Obsidian/Zettelkasten/Permanent Notes/20250818132018-predicate-logic.md

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

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)))$