# How might I notate that an entity either cannot have a quality or has all of it?

I have two friends, Charles and Louis.

I want to state that given one of the following conditions then Charles and Louis are identical:

• Charles does nothing.
• Louis does nothing.
• Charles does everything.
• Louis does everything.
• Charles knows nothing.
• Louis knows nothing.
• Charles knows everything.
• Louis knows nothing.
• How is your proposed edit related to the question as currently phrased? – Canyon Aug 23 '18 at 1:06

One way to formalize it would be

``````(forall x)(Ax & Dca -> (forall y)(Ay -> Dcy))
``````

Which means

For any x, if x is an action and Charles does x, then for any y, if y is an action then Charles does y.

So as long as there is one action that Charles does, Charles will do any action whatsoever.

Since this question is asking whether Charles and Louis are identical given a list of properties, it may be related the identity of indiscernibles.

Wikipedia describes it as

The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa; to suppose two things indiscernible is to suppose the same thing under two names.

Wikipedia provides two ways of notating this idea depending on what one is trying to say:

There are two principles here that must be distinguished (equivalent versions of each are given in the language of the predicate calculus). Note that these are all second-order expressions. Neither of these principles can be expressed in first-order logic (are nonfirstorderizable).

1. The indiscernibility of identicals
• For any x and y, if x is identical to y, then x and y have all the same properties.

∀x∀y[x = y → ∀P(Px ↔ Py)]

1. The identity of indiscernibles
• For any x and y, if x and y have all the same properties, then x is identical to y.

∀x∀y[∀P(Px ↔ Py) → x = y]

Principle 1 doesn't entail reflexivity of = (or any other relation R substituted for it), but both properties together entail symmetry and transitivity (see proof box). Therefore, Principle 1 and reflexivity is sometimes used as a (second-order) axiomatization for the equality relation.

Principle 1 is taken to be a logical truth and (for the most part) uncontroversial. Principle 2, on the other hand, is controversial; Max Black famously argued against it.

In the above two notations P ranges over the list of properties in the OP's question and any others that might apply. The variables x and y would be Charles and Louis.

If Charles is identical to Louis then Charles and Louis have the same properties. That is the "indiscernibility of identicals" in the first principle. The second principle, the "identity of indisceribles", claims that if Charles and Louis have the same properties then they are identical.

As Wikipedia notes the second principle is controversial.

Wikipedia contributors. (2019, May 14). Identity of indiscernibles. In Wikipedia, The Free Encyclopedia. Retrieved 19:44, May 16, 2019, from https://en.wikipedia.org/w/index.php?title=Identity_of_indiscernibles&oldid=896988126