Question

我一直在做离散的结构，学习真理的证明和排序（ETC.（（A→B）∨B）→C，（¬p→q）⊕¬q等，可以知道它们是如何工作的以及如何工作得出答案，但是最近类似的事情

A→B 
B→C     A∨¬B
B       ¬A
____    ____
A→C     ????
???

已经出现，我无法在Google上找到有关该格式或如何解决该问题的任何信息，也没有得到有关该格式以及如何解决它的任何信息，因此它是新的，我不知道该如何处理。我发现最接近相似来源的是有关计算，自动机和代数教科书的Google书籍，这些书籍似乎将它们称为矩阵。我尝试单独制作每个真值表，但找不到链接它的模式。

对于如何处理此类证明的资料或示例，我们表示赞赏。我以前从未遇到过这种布局，我很可能只是以一种新格式知道了一些东西。感谢您提前提供任何相关帮助。

Answer 1

There are two ways to prove things in logic:

by computing the value for all inputs and then verifying it's a tautology
by using a proof system to manipulate the formulae to show it's a tautology

You are used to approach 1. It is easy to understand and concrete. Approach 2 is more abstract and more difficult. To execute proofs in a proof system, you must first fix the proof system. This means declaring some axioms and rules of logical deduction.

For instance, a simple system can be found here:

A→(B→A)
(A→(B→C))→((A→B)→(A→C))
(¬A→¬B)→(B→A)
Deduction is by Modus Ponens: A, A → B | B

In this system, your proof might look like this:

1. A→B, B→C, B | (A→(B→C))→((A→B)→(A→C))       (2)
2. A→B, B→C, B | (B→C)→(A→(B→C))               (1)
3. A→B, B→C, B | A→(B→C)                       MP on 2. and hypothesis B→C
4. A→B, B→C, B | (A→B)→(A→C)                   MP on 1. and 3.
5. A→B, B→C, B | A→C                           MP on 4. and hypothesis A→B

At each step, you either introduce a new instance of one of your axioms or you apply your rule of deduction to produce a new line. From then on, you can use the lines so derived to derive new lines, until you derive the intended result.

Your second one doesn't make a lot of sense since you can't prove something that hasn't been claimed. If you get something like that I suggest you just fill in the blank with something easy to prove, like true, and then state true in the proof and be done with it.

了解与证明表有关的真理证明

1 个答案: