Deduction & Induction

ref:http://www.psych.utah.edu/gordon/Classes/Psy4905Docs/PsychHistory/Cards/Logic.html

Logical arguments are usually classified as either 'deductive' or 'inductive'.

Deduction: In the process of deduction, you begin with some statements, called 'premises', that are assumed to be true, you then determine what else would have to be true if the premises are true. For example, you can begin by assuming that God exists, and is good, and then determine what would logically follow from such an assumption. You can begin by assuming that if you think, then you must exist, and work from there. In mathematics you can begin with some axioms and then determine what you can prove to be true given those axioms. With deduction you can provide absolute proof of your conclusions, given that your premises are correct. The premises themselves, however, remain unproven and unprovable, they must be accepted on face value, or by faith, or for the purpose of exploration.

Induction: In the process of induction, you begin with some data, and then determine what general conclusion(s) can logically be derived from those data. In other words, you determine what theory or theories could explain the data. For example, you note that the probability of becoming schizophrenic is greatly increased if at least one parent is schizophrenic, and from that you conclude that schizophrenia may be inherited. That is certainly a reasonable hypothesis given the data. Note, however, that induction does not prove that the theory is correct. There are often alternative theories that are also supported by the data. For example, the behavior of the schizophrenic parent may cause the child to be schizophrenic, not the genes. What is important in induction is that the theory does indeed offer a logical explanation of the data. To conclude that the parents have no effect on the schizophrenia of the children is not supportable given the data, and would not be a logical conclusion.

Deduction and induction by themselves are inadequate for a scientific approach. While deduction gives absolute proof, it never makes contact with the real world, there is no place for observation or experimentation, no way to test the validity of the premises. And, while induction is driven by observation, it never approaches actual proof of a theory. The development of the scientific method involved a gradual synthesis of these two logical approaches.

For a more comprehensive discussion of deduction, and induction, read the relevant sections of the book by Copi, referenced on this page.

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### 自然推理逻辑规则及其应用 自然推理是一种形式化的证明方法,用于构建命题和谓词逻辑中的论证。这种方法允许通过一系列直观的推导规则来验证陈述的有效性。 #### 基本概念 在计算机科学中,自然推理系统通常由一组初始假设以及定义良好的引入和消去规则组成[^1]。这些规则使得可以从已知前提逐步推出新的结论: - **合取(Conjunction)** - 引入:如果 \( A \) 和 \( B \) 都成立,则可以得出 \( A ∧ B \)。 - 消除:从 \( A ∧ B \),可以直接得到 \( A \) 或者 \( B \)。 - **析取(Disjunction)** - 引入:给定任意一个命题 \( A \) 或 \( B \),则可形成 \( A ∨ B \)。 - 消除:为了证明某个目标,在两个分支下分别考虑当左边或右边为真时的情况;只要能在这两种情况下都达到相同的结果即可。 - **蕴含(Implication)** - 引入:要证 \( A → B \),假定 \( A \) 并试图在此基础上建立 \( B \)。 - 消除:如果有 \( A → B \) 及其前件 \( A \),那么就可以断言 \( B \)。 - **否定(Negation)** - 引入:若假设某项声明会引发矛盾,则该声明必然是错误的。 - 消除:利用反证法——即尝试展示接受某一命题会导致不合理之处从而间接证实它的对立面。 #### 示例代码实现 下面是一个简单的 Python 函数模拟了如何使用上述原则来进行基本的布尔表达式的求值过程: ```python def natural_deduction(expression): stack = [] operators = { '∧': lambda a, b: a and b, '∨': lambda a, b: a or b, '→': lambda a, b: not a or b, '~': lambda a: not a } for token in expression.split(): if token.isalpha(): # Assuming variables are single letters. value = input(f"Enter truth value (True/False) for {token}: ") while value.lower() not in ['true', 'false']: print("Invalid entry.") value = input(f"Enter truth value (True/False) for {token}: ") stack.append(value.lower() == 'true') elif token in operators: if token != '~': operand_b = stack.pop() operand_a = stack.pop() result = operators[token](operand_a, operand_b) else: operand = stack.pop() result = operators[token](operand) stack.append(result) return stack[-1] print(natural_deduction('p → q')) ``` 此函数接收字符串表示的逻辑运算符序列作为输入,并提示用户提供变量的实际真假值。它按照逆波兰记号的方式处理操作数栈上的元素并最终返回整个公式的计算结果。
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