Saved from the Cellar by Jan von Plato

Author:Jan von Plato
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

17 —————————————————————————————————

The inferences that arise from the schemes UI and EE have to satisfy the following “variable condition”: The variable of the inference UI resp. EE must not occur in any other proposition of the inference except in , and furthermore [cancelled in margin: Also for the inference FI a restriction is needed.]

ULS 6

in no assumption the conclusion of the inference depends on.

The definition of a proof is hereby finished. [Cancelled: (Another, essentially equivalent version of the variable condition is the following:

The variable of UI must occur only above and in the premiss of the UI, otherwise nowhere in the whole proof; the variable of the EE must occur only above the premiss of the EE, otherwise nowhere in the whole proof.)]

I shall give clarifications to the inference schemes, by explaining for some of them their contentful meaning. I try to make it evident that the calculus really presents “natural deduction.”

FI: This inference says in words: If has been proved through the use of the assumption , also the following holds, this time without the assumption: From follows . It makes no difference if has been used for the derivation of repeatedly or even not at all. – There can have naturally been further assumptions from which then also still remains dependent.

OE: If we have: , and a result follows from the assumption that holds, and the result follows also from the assumption that holds, then holds in any case, i.e., this time independently of the two assumptions. (Again, naturally not independently of eventual further assumptions made under the derivation of the 3 conditions.)

UI: If has been proved for an “arbitrary” , then follows. The condition that be “completely arbitrary” can be expressed precisely as follows: must not depend on any assumptions that contain the variable . And this is precisely the part of the above “variable condition” that is pertinent to the inference UI.

EE: One has . Then one says: Let be such an object for which holds, i.e., one assumes: Let hold. If one has proved on this basis a proposition that does not contain anymore and that does not depend on any assumption that contains , then is provable independently of the assumption . Hereby is expressed precisely the part of the “variable condition” pertinent to EE. (There is a certain analogy between OE and EE.)

ULS 7 18

The remaining inferences should be easy to explain.19

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