哥德尔：上帝存在的本体论证明。
英文本可参考 Stanford Encyclopedia of Philosophy: Gödel’s Ontological Argument.
证明涉及模态逻辑, 引入了 ‘□’ (必然)和 ‘◇’ (可能) 两个算子. 有
把证明抄在下面:
公理1: 一个性质是肯定的当且仅当它的否定是否定的.
Axiom 1: If a property is positive, then its negation is not positive.
Pos(ψ)↔ ﹁Pos(﹁ψ)
公理2: 肯定性质蕴涵的性质必肯定.
Axiom 2: Any property entailed by - i.e., strictly implied by - a positive
property is positive.
□∀x{[φ(x)→ψ(x)]∧Pos(φ)}→Pos(ψ)
定理1: 一个肯定性质是逻辑上一致的(可能有某个实例).
Theorem 1: If a property is positive, then it is consistent, i.e., possibly
exemplified.
Pos(φ)→◇∃x φ(x)
定义1: 某物是类上帝的当且仅当它具备所有的肯定性质.
Definition 1: x is God-like iff x has as essential properties those and only
those properties which are positive.
G(x)↔∀φ[Pos(φ)→φ(x)]
公理3: “是类上帝的”是一个肯定性质.
Axiom 3: The property of being God-like is positive.
Pos(G)
推论1: “是类上帝的”是一致的(可能有某个实例, 即上帝可能存在)
Corollary 1: The property of being God-like is consistent.
◇∃x G(x)
公理4: 一个肯定性质是必然肯定的.
Axiom 4: If a property is positive, then it is necessarily positive.
Pos(φ)→□Pos(φ)
定义2: 性质 φ 是 x 的本质, 当且仅当 x 满足 φ 且对 x 的任意性质 ψ , φ 蕴涵 ψ.
Definition 2: φ is an essence of x iff for every property ψ, x has ψ
necessarily iff φ entails ψ.
φ ess x ↔φ(x)∧∀ψ{ψ(x)→□∀x[φ(x)→ψ(x)]}
定理2: 如果 x 是类上帝的, 那么类上帝的是 x 是的本质.
Theorem 2: If something is God-like, then the property of being God-like is an
essence of that thing.
G(x)→G ess x
定义3: x 必然存在, 如果 x 的本质都必然有某个实例.
Definition 3: x necessarily exists iff every essence of x is necessarily
exemplified.
NE(x)↔∀φ[φ ess x → □∃xφ(x)]
公理5: “是必然存在”是肯定的.
Axiom 5: Necessary existence is positive
Pos(NE)
定理3: 必然有某个 x, x 是类上帝的.
Theorem 3: Necessarily, the property of being God-like is exemplified.
□∃xG(x)
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楼主: 徐公孰
发表于 2022-11-22 12:27
显身卡
发表于 2022-11-22 12:30
