Frege's Propositional Calculus

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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.

<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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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 />

<u>Inference Rule</u> <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...

Read More

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