[英]Representing FOL in english language
I have the following FOL formula: ∀e(S(e)) → ∃d(P(d))
我有以下FOL公式: ∀e(S(e)) → ∃d(P(d))
And the vocabulary: 和词汇:
variables: e:'exam', d:'day' functions: S:'successful', P:'party'
I initially translated that formula into: 我最初将该公式翻译成:
For every successful exam, there will be a day of party
While apparently the correct translation would be something of the sort of: 虽然显然正确的翻译会是这样的:
You party at least one day after all exams were successful.
Why does the correct one say that we only party after ALL exams were successful? 为什么正确的人说我们在所有考试成功后才参加派对?
Does ∀e(S(e))
mean: "For all exams, they will all be successful"? ∀e(S(e))
意味着:“对于所有考试,他们都会成功”? And ∃d(P(d))
mean: "there exists at least one day where we party"? 而∃d(P(d))
意思是:“至少有一天我们聚会”?
And doesn't the implication translte to "if a then b" ? 并不意味着转换为“if a then b”?
I think I can somehow see the logic of the correct translation, but there's something about the implication that makes me unsure... 我想我可以以某种方式看到正确翻译的逻辑,但是有一些关于这个含义让我不确定......
Careful here. 小心点。 This formula: 这个公式:
∀e(S(e)) → ∃d(P(d))
Really only has one precise sense, the one you acknowledge as apparently correct: 真的只有一个准确的意义,你承认显然是正确的:
If all exams are successful, then there will be a party. 如果所有考试都成功,那么就会举办派对。
Your translation is wrong for a subtle but significant reason. 由于一个微妙但重要的原因,你的翻译是错误的。 Your translation: 你的翻译:
For every successful exam, there will be a day of party 每次成功的考试,都会有一天的聚会
Corresponds to this formula: 对应这个公式:
∀e.∃d(S(e) → P(d))
These formulae are not logically equivalent, that is, the following is not a tautology: 这些公式在逻辑上不等同,即以下不是重言式:
(∀e(S(e)) → ∃d(P(d))) <=> (∀e.∃d(S(e) → P(d)))
To see this, consider what happens when you pass ten exams and fail one exam. 要看到这一点,请考虑通过十门考试并通过一门考试后会发生什么。 The original formula is vacuously true regardless of whether any party is had, since ∀e(S(e))
is not satisfied. 无论是否有任何一方,原始公式都是真实的,因为∀e(S(e))
不满足。 However, your statement is only true if you have at least one party, since you did pass at least one exam. 但是,如果您至少有一方,则您的陈述是正确的,因为您确实通过了至少一次考试。
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