How to denote the idea of nothingness in formal terms?

I was thinking about the question "Why is there something rather than nothing?" , and have read about some theories that existence is the case because non-existence is logically impossible

So, I devised a logical proof that shows nothingness is impossible, but only if we can formally denote nothingness by : "There exists x, so that for all y , y is not identical to x".

(∃x)(∀y)(y≠x)

Which violates the law of identity, therefore : nothingness is logically impossible. (there is no x that is not identical to all y)

My question : are there any references that try to denote "nothingness" in formal terms?

• Comments are not for extended discussion; this conversation has been moved to chat. Sep 10 '19 at 2:08
• Not that I know of. It seems to be an idea that takes us out of the system. To say 'Nothing exists' might be okay if we assume an x may be real and not-exist (not stand-out) but to say 'Nothing is real' is to say there is no x and logic collapses. This would be connected with Bradley's comment that subject-predicate language, while necessary, is unsuitable for metaphysics. . .
– user20253
Sep 14 '19 at 13:24
• @PeterJ agree, or put another way, if nothing exists then there is no proposition that denotes nothing exists, because if there is a true proposition that meabs nothing exists, then it would follow that the proposition is wrong since there is something which is a true proposition thank you Sep 16 '19 at 11:15

There's some recent formal work by Graham Priest on this topic, which can be found in his monograph One. Oxford 2016, p.99ff.

Priest works in a non-standard mereology, where parthood is a non-well-founded preorder. His basic idea is to to formalize nothingness as the fusion of the empty set, this fusion being denoted by the constant n. So, within this formal picture, nothingness is what you get when you take the sum of no things whatsoever.

To get his mereological theory with nothingness added going Priest has to add axioms governing the behavior of the constant n, such as ~ x < n and ~ n < x (no thing is a proper part of nothing and nothing is not a proper part of anything). Priest also gives algebraic models of this theory using Boolean algebras.

• The empty sum is zero, not nothing. Mathematically at least. The empty product is 1. Sep 9 '19 at 2:54
• @user4894 You are confusing mathematical empty sum with Graham Preist's empty sum. They are two different things. The answer is correct. Sep 9 '19 at 8:28
• @BertrandWittgenstein'sGhost I said that the empty sum is zero, mathematically. Can you explain what is incorrect about what I said? Sep 9 '19 at 8:30
• @user4894 If you said the empty sum is zero, "mathematically," then your comment is irrelevant. Why bother posting it in the first place since neither the question, and nor the answer is about mathematics. Did you believe, for a moment, that you are contributing something meaningful to the discussion by derailing a perfectly valid answer? Regards Sep 9 '19 at 8:33
• @Sequitur thank you so much for your answer, I am looking for references on Pierce's work , Best +1 ! Sep 9 '19 at 15:06

This formulation: (∃x)(∀y)(y≠x) is not terrible but...it works. Yet, it has some issues:

1. It is a statement about Nothingness, not Nothingness itself. To write Nothingness you need to write : Nothingness is such that (∀y in Existence)(y ≠ Nothingness). Or (Nothingness = x) such that (∀y in Existence)(y≠x). Or more compactly: (Nothingness = x) (∀y)(y≠x). Your formulation starts with (∃x), hence, not only it defines Nothingness, it also states that Nothingness does exist! Is this what you want ?
2. Does Nothingness exist ? It depends on what you mean by "existence". Here is the interpretation in Phenomenology:

a- It is what we do not know at all. It is not an empty and infinite space but the answer to the question: "Why is there something rather than nothing?". We say that Nothingness is the negation of Being or an existence devoid of essence BUT NOT the negation of existence. It appears to the Consciousness as one asks the question and every Consciousness guaranties the existence of its object, hence Nothingness does exist.

b- The non-Existent becomes this that escapes the Consciousness because there does not exist anything that can escape the Consciousness, if there were a such thing then too late for I have just captured it!!

Given its importance, it is remarkably strange that no formal definition is given by logicians or even philosophers to the concept of Nothingness.

• Thank you so much for this great answer +1 ... This is exactly the conclusion that I came to after some research , my Universe of discourse already presumes that nothingness exists. Thank you ! On the other hand, I have another nomological argument against 'nothing', a 'nothing' world would have zero height, width, and length, and those are the exact dimensions of the green elephant in my pocket, that's just equivalent to something that cannot exist physically. This is not a formal argument, but it is better. Aug 21 '21 at 13:49
• Sorry, I didn't not see it as it's been long time since I last checked my SE account. Best ! Aug 21 '21 at 13:50

Nothingness had to exist, and it had to exist beyond our universe. Nothingness is a kind of space, but it is very important to understand that it is different from our 3D physical space. The universe must have begun from nothingness and it must be expanding into nothingness. If the universe contracted or travelled, it would do so into nothingness.

Imagining a simple scenario with 2 universes expanding into nothingness, with their boundaries eventually coming to touch each other, then the nothingness that was initially in between them would have had to contract. Thus, nothingness, as a special kind of space may have properties, like size, or being 3-dimensional, just like normal space. Therefore, nothingness is not exactly nothing. Something also important to note is that nothingness is not stable. It is constantly producing positive and negative virtual particles, which annihilate each other continuously. The book A Universe from Nothing by Dr Krauss explains nothingness in detail but it doesn't explain nothingness as a special case of something.

• I think the OP is asking for how to represent nothingness formally in symbolic logic. Sep 11 '19 at 12:35
• Thanks for your answer, The Universe from Nothing (in the physical sense) by Lawrence Krauss is different subject. The physical nothing is actually something in the philosophical sense, something that looks like nothing, and that has the potential to bring forth everything we see around us. besides The question, as Frank pointed out, is about how to present nothingness formally. +1 Sep 13 '19 at 17:57