Limitations of First-order Logic edit
Natural languages edit
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[1] 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' |
references edit
- ^ Gamut, L.T.F. (1991), Logic, Language, and Meaning, Volume 2: Introduction to Logic, University Of Chicago Press, ISBN 0-226-28088-8page 75