1993 for a discussion of how someone might claim that theright-hand condition of an instance might imply its correspondingleft-hand condition.

17. Again, see the work by Wright cited in the Bibliography.

18. In the long footnote to §10, Frege seems to suggest that theidea of replacing the truth values with their unit classes cannot beextended to the case of every object in the domain without conflictingwith his earlier stipulations (inGg I,§§3, 9 and 20), and in particular, with Basic Law V.

19. Wehmeier (1999) shows that Frege would not have had much luckattempting to restrict the quantifiers ofGg toextensions. He considers two consistent subsytems that Frege mighthave adopted to avoid the contradiction, namely, the system Hdescribed in Heck (1996) and the system Wehmeier himself developed andlabels \(T_{\Delta}\). Both of these systems retain Basic Law V butplace restrictions on the Comprehension Principle for Concepts.However, both systems imply the existence of objects which are notextensions (or courses-of-values), and indeed, they imply an infinitenumber of such objects.