quantifiers

Quantifiers

FOL provides two quantifiers, $\forall$ and $\exists$ which stands for $forall$ and $exists$ respectively.

$\forall$

$\forall x(\alpha )$

This implies that for all $x$, $\alpha$ formula holds ($true$ or $false$).

In lisp, it can be represented as, $(forall(x)(\alpha ))$

$\exists$

$\exists x(\alpha )$

This implies that there exists a $x$ for which $\alpha$ formula holds ($true$ or $false$).

In lisp, it can be represented as, $(exists(x)(\alpha ))$

Examples

All Men are Mortal.

$\forall x(Man(x)\to Mortal(x))$

In lisp,

(forall (x) (if (man x) (mortal x)))

Some Men are tall.

$\exists x(Man(x)\wedge Tall(x))$

(exists (x) (and (man x) (tall x)))

Info

Notice the connective with $\forall$ and $\exists$.

Todo

• Find why different connective for universal and existential quantifier (@2024-06-20)

Why different connectives for universal and existential quantifier?

We use $\to$ with $\forall$ and $\wedge$ with $\exists$. Why?

Not enough clear about why? As of now just with the thumb rule. I need to figure out why or I’m not satisfied or didn’t understand it.

See these conversations

1. Why can’t we use implication for the existential quantifier?
2. “Why can’t we use implication for the existential quantifier?” (From MSE)