- Frege's propositional calculus
mathematical logicFrege's propositional calculus was the first axiomatizationof propositional calculus. It was invented by Gottlob Frege, who also invented predicate calculus, in 1879as part of his second-order predicate calculus(although Charles Peircewas 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.
"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
"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 FRG-1*: A→B ⊢ ¬B→¬A
Theorem TH2: A→(¬A→¬B)
Theorem TH5: (A→¬B)→(B→¬A)
Theorem TH8: A→((A→B)→B)
Note: ¬(A→¬B)→A (TH4), ¬(A→¬B)→B (TH6), and A→(B→¬(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→¬B)→¬A)
Rule TH14*: A→(B→P), P→Q ⊢ A→(B→Q)
Theorem TH17: (¬A→B)→(¬B→A)
Note: A→((A→B)→B) (TH8), B→((A→B)→B) (TH9), and(A→C)→((B→C)→(((A→B)→B)→C)) (TH19), so ((A→B)→B) behaves just like A∨B. (Compare with axioms OR-1, OR-2, and OR-3.)
Theorem TH20: (A→¬A)→¬A
Theorem ST2: A→¬¬A
ST4 is axiom FRG-2 of Frege's PC.
Prove ST5: (A→(B→C))→(B→(A→C))
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.
(For if the theories should be different, then one of them should contain theorems not contained by the other theory. These theorems can be derived from their own theory's axiom set: but as has been shown this entire axiom set can be derived from the other theory's axiom set, which means that the given theorems can actually be derived solely from the other theory's axiom set, so that the given theorems also belong to the other theory. Contradiction: thus the two axiom sets span the same theorem-space. By construction: Any theorem derived from the standard axioms can be derived from Frege's axioms, and vice versa, by first proving as theorems the axioms of the other theory as shown above and then using those theorems as lemmas to derive the desired theorem.)
*cite conference | first = Samuel | last = Buss | title = An introduction to proof theory | booktitle = Handbook of proof theory | pages = 1–78 | publisher = Elsevier | date = 1998 | id = ISBN 0-444-89840-9
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