Gottlob Frege's magnum opus, "The Basic Laws of Arithmetic" (Die Grundgesetze der Arithmetic in German) constitutes one of most impressive and meticulous attempts at developing a rigorous foundation for mathematics based on logic. Unfortunately, Bertrand Russell derived a contradiction from Basic Law V, one of the axioms of Frege's system, and so mathematicians had to search for new foundations, moving from Frege's naive set theory to the system of axiomatic set theory that we use today.

But I have a question of another one of Frege's Basic Laws. Now Frege originally worked in a strange diagrammatic language that very few people have ever taken the time to learn. But Richard Heck, in the introduction to his new book "Reading Frege's Grundgesetze", puts Frege's Basic Laws into modern logical symbolism, and he states Basic Law IIb as follows: $\forall F (M_x Fx) \rightarrow M_x Gx$. That seems like the universal instantation (AKA universal elimination) axiom of second-order logic, except for the part about the $M_x$. What does the $M_x$ mean?

Any help would be greatly appreciated.

Thank You in Advance.


1 Answer 1


$M_x$ stands for a second level concept (in Frege's sense), and yes Law IIB can here be taken to be the UE law at the second level. See Grundgesetze §25 (p. 42 in the new translation).

For a bit more explanation: remember, a first level concept takes an object and yields a truth. So, for Frege, the expression for a first level concept will be something like $F^1(\ )$ -- with a gap in it to indicate that the expression needs completion by a name to yield a sentence.

Likewise a second level concept takes a first-level concept and yields a truth. So, for Frege, the expression for a second level concept will be -- at a first shot -- something like $M^2(\ )$, with a gap in it to indicate that the expression needs completion by an expression for first-level concept to yield a sentence.

But hold on! Plugging the first level expression into the second level one gives us $M^2(F^1(\ ))$, which is still gappy and not a sentence! Ooooops!!

But it is plain what is missing. The second-level expression needs to supply something to fill the gap (in the first-level concept expression)! Paradigmatically, it will supply and then bind an object variable. Let's indicate the the supplied variable by subscripting $M^2_x(\ )$. Then then idea is that applying this to $F^1(\ )$ yields $M^2_x(F^1(x))$, which is a sentence as required.

Does that look weird? No, we are in fact very familiar with such variable-binding second-level concepts -- our old friend the universal quantifier is one. The second-level concept-expression $\forall_x$ (as Frege would regard is) is applied to $F(\ )$ to get the familiar $\forall_xF(x)$. Frege's and Heck's $M_x$ is just standing in for second-level concept expressions like $\forall_x$.

  • $\begingroup$ Thanks, @Peter. I find it awesome that some people today can still read and comment on Frege. $\endgroup$ Dec 31, 2013 at 9:33
  • $\begingroup$ It can even be pleasant... $\endgroup$
    – mmw
    Dec 31, 2013 at 17:12
  • $\begingroup$ Thanks, the $\forall_x$ example really clarified things. By the way, now that you mention it, how is the new Oxford Grundgesetze translation? Does Frege's notation make it too difficult to read? $\endgroup$ Dec 31, 2013 at 18:48

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