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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. Axioms THEN-1: A → (B → A) THEN-2: (A →) → (→) THEN-3: (A →) → (B →) FRG-1: (A → B) → (¬B → ¬A) FRG-2: ¬¬A → A FRG-3: A → ¬¬A Inference Rule MP: P, P→Q ⊢ Q 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 which 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. ) Rule THEN-1*: A ⊢ B→A Rule THEN-2*: A→(B→C) ⊢ (A→B)→(A→C) Rule THEN-3*: A→(B→C) ⊢ B→(A→C) Rule FRG-1*: A→B ⊢ ¬B→¬A Rule TH1*: A→B, B→C ⊢ A→C Theorem TH1: (A→B)→(→) Theorem TH2: A→(¬A→¬B) Theorem TH3: ¬A→(A→¬B) Theorem TH4: ¬(A→¬B)→A Theorem TH5: (A→¬B)→(B→¬A) Theorem TH6: ¬(A→¬B)→B Theorem TH7: A→A Theorem TH8: A→(→B) Theorem TH9: B→(→B) Theorem TH10: A→(B→¬) Note: ¬(A→¬B)→A (TH4), ¬(A→¬B)→B (TH6), and A→(B→¬) (TH10), so ¬(A→¬B) behaves just like A∧B (compare with axioms AND-1, AND-2, and AND-3). Theorem TH11: (A→B)→(→¬A) TH11 is axiom NOT-1 of standard PC, called "reductio ad absurdum". Theorem TH12: (→C)→(A→) Theorem TH13: (B→)→(B→C) Rule TH14*: A→(B→P), P→Q ⊢ A→(B→Q) Theorem TH15: (→)→(A→) Theorem TH15 is the converse of axiom THEN-2. Theorem TH16: (¬A→¬B)→(B→A) Theorem TH17: (¬A→B)→(¬B→A) Compare TH17 with theorem TH5. Theorem TH18: (→B)→(¬A→B) Theorem TH19: (A→C)→ (→) Note: A→(→B) (TH8), B→(→B) (TH9), and (A→C)→(→) (TH19), so (→B) behaves just like A∨B. (Compare with axioms OR-1, OR-2, and OR-3. ) Theorem TH20: (A→¬A)→¬A TH20 corresponds to axiom NOT-3 of standard PC, called "tertium non datur". Theorem TH21: A→(¬A→B) TH21 corresponds to axiom NOT-2 of standard PC, called "ex contradictione quodlibet". All the axioms of standard PC have be derived from Frege's PC, after having let A∧B := ¬(A→¬B) and A∨B := (A→B)→B. These expressions are not unique, e.g. A∨B could also have been defined as (B→A)→A, ¬A→B, or ¬B→A. Notice, though, that the definition A∨B := (A→B)→B contains no negations. On the other hand, A∧B cannot be defined in terms of implication alone, without using negation. In a sense, the expressions A∧B and A∨B can be thought of as "black boxes". Inside, these black boxes contain formulas made up only of implication and negation. The black boxes can contain anything, as long as when plugged into the AND-1 through AND-3 and OR-1 through OR-3 axioms of standard PC the axioms remain true. These axioms provide complete syntactic definitions of the conjunction and disjunction operators. The next set of theorems will aim to find the remaining four axioms of Frege's PC within the "theorem-space" of standard PC, showing that the theory of Frege's PC is contained within the theory of standard PC. Theorem ST1: A→A Theorem ST2: A→¬¬A ST2 is axiom FRG-3 of Frege's PC. Theorem ST3: B∨C→(¬C→B) Theorem ST4: ¬¬A→A ST4 is axiom FRG-2 of Frege's PC. Prove ST5: (A→)→(B→) ST5 is axiom THEN-3 of Frege's PC. Theorem ST6: (A→B)→(¬B→¬A) ST6 is axiom FRG-1 of Frege's PC. Each of Frege's axioms can be derived from the standard axioms, and each of the standard axioms can be derived from Frege's axioms. This means that the two sets of axioms are interdependent and there is no axiom in one set which is independent from the other set. Therefore the two sets of axioms generate the same theory: Frege's PC is equivalent to standard PC.
- A Frege-kalkulus egy matematikai logikai kalkulus (levezetőrendszer), azaz egy alapjelekből, axiómákból, levezetési szabályokból álló formális nyelv vagy elmélet, melyet Gottlob Frege jénai matematikus alkotott meg 1879-ben megjelent, Fogalomírás (Begriffsschrift) c. könyvében.
- 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。 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 &rarr) → (&rarr) THEN-3: (A &rarr) → (B &rarr) 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。
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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).
- A Frege-kalkulus egy matematikai logikai kalkulus (levezetőrendszer), azaz egy alapjelekből, axiómákból, levezetési szabályokból álló formális nyelv vagy elmélet, melyet Gottlob Frege jénai matematikus alkotott meg 1879-ben megjelent, Fogalomírás (Begriffsschrift) c. könyvében.
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