In mathematical logic, Frege's propositional calculus was the first axiomatization of propositional calculus. It was invented by Gottlob Frege, who also invented predicate calculus, in 1879 as part of his second-order predicate calculus (although Charles Peirce was the first to use the term "second-order" and developed his own version of the predicate calculus independently of Frege). It makes use of just two logical operators: implication and negation, and it is constituted by six axioms and one inference rule: modus ponens.
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| - Cálculo proposicional de Frege (es)
- Frege's propositional calculus (en)
- 弗雷格命题演算 (zh)
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| - Cálculo proposicional de Frege, en la Lógica matemática, el cálculo proposicional de Frege fue la primera axiomatización del cálculo proposicional. Fue inventado por Gottlob Frege, quien también inventó el cálculo de predicados, en 1879, como parte de su (a pesar de que Charles Peirce fue el primero en utilizar el término "segundo orden" y desarrolló su propia versión de forma independiente del cálculo de predicados de Frege). Hace uso de sólo dos operadores lógicos: Implicación y la negación, y está constituida por seis axiomas y una regla de inferencia: modus ponens. (es)
- 在数理逻辑中弗雷格命题演算是第一个公理化的命题演算。它由弗雷格发明，他还在1879年发明了谓词演算，作为他的二阶谓词逻辑的一部分(尽管查尔斯·桑德斯·皮尔士首次使用了术语“二阶”并独立于 Frege 开发了自己版本的谓词演算)。 它只使用两个逻辑算子: 蕴涵和否定，并且由六个公理和一个推理规则肯定前件构成。 公理
* THEN-1: A→(B→A)
* THEN-2: (A→(B→C))→((A→B)→(A→C))
* THEN-3: (A→(B→C))→(B→(A→C))
* FRG-1: (A→B)→(¬B→¬A)
* FRG-2: ¬¬A→A
* FRG-3: A→¬¬A 推理规则
* MP: P, P→Q ├ Q Frege 的命题演算等价于任何其他经典的命题演算，比如有 11 个公理的“标准 PC”。Frege 的 PC 和标准的 PC 共享两个公共的公理: THEN-1 和 THEN-2。注意从 THEN-1 到 THEN-3 的公理只使用(和定义)蕴涵算子，而从 FRG-1 到 FRG-3 的公理定义否定算子。 (zh)
- In mathematical logic, Frege's propositional calculus was the first axiomatization of propositional calculus. It was invented by Gottlob Frege, who also invented predicate calculus, in 1879 as part of his second-order predicate calculus (although Charles Peirce was the first to use the term "second-order" and developed his own version of the predicate calculus independently of Frege). It makes use of just two logical operators: implication and negation, and it is constituted by six axioms and one inference rule: modus ponens. (en)
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| - In mathematical logic, Frege's propositional calculus was the first axiomatization of propositional calculus. It was invented by Gottlob Frege, who also invented predicate calculus, in 1879 as part of his second-order predicate calculus (although Charles Peirce was the first to use the term "second-order" and developed his own version of the predicate calculus independently of Frege). It makes use of just two logical operators: implication and negation, and it is constituted by six axioms and one inference rule: modus ponens. Frege's propositional calculus is equivalent to any other classical propositional calculus, such as the "standard PC" with 11 axioms. Frege's PC and standard PC share two common axioms: THEN-1 and THEN-2. Notice that axioms THEN-1 through THEN-3 only make use of (and define) the implication operator, whereas axioms FRG-1 through FRG-3 define the negation operator. The following theorems will aim to find the remaining nine axioms of standard PC within the "theorem-space" of Frege's PC, showing that the theory of standard PC is contained within the theory of Frege's PC. (A theory, also called here, for figurative purposes, a "theorem-space", is a set of theorems that are a subset of a universal set of well-formed formulas. The theorems are linked to each other in a directed manner by inference rules, forming a sort of dendritic network. At the roots of the theorem-space are found the axioms, which "generate" the theorem-space much like a generating set generates a group.) (en)
- Cálculo proposicional de Frege, en la Lógica matemática, el cálculo proposicional de Frege fue la primera axiomatización del cálculo proposicional. Fue inventado por Gottlob Frege, quien también inventó el cálculo de predicados, en 1879, como parte de su (a pesar de que Charles Peirce fue el primero en utilizar el término "segundo orden" y desarrolló su propia versión de forma independiente del cálculo de predicados de Frege). Hace uso de sólo dos operadores lógicos: Implicación y la negación, y está constituida por seis axiomas y una regla de inferencia: modus ponens. (es)
- 在数理逻辑中弗雷格命题演算是第一个公理化的命题演算。它由弗雷格发明，他还在1879年发明了谓词演算，作为他的二阶谓词逻辑的一部分(尽管查尔斯·桑德斯·皮尔士首次使用了术语“二阶”并独立于 Frege 开发了自己版本的谓词演算)。 它只使用两个逻辑算子: 蕴涵和否定，并且由六个公理和一个推理规则肯定前件构成。 公理
* THEN-1: A→(B→A)
* THEN-2: (A→(B→C))→((A→B)→(A→C))
* THEN-3: (A→(B→C))→(B→(A→C))
* FRG-1: (A→B)→(¬B→¬A)
* FRG-2: ¬¬A→A
* FRG-3: A→¬¬A 推理规则
* MP: P, P→Q ├ Q Frege 的命题演算等价于任何其他经典的命题演算，比如有 11 个公理的“标准 PC”。Frege 的 PC 和标准的 PC 共享两个公共的公理: THEN-1 和 THEN-2。注意从 THEN-1 到 THEN-3 的公理只使用(和定义)蕴涵算子，而从 FRG-1 到 FRG-3 的公理定义否定算子。 (zh)
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