If P is a property, then is (not P) a property?

For a proposition, such as:

P: Socrates is a man

Then not P is also a proposition:

Not P: Socrates is not a man

But do the same goes for properties? One can argue in the following way:

red is a property of red flags, or of red apples; but not red is not a property; for no thing instantiates not redness; that is not redness does not cohere in some substrate - here a flag, or apple; even though whiteness, which is an example of not redness, coheres in a white wall.

Q. Is this right?

All this, I think, takes it for granted that properties cohere in some substrate or substance - this is a commitment to a specific ontology; we may choose something else; the obvious choice being a world where all that there is are bundles of properties, and no substrate or substance to which to hang them on; there, existence is also a property - a property which all existing entities have.

Q. Given this ontological commitment; then is not P also a property?

• A black object is not red. Depending on if you are using additive or subtractive color theory. (Light or pigment) Sep 5, 2015 at 7:37
• You might also be able to say that you have the property of not having the property, for any arbitrary property p which you do not have. Sep 5, 2015 at 7:38
• You can hardly make sense of such concept without formal definitions about what is a property, what is a proposition, and what it is not. Otherwise, suppose that "to be a property" is a property. Then, if the negation of a property is a property, "not to be a property" is a property, but its not a property, so there's a contradiction. This make you want to say that the negation of a property is not a property, but the negation of a non property can then become a property, which could again be sensitive to diagonal argument if you think a bit.
– sure
Sep 5, 2015 at 10:20
• @sure: there are ongoing attempts at formalisations; but I'd argue that one needs to understand also the concepts at a naive level too; consider for example the relationship between naive set theory (which I'd argue that most mathematicians are familiar with, and if they think of set theory - actually think of this, as opposed to fully formal ZFC); interestingly the SEP gives Sep 5, 2015 at 10:39
• Formalisations that drop LEM, in favour of a three-valued logic, or drop Contradiction ie are paraconsistent, or drop the transitivity of consequence. Sep 5, 2015 at 10:41

It is a great question, and the subject of current debate among metaphysicians.

Proponents of sparse properties deny that there is a property (i.e. a real metaphysical constituent of an object) named by every predicate of language. So sparse property theorists are going to say that things like "negative properties" (e.g. "being non-Socrates") or "disjunctive properties" (e.g. "being a bat or not") are not genuine properties at all.

Proponents of abundant properties, on the other hand, would deny this and claim that there must be some feature of, e.g. the cup on my desk, that makes the sentence "The cup is either a bat or not" true, and that thing is the property of being either a bat or not.

The debate is, in some sense semantic. It is about what it features in reality are required to make the sentence "the cup is either a bat or not" true.

It depends on what objects you want to apply the property to. A property is formally defined as just a set, and so you'd think you can just take the compliment of the set defining P, and have the set defining "not-P". This works, but only if the resultant object is actually a set. Set compliments don't exist in a vacuum, they exist relative to the universe we are working in. "the numbers that aren't 1" means something very different in the Natural Numbers than in the Real Numbers. It turns out that it is necessary and sufficient for the universe of objects that you are working with to be a set.

When the universe is not a set, it's a proper class (sets are classes, proper classes are classes that are not sets), which you can read about here: https://en.wikipedia.org/wiki/Class_(set_theory)

• A property as a relation may be defined (or modelled) as a set, however from a philosophical point of view this is too narrow a definition. For example, a real-world property may change over time. What is a red apple today may be a grey, mouldy mess tomorrow. Other properties vary continuously over time, etc..
– nwr
Sep 7, 2015 at 16:57
• I'm not sure that this is the best way to look at this; since using set theory in the way you suggest is already committing oneself to a very specific ontology; according to the SEP, when properties are instantiated by a predicate or a proposition then one might decide to go down the set theory route; having said that this is one way I was thinking about it. Sep 7, 2015 at 21:55
• Another way, which is a mathematical strategy is simply to use restriction for example: I will consider only those properties whose negation are properties; or I consider them as an important kind of properties - so I will distinguish them and name them. Sep 7, 2015 at 21:58