How did symbolic logic show that Heidegger's assertions about the nothing were illogical?

Question

In his inaugural address at Freiburg University in 1929, Heidegger explicitly challenged the central place given to logical principles in neo-Kantianism, on the basis of a radical account of ‘the nothing’. Two years later, Carnap used the tools of symbolic logic to show how Heidegger’s assertions about the nothing were illogical and thus meaningless, like much of traditional metaphysics.

From Jonah Wilberg’s review of Heidegger and Logic: The Place of Lógos in Being and Time, by Greg Shirley.

How did symbolic logic show that Heidegger’s assertions about the nothing were illogical? What is symbolic logic and is it related to mathematics? What are the claims that were proved to be wrong?

Answer 1

I guess it refers to this passage later in the article/review you were quoting from:

the debate between Heidegger and Carnap -- Shirley's next topic -- precisely turns on whether Heidegger's account is compatible with a different aspect of mathematical logic: the use of existential quantification in first-order predicate calculus. As Shirley presents it, Carnap's charge is that in making assertions like 'the nothing itself nihilates', Heidegger illegitimately uses the word 'nothing' to refer to an entity, as opposed to using it in its ordinary sense, the sense expressed in predicate logic by negative existential quantification (for example, the English sentence 'Nothing is outside' is formalized as: ~∃x (Ox) where '~' is the operator for negation, '∃' is the operator for existential quantification, and 'Ox' means: x is outside).

Shirley rebuts Carnap's arguments in a few lines. He correctly points out that Heidegger's account of the nothing as an aspect of the being of beings means that he intends to use the word 'nothing' neither to refer to an existing entity nor in the sense of negative existential quantification. He briskly -- and again correctly - concludes that Carnap is wrong to accuse Heidegger of using the word to refer to an entity. But by taking this to be the end of the story, Shirley fails to treat Carnap with the same hermeneutic charity he shows to Heidegger. For it requires only a little rational reconstruction of Carnap's position to see that on his view the two uses of the word 'nothing' he discusses -- negative existential quantification on the one hand and reference to an entity on the other -- jointly exhaust the legitimate usage of the word, essentially because these are the only uses that are formalizable using predicate logic. So the fact that Heidegger is trying to use the word in some mysterious third way, far from constituting a rebuttal of Carnap's case, is rather precisely what indicts him on Carnap's view.

Answer 2

Building on Conifold's helpful comment where he quotes Carnap:

"For even if it were admissible to introduce "nothing" as a name or description of an entity, still the existence of this entity would be denied in its very definition..."

That is exactly what Heidegger says in "What is Metaphysics?":

The Elaboration of the Question

... In our asking we posit the nothing in advance as something that "is" such and such; we posit it as a being. But that is exactly what it is distinguished from. ... Accordingly, every answer to this question is also impossible from the start.

So later Heidegger is saying the idea of the nothing extinguishes itself in our mind as we think it. ("The nothing itself nihilates.") This on his way to describing angst in the mind on anticipation of becoming nothing in death, (overlooking here the nuanced problem in becoming nothing).

Carnap is being a bit previous in trying to shut down the metaphysics with such literalism.