Internalizing vacuous truth

Looking over the notes from last week, I’ve stumbled upon the notion of vacuous truth again. While I had intuitive understanding of it, I felt I was not entirely comfortable with it. It looks like there’s a bit of confusion about it too.

Conventional explanation says that the result of a vacuous statement is related to the way quantified claims are evaluated. That is,

• a universal claim is false if and only if $P(x)$ is true and $Q(x)$ is false;
• while an existential statement is false if $P(x)$ is false for every $x$ in the domain.

This still sounds too convoluted to remember, so I’ve decided on a shorthand to remember it:

• A universal statement is a promise in the space of possibilities.
• An existential statement is an event in the space of events that have occurred.

This seems to work because evaluating a statement is always based on what we know about the world (or a given domain).

To illustrate this, I’ll use the following sentence: “If John answers questions truthfully, he receives a reward.” If John is never asked any questions, the promise remains unbroken, so a universal statement would evaluate to true.¹ However, if we are looking for an instance, an event where John answered a question truthfully and received a reward, we’ll find none, and our existential statement would evaluate to false.

Here are a few links to help a fellow student arrive at their own conclusion:

1. Assumed True until proven False. The Curious Case of the Vacuous Truth — math.stackexchange.com
2. In classical logic, why is $(p⇒q)$ True if both p and q are False? — math.stackexchange.com
3. Blog post: Meaningless Truth — The True Beauty of Math
4. Vacuous truth — Wikipedia

Hope this helps someone!