In the paper,

the authors give the following quote of Frege, from his paper "Über die Grundlagen der Geometrie"*:

What I call a proposition tout court or a real proposition is a group of of signs that expresses a thought [has a sense—my comment]; however, whatever only has the grammatical form [i.e. being a well-formed formula—my comment] of a proposition I call a pseudo-proposition.

Since it is well-known that $$R \notin R \Leftrightarrow R \in R$$ (this is actually true if both $R \notin R$ and $R \in R$ are false, but then this seems to contradict the Law of Excluded Middle), can one rightly attribute either a reference (since they are 'propositions', their reference must be, according to Frege, either The True, or The False) or a sense (what would that be, exactly) to "$R \notin R$" or "$R \in R$"? And if not, are "$R \notin R$", "$R \in R$" pseudo-propositions?

Furthermore, since $$R \notin R \Leftrightarrow R \in R$$ can be derived from Basic Law V, shouldn't Basic Law V be restricted to apply only to "real propositions"? And how might that be done in the context of the Grundgesetze?

*In Jahresbericht der Deutschen Mathematiker Vereinigung, vol. 15 (1906), pp. 293-309, 377-403, 423-30.
English translation by E-H. W. Kluge in Gottlob Frege, On the Foundations of Geometry and Formal Theories of Arithmetic, Yale University Press, 1971, p.69.

  • $\begingroup$ Don’t you need to resolve first what are the sense and reference of $R$? $\endgroup$ May 6, 2016 at 14:10
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    $\begingroup$ You confused Excluded middle ($P \lor \lnot P$) with Law of non-contradiction ($\lnot (P \land \lnot P)$). $\endgroup$ May 6, 2016 at 20:16
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    $\begingroup$ Excluded middle is stronger than the law of non-contradiction. The tautology you tried to write out requires the assumption of excluded middle (take $q = 0$ to derive $p = \neg \neg p$). It might not be an important distinction for your purposes, but it's a distinction worth bearing in mind in other contexts and is generally worth preserving. $\endgroup$
    – Todd Trimble
    May 7, 2016 at 16:01
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    $\begingroup$ By your reasoning, any two equivalent propositions could be interchanged. Since $p \lor \lnot p$ is equivalent to $\forall n, m \in \mathbb{Z} . n + m = m + n$ (because they are both true), you should agree that referring to "the law of excluded middle" is the same thing as referring to "the commutativity of addition". I think Frege wouldn't be so happy. (And by the way, this is a common misconception among mathematicians, namely that "$p$ and $q$ are equivalent" is the same as "$p$ and $q$ have the same meaning".) $\endgroup$ May 7, 2016 at 19:37
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    $\begingroup$ Let $R$ be a one-place predicate and $E$ a two-place relation (we secretly think of $R$ as Russell's paradoxical set and $E$ as the elementhood relation). Then we can get rid of set theory like this: $(\forall x . R(x) \Leftrightarrow \lnot E(x, x)) \Rightarrow \lnot \exists r . \forall y . R(y) \Leftrightarrow E(y,r)$. This is provable in first-order intuitionistic logic. It says: "if $R$ is the property of not being in $E$-relation with itself, then there is no individual $r$ such that the (extension of) predicate "is in $E$-relation with $r$" coincides with $R$. $\endgroup$ May 9, 2016 at 13:42

2 Answers 2


See Frege's Theorem: First Derivation of the Contradiction for reference.

Let "$P$" abbreviate the name for the concept being the extension of a concept which you don't fall under:

$[λx ∃F(x = εF \land ¬Fx)]$.

In Grundgesetze's logic it is proved that $εP$ exists, where for each value of the variable $F$, $εF$ denotes the extension of $F$.

Now supposing $P(εP)$, by Basic Law V, we have $¬P(εP)$, contrary to hypothesis.

Thus, from $(\varphi \to \lnot \varphi) \to \lnot \varphi$, we have: $\lnot P(εP)$. But then we have that $P(εP)$, again contrary to hypothesis.

Having derived a contradiction (which is a purely syntactical fact), we have to reject (as done by Frege) Basic Law V, i.e. to conclude with $\vdash \lnot \text { Basic Law V}$.

This means that, using Frege's "horizontal" fucntion $-$:

$- \Delta =$ the True if $\Delta$ is the True; the False otherwise,

we have that:

$- \text { Basic Law V}$ denotes the False.

See Frege's letter to Russell (1902) on the contradiction:

Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic. It seems, then, that transforming the generalization of an equality into an equality of courses-of-values is not always permitted, that my Rule V is false, and that my explanations are not sufficient to ensure that my combinations of signs have a meaning in all cases [emphasis added].

This can be interpreted as meaning that it is not true in general that:

for each value of the variable $F$, the extension of $F$: $εF$, exists.

And this in turn implies that the "combinations of signs" $εF$ is not always meaningful.

