▁each ▁clause ▁consists ▁of ▁two ▁liter als , ▁so ▁does ▁each ▁possible ▁resol vent . ▁Therefore , ▁by ▁resolution ▁without ▁factor ing , ▁the ▁empty ▁clause ▁can ▁never ▁be ▁obtained . ▁Using ▁factor ing , ▁it ▁can ▁be ▁obtained ▁e . g . ▁as ▁follows : ▁ ▁Non - c laus al ▁resolution ▁General izations ▁of ▁the ▁above ▁resolution ▁rule ▁have ▁been ▁de vised ▁that ▁do ▁not ▁require ▁the ▁origin ating ▁formulas ▁to ▁be ▁in ▁cla us al ▁normal ▁form . ▁ ▁These ▁techniques ▁are ▁useful ▁mainly ▁in ▁interactive ▁theorem ▁prov ing ▁where ▁it ▁is ▁important ▁to ▁preserve ▁human ▁read ability ▁of ▁intermediate ▁result ▁formulas . ▁Besides , ▁they ▁avoid ▁combin atorial ▁explos ion ▁during ▁transformation ▁to ▁clause - form , ▁and ▁sometimes ▁save ▁resolution ▁steps . ▁ ▁Non - c laus al ▁resolution ▁in ▁propos itional ▁logic ▁For ▁propos itional ▁logic , ▁Murray ▁and ▁Mann a ▁and ▁Wald inger ▁use ▁the ▁rule ▁, ▁where ▁ ▁denotes ▁an ▁arbitrary ▁formula , ▁ ▁denotes ▁a ▁formula ▁containing ▁ ▁as ▁a ▁sub form ula , ▁and ▁ ▁is ▁built ▁by ▁replacing ▁in ▁ ▁every ▁occurrence ▁of ▁ ▁by ▁; ▁like wise ▁for ▁. ▁The ▁resol vent ▁ ▁is ▁intended ▁to ▁be ▁simplified ▁using ▁rules ▁like ▁, ▁etc . ▁In ▁order ▁to ▁prevent ▁generating ▁useless ▁trivial ▁resol vent s , ▁the ▁rule ▁shall ▁be ▁applied ▁only ▁when ▁ ▁has ▁at ▁least ▁one ▁" negative " ▁and ▁" pos itive " ▁occurrence ▁in ▁ ▁and ▁, ▁respectively . ▁Murray ▁has ▁shown ▁that ▁this ▁rule ▁is ▁complete ▁if