Equivalence relations and ${\rm S}5$.
G. E. Hughes · Notre Dame Journal of Formal Logic · 1980
An equivalence relation is commonly defined as one which is reflexive, symmetrical, and transitive.This paper* starts from the problem of finding a pair of conditions on a dyadic relation which together yield equivalence but neither of which by itself yields either reflexiveness or symmetry or transitivity.It will be shown that there are infinitely many such pairs of conditions.There is a parallel problem in modal logic, that of finding a pair of formulas which, if added to the minimal normal modal logic K, yield precisely 55, but neither of which, when added to K, yields either Lp D p or p D LMp or Lp D LLp as a theorem.It will be shown that there are infinitely many such pairs of formulas. 2One solution to the second problem is provided by the following formulas:Since in S5 an affirmative modality is equivalent to its last member, it is clear that A and B are theorems of S5 and hence that S5 contains K + A + B. For the converse it is sufficient to derive MLp D Lp and Lp D p.We first note that A is interdeducible in the field of K with its dual:pDMLMp.We then have:*I acknowledge with gratitude the help given to me by Dr. R. L. Epstein in preparing this paper.In particular, he is responsible for the generalizations of conditions Y and Z in Section 4.