Vacuous truth
Vacuous truth
In logic, a vacuous truth is a conditional statement that is true because the condition is false or logically unrelated to the assertion.
For example, the statement “all cats in the room are black” will be true when no cats are in the room. At the same time, the statement “all cats in the room are white” would also be vacuously true, because it does not really say anything.
Alice: Which way should I go?
Cat: That depends on where you are going.
Alice: I don’t know.
Cat: Then it doesn’t matter which way you go.
― Lewis Carroll, Alice in Wonderland
It’s all good until your condition says otherwise.
This logic described as the material condition (also known as material implication) P → Q, which is true unless P is true and Q is false:
| P | Q | P → Q |
|---|---|---|
| ✅ | ✅ | ✅ |
| ✅ | ❌ | ❌ |
| ❌ | ✅ | ✅ |
| ❌ | ❌ | ✅ |
For example:
| P | Q | P → Q |
|---|---|---|
| If it’s a duck | it quacks | ✅ |
| If it’s a duck | it doesn’t quack | ❌ |
| If it’s not a duck | it quacks | 🤔 |
| If it’s not a duck | it doesn’t quack | 🤯 |
If the premise is false or unrelated (3rd and 4th cases), then any further (even absurd) conclusions seem valid.
In Ruby Enumeration#all? of empty collection returns true:
[].all? # => true
You can pass anything you want in the block, and it will always return true and the block will not be executed:
[].all?(&:absurd?) # => true
On the other hand, #any? of empty collection returns false, however the passed block will not be executed either:
[].any? # => false
[].any?(&:absurd?) # => false