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

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.

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?

• Read for yourself: Carnap, The Elimination of Metaphysics Through Logical Analysis of Languages, pp.69-71:"The construction of sentence (1) is simply based on the mistake of employing the word "nothing" as a noun... But in addition it involves a contradiction. 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..." Mar 13 at 8:00

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.

• Seems to be an insightful article. Maybe it is worth pointing out that the takeaway is that Carnap has "shown" that Heidegger's assertions are illogical under the assumption that his own take on what is logical/legitimate and what's not is the correct one, ie. the conclusion only holds within Carnap's own system, which itself is Neo-Kantian. That, in turn, is not that surprising, given that Heidegger explicitly uses the concept of nothing against logical foundationalism as a whole. Mar 13 at 9:28
• @PhilipKlöcking: I actually wondered whether there's an implicit bivalence present in Carnap's argument and apparently there is one. In that programme Heidegger's viewpoint can actually be framed in some kind of paraconsistent logic framework. I haven't read the latter paper though.
– Fizz
Mar 13 at 18:38
• Actually, skimming through the latter paper, it only formalizes what "A grounds B" means and thus related notions of Sein (Being) etc. However it doesn't touch on what one could call their "Heidegger duals", i.e. what "A nihilates B" could mean, or how "nothing" would be formalized. Actually Priest himself (who edited that volume) has written on that, but it's obviously a difference conception he has in One[ism], as nothing belongs to itself in that setup.
– Fizz
Mar 13 at 19:47
• I came across this article which says 'Neither Nishida nor Heidegger means by “nothing” a literal nothing, but rather that which permits beings in their relative determinacy to be what they are'. So, if one was trying to translate Heidegger's claims into predicate logic, I wonder if it could make sense to translate "nothing" neither as an object nor a negative existential quantifier, but as a property of all objects akin to Buddhist "emptiness", or other possibly universal properties like "incompleteness", "dependence" etc. Mar 27 at 16:29

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.