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|Deduction Theorem||We I 109
Definition 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. That can be continued backwards: The penultimate premise follows from the totality of previous premises, the third last, etc. until the second, which follows from the first premise.)
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: .. ++ ..
in this evidence, 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. 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)).
Logik Berlin 1999