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Extra resources for Astronomy Geometrical and Statistical Methods of Analysis of Star Configurations Dating Ptolemys Almagest
The induction hypothesis, likewise extended to ((x : X)(P x) → R) is present. Further Ex itself occurs in the hypothesis; for its type we have Ex : (X : Set)(X→ Prop) → Prop, as in Chap. 2. R is the entity in Prop for which a resident must be constructed in proof mode. In order to illustrate the use of Ex-ind we shall now verify separately The Drinker´s Paradox discussed already in Chap. 2. We assume given declarations of S : Set (the bar), s : S (there is a drinker in the bar) and Q : S → Prop.
Here are a few choices for P used above to illustrate the use of ELIM in the construction explained. A special case, using * to indicate “don’t care” for an argument, is [l h : (and A B)](ELIM h [l * :A; * :B]h) : (and A B) → (and A B). Inductive Entities 13 Note that the two projection rules of Chap. 1 are verified immediately by using ELIM [l h (and [l h (and : A : A (and A B)](ELIM h [l a:A; * :B]a ) : B) → A. (and A B)](ELIM h [l * :A; b:B]b ) : B) → B. Note further [l h : (and A B)](ELIM h [l a:A; b:B](Conj B A b a) ) : (and A B) → (and B A).
We owe the view adopted here (and throughout the treatise) to L. Wittgenstein’s: “Der Sinn des Satzes ist seine Verification”. Transposed to the LINCOS system this would mean that facts should be accompanied by their verifications, the formal proofs. The examples provided hitherto have already shown that verifications are usually lengthy. Is that an argument for giving proofs a lesser status in the annotations part? The sizes of proofs or the intricacies often present would be one reason for giving them less attention or even considering omitting them in messages for ETI using the LINCOS system.