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• Freges_propositional_calculus abstract "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:\tA → (B → A) THEN-2:\t(A → (B → C)) → ((A → B) → (A → C)) THEN-3:\t(A → (B → C)) → (B → (A → C)) FRG-1:\t(A → B) → (¬B → ¬A) FRG-2:\t¬¬A → A FRG-3:\tA → ¬¬AInference Rule MP: P, P→Q ⊢ QFrege'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→ARule THEN-2*: A→(B→C) ⊢ (A→B)→(A→C)Rule THEN-3*: A→(B→C) ⊢ B→(A→C)Rule FRG-1*: A→B ⊢ ¬B→¬ARule TH1*: A→B, B→C ⊢ A→CTheorem TH1: (A→B)→((B→C)→(A→C))Theorem TH2: A→(¬A→¬B)Theorem TH3: ¬A→(A→¬B)Theorem TH4: ¬(A→¬B)→ATheorem TH5: (A→¬B)→(B→¬A)Theorem TH6: ¬(A→¬B)→BTheorem TH7: A→ATheorem TH8: A→((A→B)→B)Theorem TH9: B→((A→B)→B)Theorem TH10: A→(B→¬(A→¬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)TH11 is axiom NOT-1 of standard PC, called \"reductio ad absurdum\".Theorem TH12: ((A→B)→C)→(A→(B→C))Theorem TH13: (B→(B→C))→(B→C)Rule TH14*: A→(B→P), P→Q ⊢ A→(B→Q)Theorem TH15: ((A→B)→(A→C))→(A→(B→C))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: ((A→B)→B)→(¬A→B)Theorem TH19: (A→C)→ ((B→C)→(((A→B)→B)→C))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)→¬ATH20 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→ATheorem ST2: A→¬¬AST2 is axiom FRG-3 of Frege's PC.Theorem ST3: B∨C→(¬C→B)Theorem ST4: ¬¬A→AST4 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.)".
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• Freges_propositional_calculus wikiPageWikiLink Axiom.
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• Freges_propositional_calculus wikiPageWikiLink Begriffsschrift.
• Freges_propositional_calculus wikiPageWikiLink Category:Logical_calculi.
• Freges_propositional_calculus wikiPageWikiLink Category:Propositional_calculus.
• Freges_propositional_calculus wikiPageWikiLink Category:Systems_of_formal_logic.
• Freges_propositional_calculus wikiPageWikiLink Charles_Sanders_Peirce.
• Freges_propositional_calculus wikiPageWikiLink Converse_(logic).
• Freges_propositional_calculus wikiPageWikiLink First-order_logic.
• Freges_propositional_calculus wikiPageWikiLink Generator_(mathematics).
• Freges_propositional_calculus wikiPageWikiLink Gottlob_Frege.
• Freges_propositional_calculus wikiPageWikiLink Law_of_excluded_middle.
• Freges_propositional_calculus wikiPageWikiLink Logical_conjunction.
• Freges_propositional_calculus wikiPageWikiLink Logical_disjunction.
• Freges_propositional_calculus wikiPageWikiLink Mathematical_logic.
• Freges_propositional_calculus wikiPageWikiLink Modus_ponens.
• Freges_propositional_calculus wikiPageWikiLink Principle_of_explosion.
• Freges_propositional_calculus wikiPageWikiLink Propositional_calculus.
• Freges_propositional_calculus wikiPageWikiLink Reductio_ad_absurdum.
• Freges_propositional_calculus wikiPageWikiLink Rule_of_inference.
• Freges_propositional_calculus wikiPageWikiLink Second-order_logic.
• Freges_propositional_calculus wikiPageWikiLink Well-formed_formula.
• Freges_propositional_calculus wikiPageWikiLinkText "Frege's propositional calculus".
• Freges_propositional_calculus wikiPageUsesTemplate Template:Cite_conference.
• Freges_propositional_calculus subject Category:Logical_calculi.
• Freges_propositional_calculus subject Category:Propositional_calculus.
• Freges_propositional_calculus subject Category:Systems_of_formal_logic.
• Freges_propositional_calculus hypernym Axiomatization.
• Freges_propositional_calculus type Method.
• Freges_propositional_calculus comment "In mathematical logic Frege's propositional calculus was the first axiomatization of propositional calculus.".
• Freges_propositional_calculus label "Frege's propositional calculus".
• Freges_propositional_calculus sameAs Q714691.
• Freges_propositional_calculus sameAs Cálculo_proposicional_de_Frege.
• Freges_propositional_calculus sameAs Frege-kalkulus.
• Freges_propositional_calculus sameAs m.06cfwf.
• Freges_propositional_calculus sameAs Q714691.
• Freges_propositional_calculus sameAs 弗雷格命题演算.
• Freges_propositional_calculus wasDerivedFrom Freges_propositional_calculus?oldid=695749317.
• Freges_propositional_calculus isPrimaryTopicOf Freges_propositional_calculus.