Frege's propositional calculus

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.

<u>Axioms</u> <br />THEN-1: A → (B → A) <br />THEN-2: (A → (B → C)) → ((A → B) → (A → C)) <br />THEN-3: (A → (B → C)) → (B → (A → C)) <br />FRG-1: (A → B) → (¬B → ¬A) <br />FRG-2: ¬¬A → A <br />FRG-3: A → ¬¬A <br />

<u>Inference Rule</u> <br>MP: P, P→Q ⊢ Q <br />

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...
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