Can combining elements of a binary property result in the opposite property?

For non-binary properties, like size, we can have states like "large" and "small", and elements with the state "small", like atoms, can sum to an element with the state "large", like an elephant. In this way, the whole is in a state that its constituents are not.

For binary properties, like life, there is no scalar. Something is either 100% alive or 100% dead. No amount of corpses can make up a living being. Intuition gives that the difference lies in the discreteness: for continuous systems whereby things can potentially take on an infinite number of states, it may be possible to combine elements to obtain an emerging property, but for discrete systems like ternary or binary systems, where things can take on a finite number of states, it is impossible for any number of elements of state A to collectively be, or to combine to be, of state B or C.

Is my intuition correct, and if so, is there a formal proof available? If not, how can we formalize this?

A simple attempt:

In a system with two binary qualities $A$ and $B$, these qualities are mutually exclusive, meaning that no element can simultaneously have both qualities. If something has quality $A$ it cannot simultaneously have quality $B$, and vice versa. $(\text{Axiom $1$})$

The binary qualities are exhaustive, meaning that in any binary system, all elements within the defined set $S$ (i.e., the set of all things that can be said to have quality $A$ or $B$) must either have quality $A$ or quality $B$, and there is no "in-between" quality, nor can an element in $S$ have neither quality. $(\text{Axiom $2$})$

1. Suppose you have a set $S$ of elements, each of which has either quality $A$ or $B$, but not both (from Axiom 1).

2. Take any two elements, $x$ and $y$, both of which have quality $A$. By definition, neither $x$ nor $y$ has quality $B$, as they are mutually exclusive.

3. Now consider the combination of $x$ and $y$, denoted ($x + y$). $x+y$ is necessarily an element with property $A$. To show this, let us say quality $A$ is "true," and quality $B$ is "false", and denote "true" as $1$ and "false" as $0$. Then:

   • $x + y = 1 + 1 = 1$
   • So, $x + y$ has property $A$. Let now $a$, $b$ be elements with property $B$, being false.
   • $a + b = 0 + 0 = 0$
   • So, $a + b$ has property $B$.

4. If a combination of elements with quality A have quality B or a combination of elements with quality B have quality A, then mutual exclusivity ($\text{Axiom $1$}$) has been violated.

5. Therefore, combining $A$-quality elements cannot result in something with quality $B$.

The above was very flimsily put together, and it only accounts for discrete systems with two states, and IMO isn't convincing enough, especially when putting it into real world situations like "can enough corpses encompass a living being", which was the original question I was trying to solve. Is there a name for this problem, and has it been solved?