Philosophy Dictionary of ArgumentsHome
| |||
|
| |||
| Deduction theorem: The deduction theorem is a metatheorem in logic that states that if a proposition B can be deduced from a proposition A, then the implication A → B is also deducible. In other words, if we can prove that B is true given that A is true, then we can also prove that A implies B.
The deduction theorem is a useful tool for proving theorems, as it allows us to break down complex proofs into smaller, more manageable steps._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. | |||
| Author | Concept | Summary/Quotes | Sources |
|---|---|---|---|
|
H. Wessel on Deduction Theorem - Dictionary of Arguments
We I 109 Def Deduction Theorem/calculus NS/Wessel: MT 1. If A1 ... An l B, so A1..An 1 l An> B. ((S) If the conclusion follows from the totality of the premises, so the last premise follows from the totality of the previous premises and from the last premise follows then the conclusion...) I 110 Induction proof/calculus NS/Wessel: in the conclusion B1 can stand an assumption formula (a.f.) or an axiom variant (a.v.). Is it an assumption formula, there are again two possible cases: it may be the assumption formula An or an assumption formula different from An. Deduction theorem/proof/Wessel: .. ++ .. I 111 in this proof, only the following three theorems were used: p > (q > p), p > (q > r)> (p > q> (P > r)) and p > p. Deduction theorem/Wessel: as a conclusion we get: MT 2. If A1 ... An l B, so l A1 > (A2> ..> (An> B) ...). The deduction theorem states an essential relationship between proofs and derivations. >Proofs, >Provability, >Derivation, >Derivability. In the future, it is sufficient, when proving a theorem, to prove a derivational relationship and to apply to it the deduction theorem. E.g. from the derivational relationship p > q, q > r, p l r we get by three-time application of MT 1: T3. l p > q> (q > r> (p > r))._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Wessel I H. Wessel Logik Berlin 1999 |
||
> Counter arguments against Wessel
> Counter arguments in relation to Deduction Theorem
Ed. Martin Schulz, access date 2024-04-27