The Limitation of Propositions
Remember what a proposition is?
A statement that is either true or false.
But what about this:
“x is greater than 5”
Is this true or false? It depends on x.
- If x = 7, it’s true
- If x = 3, it’s false
This isn’t a proposition yet. It has a hole in it.
What is a Predicate?
A predicate is a statement with a hole. Fill the hole, get a proposition.
Example:
- P(x) = “x is greater than 5”
| Fill with… | You get… | True? |
|---|---|---|
| P(7) | “7 is greater than 5” | true |
| P(3) | “3 is greater than 5” | false |
Predicate = proposition with a blank. Fill the blank, get true or false.
Predicates with Multiple Blanks
Predicates can have more than one hole.
Example:
- L(x, y) = “x likes y”
| Fill with… | You get… |
|---|---|
| L(Alice, coffee) | “Alice likes coffee” |
| L(Bob, tea) | “Bob likes tea” |
| L(Carol, Mondays) | “Carol likes Mondays” |
More blanks = more things to specify.
The Problem
What if you want to talk about everyone or someone without naming them?
- ”Everyone likes water”
- ”Someone likes coffee”
You can’t fill in every name one by one. We need a shortcut.