Superconstructive Propositional Calculi with Extra Axiom Schemes Containing One Variable
✍ Scribed by J. G. Anderson
- Book ID
- 102939649
- Publisher
- John Wiley and Sons
- Year
- 1972
- Tongue
- English
- Weight
- 943 KB
- Volume
- 18
- Category
- Article
- ISSN
- 0044-3050
No coin nor oath required. For personal study only.
✦ Synopsis
We aim to show that the elements of the sequence {xn) are all the 1-assignment L e m m a 2.1. If i E { 1 , 2 , 3 , . . .} then for all j , j > i + 1, CKr-models. (i) x, is a submodel of x,, (ii) a sentence of one variable rejected by x, is also rejected by x,.
Proof. (i): Direct from the definition of the sequence {xn}. (ii): From (i) and the fact that if a formula is rejected on a submodel of a Kr-model it will be rejected by the Kr-model itself (see lemma 3.2*).
L e m m a 2.2. For each n E (1 , 2 , 3 , . . .} x,, is contracted.
Proof. First of all we take it as clear that
Then the proof of the lemma is by course of values induction on n .
Consider an element x,, of the sequence. If n 5 5 then by inspection xn is contracted. If n > 5 , by definition, x,, has just two maximal submodels x ~-~ and xn-2 and from the induction hypothesis x ~-~ and are both contracted. We show in turn that neither of the contraction operations is applicable to xn.