While propositional logic assumes the world contains facts, first-order logic assumes the world contains objects, relations, functions, like natural language.

Syntax

Constants: Poly, 2, refer to objects
Predicates: brother, larger than, refer to relations
functions: sqrt, refer to functional relations
variables: x, y
connectives: ^ $\Rightarrow$
equality: =
quantifiers: $\exists$ , $\forall$

Sentences

atomic sentence = predicate(term1, …, termN) or term1 = term2. where term = function(term1, …, termN) or constant or variable
complex sentences are made from atomic sentences using connectives

Universal quantification

$\forall <variables> <sentence>$ Typically, $\Rightarrow$ is the main connective with $\forall$ , avoid $\wedge$ .

Existential quantification

$\exists <variables> <sentence>$ Typically, $\wedge$ is the main connective with $\forall$ , avoid $\Rightarrow$ .

Properties of quantifiers

$\forall x \forall y$ is same as $\forall y \forall x$
$\exists x \exists y$ is same as $\exists y \exists x$
duality:
$\forall x Relation(object)$ is same as $\neg\forall x \neg Relation(object)$

Universal instantiation

Summary

objects and relations are semantic primitives
increase expressive power

AI: first-order logic