Herbrand’s Theorem for a Modal Logic
نویسنده
چکیده
ion X → X ′ 〈λx.X〉(t)→ 〈λx.X ′〉(t) +Lambda ¬X → ¬X ′ ¬〈λx.X〉(t)→ ¬〈λx.X ′〉(t) −Lambda Quantification For new variables x1, . . . , xn, ¬φ(x)→ ¬φ1(x) . . . ¬φ(x)→ ¬φn(x) ¬(∀x)φ(x)→ ¬[φ1(x1) ∧ . . . ∧ φn(xn)] −Quant Binding For x not free in X, X → X ′ X → 〈λx.X ′〉(t) +Bind ¬X → ¬X ′ ¬X → ¬〈λx.X ′〉(t) −Bind Definition 5.2 We say Y is a modal Herbrand expansion of X provided there is a formula X∗ that is a validity functional form of X, Y is a modal Herbrand transform of X∗, and Y is closed. Example 5.3 Consider modal formula 2(∀x)¬(∀y)Rxy ⊃ (∀x)2¬(∀y)Rxy. For it, 2(∀x)¬〈λy.Rxy〉(fx) ⊃ 〈λx.2¬(∀y)Rxy〉(c) is a validity functional form, and a closed modal Herbrand transform of it is: 〈λz.2〈λx.¬〈λy.Rxy〉(fx)〉(z)〉(c) ⊃ 〈λx.2¬〈λy.Rxy〉(fx)〉(c). Consequently, this is a modal Herbrand expansion of 2(∀x)¬(∀y)Rxy ⊃ (∀x)2¬(∀y)Rxy.
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