4

Usually, whether a statement $P(x)$ about a free variable $x$ is true or false depends on the value or meaning of $x.$

For example, if $P(x)$ means $x\in \{ 1, -9, 15 \},$ we can neither say that $P(x)$ is true nor that it is false if we don't know what $x$ is.

But can we say that the statement $$x\in\emptyset$$ is false? After all, it does not have any quantifier.

On the other hand, these five examples all have quantifiers (either $\forall x$ or $\exists x$), and

  1. $\forall x(x\in \emptyset \Rightarrow P(x))$ is vacuously true.

  2. $\forall x(x \in U \Rightarrow x \in \emptyset)$ is false.

  3. $\exists x(x \in U \land x\in \emptyset)$ is false.

  4. $\exists x(x \in \emptyset)$ is false.

  5. $\forall x(x \in \emptyset)$ is false.

ryang
  • 38,879
  • 14
  • 81
  • 179
Stats Cruncher
  • 381
  • 2
    I think every time $x\in \emptyset$ has a meaning, it is false. But a logician's answer is needed. – Gribouillis Aug 20 '22 at 10:33
  • 3
    Yes we can. Usually we read formulas with free variables as implicitly universally quantified. – Mauro ALLEGRANZA Aug 20 '22 at 10:36
  • 3
    But in this case, also every assignment of a value to the free variable x will produce a false sentence. – Mauro ALLEGRANZA Aug 20 '22 at 10:37
  • 3
    @MauroALLEGRANZA: I think what Stats Cruncher is asking is whether we can regard $x\in\emptyset$ as a false statement, even when $x$ is not considered to be universally quantified, either implicitly or explicitly. – Joe Aug 20 '22 at 10:40
  • 1
    @Joe - answered in my second comment above. But we may compare it with e.g. $x=0$. – Mauro ALLEGRANZA Aug 20 '22 at 10:53
  • @Joe Indeed, What Joe says is what I intended to ask about or what this question was intended for. Appreciate it, Joe. – Stats Cruncher Aug 20 '22 at 11:07

2 Answers2

5

As usual, we are not admitting an empty domain of discourse. Then

$$x\in \emptyset$$ is an unsatisfiable open formula:

  • open formula

    it is technically not a statement (closed formula), as it has a free variable;

  • unsatisfiable

    it has a definite truth value False regardless of interpretation.

ryang
  • 38,879
  • 14
  • 81
  • 179
0

Is it correct to say that the statement x∈∅ is false?

Yes. We usually define $\emptyset$ as follows: $~~\forall a:a\notin \emptyset$.

Postulating $x\in \emptyset$ leads to the obvious contradiction, so $x\in \emptyset$ must be false.

Dan Christensen
  • 14,529