▁augment ed ▁by ▁appropriate ▁logical ▁transformation ▁rules . ▁ ▁Tra ug ott ▁uses ▁the ▁rule ▁, ▁where ▁the ▁ex ponents ▁of ▁ ▁indicate ▁the ▁polar ity ▁of ▁its ▁occurr ences . ▁While ▁ ▁and ▁ ▁are ▁built ▁as ▁before , ▁the ▁formula ▁ ▁is ▁obtained ▁by ▁replacing ▁each ▁positive ▁and ▁each ▁negative ▁occurrence ▁of ▁ ▁in ▁ ▁with ▁ ▁and ▁, ▁respectively . ▁Similar ▁to ▁Murray ' s ▁approach , ▁appropriate ▁simpl ifying ▁transformations ▁are ▁to ▁be ▁applied ▁to ▁the ▁resol vent . ▁Tra ug ott ▁proved ▁his ▁rule ▁to ▁be ▁complete , ▁provided ▁ ▁are ▁the ▁only ▁connect ives ▁used ▁in ▁formulas . ▁ ▁Tra ug ott ' s ▁resol vent ▁is ▁stronger ▁than ▁Murray ' s . ▁Moreover , ▁it ▁does ▁not ▁introduce ▁new ▁binary ▁jun ctors , ▁thus ▁avoid ing ▁a ▁t endency ▁towards ▁cla us al ▁form ▁in ▁repeated ▁resolution . ▁However , ▁formulas ▁may ▁grow ▁longer ▁when ▁a ▁small ▁ ▁is ▁replaced ▁multiple ▁times ▁with ▁a ▁larger ▁ ▁and / or ▁. ▁ ▁Pro pos itional ▁non - c laus al ▁resolution ▁example ▁As ▁an ▁example , ▁starting ▁from ▁the ▁user - g iven ▁assumptions ▁ ▁the ▁Murray ▁rule ▁can ▁be ▁used ▁as ▁follows ▁to ▁infer ▁a ▁contradiction : ▁ ▁For ▁the ▁same ▁purpose , ▁the ▁Tra ug ott ▁rule ▁can ▁be ▁used ▁as ▁follows ▁: ▁ ▁From ▁a ▁comparison ▁of ▁both ▁dedu ctions , ▁the ▁following ▁issues ▁can ▁be ▁seen : ▁ ▁Tra ug ott ' s ▁rule ▁may ▁yield ▁a ▁sh ar per ▁resol vent :