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Author: Admin | 2025-04-28
Be implemented in single-sorted first-order logic: ∀X(∀x(Sx → Xx) → Xs).Predicate adverbialJohn is walking quickly.Example cannot be analysed as Wj ∧ Qj;predicate adverbials are not the same kind of thing as second-order predicates such as colour.Relative adjectiveJumbo is a small elephant.Example cannot be analysed as Sj ∧ Ej;predicate adjectives are not the same kind of thing as second-order predicates such as colour.Predicate adverbial modifierJohn is walking very quickly.—Relative adjective modifierJumbo 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".PrepositionsMary is sitting next to John.The preposition "next to" when applied to "John" results in the predicate adverbial "next to John".Restrictions, extensions, and variations[edit]There are many variations of first-order logic. Some of these are inessential in the sense that they merely change notation without affecting the semantics. Others change the expressive power more significantly, by extending the semantics through additional quantifiers or other new logical symbols. For example, infinitary logics permit formulas of infinite size, and modal logics add symbols for possibility and necessity.Restricted languages[edit]First-order logic can be studied in languages with fewer logical symbols than were described above:Because can be expressed as , and can be expressed as , either of the two quantifiers and can be dropped.Since can be expressed as and can be expressed as , either or can be dropped. In other words, it is sufficient to have and , or and , as the only logical connectives.Similarly, it is sufficient to have only and as logical connectives, or to have only the Sheffer stroke (NAND) or the Peirce arrow (NOR) operator.It is possible to entirely avoid function symbols and constant symbols, rewriting them via predicate symbols in an appropriate way. For example, instead of using a constant symbol one may use a predicate (interpreted as ) and replace every predicate such as with . A function such as will similarly be replaced by a predicate interpreted as . This change requires adding additional axioms to the theory at hand, so that interpretations of the predicate symbols used have the correct semantics.[32]Restrictions
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