Q&A: What’s the Intuition for these Weird Logical Equivalences?

by Justin Skycak on

Cross-posted from here.


The phrases “if P then Q”, “Q if P”, and “P only if Q” are logically equivalent. I know this can be proved using a truth table, but it still feels counterintuitive and hard to remember. Is there any way to develop more comfort with this concept?

I have the same question abut the phrases “P unless Q” and “P if not Q”.


Regarding the equivalence of “P unless Q” and “P if not Q” – just look at the following sentences, which obviously mean the same thing:

  • Buy a gallon of milk unless we already have one.
  • Buy a gallon of milk if we do not already have one.

The equivalence of “if P then Q”, “Q if P”, and “P only if Q” is a bit tougher to intuit, but here’s what feels intuitive to me:

  • Suppose we're sitting next to each other in class. If you say "hello" then I hear "hello." You say "hello" only if I hear "hello" (because if I didn't hear it then you didn't say it). But other people can also say hello to me so it's not true that I hear "hello" only if you say "hello."

Personally, though, that intuition isn’t what I use in practice. In practice, I just think about this visually with inference arrows:

  • the phrase "if X" tells you the arrow starts at X,
  • but when the word "only" precedes it, that indicates to flip the arrow.

For the phrases you mentioned:

  • "if P then Q" starts at P: $P \rightarrow Q$
  • "P if Q" starts at Q: $P \leftarrow Q$
  • "P only if Q" is like the above but flipped: $P \rightarrow Q$

Conveniently, these inference arrows can be replaced with implications:

  • "if P then Q" means $P \Rightarrow Q$
  • "P if Q" means $P \Leftarrow Q$
  • "P only if Q" means $P \Rightarrow Q$