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# There are many extensions to rst order logic higher

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Unformatted text preview: rybody loves somebody ∀x∃yLoves( x, y ) •  Everybody loves everybody € ∀x∀yLoves( x, y ) •  Everybody loves Raymond € ∀xLoves( x, Raymond ) •  Raymond loves everybody ∀xLoves(Raymond, x ) € € 4 Equality •  Equality states that two objects are the same –  Son_of(Barbara) = Ron •  Equality is a special rela<on that holds whenever two objects are the same •  We can imagine that every interpreta<on comes with its own iden<ty rela<on –  Iden<cal(object27, object58) What’s Missing? •  There are many extensions to ﬁrst order logic •  Higher order logics permit quan<ﬁca<on over predicates: ∀x, y ( x = y ) ⇔ (∀p( p( x ) ⇔ p( y ))) € •  Uniqueness •  Extensions typically replace a poten<ally long series of conjuncts with a single expression 5 Inference •  All rules of inference for proposi<onal logic apply to ﬁrst order logic •  We need extra rules to handle subs<tu<on for quan<ﬁed variables SUBST ({x / Harry, y / Sally}, Loves( x, y )) = Loves(Harry, Sally ) € Inference Rules •  Universal Elimina<on ∀v : α (v) SUBST ({v / g}, α (v)) •  How to read this: –  We have a universally quan<ﬁed variable v in α –€ subs<tute any g for v and α will s<ll be true   Can 6 Inference Rules •  Existen<al Elimina<on ∃v : α (v) SUBST ({v / k}, α (v)) •  How to read this: –  We have a universally quan<ﬁed variable v in a –  Can subs<tute any k for v and α will s<ll be true € –  IMPORTANT: k must be a previously unused constant (skolem constant). Why is this OK? Skolemiza<on within Quan<ﬁers •  Skolemizing w/in universal quan<ﬁer is tricky •  Everybody loves somebody ∀x∃y : loves( x, y ) •  With Skolem constants, becomes: ∀x : loves( x, object34752) € •  Why is this wrong? •  Need to use skolem func<ons: € ∀x : loves( x, p...
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