Vacuous Truths

Something that has been bothering me greatly about Zeroth Order Logic was that the truth table of the material implication results in true regardless of the consequent whenever the antecedent is false. That is, p => q is always true when p is false, regardless of the value of ** q**:

p q p => q

This is bothersome because it means that you can construct implications that are true whenever you know something is absolutely false. For example:

  • If the earth is flat then dinosaurs are alive
  • If California is in Brazil then I'm the smartest person in the world
  • If red is blue then mathematics is wrong

These are all true statements, because the antecedents are known to be false making the entire implication true.

It also bothered me that if you know something p to be false and you asked whether the implication p => q was true, you'd always get true regardless of the q you used.

So, I was really happy when a friend taught me the concept of vacuous truths because it describes exactly what I was struggling with, and more.

Specifically, vacuous truths also appear in universal quantifications of empty sets.

For example: all my grandkids are tall is true if I don't have any grandkids, as would the otherwise apparently conflicting conjunction all my grankids are tall and short.

Now that I know there is a name to what I was feeling, I can dig into the ways people have addressed this! What a relief!

More to follow!