Logical syntax of vacuity

Question

I was taught that vacuously true statements are of the form $P \implies Q$ where $P$ is a false statement. This property is routinely used to make claims about the empty-set.

Consider the function $f: \emptyset \to A$ where $A$ is a non-empty set. We have that $f$ is a function. Consequently, it satisfies the condition that $(\forall x \in \emptyset)(\exists y \in A) (x,y) \in f$. It is contended that this condition is satisfied vacuously.

How would I parse the above condition in an if-then statement so as to be able to apply the properties of material implication and conclude the vacuousness of the statement?

Answer 1

Let $A,B,f$ be sets.

Then $f$ is a function $A\to B$ if and only $f\subseteq A\times B$ and $\forall x\left[x\in A\to\exists!y\left[\left(x,y\right)\in f\right]\right]$

If $A=\varnothing$ then $f=\varnothing$ acccording to the first condition.

Further we observe that: $$x\in\varnothing\to\exists!y\left[\left(x,y\right)\in f\right]$$ is vacuously true, and this for every $x$.

Answer 2

Theorem: $\forall A: \exists f: \forall a: [a\in \emptyset \implies f(a)\in A]$

Hint: Use $f=\emptyset \times A$. Prove (vacuously):

1. $\forall a, b: [(a,b)\in f \implies a\in \emptyset \land b\in A]$
2. $\forall a: [a\in \emptyset \implies \exists b: [b\in A \land(a,b)\in f]]$
3. $\forall a, b, c: [a\in \emptyset \land b\in A \land c \in A \implies [(a,b)\in f \land (a,c)\in f \implies b=c]]$

Answer 3

• Your question comes from the fact that your proposition is stated in abbreviated form " for all x belonging to the empty set" ( in the frst parenthesis). This " present participle" ( namely: " belonging") hides an " if...then ".

• Example : "All man having a son or a daughter is a father" has to be translated, in logical language ( with the set of male men as universal set) as :

         for all x [( IF x has a son or a daughter) THEN (x is a father)]
  

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The statement defining a relation R as function from A to B is :

for all x, [ ( x belongs to A) --> ( there is some unique y in B such that xRy) ].

Substitute the empty set for A.

In that case, the antecedent of the conditional, namely, " x belongs to A" , is necessarily false.

So, whatever the truth value of the consequent might be, the whole conditional is true.