# How can we represent 'nothing' or 'nothingness' in predicate logic?

The question says it all. I'm at lost for how to do this

edit: what had motivated this was Heidegger's "What is Metaphysics." This was my first stab at philosophy of a continental persuasion and recently finishing advanced logic courses, I was curious at first reading, especially when our professor decided to try and represent 'nothingness' in predicate logic.

I would vote up, but I do not have enough reputation points yet, my apologies

• Why you want to do so ? If "nothing" is an object, you must use a term (like : noth); if it is a property, you must use a predicate (like noth(x))... Mar 18, 2014 at 13:52
• Maybe you can tell us a little bit more to motivate the question. What sentence or sentences are you trying to say?
– user5172
Mar 18, 2014 at 14:04
• If you have identity available to you, you could define a predicate P to be true of an object x just in case Not-(x=x). This will be a (purely logical) predicate that holds of nothing and so you could identify it with the property of "nothingness" if you chose to. Without knowing your motivations, though, it's hard to see what you're after. Mar 18, 2014 at 16:29
• @dennis: I'd unreflectively considered that x=x, is a tautology that must hold always; but of course it doesn't hold for nothingness, as it can't be compared to anything, including itself - if I understand you correctly. Mar 19, 2014 at 6:51
• @allegranza: I think that works, ie noth(x), if you identify nothingness with the empty set; but must one identify nothingness with the empty set, or are there any other options? Mar 19, 2014 at 6:53

## 3 Answers

With only first-order quantifiers, you can express things like:

(1)   There is nothing in the fridge.         ¬∃x( In(x, f) )

If you have higher-order quantifiers, you can also express things like:

(2) I like nothing about Hegel.         ¬∃P( P(h) ∧ ¬LikesAbout(i, P, h) )

But there are situations where you want to explain 'nothing' or 'nothingness' away, e.g.:

(3) Ruby is nothing special.

(4) Nothingness is characteristic of the contents of my bank account.

In (3) you can say that Ruby is not special, and in (4), that your bank account is empty.

The general concept of 'nothingness' is unlike concepts like whiteness and friendship, which following Frege we can call first-order concepts, that is, functions from individual things to truth-values. The concept of 'nothingness' seems to me to hold of things that contain other things: collections, classes, sets, and so on. Nothingness holds of such set-like objects just in case their contents are empty.

e.g. (1) really means: the fridge as a set of products is empty

e.g. (2) really means: the set of properties of Hegel that I like about him is empty

In other words: nothingness-in-general is a higher-order concept, and it holds of objects that can have members but for one reason or another do not.

• Can you say "nothing is an entity that doesn't need a predicate"? Mar 18, 2014 at 22:56
• I'm not sure I understand what that means. In particular: what do you mean by 'x doesn't need a predicate'? If that's defined, we can try to paraphrase your sentence in a way that no reference is made to 'nothing' as a something, as an object. Mar 18, 2014 at 23:11
• It's really in relation to medieval mysticism: in a simple tone, if there is the idea of something, normally it needs production to exist; except nothing which does not need production to exist. It all depends on the definition, so it would be interesting to see how predicate logic handles it. Mar 18, 2014 at 23:18
• I see. If you want to take nothing a little more seriously than I here have, I think @Dennis 's suggestion in the question comments might be worth serious consideration. Mar 18, 2014 at 23:30
• @Degnan: Or, perhaps, for which all predicates are false? Mar 19, 2014 at 6:47

If your 'nothingness' is akin (but not identical) to the Parmenidian Void then no predicate can apply;(for what is not, is not); so using second order logic, in which one can quantify over predicates, one can say x is nothingness, if for all predicates p, it never holds that p(x) is true.

But what happens when *p(x) is the predicate: x is nothingness*? It doesn't apply - you can't compare nothing to anything, and that ought to include to itself.

• One might object that the following should be trivially true: "nothingness is nothingness". An obvious way to counter this (and in general, other self-predications) is to appeal to the fact that expressions of form x(x) are not well-typed. (This might not be a fact, but it can surely be added as a restriction to the underlying system). Mar 19, 2014 at 7:08
• @Rostomyan: I'm not sure I follow you, isn't 2=2 a self-predication, which according to you prescription would be banned? But Isuppose 2 isn't a type, whereas nothingness is a type, and can also be a subject of a predicate - is that right? Is banning self-predications what is meant by impredicative mathematics? Mar 19, 2014 at 7:21
• I dragged Parmenides in, because for him, the void isn't conceptualisable, so we can't say anything of it, including that it is identical to itself. Strangely enough, this sounds awfully similar to Kants noumena, of which one cannot say anything at all either, except that it exists - but is existance a predicate? Mar 19, 2014 at 7:24
• 2 = 2 isn't self-predication, 2(2) is. I meant the 'is' of predication/membership rather than that of identity. If =2 is the predicate, then =2(2) is perfectly fine; it is, in fact, the curried form of the usual 2 = 2 (= 2 2 in polish notation). And yes, putting such a type-restriction will result in a system without fixed-point ("Y") combinators (because these rely on terms of form xx), which are generally used to represent recursive functions. I believe there is another way of representing them, but I'm not too sure. So the cost might be too high, you're right. Mar 19, 2014 at 8:02
• Ok, I follow what you mean by =2(2), thanks; presumably, then x(x) is short for (=x)x? Mar 19, 2014 at 8:44

Take a look at the way Spencer-Brown applies his Laws of Form to logic. If you equate 'nothing' with an unmarked state, a mark cancelling another out = the unmarked state. If you equate it with an act of distinction per se, then it becomes a mark. See Appx 2 p 90 of the 2010 edition available here.

A good layman's guide which links to more advanced work can be found here.