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.
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.
