Editing First-order logic (section)

===Formalizing natural languages===
{{Main|Logic translation#Natural language formalization}}
First-order logic is able to formalize many simple quantifier constructions in natural language, such as "every person who lives in Perth lives in Australia". Hence, first-order logic is used as a basis for [[knowledge representation language]]s, such as [[FO(.)]].

Still, there are complicated features of natural language that cannot be expressed in first-order logic. "Any logical system which is appropriate as an instrument for the analysis of natural language needs a much richer structure than first-order predicate logic".{{sfn|Gamut |1991 |p=75}}

{| class="wikitable"
|-
! Type !! Example !! Comment
|-
! {{rh}} | {{nowrap|Quantification over properties}} 
|If John is self-satisfied, then there is at least one thing he has in common with Peter.  || Example requires a quantifier over predicates, which cannot be implemented in single-sorted first-order logic: {{nowrap|{{mono|Zj → ∃X(Xj∧Xp)}}}}.
|-
! {{rh}} | Quantification over properties 
| Santa Claus has all the attributes of a sadist. || Example requires quantifiers over predicates, which cannot be implemented in single-sorted first-order logic: {{nowrap|{{mono|∀X(∀x(Sx → Xx) → Xs)}}}}.
|-
! {{rh}} | Predicate adverbial 
| John is walking quickly. || Example cannot be analysed as {{nowrap|{{mono|Wj ∧ Qj}}}};<br/>predicate adverbials are not the same kind of thing as second-order predicates such as colour.
|-
! {{rh}} | Relative adjective
| Jumbo is a small elephant. || Example cannot be analysed as {{nowrap|{{mono|Sj ∧ Ej}}}};<br/>predicate adjectives are not the same kind of thing as second-order predicates such as colour.
|-
! {{rh}} | Predicate adverbial modifier 
| John is walking very quickly. || {{sdash}}
|-
! {{rh}} | Relative adjective modifier 
| Jumbo is terribly small. || An expression such as "terribly", when applied to a relative adjective such as "small", results in a new composite relative adjective "terribly small".
|-
! {{rh}} | Prepositions 
| Mary is sitting next to John. || The preposition "next to" when applied to "John" results in the predicate adverbial "next to John".
|}