Vacuous truth and empty set properties I have a flaw in my reasoning, could you help me figure out what I am doing wrong?
To proof that $\emptyset$ is closed under addition we need to evaluate the following logical statement:
$$
\forall x,y \in \emptyset: Q(x,y)
$$
where $Q(x,y)$ is "$x+y \in \emptyset $"
We can transform this into a conditional statement like so:
$$
\forall x,y \in A:P(x,y)\implies Q(x,y)
$$
where $A$ is a non-empty set with at least two elements and $P(x,y)$ is "$x,y \in \emptyset$".
Here we can see that the first statement will always be false since $\emptyset$ has no elements and then by conditional logic this whole statement evaluates to true and so we conclude that $\emptyset$ is closed under addition. This is a vacuous truth.
But can we not use the same conditional logic to proof that $\emptyset$ is not closed under addition by using $R(x,y)$ to be "$x+y \notin \emptyset$"?
$$
\forall x,y \in A:P(x,y)\implies R(x,y)
$$
Since the antecedent $P(x,y)$ will always be false the whole statement will always evaluate to true and then we can conclude that $\emptyset$ is not closed under addition.
 A: For addition not to be closed on the empty set, you would need to show $\exists \,x,y \in \emptyset: x+y \notin \emptyset$.
This is not a consequence of the vacuous truth $\forall x,y \in \emptyset: x+y \notin \emptyset$, so there is no contradiction with the other vacuous truth $\forall x,y \in \emptyset: x+y \in \emptyset$. Assuming you have a suitable definition of addition, it remains the case that addition is closed on the empty set.
A: You have almost "reinvented" Proof By Contradiction !
We want to know whether $P$ is true or not.
You have :
$ P \rightarrow Q $
$ P \rightarrow R $
You know that :
$Q = \lnot R$ [ Equivalently $R = \lnot Q$ ]
Putting these together , we have a Contradiction.
Hence , we know that $P$ can not be true !
Example :
$M$ is the Empty Set of Martians.
$P$ : "Martians Exist" : $\exists m \in M$
$Q$ : "All Martians are born on Mars" : $\forall m \in M : BornOnMars(m)$
$R$ : "All Martians are not born on Mars" : $\forall m \in M : \lnot BornOnMars(m)$
$Q = \lnot R$ [ Equivalently $R = \lnot Q$ ]
$P \rightarrow Q$
$P \rightarrow R$
We can conclude ( Proof By Contradiction ) that "Martians Exist" is not true : $P$ is not true.