But note also Frege's clarification of the "form" of the contradiction in Grundgesetze's system:

Incidentally, it seems to me that the expression "a predicate is predicated of itself" is not exact. A predicate is as a rule a first-level function, and this function requires an object as argument and cannot have itself as argument (subject). Therefore I would prefer to say "a notion is predicated of its own extension". If the function $\Phi(\xi)$ is a concept, I denote its extension (or the corresponding class) by "$ἐ \Phi(ε)$". In "$\Phi(ἐ \Phi(ε))$" we then have a case in which the concept $\Phi(\xi)$ is predicated of its own extension.

See On Sense and Reference (1892) :

[page 32] Is it possible that a sentence as a whole has only a sense, but no meaning? At any rate, one might expect that such sentences occur, just as there are parts of sentences having sense but no meaning. And sentences which contain proper names without meaning will be of this kind.

[page 41] This arises from an imperfection of language, from which even the symbolic language of mathematical analysis is not altogether free; even there combinations of symbols can occur that seem to mean something but (at least so far) do not mean anything, e.g. divergent infinite series. [...] A logically perfect language (Begriffsschrift) should satisfy the conditions, that every expression grammatically well constructed as a proper name out of signs already introduced shall in fact designate an object, and that no new sign shall be introduced as a proper name without being secured a meaning. The logic books contain warnings against logical mistakes arising from the ambiguity of expressions. I regard as no less pertinent a warning against apparent proper names without any meaning.

So, if the set $R$ does not exist, the name for it lacks reference, and thus any sentence containing it lacks truth-value.

But in a logically perfect language every expression grammatically well constructed must have reference: an object for a name and a truth-value for a sentence.

Conclusion: R \notin R$ is not "an expression grammatically well constructed" in Frege's system.

Frege's first (and unsuccessful) attempt to salvage his system was precisely to restric the application of Basic Law V (which does not apply to propositions, but to functions: $ἐf(ε) = ἀg(α) ≡ ∀x[f(x) = g(x)]$ ); see:

It may be interesting to consider that, in his "modified" system sketched into the Appendix to GG II, Frege claims to have proved, with its modification to BL V, that:

$\vdash \lnot x \text ^ x$

which amounts (approximately) to: $\vdash R \notin R$.

Regarding Klement's paper, it discuss a very subtle point, which supports your previous concern that Basic Law V is (presumably) not the only source of trouble in Frege's system.

As Klement notes:

A few months after reporting Russell’s paradox, Russell despaired in a letter to Frege that "from Cantor’s proposition that any class contains more subclasses than objects we can elicit constantly new contradictions", and reports on his discovery of a Cantorian paradox regarding propositions, suggesting that, mutatis mutandis, this should be a worry for Frege’s theory of thoughts. In the ensuing correspondence, Russell formulates the paradox in terms of both generating a distinct proposition for every class, as well as generating a distinct proposition for every propositional function. Frege never quite fully appreciated Russell’s point, but nevertheless Russell had put his finger on a genuine issue Frege should have been concerned with, as I have argued elsewhere.

If we apply modern semantics to Frege's system, we have that every concept "identify" a subset of the universe. Thus, by Cantor's theorem, there are more concepts than objects in the universe.

But in Frege's system - and this assumption is prior to Basic Law V - for every concept must exists its extension, and extensions are objects; thus, we have already an inconsistency, as suggested by Russell.

  • $\begingroup$ Thank you for your answer. Are you aware that for restricted classes of wff's, Basic law $V$ is consistent with second-order logic? This leads me to believe that what engenders Russell's paradox is his "explanations", and not Basic law $V$. Regarding Basic Law $V$ as Frege conceived it (that is, for functions) , one could substitute a proposition for $x$ in "$V$", provided f($p$)={ The True, The False}. If one substituted $R$$\notin$$R$ or $R$$\in$$R$ for $x$ in Basic Law $V$, what would be the truth-value of Basic Law $V$? $\endgroup$ May 8, 2016 at 3:07
  • $\begingroup$ I would gladly accept your answer if you would discuss (in short form, of course) Kevin Klement's example in his paper "Does Frege have too many thoughts. A Cantorian problem revisited" (the cite is in my latest comment to Andrej Bauer). $\endgroup$ May 9, 2016 at 12:50
  • $\begingroup$ Since for $\epsilon$$P$, one has $\lnot$$P$($\epsilon$$P$) $\Leftrightarrow$ $P$($\epsilon$$P$), one can form the following subset of $R$(={$x$|$x$$\notin$$x$}): {$R$|$R$$\notin$$R$}. Though a well-formed formula in set theory, can it be said to have a coherent "sense" as Frege understood the term? $\endgroup$ May 10, 2016 at 11:33

Consider the following passage from Frege's lecture "Funktion und Begriff" (English translation by Peter Geach with the title "Function and Concept" [Note: this is the lecture which introduced the notions "value-range" and "Basic Law $V$"]):

