# What is the purpose of quantification over the empty set?

This question is rather short, but I couldn't find the specific answer I was looking for anywhere.

My understanding so far: (Please correct me if I am wrong on some of this)

• I can assume any statement $$P(x)$$ for all $$x \in \emptyset$$ to be true because the empty set doesn't contain any elements in the first place.
• I can also assume any statement $$Q(x)$$ to be false for all $$x \in \emptyset$$ because of the same reason.
• An example would be $$\forall M \subseteq \emptyset$$ where $$M \neq \emptyset$$, it holds that $$M = \emptyset$$ . This statement has to be true for all subsets $$M$$ because the premise (there are no subsets of $$\emptyset$$ which are not the empty set themselves) is false.

My question: What is this good for? Is this behavior just a trivial property of the empty set? The only usage that comes to my mind is if I want to prove something specific for the empty set itself (for example that $$\emptyset$$ is an ordinal).

• What do you mean with a function to be "true" or "false" ? May 2 at 14:47
• @Peter I assumed the word "function" here should actually be (logical) "predicate". May 2 at 14:52
• I edited the post and changed the word function to statement. I also tried to make clear what I wanted to say in the example. May 2 at 14:53
• No you can not assume any statement to be false for all $Q(x)$. You MUST assume all statements are true for all $x \in \emptyset$. That does mean that both a statement and the negation of a statement are both true for all $x \in \emptyset$ but an negation of a statement being true over an empty does not imply a statement is false over an empty set. May 2 at 16:10
• It happens that quantifying over the empty set can lead to some jokes, e.g. "during all my years as King of England, I was a perfect ruler". May 2 at 16:38

Be careful: $$P(x)$$ is not a statement, unless $$x$$ is a constant. If $$x$$ is a variable, then, $$P(x)$$ is a propositional function and its truth value is, in general, undetermined.

The statement: $$\forall xP(x)$$, is false, if there are values of $$x$$, for which, $$P(x)$$ is false. The statement: "$$\forall x\in\varnothing P(x)$$", is always true, since $$\varnothing$$ has no members. But, this is actually an abbreviation for: $$\forall x\left[x\in\varnothing\rightarrow P(x)\right]$$ This is true, because, if $$A$$ is always false, $$A\rightarrow B$$ is always true. So, when you say:

I can assume any statement $$P(x)$$ for all $$x\in\varnothing$$ to be true because the empty set doesn't contain any elements in the first place.

This is not right. You should not assume $$P(x)$$ to be true. You should only assume "For all $$x\in\varnothing$$, $$P(x)$$" to be true. Similarly, you should not assume $$Q(x)$$ to be false. It has no truth value. Furthermore, if you assume that "For all $$x\in\varnothing$$, $$Q(x)$$" is false, your wrong. This is always a true statement.

Your example, formally expressed, gives the following: $$\forall M\left[M\subseteq\varnothing\wedge M\neq\varnothing\rightarrow M=\varnothing\right]$$

Since "$$M\subseteq\varnothing\wedge M\neq\varnothing$$" is always false (for every value of $$M$$), then "$$M\subseteq\varnothing\wedge M\neq\varnothing\rightarrow M=\varnothing$$" is always true. Thus, the above is a true statement. But from this, you could never conclude: "$$M\neq\varnothing$$ and $$M=\varnothing$$".

• Thank you for your answer. I now have a better understanding of using the statements P(x), Q(x), and so on. I just included the example as an extreme case of using the empty set quantification. May 2 at 15:47