A Descending Chain of Incomplete Extensions of Implicational S 5

On the Incompleteness of a Descending Chain of Extensions of Implicational S5

“…Where u and T contain arrows, their quasidisjunction d v T is the wff ( a -T) + T. Understanding omitted parentheses in extended quasidisjunctions to be restored from the left, [2] obtains for each positive integer n the extension C 5.n of C 5.w by adding to the latter the instances of the corresponding wf€ (Gn) below:…”

“…The general strategy, which is sound, is to show that any frame validating the theorems of C5.w validates also p + ( q + p ) , though the latter is not provable even in CS.3. The details of the overly-simplified proof sketched in [2] are, however, embarrassingly garbled and cover at best one extremely special case. The following proof, though a bit more involved, covers all cases.…”

“…axioms the instances of In [2], the present author introduced the implicational calculus C5.w by adding to C5's…”

For Want of an ‘And’: A Puzzle about Non-Conservative Extension

“…One might think that because of its considerably greater strength, S5 is the last place to look for such examples. Surprisingly enough, Dolph Ulrich (1985, with corrections in Ulrich 1992 has found that the pure strict implication fragment of S5 can be extended in such a way that other strict implication formulas are not deducible, while these latter are deducible in the corresponding of full (non-fragmentary) S5-see note 1 on page 202 of Ulrich 1985. In fact, Ulrich finds an infinite sequence of extensions of strict implicational S5 answering to the above description, all containing the formula (a) here but not the formula (b), in which we write strict implication as '?…”