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- Deduction_theorem abstract "In mathematical logic, the deduction theorem is a metatheorem of first-order logic. It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then deriving B from this assumption conjoined with known results. The deduction theorem explains why proofs of conditional sentences in mathematics are logically correct. Though it has seemed "obvious" to mathematicians literally for centuries that proving B from A conjoined with a set of theorems is sufficient to proving the implication A → B based on those theorems alone, it was left to Herbrand and Tarski to show (independently) this was logically correct in the general case—another instance, perhaps, of modern logic "cleaning up" mathematical practice. The deduction theorem states that if a formula B is deducible from a set of assumptions , where A is a closed formula, then the implication A → B is deducible from In symbols, implies . In the special case where is the empty set, the deduction theorem shows that implies The deduction theorem holds for all first-order theories with the usual deductive systems for first-order logic. However, there are first-order systems in which new inference rules are added for which the deduction theorem fails.The deduction rule is an important property of Hilbert-style systems because the use of this metatheorem leads to much shorter proofs than would be possible without it. Although the deduction theorem could be taken as primitive rule of inference in such systems, this approach is not generally followed; instead, the deduction theorem is obtained as an admissible rule using the other logical axioms and modus ponens. In other formal proof systems, the deduction theorem is sometimes taken as a primitive rule of inference. For example, in natural deduction, the deduction theorem is recast as an introduction rule for "→".".
- Deduction_theorem wikiPageExternalLink ml.htm.
- Deduction_theorem wikiPageID "480010".
- Deduction_theorem wikiPageRevisionID "598321282".
- Deduction_theorem hasPhotoCollection Deduction_theorem.
- Deduction_theorem subject Category:Deductive_reasoning.
- Deduction_theorem subject Category:Metatheorems.
- Deduction_theorem subject Category:Proof_theory.
- Deduction_theorem subject Category:Theorems_in_the_foundations_of_mathematics.
- Deduction_theorem type Abstraction100002137.
- Deduction_theorem type Communication100033020.
- Deduction_theorem type Message106598915.
- Deduction_theorem type Proposition106750804.
- Deduction_theorem type Statement106722453.
- Deduction_theorem type Theorem106752293.
- Deduction_theorem type TheoremsInTheFoundationsOfMathematics.
- Deduction_theorem comment "In mathematical logic, the deduction theorem is a metatheorem of first-order logic. It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then deriving B from this assumption conjoined with known results. The deduction theorem explains why proofs of conditional sentences in mathematics are logically correct.".
- Deduction_theorem label "Deduction theorem".
- Deduction_theorem label "Deduktionstheorem".
- Deduction_theorem label "Teorema de la deducción".
- Deduction_theorem label "Teorema di deduzione".
- Deduction_theorem label "Twierdzenie o dedukcji".
- Deduction_theorem label "演繹定理".
- Deduction_theorem label "演绎定理".
- Deduction_theorem sameAs Deduktionstheorem.
- Deduction_theorem sameAs Teorema_de_la_deducción.
- Deduction_theorem sameAs Teorema_di_deduzione.
- Deduction_theorem sameAs 演繹定理.
- Deduction_theorem sameAs Twierdzenie_o_dedukcji.
- Deduction_theorem sameAs m.02fkch.
- Deduction_theorem sameAs Q1182249.
- Deduction_theorem sameAs Q1182249.
- Deduction_theorem sameAs Deduction_theorem.
- Deduction_theorem wasDerivedFrom Deduction_theorem?oldid=598321282.
- Deduction_theorem isPrimaryTopicOf Deduction_theorem.