古典邏輯的子邏輯以及"常義子邏輯"

Basic idea: In classical logic, there has valid inferences and logical truth. And we want to give a proof to explain what is a sublogic of classical logic.

我們的基本想法是: 在古典邏輯當中有一些推論是有效的,並且我們可以從古典邏輯系統當中,找到一些邏輯真理。而我們便是要使用所說的「有效推論」以及「邏輯真理」來定義,當一個邏輯系統,它同時也是古典邏輯的子系統(子邏輯)的時候,它必須具備哪些條件。

First, classical logic is an normal logic, a normal logic (1)assigned two value, {1,0}≤V, 1 is designated and 0 is not. (2)Every truth-functions, ⋀, ⋁, ¬, ⊃,...,and so on, input them to a function " f " like f⋀, f⋁, f¬, f⊃,...,and so on, we have many value V1,V2,...,Vn, and they all ∈{1,0} ,(3)after then output that we have V1,...,Vn ,it is the same that truth-functions in classical logic, means a function called " f' ". 

首先,古典邏輯是一種標準邏輯,(1)一個標準邏輯必須被指定二個值(也就是有兩個指定值),{1,0}≤V,1是真而0是假,(2)一個標準邏輯的所有真值函項,⋀, ⋁, ¬, ⊃,...等等,代入函數 f (寫成f⋀, f⋁, f¬, f⊃,...等等)會得到許多的值V1,V2,...,Vn,這些值也都屬於{1,0},換言之所得的值為真或為假,(但不會有第三值)(3)而這些真值函項所得的值也會是古典邏輯的真值函項所帶入的值。

Second, assume a logic called Logic 3, written L3. L3 is a sublogic of classical logic:

第二,假設一個我們自創的邏輯叫做L3,L3是古典邏輯的一個子邏輯:

⊨A (in L3) ⊃ ⊨A (in classic)

and a premise is a set of sentences called ∑ will satisfy classical logic:

而∑是語句集合所形成的前提,它會滿足古典邏輯:

∑⊨A(in L3)⊃ ∑⊨A(in classic) .

Finally, we have to proof that : there is a model of classical logic can not semantics entail A, A is the conclusion of inference. If it were been finished, we can understand that L3 is a normal logic also a sublogic for classical logic. But we know that assume again L3 is a three-value logic, we can find a simple sentence from L3 called S', L3 is still a normal logic. Something is interesting that S' is true in classic logic and is true in three-value logic. It means that, there is a three-value model M semantics entail S' but not semantics entail A. Then we call L3 is a sublogic of classical logic and classical logic is a proper sublogic of L3.  

最後,我們必須證明,存在有一個古典邏輯的模型不能使A為真,而A是推論最後的結論。假如我們可以完成這個證明,我們便可以說L3是一個標準邏輯而且是古典邏輯的一個子邏輯(子系統)。但我們繼續假設,K3是一個三值邏輯(三值邏輯包含的指定值是: 真,假,既不真也不假),我們很容易可以找到一個簡單句S'在K3當中,而K3仍是一個標準邏輯。有趣的是,S'在古典邏輯當中為真的話,在三值邏輯系統K3裡也會為真,這就意味著,存在有一個模型M使得K3為真(使S'為真)但使得古典邏輯不為真(A不為真)。這時候,我們便稱K3是古典邏輯的子邏輯,而古典邏輯是K3的一個常義子邏輯。

碎念:我的英文還在磨練當中,就一併附上中文以避免語意不清。:p