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Limitations of First-order Logic

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Natural languages

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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

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  1. ^ Gamut, L.T.F. (1991), Logic, Language, and Meaning, Volume 2: Introduction to Logic, University Of Chicago Press, ISBN 0-226-28088-8page 75