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1. 命题
在现代哲学、数学、逻辑学、语言学中，命题是指一个判断（陈述）的语句（实际表达的概念），这个概念是可以被定义并观察的现象。命题不是指判断（陈述）本身，而是指所表达的语义。但相异判断（陈述）具有相同语义的时候，它们表达相同的命题。
在数学中，一般把判断某一事情的陈述句叫做命题。
The term proposition has a broad use in contemporary analytic philosophy. The most basic meaning is a statement proposing an idea that can be true or false.
A proposition is a theorem of lesser importance. This term sometimes connotes a statement with a simple proof, while the term theorem is usually reserved for the most important results or those with long or difficult proofs. Some authors never use "proposition", while some others use "theorem" only for fundamental results.
2. 定理
定理是指经受逻辑限制的证明为真的陈述（通过真命题出发，经过受逻辑限制的演绎推导，证明为正确的结论的命题或公式）。
在数据中，定理是指在既有命题的基础上证明出来的命题，这些既有命题可以是别的定理，或这广为接受的陈述，比如公理。
一般来说，在数学中，只有重要或有趣的陈述才能叫定理。证明定理是数学的中心活动。定理证明通常被诠释为对其真实性的验证。
有许多数学定理都是条件语句，此时定理的证明是从假设出发，推出结论。定理一般都是假设----即一些条件。然后它有结论-----一个在条件下成立的数学陈述。通常写作“若条件，则结论”。
定理是建立在公理和假设基础上，经过严格的推理和证明得到的，它能描述事物之间的内在关系，定理具有内在的严密性，不能存在逻辑矛盾。
In mathematics, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems[1][2][3].
[1] "Definition of THEOREM". www.merriam-webster.com. 2019-11-02.
[2] "The Definitive Glossary of Higher Mathematical Jargon – Theorem". Math Vault. 2019-08-01. 2019-11-02.
[3] "Theorem | Definition of Theorem by Lexico". Lexico Dictionaries | English. 2019-11-02.
3. 推论
推论指的是从一个或一些已知的命题得到新命题的思维过程或思维形式。其中已知的命题是前提，得出的命题为结论。
推论是指能够简单明了地从前述命题推出的论断。
推论往往在定理后出现；如果命题B能够简单明了地从命题A推导出，则B为A的推论。
In mathematics and logic, a corollary is a statement which can be readily deduced from a previous, more notable statement, but whose importance tends to be secondary in nature[1].
A corollary could for instance be a proposition which is incidentally proved while proving another proposition[2], while it could also be used more casually to refer to something which naturally or incidentally accompanies something else (e.g., violence as a corollary of revolutionary social changes) [3][4].
A corollary is a proposition that follows with little proof from another theorem or definition. Also, a corollary can be a theorem restated for a more restricted special case. For example, the theorem that all angles in a rectangle are right angles has as corollary that all angles in a square (a special case of a rectangle) are right angles.
[1] "The Definitive Glossary of Higher Mathematical Jargon — Corollary". Math Vault. 2019-08-01.
[2] "Definition of corollary | Dictionary.com". www.dictionary.com. 2019-11-27.
[3]"Definition of COROLLARY". www.merriam-webster.com. 2019-11-27.
[4] "COROLLARY | meaning in the Cambridge English Dictionary". dictionary.cambridge.org. 2019-11-27.
4. 引理
引理是为了证明某个定理或解释某个问题所要用到的命题。
引理是数学中为了取得某个更好的结论而作为步骤被证明的命题，其意义并不在于自身被证明，而在于为达成最终目标而做出的贡献。
一个引理可用于证明多个结论。
引理和定理没有严格的区分，如果论证某个命题时，还没有直接根据，需要某些还没有被证明的结论，把它提出来加以证明，就是所谓的构造引理。
[1]沈以淡．简明数学词典：北京理工大学出版社，2003.8
[2] 王爱民，徐扬，唐虎，李仲来．中国高等教育研究论丛：成都科技大学出版社，1994年08月第1版
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