Relation of the empty set to vacuous truth?

[Mr Davis 97]

I need a clear-cut explanation of vacuous truth, as I can't seem to wrap my head around it. I guess this more precisely comes down to why we organize the truth table for the conditional statement the way that we do. Also, in connection to this, I'm wondering why the empty set is a subset of all sets, even though it has no elements. Also, why is the empty set even allowed to be a set, when informally a set is defined as a collection of objects?

 

[fresh_42]

I need a clear-cut explanation of vacuous truth, as I can't seem to wrap my head around it. I guess this more precisely comes down to why we organize the truth table for the conditional statement the way that we do. Also, in connection to this, I'm wondering why the empty set is a subset of all sets, even though it has no elements. Also, why is the empty set even allowed to be a set, when informally a set is defined as a collection of objects?

This is a matter of consistency for the extreme ends (and convenience). Why is the entire set a subset? Why is zero a number? It doesn't count anything. In the end, it turned out, that zero is incredibly useful, although it numerates the nothing. One vacuous truth is one of my favorite phrases: All elements of the empty set have purple eyes. This is a true statement, as there is nothing to prove and a counterexample cannot be given. It is simply easier to say the number of elements of the power set of a set $S$ with $|S|=n$ is $2^n$ instead of $2^n-1$ or $2^n-2$.

All elements of the empty set are also elements of any other set (vacuous truth), so the empty set $\{\}$ is a subset of any set. Same as $0$ is a number, $\sum_{n \in\{\}}a_n=0$ and $S \subseteq S$. A group $G$ is simple, if $\{e\}$ and $G$ are the only normal subgroups, a prime integer can only be divided by $\pm 1$ and itself, and so on. To include the extreme points of possibilities simply doesn't create (artificial) exceptions at the boundaries. The example with the sum becomes more obvious in special cases. E.g. for a prime number $p$ we have
$$
\sum_{\stackrel{1<k<p}{k\mid p}}k = 0
$$
This convention is useful when we deal with sums, as sometimes the index set over which is summed is simply empty.

 

[Mr Davis 97]

This is a matter of consistency for the extreme ends (and convenience). Why is the entire set a subset? Why is zero a number? It doesn't count anything. In the end, it turned out, that zero is incredibly useful, although it numerates the nothing. One vacuous truth is one of my favorite phrases: All elements of the empty set have purple eyes. This is a true statement, as there is nothing to prove and a counterexample cannot be given. It is simply easier to say the number of elements of the power set of a set $S$ with $|S|=n$ is $2^n$ instead of $2^n-1$ or $2^n-2$.

All elements of the empty set are also elements of any other set (vacuous truth), so the empty set $\{\}$ is a subset of any set. Same as $0$ is a number, $\sum_{n \in\{\}}a_n=0$ and $S \subseteq S$. A group $G$ is simple, if $\{e\}$ and $G$ are the only normal subgroups, a prime integer can only be divided by $\pm 1$ and itself, and so on. To include the extreme points of possibilities simply doesn't create (artificial) exceptions at the boundaries. The example with the sum becomes more obvious in special cases. E.g. for a prime number $p$ we have
$$
\sum_{\stackrel{1<k<p}{k\mid p}}k = 0
$$
This convention is useful when we deal with sums, as sometimes the index set over which is summed is simply empty.

But it doesn't seem to be just a matter of convenience if logic dictates the result. It seems that the empty set being a subset of every other subset is somehow important a priori, because of the rules of logic and the truth table for conditionals, as opposed to, for example, defining $0! = 1$ just out of convenience.

I guess then my main question hinges on why is the truth table for conditionals structured the way that it is? And more particularly, why does $F \rightarrow T = T$?

 

[fresh_42]

This is part of the logic system we use: first order predicate logic, and especially how we treat the empty set. A short discussion is here. I'm not sure how other logic systems handle the case. It's a fundamental principle we use: truth can only lead to truth whereas false can lead to everything. It fits well to our intuition, I think. I guess we need a list of axioms to see where it comes from.

 

[Stephen Tashi]

I guess then my main question hinges on why is the truth table for conditionals structured the way that it is? And more particularly, why does $F \rightarrow T = T$?

What's the alternative? Suppose we say $F \rightarrow T$ is $F$. Consider the theorem: For each number $X$, if $X > 2$ then $X^2 > 4$. We don't want to let the case $X = 1$ be a counterexample to that theorem.

Your question is discussed at length in the recent thread: https://www.physicsforums.com/threads/the-truth-value-of-p-x-q-x.924051/#post-5831965