---
id: 20250818132018
title: Predicate Logic
type: permanent
created: 2025-08-18T17:20:18Z
modified: 2025-08-18T20:20:03Z
tags: []
---

# Predicate Logic

Predicate logic extends propositional logic by using
predicates and quantifiers. 

- **Predicates** express properties or relations (e.g., 
$Pump(x)$ or $Connected(v,c)$).

- **Quantifiers** let us say something about *all* or *some*
in the domain.
    - $\forall$ = for all 
    - $\exists$ = exists, at least one
    - $\exists!$ = there exists exactly one

Predicates kind of look like functions: $Pump(x)$ means $x$ is
a pump.

**Relational Predicates** have two inputs:
$Connected(x,y)$ means pump $x$ is connected to valve $y$

## Examples

Predicate logic requires you to *introduce the
variable first*. This feels weird at first, but is
mathematically explicit. The predicate does not define the
input, it only defines the property! 

Here's an example:

$\forall x (Pump(x) \rightarrow Working(x))$

Means that 'every pump works'. But, we define first a set x,
then specify it is a pump for all x, and that they all of x,
which are all pumps, work.

What if we have one pump that doesn't work?

$\exists x(Pump(x) \wedge \neg Working(x))$

## Nested Qualifiers

We can stack qualifiers and do some nifty things. 
How do we say: There exists one controller that stabilizes
every plant?

$ \exists k \forall G (Plant(G) \rightarrow Stabilizes(k,G))$

This is obviously not possible. So how do we say the
opposite, that for any plant there exists at least one
stabilizing controller:

$ \forall G \exists k (Plant(G) \rightarrow (Controller(k)
\wedge Stabilizes(k,G)))$