User:Philogo/BeyondFOLSandbox

Limitations of First-order Logic

Natural languages

Any logical systmen which is appropriate as an instrument for the analysis of natual language needs a much richer structure than first-order predicate logic The following are examples of various kinds of sentences lacking direct translation into first-order predicate logic.

Type	Example	Comment
Quantification over properties	If John is self-satisfied , then there is at least one thing he has in common with Peter	Requires second order logic: Zj→ ∃X(Xj∧Xp)
Quantification over properties	Santa Claus has all the attributes of a sadist	Requires second order logic: ∀X(∀x(Sx → Xx)→Xs)
Predicate adverbial	John is walking quickly	Cannot be analysed as Wj ∧ Qj; predicate adverbials are not the same kind of thing as second-order predicates like colour
Relative adjective	Jumbo is a small elephant	Cannot be analysed as Sj ∧ Ej; predicate adjectives are not the same kind of thing as second-order predicates like colour
Predicate adverbial modifier	John is walking very quickly	-
Relative adjective modifier	Jumbo is terribly small	An expression like 'terribly' when applied to a relative ajective like 'small' results in a new composite relative adjective 'terribly small'
Prepositions	Mary is sitting next to John	The preposition 'next to' when applied to 'John' reults in the predicate adverbial 'next to John'