量詞邏輯量詞里面的v表示？

As we know that in an AI-based agent, the knowledge is represented through two types of logic: The propositional logic and the predicate logic. In the propositional logic, we have declarative sentences, and in the predicate logic, we have a predicate defining a subject. But in both these systems, we were not able to define the quantity of any subject.

我們知道，在基于AI的代理中 ，知識通過兩種邏輯表示： 命題邏輯和謂詞邏輯 。 在命題邏輯中，我們有陳述句，在謂詞邏輯中，我們有謂詞定義主語。 但是在這兩個系統中，我們都無法定義任何主題的數量。

For example:

例如：

The predicate logic: like(boy, apple) defines that boy likes apple. But take a look at the following two statements:

謂詞邏輯： like(boy，apple)定義男孩喜歡蘋果。 但是，請看以下兩個語句：

• Some boys like apple

  有些男孩喜歡蘋果

• All boys like apple

  所有男孩都喜歡蘋果

These sentences cannot be defined completely with the help of first-order predicate logic. So, to solve this issue, the quantifiers were used.

這些句子不能借助一階謂詞邏輯來完全定義。 因此，為了解決這個問題，使用了量詞 。

Quantifiers are the quantity defining terms which are used with the predicates.

量詞是與謂詞一起使用的數量定義術語。

量詞的類型 (Types of quantifiers)

There are two types of quantifiers:

量詞有兩種：

1)通用量詞 (1) Universal Quantifier)

The universal quantifier is used to define the whole subject population under the predicate. It can be used anywhere where the phrases like: 'for all', 'for each', 'for every' are used.

通用量詞用于定義謂詞下的整個主題種群。 它可以在使用諸如“ for all” ， “ for each” ， “ for every”之類的短語的任何地方使用。

The symbol '?' is used to represent universal Quantifier. To combine the universal quantifier with the predicate and the subject, implication sign, '->' is used.

符號“?”用于表示通用量詞。 為了將通用量詞與謂詞和主語結合使用，使用了隱含符號“->” 。

Example:

例：

    ?x: Boy(x) -> like(x,Apple) 

The above statement says that: 'All boys like apple'.

上面的陳述說： '所有男孩都喜歡蘋果' 。

2)存在量詞 (2) Existential Quantifier)

The Existential Quantifier is used at the places where only some part of the subject's population is to be defined under the predicate. It can be used at all the places where the following phrases are used: 'There exist', 'For some', 'For at least', etc.

存在量詞用于在謂詞下僅定義對象總體一部分的地方。 它可以在使用以下短語的所有地方使用： “存在” ， “用于某些” ， “至少用于”等。

The Existential Quantifier is represented by the symbol '?'. To combine the Existential quantifier with the predicate and the subject, the conjunction symbol, '^' is used.

存在量詞用符號“?”表示 。 為了將存在量詞與謂詞和主語結合起來，使用了連字符“ ^” 。

Example:

例：

    ?x: Boy(x) ^ like(x,apple)

The above statement depicts that there exists a boy who likes apple. Or we can say that there are some boys who like an apple.

上面的陳述描述了存在一個喜歡蘋果的男孩。 或者我們可以說有些男孩喜歡蘋果。

量詞的性質 (Properties of Quantifiers)

• ?x.?y is the same as ?y.?x

  ?x.?y與?y.?x相同

• ?x.?y is the same as ?y.?x

  ?x.?y與?y.?x相同

• ?x.?y is not the same as ?y.?x

  ?x.?y與?y.?x不同

• Quantifier duality: Each quantifier can be expressed using the other one. This is done by complementing and changing the symbols.

  量詞對偶：每個量詞都可以用另一個來表達。 這可以通過補充和更改符號來完成。

Example:

例：

    ?x likes(x, Ice-cream) is equivalent to ~?x ~likes(x, Ice-cream).
?x likes(x, Chocolate) is equivalent to ~?x ~likes(x, chocolate). 

翻譯自: https://www.includehelp.com/ml-ai/introduction-to-quantifiers-in-the-knowledge-representation-in-an-agent-based-on-artificial-intelligence.aspx

量詞邏輯量詞里面的v表示？