It seems to be demanded by scientific rigour that we ensure that an expression never becomes bedeutungslos; we must see to it that we never perform calculations with empty signs in the belief that we are dealing with objects. People have in the past carried out invalid procedures with divergent infinite series. It is thus necessary to lay down rules from which it follows, e.g., what


stands for [bedeut], if '$\oplus$' stands for [bedeut] the sun [not Frege's sun-symbol, but an inadvertent sun-symbol nonetheless--my comment]. What rules we lay down is a matter of | indifference; but it is it is essential that we should do so --that 'a+b' should always have a Bedeutung, whatever signs for definite objects may be inserted in place of 'a' and 'b'. This involves the requirement that as regards concepts, that, for any argument they should have a truth-value as their value; that it shall be determinate, for any object, whether it falls under the concept or not. In other words: as regards concepts we have a requirement of sharp delimitation; if this were not satisfied it would be impossible to set forth logical laws about them. For any argument x for which 'x+1' were bedeutungslos, the function x+1=10 would likewise have no value, and thus no truth-value either, so that the concept:

'what gives the result 10 when increased by 1'

would have no sharp boundries. The requirement of sharp delimitation of concepts thus carries along with it this requirement for functions in general that they must have a value for every argument.

Note what this passage does not say (if Geach's translation correctly captures the sense of Frege's German). It does not say that every wff has a bedeutung; rather, it says that one should restrict one's attention only to those wffs that have a bedeutung--only these are the subject-matter of logic (for example, how could logic handle the following 'concept'

Colorless green ideas sleep furiously


With this in mind, let me consider Mauro Allegranza's example (actually Zalta's and Russell's):

$\lambda$$x$$\exists$$X$($x$=$\hat {\epsilon}$$X$ $\land$ $\lnot$$X$($x$)), where $X$ is a second-order variable.

Of this, Russell says, in his letter to Frege:

Therefore we must conclude that $w$ [i.e. $\lambda$$x$$\exists$$X$($x$=$\hat {\epsilon}$$X$ $\land$ $\lnot$$X$($x$)) when applied to itself in the manner Frege seems to allow--my comment] is not a predicate [concept, since at that point it has no reference or truth-value--my comment also].

It should be noted that if one takes Frege's prescription in "Function and Concept" seriously, the system in his Grundgesetze only applies to wffs in which variables always have references, and which only have sharp boundaries (with these restrictions, Antonelli and May show in their paper "Frege's Other Program" that the system of the Grundegesetze is consistent). All Russell's challenge to Frege's system amounts to is that it challenges the notion that all wffs of the Grundgesetze are such. If this is the case, then the following question must be asked:

Did Frege, in the system of the Grundgesetze, allow substitutions of any wff for latin letters (free variables--this comes from J$\ddot o$nne Speck's preprint "The Russell Paradox in the Theory of the Grundgesetze and Why Frege could not have done better" ) and why ( given the restrictions in "Function and Concept" make that system consistent)?

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    $\begingroup$ Comment to bedeuting; see Grungesetze (new Engl.transl.), footnote 3 to §5 (page 9), regarding the "horizontal": "Evidently, the sign "$\Delta$" must not be without reference, but it has to refer to an object. Names without reference must not occur in concept-script. The stipulation is made such that under all circumstances "$-- \Delta$" refers to something, provided only that "$\Delta$" refers to something. Otherwise $-- \xi$ would not be a concept with sharp boundaries, thus in our sense not be a concept at all. ... 1/2 $\endgroup$ May 17, 2016 at 9:06
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    $\begingroup$ ... I here use the capital Greek letters as if they were names referring to something, without stating their reference." 2/2 $\endgroup$ May 17, 2016 at 9:06
  • $\begingroup$ @MauroALLEGRANZA: Very interesting--Thanks! Since for Frege, a 'concept without sharp boundaries' is, in his "sense", "not a concept at all", what forbids him to state this to Russell as a rejoinder to Russell's claim of an inconsistency in the Grundegesetze? $\endgroup$ May 17, 2016 at 12:43
  • $\begingroup$ For a (very) different point of view on the "correct" way to read Frege's logical symbolism, see: Gregory Landini, Frege's Notations: What They Are and How They Mean (2012) (but I've haven't read it yet). $\endgroup$ May 19, 2016 at 13:38
  • $\begingroup$ Regarding your last comment, I think that your suggested "way out" from the paradox is not viable by the fact that (Frege's symbolic equivalent) of Russell's "contradictory" property is a well-formed formula in F's system... In Appendix to vol 2 of Grundgesetze, F tried to fix his system "limiting" BLV; it is interesting to note F's claim that, with this modification, he can now prove (the symbolic equivalent of): $\vdash \lnot (x \in x)$. $\endgroup$ May 19, 2016 at 14:18

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