一、Basic Concepts in Logic

1. Proposition (命题)

A proposition asserts that something is the case or it asserts that something is not. Every propositon is either true or false.

Proposition是逻辑学中最基础的概念，在此基础上有一些衍生概念，例如Premises(前提)、Conlusion(结论)、Simple Proposition、Compound Proposition等；Statement(陈述)通常被视为Proposition的同义词。

Simple Proposition指不含其它Proposition的Proposition，例如”天是蓝的”；Compound Proposition指的是含有其它Proposition的Proposition，例如”天是蓝的且水是蓝的”。

2. Argument (论证)

Argument是推导Proposition的过程，它是逻辑学的主要研究对象。

In any argument we affirm one proposition on the basis of some other propositions. In doing this, an inference is drawn. Inference is a process that may tie together a cluster of propositions. Some inferences are warranted (or correct); others are not. The logician analyzes these clusters, examining the propositions with which the process begins and with which it ends, as well as the relations among these propositions. Such a cluster of propositions constitutes an argument. Arguments are the chief concern of logic.

3. Deductive and Inductive Argument (演绎论证与归纳论证)

A deductive argument makes the claim that its conclusion is supported by its premises conclusively (不容置疑地). An inductive argument, in contrast, does not make such a claim. Because every argument either makes this claim of conclusiveness (explicitly or implicitly) or does not make it, every argument is either deductive or inductive.

Inductive Argument所做的断言弱于Deductive Argument。

4. Truth and Validity (真实性与有效性)

Truth用来形容Proposition是否与事实相符。前面提到过，一个Proposition要么是正确的，要么是错误的。

Validity用来形容Deducive Argument的论证是否正确，一个Deductive Argument要么有效，要么无效。它只适用于Deductive Argument，不适用于Inductive Argument，因为Deductive Argument是确定的，而Inductive Argument所做的断言更弱，并非百分百确定，通常我们可以说一个Inductive Argument较强、较弱等。


关于Proposition和Argument的属性总结为如下表格：

Proposition true / false
Deductive Argument effective / ineffective
Inductive Argument strong / weak

二、Deductive Argument

Deductive Aegument有两大流派：古典逻辑(Classical Logic)和符号逻辑(Symbolic Logic)，二者的关系有点类似于概率论中Classical Method和现代以Probability Space为基础的概率模型的关系，Symbolic Logic是Classical Logic的超集，这里主要介绍Symbolic Logic。

我们需要建立一个为Logic服务的Domain Specific Language (DSL)。

前面说过，在逻辑学中，Proposition和Statement通常被视为同义词，本节更多使用Statement。

1. Simple / Compound Statement and Component

  • Simple Statement: A statement that does not contain any other statement as a component.
  • Compound Statement: A statement that contains two or more statements as components.
  • Component: A part of a compound statement that is itself a statement, and is of such a nature that, if replaced in the larger statement by any other statement, the result will be meaningful.

2. Conjunction, Negation and Disjunction

2.1 Conjunction

\(p\) and \(q\)

\(p\) \(q\) \(p \cdot q\)
\(T\) \(T\) \(T\)
\(T\) \(F\) \(F\)
\(F\) \(T\) \(F\)
\(F\) \(F\) \(F\)

2.2 Negation

not \(p\)

\(p\) \(\sim p\)
\(T\) \(F\)
\(F\) \(T\)

2.3 Disjunction

\(p\) or \(q\)

\(p\) \(q\) \(p\vee q\)
\(T\) \(T\) \(T\)
\(T\) \(F\) \(T\)
\(F\) \(T\) \(T\)
\(F\) \(F\) \(F\)

3. Conditional Statement (Hypothetical Statement)

if \(p\) then \(q\)，表示为\(p\supset q\)，还有两种等价的说法是\(p\)是\(q\)的充分条件(sufficient condition)，\(p\)是\(q\)的必要条件(necessary condition)；此外，我们称\(p\)为Antecedent，\(q\)为Consequent。

\(p\supset q\) 被严格定义为\(\sim (p\cdot \sim q)\)。

\(p\) \(q\) \(\sim q\) \(p\cdot\sim q\) \(\sim (p\cdot \sim q)\) \(p\supset q\)
T T F F T T
T F T T F F
F T F F T T
F F T F T T

4 Argument Forms

Variable or Statement Variable

A place-holder; a letter (by convention, any of the lower case letters, beginning with \(p\), \(q\), etc.) for which a statement may be substituted.

Argument Form

An array of symbols exhibiting logical structure; it contains no statements but it contains statement variables. These variables are arranged in such a way that when statements are consistently substituted for the statement variables, the result is an argument.

Specific Form

When referring to a given argument, the argument form from which the argument results when a different simple statement is substituted consistently for each different statement variable in that form.

Sbusitution Instance

Any argument that results from the substitution of statements for the statement variables of a given argument form.

4. Invalid and Valid (如何定义有效)

Every deductive argument is either valid or invalid; every deductive argument form is also either valid or invalid.

Invalid Argument Form

An argument form is invalid if and only if it has at least one substitution instance with true premises and a false conclusion. An argument is invalid if and only if the specific form of that argument is a invalid argument form.

Valid Argument Form

An argument form is valid if and only if it has no substitution instances with true premises and a false conclusion. An argument is valid if and only if the specific form of that argument is a valid argument form.

True Value Table (真值表)

我们可以通过构建True Value Table来验证一个Argument Form是否有效。

5. 量化理论

Universal and Existential Quantifiers

三、Inductive Argument

TODO