'Vacuous truth' step by step

We accept these four rules of standard mathematical logic:

1. If P is true and Q is true, then 'P implies Q' is true (takes a binary value denoted T).

2. If P is true and Q is false, then 'P implies Q' is false (takes the value F).

3. If P is false and Q is true, then 'P implies Q' is true.

4. If P is false and Q is false, then 'P implies Q' is true.

By setting up truth tables for these rules, one can see their reasonableness. For example, if we re-express P --> Q as ~P v Q, we see that the truth table is

    ~P      v        Q
 
1.  F                  T         T
2.  F                  F         F
3.  T                  T         T
4.  T                  F         T

Yet, rule 3 can strike people as odd. The statement may be called "true," but vacuously so.

So x e ∅ --> x e S always holds vacuously (and so is a tautology).

That is, if x e S, then rule 3 applies. But if ~(x e S), then rule 4 applies.

Note that this expression justifies the accepted concept that the null set is a subset of every set.

Observe that x e S --> x e ∅ does not necessarily hold. If x e S is true, then the asserted implication is false, by rule 2.

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A curious result

In modern set theories, x e x is ruled out in order to avoid Russell's contradiction, which proves naive set theory to be inconsistent (a false theory).

We can use the fact that in modern theory, x e x is false to prove that x = x.

First we note that in a goodly number of modern logic systems, 'P --> P' is either an axiom, or immediately derivable from axioms. So we accept 'P --> P' and let P be the assertion 'x is an element of x' , or 'x e x'.

Thence, x e x --> x e x.

Proof:

-->

1a. x e x --> x e x

(1a) is true by rule 3, such that a falsehood implies a falsehood.

2a. so x ⊆ x

<--

3a. x e x --> x e x

4a. x ⊆ x

In other words, the implication arrow works in both directions. Or

5a. x e x <--> x e x

Well, here x is defined vacuously by its purported elements and so

6a. x = x

Note that x e x --> ~(x e x) is also true since, by rule 4, a falsehood also implies a truth. Still, the proof above holds, by rule 3.

Well, yes, but there is no set x such that x e x.

So are we allowed to say this non-set is defined by its purported elements?

Maybe we can get away with saying that a set that doesn't exist in some system even so exists as a non-entity in a larger sense.

Nonexistence has long been a philosophical puzzler, as one can see from Russell's 'A - B = 0 ' problem in his Theory of Descriptions. For more on the nonexistence issue, please see

Philosophy Professor Thomas C. Ryckman's excellent page
https://www2.lawrence.edu/fast/ryckmant/Russell's%20Theory%20of%20Descriptions.htm

We caution that this proof of 'x = x' would not be accepted in some logic texts before quite a bit of groundwork had been done.