# Classical Logic

> 【EN】Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics. The language has components that correspond to a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record arguments that are valid for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions for at least part of the language. The following sections provide the basics of a typical logic, sometimes called “classical elementary logic” or “classical first-order logic”. Section 2 develops a formal language, with a rigorous syntax and grammar. The formal language is a recursively defined collection of strings on a fixed alphabet. As such, it has no meaning, or perhaps better, the meaning of its formulas is given by the deductive system and the semantics. … 【中】词条给出典型逻辑即“经典基础逻辑”或“经典一阶逻辑”的基础。一个逻辑通常包含语言、演绎系统与模型论语义：语言对应自然语言的一部分，演绎系统记录有效的论证，语义记录意义或真值条件。文章依次展开形式语言的句法与语法、原子公式、复合公式、句法的特征，然后讨论演绎、语义与元理论。

- ID: m13240
- Category: structure
- Domain: 逻辑学

## Definition

通常一个逻辑由形式的或非形式的语言，加上一个演绎系统和（或）模型论语义构成。语言的成分对应于英语或希腊语等自然语言的某一部分，演绎系统用以刻画该语言中有效的论证，语义则刻画语言的意义或真值条件。 脚手架作用：- 论证检验：把论证译为形式语言，用有效性判定其是否保真。 - 歧义消除：借形式化暴露自然语言中隐含的量词与结构。 - 系统选择：据可靠性与完全性判断某形式系统是否够用。

## How it works

经典逻辑的两个支柱是句法与语义：演绎系统从公理或规则出发给出可证性，模型论语义用结构与赋值给出真值条件，有效性则由在所有模型中保真来定义。元理论把二者联系起来，可靠性与完全性定理说明可证与有效恰好重合。

## Practice

1) 确定待分析的语言片段，列出非逻辑词项与逻辑常项；2) 按语法把语句译为公式，注意量词辖域；3) 明确要检验的是语义后承还是句法可推；4) 用模型或证明加以验证，并说明用到了排中律等经典假设。

## Use

- 论证检验：把论证译为形式语言，用有效性判定其是否保真。 - 歧义消除：借形式化暴露自然语言中隐含的量词与结构。 - 系统选择：据可靠性与完全性判断某形式系统是否够用。

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